Non-linear behaviour of lattice panel of angle towers - · PDF file43 Non-linear behaviour of...

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34 Journal of Constructional Steel Research �� (2001) ��–��37

www.elsevier.com/locate/jcsr38

42

43 Non-linear behaviour of lattice panel of angle44 towers

45 N. Prasad Rao a, V. Kalyanaraman b,*

46a Structural Engineering Research Centre, Chennai 600036, India

47b Department of Civil Engineering, Indian Institute of Technology, Madras, Chennai 600036, India

48 Received 8 September 2000; accepted 18 September 2001

49

50 Abstract

51 Lattice microwave towers and transmission towers are frequently made of angles bolted52 together directly or through gussets. Such towers are normally analysed to obtain design forces53 using the linear static methods, assuming the members to be subjected to only axial loads and54 the deformations to be small. The effects of the end restraints, eccentricity of connections55 and secondary bracings (redundants) on the strength of the compression members are usually56 accounted for in the codal recommendations by modifying the effective length of the members57 and thus the design compressive strength. Hence, forces in the redundants are not known from58 the analysis and their design is empirical. In this study, non-linear analysis of angle com-59 pression members and the single panel of angle planar as well as three-dimensional lattice60 frames, as in typical lattice towers, are carried out using MSC-NASTRAN software. Account61 is taken of member eccentricity, local deformation as well as rotational rigidity of joints, beam-62 column effects and material non-linearity. The analytical models are calibrated with test results.63 Using this calibrated model, parametric studies are carried out to evaluate the forces in the64 redundants. The results are compared with codal provisions and recommendations for the65 design of redundants are presented. 2001 Published by Elsevier Science Ltd.

66 Keywords: Lattice towers; Non-linear analysis; Compression members; Secondary bracings67

68

1

27 *28 Corresponding author.2930 E-mail address: [email protected] (V. Kalyanaraman).1

DTD v4.1.0 / JCSR2045

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3 0143-974X/01/$ - see front matter 2001 Published by Elsevier Science Ltd.4 PII: S0143-97 4X( 01 )0 0054-25

1 JCSR: j. of constructional steel research - ELSEVIER2 18-10-01 09:00:35 Rev 16.03x JCSR$$045P

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1 22 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research �� (2001) ��–��3

69 1. Introduction

70 Microwave and overhead electric transmission line towers are usually fabricated71 using angles for the main legs and the bracing members. The members are bolted72 together, either directly or through gusset plates. In order to reduce the unsupported73 length and thus increase their buckling strength, the main legs and the bracing mem-74 bers are laterally supported at intervals in between their end nodes, using secondary75 bracings or redundants (Fig. 1).76 The lattice towers are usually analysed assuming the members to be concentrically77 connected using hinged joints so that the forces in the angle members are only axial.78 Under this assumption, the forces in the redundants are negligibly small or zero and79 hence are not included in the linear analysis models. However, the main legs and80 the bracing members are not axially loaded and the redundant forces are not negligi-81 bly small, due to the following reasons:

82 �83 The main legs are usually continuous through the joint.

516517

518519520

521 Fig. 1. Tower configuration.522

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84 �85 Usually more than one bolt is used in the connections and hence the joints are86 semi-rigid.87 �88 The angle members are normally bolted through only one of their legs and hence89 the force transfer in the members is eccentric.90 �91 The joints are flexible due to the local deformation of the leg of the angles under92 the concentrated bolt forces.93 �94 The towers with high electric ratings tend to be flexible and hence equilibrium95 in the deformed configuration has to be considered (large deformation effects).96 �97 The compression member deformation increases the bending moments (P–d98 effect).

99 Therefore, the angle members of the tower experience both axial force and bending100 moments, even well before the tower fails. This also produces forces in the redundant101 members due to their participation in overall frame action, which are not negligible102 as often assumed in designs.103 Roy et al. [1] studied the effects of joint rigidity and large deformation of tall104 high-power electric transmission towers and concluded that these towers experienc-105 ing heavier loads are more flexible and the secondary effects are more pronounced.106 Al-Bermani and Kitipornchai [2] evaluated the ultimate strength of towers consider-107 ing the material (lumped plasticity) and geometric non-linearity, joint flexibility and108 large deflection, using an equivalent tangent stiffness matrix for the members. They109 concluded that the material and geometric non-linearity have a major effect on the110 ultimate strength of towers. They attributed the larger difference between their analy-111 sis and experimental results to the bolt slippage, not modeled in the analysis. Hui112 et al. [3] presented details of geometric non-linear analysis of transmission towers113 to trace the load deformation behaviour, treating the main legs as beam-columns and114 the bracings as truss members, using updated Lagrangian formulation.115 Chuenmei [6] and Shan et al. [7] used rectangular plate elements to model the116 lattice tower members, which is impractical in the analysis of full towers. Rajmane117 [8] used the beam-column element with seven degrees of freedom per node118 (including the warping deformation) to analyse the braced frames including the119 effects of eccentricity. Stoman [9] used minimisation of total potential energy to120 study the plastic stability of X-braced systems and demonstrated the restraining121 effects of tension diagonals.122 Experimental studies have been conducted on concentrically and eccentrically123 loaded single angles [6,8,10–12], planar and three-dimensional lattice frames made124 of angles [9,13–15] and full-scale towers [22].125 It is seen that the analytical studies reported have not considered all the important126 factors that may influence the behaviour of lattice towers before failure, particularly127 the eccentricity of connections, and the flexibility of the joints due to the local defor-128 mation of the bolted leg of the angles. Rao and Kalyanaraman [18] presented details129 of a non-linear analysis of a panel of lattice towers, considering the effects listed130 earlier, which affect the tower member forces. In their study, plate elements were131 used at joints and at plastic hinge locations, and beam-column elements at the rest132 of the locations of members, to model the angle members in the towers.

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133 This paper initially presents details of non-linear analyses of angle members and134 lattice towers made of angles, using MSC-NASTRAN. These analysis models con-135 sider all the factors listed earlier, which affect the tower behaviour, and the analysis136 results are calibrated against test results. Using the model thus developed, a para-137 metric study has been done in order to understand the effects of the various factors138 that influence the strength of lattice towers and the design of redundant members.139 Finally, the analysis results are compared with the empirical methods recommended140 in codes of practice for the design of members. Based on this approach, a method141 for designing redundants in lattice towers is recommended.

142 2. Calibration of non-linear analysis model

143 Initially, single angle compression test specimens, loaded through the centroid or144 through one of the legs, are modeled and analysed. Subsequently, latticed plane145 frame and space frame tests using angle members are also modeled and analysed.146 These analyses help to calibrate the models used in the subsequent parametric studies.

147 2.1. Single angles under compression

148 Concentrically loaded, ideal single angle compression members theoretically149 should fail by bifurcation buckling about their weak axis, at the Euler buckling load.150 However, due to imperfections they undergo a beam–column type of failure at loads151 below the Euler buckling load. At some stage, a part of the section subject to152 maximum stress under combined bending and compression and residual stress yields.153 The final member failure may be by progressive yielding and plastic hinge formation154 or partial yielding and local plate buckling, depending upon the width to thickness155 ratio of legs and the overall slenderness ratio of the member.156 In practice, the angle members in towers are usually loaded eccentrically through157 only one leg, which is connected to gussets or directly to a leg of adjacent angle158 members. Consequently, they undergo bi-axial bending in addition to axial com-159 pression. Under this loading, the cross section of the angle progressively yields and160 fails by the formation of a plastic hinge under the combined action of axial load161 and magnified biaxial bending. Further, the bolted leg of the angle undergoes local162 deformation under the bearing force of the bolts, causing flexibility in the connection,163 and shear lag in the member.164 Rajmane [8] tested single angles under concentric compression and eccentric com-165 pression by loading through end gussets. Chuenmei [6] presented test results of166 angles loaded through end gussets, covering a range of slenderness ratio, size and167 yield strength. Natarajan et al. [22] tested angles as part of a plane lattice. These168 test results are compared with strengths obtained from design equations and numeri-169 cal analysis in the following sections.

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170 2.1.1. Design equations171 The British Standards Institute [16], the American Society of Civil Engineers [19]172 and the Bureau of Indian Standards [20] specify essentially the same method for173 evaluating the compressive strength of angle members in lattice towers, accounting174 for the effects of residual stresses, imperfections and end conditions. This method175 involves modifying the effective slenderness ratio of the member, depending upon176 the location of the member in the tower and the eccentricity of connection. The177 strength of the angle members tested is compared with the results based on the code178 recommendations, in Tables 1 and 2 under the column Code.179 It is seen that the theoretical strengths evaluated based on code provisions are180 conservative compared to concentric compression test results, which is under-181 standable, since these code provisions are for the design of angle members with182 eccentric end connections through one leg. However, the angle strengths based on183 code provisions are highly unconservative compared to eccentric compression test184 results. It is also seen that the extent of the unsafe nature of code provisions decreases185 with increases in the slenderness ratio. This comparison indicates that code pro-186 visions do not seem to adequately account for eccentricity, imperfection and residual187 stress effects, which have a major influence on the strength of compression members188 in the intermediate slenderness ratio ranges (60�l/r�120).

189 2.1.2. Numerical method190 The non-linear finite element analysis methods are effective for evaluating the191 behaviour and strength of compression members and space frames, considering vari-192 ous effects discussed earlier. The angles under compression were analysed with the193 help of MSC-NASTRAN. The non-linear analysis capability of the software was194 used for the strength evaluation. In the case of concentrically loaded members, sinus-195 oidal initial imperfection amplitude of 1/1000 of the length of the member was196 assumed in the analysis, to trace the non-linear large deformation behaviour. In197 eccentrically loaded members the effect of member imperfection was neglected, since198 the eccentric load caused much larger lateral deflection of members. Three different199 models, with increasing elaboration, as given below, were used for angle members200 under compression.

201202 Model 1 (Fig. 2(a)). A number of beam-column line elements (six in total) along203 the centroid of the section were used to model each angle in this model (M1).204 The eccentric loading was applied through a rigid link between the centroid of205 the member and the point of application of the load. The limit load in this model206 is reached in the MSC-NASTRAN analysis when the stress at the maximum207 stressed point in the member reaches the yield stress. This is obviously conserva-208 tive, especially in slender members, since it does not account for the post-first-209 yield plastification of the maximum stressed section before failure.210211 Model 2 (Fig. 2(b)). In this model (M2) a major segment of the member is mod-212 elled using the beam-column elements as before. However, over a short length at213 the center of the member (0.2 times the length), where the member plastification214 is expected to occur, the two legs of the angles were modeled using flat-shell

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526527528

529 Fig. 2. Eccentrically compressed single angle model.530

215 elements. This enabled modeling of the progressive yielding at the point of plastic216 hinge formation and the subsequent failure by local buckling of the elements. At217 the transition between the beam-column element and the flat-shell elements, rigid218 elements were used to connect the beam column node to the nodes of the flat-219 shell elements. Whenever the load is transferred through gussets at the ends, the220 gussets and the legs of the angle over 0.2 times the length at the ends were221 modeled using the flat-shell elements. Beam elements were used to represent the222 bolts, connecting the gussets and the beam-columns/flat-shell elements.223224 Model 3 (Fig. 2(c)). In this case (M3), the entire length of the angle member is225 modeled using a number of flat-shell elements. Whenever the load is transferred226 through gussets at the ends, the gusset plates also are modeled using the flat-shell227 elements and the connections between the gussets and the angles are modeled228 using the gap elements available in MSC-NASTRAN. The bolts are modeled using229 beam elements.

230 The non-linear analysis capability of MSC-NASTRAN, accounting for the geo-231 metric and material non-linearity, was used to analyse the models and obtain their232 pre-ultimate behaviour and the limit loads. The elastic–plastic material property of233 steel was represented by a bi-linear model, having modulus of elasticity up to a yield234 stress equal to 2.0×105 MPa and 2000 MPa beyond yield stress. The incremental235 load and predictor–corrector iteration under each load increment were used in the236 non-linear range. The Von-Mises criterion was used to define yielding. The isotropic237 hardening model was used in the post-yield range. The load increments were carried

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238 out in 20–30 steps, until the limit point was reached in the load deformation behav-239 iour.240 The test results are compared in Tables 1 and 2 with the strength evaluated based241 on the three MSC-NASTRAN models. The percentage error242 ((Theory�Test)×100/Test), the mean and standard deviation of the errors also are243 presented in Tables 1 and 2, corresponding to the three models (M1, M2 and M3,244 respectively). It is seen that the MSC-NASTRAN model results compare well with245 the test results. The model M3 comparison with the test results is the best of the246 three, although model M2 is quite adequate. The model M1 has the largest mean247 error among the three models, particularly in the eccentrically compressed cases. For248 the further study of lattice frame models M1 and M2 are used, since the model M3249 consumes a large amount of time and memory due to the large number of degrees250 of freedom.

251 2.2. Behaviour of lattice frames

252 Rajmane [8] tested planar angle lattice frames, and Natarajan [13] tested planar253 and three-dimensional angle lattice frames, consisting of X bracings and K bracings.254 Details of the test specimens and results are presented in Fig. 3 and Tables 3–5. The255 experimental strengths of these frames are compared with the code based and numeri-256 cal analysis based strengths as discussed below.

257 2.2.1. Code equations258 These lattice frame test results are compared with the strengths based on code259 provisions by the following procedure. The member forces are obtained from a linear260 elastic analysis of concentrically connected lattice truss models of the frame, as com-261 monly done in practice. The design strength of the critical angle member as obtained262 from the code provisions and linear analysis member forces are used to calculate263 the frame strength. It is seen (Tables 3–5) that the code provisions either under- or264 overestimate the actual strength of the lattice frame by as much as �18%265 (conservative) to +29% (unconservative). It is clear from this study that the error in266 the code based design of members, for forces obtained from the linear elastic analysis267 of a concentrically connected truss model, could be high, particularly in the case of268 slender bracing members.

269 2.2.2. Numerical analysis270 The conventional assumption of hinged joints does not represent the real joint271 behaviour in latticed towers. Two types of joint models given below, to represent272 the bolted connections between angles in the frames, were evaluated in the numeri-273 cal study.274 In the rigid joint model, the flexibility of the bolt and the legs of the angle at the275 joint were disregarded and the joints were assumed to be rigid by enforcing the276 compatibility of translations and rotations in all the members meeting at the joint.277 However, the effect of an eccentric bolted connection between members was278 accounted for by using rigid elements between the bolt lines and the centroid of the

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534535536

537 Fig. 3. Joint models. (a) Rigid bolted joint model. (b) Flexible bolted joint model.538

1059

1060 Table 31061 ‘X’-braced plane framesa10621063

106810731078

Reference Bracing L/r Panel failure load (kN) % Difference with test results:100×(Theory�Test)/Test

1089

Test Code Model Model Code Element ElementM1 M2 model M1 model M21103

111311231133

8 117 157.0 144.7 154.0 155.0 �7.8 �1.9 �1.31143

8 106 159.0 155.5 166.0 173.0 �2.2 4.4 8.81153

14 177 60.0 60.6 59.0 62.0 +1.0 �1.7 3.31163

Mean �3.0 0.3 3.61168

Standard deviation 4.4 3.6 5.01173

11781183

1188a Model 1: beam-column model; model 2: beam-column and flat-shell model.

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1451 Table 51452 K-braced panel of a 6 m extension of a 220 kV tower [15]14531454

145814621466

Bracing L/r Panel failure load (kN) % Difference with respect to testresult: 100×(Theory�Test)/Test

1476

Test Code Analysis Code Member model11486

Member model11494

150115081515

191 51.6 66.4 54.3 28.6 5.231522

1529

279 angle members (Fig. 3, elements 1 and 3 for members A and B, respectively) and280 a beam element (element 2) joining these rigid elements was used to represent the281 bolts. Freedom of relative rotation of the members about the axis of the single bolt282 was modeled by keeping the torsional stiffness of the beam element very low (60283 mm4).284 The above rigid joint model does not account for the flexibility and the local285 deformation of the legs of the angles at the bolted joint. For evaluating these effects,286 a finite element analysis of the joint region alone was carried out using the model287 shown in Fig. 3(a). In this flexible joint model (FJM) a short segment of angles288 joining at the node along with the bolts were studied. The angles were modeled289 using flat-shell elements. The contact force transfer between the legs of the angles290 was modeled using the gap elements, available in MSC-NASTRAN. The bolts in291 the joint were modeled using a rod element.292 Static analyses of the joint model were carried out to obtain the joint stiffness293 considering the local deformation effects. The analyses were carried out for two294 different sets of member sizes to obtain the joint stiffness values in the practical295 range of member sizes. These joint analyses results were used to evolve a beam296 element connecting the centroidal lines of the two angles, with an equivalent stiff-297 nesses. The flexural stiffnesses of the connecting equivalent beam elements are given298 in Table 6. The equivalent link elements were used in the full frame model, to299 represent the joint flexibility and eccentricity. Such full frames with equivalent beam300 elements corresponding to the flexible joint model are referred to as FJM. The FJM301 has been used in the analyses of K-braced frames only.

1537

1538 Table 61539 Joint flexibility model results15401541

154515491553

Angle member Joint rotation/unit moment Equivalent moment of(rad/N mm) inertia of joint member

(mm4)1562

Leg Bracing1567

157215771582

90×90×8 45×45×3 1.78×10�8 10,2571587

45×45×3 45×45×3 1.93×10�7 6211592

1597

1


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542543544

545 Fig. 4. X-braced plane frame.546

302 Two types of models were used to represent the angle members in the frame, as303 discussed earlier. In one, the entire length of the angle is represented by a number304 of beam-column elements (model M1). In the second model 20% of the central305 length of the angle compression members and 20% of the length closer to the joints306 in the angle tension members were modeled using flat-shell elements (model M2),307 as discussed earlier.308 The non-linear analyses were carried out assuming an initial bow of member309 length/1000 in a few cases, to study their effects. The eccentricity of connections310 had greater influence than the initial bow in these frames. The K-braced latticed311 space frames were tested for different patterns of secondary bracings and in the312 analytical model of these frames, the different secondary bracing patterns were rep-313 resented.314 Only plane frame analyses were carried out in X-braced frames, whereas K-braced315 frames, tested as three-dimensional lattices, were analysed as both plane and space316 frames. The angle member models (M1) and (M2) were used in the case of the317 flexible joint model of space frames and only the angle member model (M1) was318 used in the case of the rigid model. Typical analytical models are shown in Figs. 4319 and 5. Some of the failure mode shapes are shown in Fig. 6.320 The strength of the frames as obtained for different frames tested and different321 nonlinear analysis models are presented in Tables 3–5. These non-linear analysis322 results when compared with the test results indicate the following:

548549

550551552

553 Fig. 5. K-braced frame [13].554

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558559560

561 Fig. 6. Three-dimensional model of ‘6 m extension’ of a 220 kV tower [15]. (a) Secondary bracing562 pattern (I); (b) secondary bracing pattern (II).563

323 �324 The numerical analyses results for X- and K-braced lattice frames compare well325 with the test results. The maximum error is 8%.326 �327 The flexible joint frame models generally compare better with test results in the328 case of K-braced frames.329 �330 There is not much of a difference in the results obtained using the two angle331 member element models, M1 and M2.332 �333 The mean and the standard deviation of the error in numerical analysis results334 are less than 5%.

335 The results of the finite element analysis using member model M1, considering336 the eccentricity and flexibility of connection as well as material and geometric non-337 linearity, compare fairly well with the test results. Hence this model is used for338 parametric studies in the following sections without incurring the high expenses of339 experimental studies.

340 3. Behaviour of secondary bracings

341 The secondary bracing members are provided to reduce the unsupported length342 and thus increase the buckling strength of the main compression members. Linear343 elastic analysis of lattice towers with secondary bracings, assuming the member con-344 nectivity to be concentric and hinged, would normally indicate zero or near zero345 force in the secondary members. Hence no force for the design of secondary bracings346 can be obtained from such analyses. However, secondary bracings should have some347 minimum strength and stiffness to perform intended functions.

348 3.1. Code methods

349 Codes of practice suggest provisions for the design of the secondary bracings as350 given below.

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351352 British code. The British code prescribes the application of a fictitious load acting353 transverse to the main member being stabilized by the secondary member, at the354 node of attachment of the secondary member to the main member. This force to355 be applied is prescribed as a percentage of the main leg or other main bracing356 member force, depending upon the slenderness ratio of the member (Table 7) .357 This force should be applied in the plane of the bracings in turn at each node358 where the secondary members meet the main member. The secondary bracing359 forces should also be analysed separately, by applying 2.5% of the force in the360 main leg distributed equally at all the interior nodal points along the length of361 the leg excluding the first and the last node. The nodal forces should be applied362 transverse to the leg member in the plane of the bracing.363364 ASCE Manual 52. The maximum slenderness ratio of the secondary bracing mem-365 bers is restricted to be below 330. This manual does not require calculation of366 forces for which the secondary bracing members have to be designed. However,367 it suggests that the magnitude of the load in the redundant members can vary368 from 0.5 to 2.5% of the force in the supported member.369370 IS: 802 (1992). This standard specifies the maximum limit on the slenderness371 ratio of the redundants to be equal to 250.

372 Thus, it is seen that some variations in the design requirements of the secondary373 bracings exist in codes. The non-linear finite element analysis method, discussed in374 the earlier section, can be used to evaluate the forces in the secondary bracings prior375 to failure. The forces in the secondary bracings so evaluated could serve as a guide-376 line for the design of secondary bracing members.

377 3.2. Numerical parametric study

378 For this purpose a parametric study was carried out to evaluate the forces in the379 secondary bracings in a typical bottom panel of a K-braced three-dimensional latticed380 frame (Fig. 6). In a typical tower the force resultants in the form of vertical force381 V, the shear force H and the over turning moment M vary over the height of the382 tower. In the parametric study of the single panel of the tower, the force resultants383 at the top of the panel were applied corresponding to different values of V/H and384 M/bH ratios in the practical range, where V, H and M are vertical force, shear force

1603

1604 Table 71605 Secondary member forces calculation BSI DD 13316061607

160916111613

Applied force as percentage of leg load, F1615

161716191621

Slenderness 0 to 45 50 55 60 65 70 75 80 85 90 95 100ratio (L/r) 401638

Applied 1.02 1.15 1.28 1.42 1.52 1.60 1.65 1.70 1.75 1.80 1.85 1.92 2.0force(percentageof FL)1656

1671

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385 and over turning moment resultants acting at the centre at the top of the panel. The386 forces in the four corner nodes in the model were evaluated corresponding to these387 force resultant values and were applied in the three orthogonal directions at the four388 top nodes, so as to obtain the desired ratio of the force resultants as given in Table 8389 The sections of the main bracing, secondary bracing and leg members were kept390 constant in most cases. Changes in the size of these members were made in a few391 of the analysis cases (Sl. nos. 2, 3, 4, 5 and 7), to understand the impact of such392 changes. In all cases secondary ties joining the main bracings on two adjacent faces393 of the three-dimensional latticed tower were provided. A typical displacement con-394 figuration prior to failure is shown in Fig. 7(b). The shear H, corresponding to the395 failure of the structure as obtained from the non-linear analysis is given in Table 8,396 in addition to the corresponding maximum compressive forces in the main leg, FL,397 main bracing, Fb, and the secondary bracings Fsb. Further, the maximum values of398 equivalent panel shear, tmax, corresponding to the secondary bracing forces, Fsb, from399 the non-linear analysis at limit load, are also presented in Table 8.400 The parametric study results in Table 8 indicate the following:

401 �402 The leg forces, FL, obtained from linear and non-linear analyses are nearly the403 same in all the cases, the maximum difference being 4%.404 �405 The non-linear analysis results indicate appreciable increase in the maximum axial406 force in the bracing. The increase can be as high as 38%. This is usually more407 in cases where secondary bracings are very light or type 2 secondary bracings408 are used. It is therefore essential to design the bracing members conservatively409 for the force obtained from the linear analysis.410 �411 As the size of the secondary bracings decreases from the standard value (45×35×5412 having l/r�250) to a lesser value (25×25×4 having l/r�330), the strength of the413 panel is appreciably decreased (Sl. no. 1 versus Sl. nos. 2 and 6 versus Sl. no.7414 in Table 8). However, increases in the size of the secondary bracing above the415 standard value do not seem to improve the strength of the panel appreciably (Sl.416 no. 1 versus Sl. no. 3 in Table 8). This indicates the importance of the minimum417 stiffness requirement of secondary bracings.418 �419 It is seen from the results of Sl. no.4 in Table 8 that the same secondary bracings420 (45×35×5) are able to sustain even a larger panel force without initiating failure421 when the other (leg and main bracing) member sizes are increased. Similarly,422 reduction in the main leg size (Sl. no. 5 in Table 8) causes reduction in the strength423 of the panel, due to the strength being governed by the leg buckling.

424 The design recommendations of various codes are compared with the parametric425 study results in Table 9. The following conclusions can be drawn based on this com-426 parison:

427 �428 The secondary bracing forces calculated based on BS recommendations, Fsb, are429 compared with the secondary bracing forces obtained from the non-linear analysis430 results, Fsb,NLA, in terms of their ratios in Table 9. It is seen that the correlation431 is very poor, with the mean value of the ratio equal to 0.86 and the coefficient432 of variation equal to 0.38.

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1690

1691

1692

Tab

le8

1693

Para

met

ric

stud

yre

sults

a16

94

1702

1710

1718 Sl.

no.

V/H

M/b

HSe

ctio

nFa

ilure

HFo

rces

inm

embe

rsat

failu

reH

t max

1730

Lin

ear

anal

ysis

Non

-lin

ear

anal

ysis

1743

Leg

Mai

nbr

ace

Bel

tR

edun

dant

FL

Fb

FL

Fb

Fsb

1758

1773

1788

1803 Se

cond

ary

brac

ing

patt

ern

I18

05 10.

53

100×

100×

875

×75×

675

×75×

645

×35×

525

843

520

.943

321

.19.

04.

0018

20 20.

53

100×

100×

875

×75×

675

×75×

625

×25×

420

4.2

345

16.6

343

19.4

4.7

3.55

1835 3

0.5

310

0×10

0×8

75×7

5×6

75×7

5×6

60×6

0×5

259.

243

821

.043

620

.611

.74.

6618

50 40.

53

150×

150×

1275

×75×

610

0×10

0×8

45×3

5×5

448

757

36.3

754

35.1

17.0

9.75

1865 5

0.5

365

×65×

1075

×75×

675

×75×

645

×35×

519

4.5

328

15.8

327

15.8

5.3

3.60

1880 6

1.3

310

0×10

0×8

75×7

5×6

75×7

5×6

45×3

5×5

229

433

18.6

431

18.5

6.8

3.40

1895 7

1.3

310

0×10

0×8

75×7

5×6

75×7

5×6

25×2

5×4

153.

629

012

.527

817

.33.

72.

1719

10 80.

54

100×

100×

875

×75×

675

×75×

645

×35×

519

5.2

409

7.3

407

9.3

10.0

4.60

1925 9

1.3

410

0×10

0×8

75×7

5×6

75×7

5×6

45×3

5×5

178.

841

06.

740

88.

88.

04.

5519

40 104.

53

100×

100×

875

×75×

675

×75×

645

×35×

513

3.2

426

12.8

428

14.3

3.7

3.60

1955 11

63

100×

100×

875

×75×

675

×75×

645

×35×

513

9.3

427

11.3

426

10.9

3.7

3.65

1970 Se

cond

ary

brac

ing

patt

ern

II19

72 10.

53

100×

100×

875

×75×

675

×75×

645

×35×

525

9.2

438

16.6

436

21.4

4.2

3.55

1987 2

1.3

310

0×10

0×8

75×7

5×6

75×7

5×6

45×3

5×5

229.

843

315

.843

320

.94.

13.

4720

02 31.

34

100×

100×

875

×75×

675

×75×

645

×35×

517

8.4

410

6.7

408

6.9

4.6

4.55

2017 4

0.5

410

0×10

0×8

75×7

5×6

75×7

5×6

45×3

5×5

195.

340

97.

340

710

.74.

84.

5520

32

2047

2062

2077

aF

L=f

orce

inth

ele

g;F

b=f

orce

inth

em

ain

brac

ing;

H=t

otal

shea

rin

the

stru

ctur

e,F

sb=f

orce

inth

ese

cond

ary

brac

ing;t m

ax=h

oriz

onta

lco

mpo

nent

ofsh

ear

2078

inth

ese

cond

ary

brac

ings

.20

79

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567568569

570 Fig. 7. Failure modes of X- and K-braced frames.571

433 �434 The ratio of the maximum value of secondary bracing forces obtained from non-435 linear analyses to the maximum leg forces, (Fsb/FL) is also presented in Table 9.436 The ratio is in the range of 0.9–2.7%, comparable to the ASCE recommended437 range of 0.5–2.5%. The mean value of the ratio is equal to 1.6% and the coefficient438 of variation is equal to 0.375.439 �440 The ratio of tmax to leg force, FL, as a percentage is also given in Table 9. Usual441 design practice has been to use a value of 2.5%. It is seen that the mean value442 of tmax/F expressed as a percentage is equal to 1.01% with a coefficient of vari-443 ation of 0.16. It is seen that designing the secondary bracings for a characteristic444 panel shear of 1.3% of the leg force is the most consistent method for designing445 secondary bracings in addition to prescribing a limiting slenderness ratio in the446 range of 250–330.

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2083 Table 92084 Comparison of non-linear analysis results with code provisions20852086

209220982104

Sl. V/H M/bH Maximum force in Comparisonno. redundants, FSb (kN)2112

211521182121

FSb,NAL FSb,Code FSb,Code/FSb,NAL tmax/FL×100 Fsb,NLA/FL×1002130

213921482157

Secondary bracing pattern I2159

1 0.5 3 9.0 57 0.63 0.93 2.072168

3 0.5 3 11.7 6 0.51 1.07 2.682177

4 0.5 3 17.0 8 0.47 1.29 2.252186

5 0.5 3 5.3 5.4 1.02 1.10 1.622195

6 1.3 3 6.8 5.7 0.84 0.80 1.582204

7 1.3 3 3.7 3.5 0.95 0.78 1.332213

8 0.5 4 10.0 5.7 0.57 1.13 2.452222

9 1.3 4 8.0 5.7 0.71 1.11 1.962231

10 4.5 3 3.7 5.7 1.54 1.00 0.872240

11 6 3 3.7 5.7 1.54 0.86 0.872249

Secondary bracing pattern II2251

1 0.5 3 4.23 3.7 0.87 0.81 0.962260

2 1.3 3 4.14 3.7 0.89 0.80 0.952269

3 1.3 4 4.55 3.7 0.81 1.12 1.132278

4 0.5 4 4.80 3.7 0.77 1.12 1.182287

2296

447 4. Summary and conclusions

448 Non-linear FEM models were developed for the analysis of panels of latticed angle449 towers by calibration with test results. It is found that the current methods of design450 of main leg members based on the forces obtained from a linear analysis are not451 consistent with test results. The results obtained using non-linear analyses compare452 well with test results. Using such a model, full tower analysis can be done to obtain453 more accurate values of member forces including secondary bracing forces prior to454 failure and the strength of a tower.455 This analysis model was used to perform a parametric study to obtain forces in456 the secondary bracing members prior to failure. Based on this study it is rec-457 ommended that the secondary bracing member designs should meet both strength458 requirements (tmax�1.30FL/100) and stiffness requirements (l/r�250–330) to per-459 form their functions adequately.

460 5. Uncited references

461 [4,5,17,21].

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462 Acknowledgements

463 The authors acknowledge the constant support given by Dr. T.V.S.R. Appa Rao,464 Director, Dr. R. Narayanan, DGS, Structural Engineering Research Centre, Madras.465 The authors also wish to thank Mr. P.R. Natarajan, former Head, Tower Testing &466 Research Station, SERC, Madras for the technical support during the work.

467 References

468 [1] Roy S, Fang S-J, Rossow EC. Secondary stresses on transmission tower structures. J Energy469 Engng 1984;110(2).470 [2] Al-Bermani FGA, Kitipornchai S. Nonlinear analysis of transmission towers. J Engng Struct471 1992;14(3):139–51.472 [3] Yan H, Liu Y, Zhao D. Geometric nonlinear analysis of transmission tower with continuous legs.473 In: Advances in steel structures—volume I, Proceedings of International Conference on Advances474 in Steel Structures, Dec 11–14; Hong Kong, 1996:339–44.475 [4] Kitipornchai S, Lee HW. Inelastic buckling of single-angle, tee and double angle struts. J Construct476 Steel Res 1986;(6):3–21.477 [5] Chen WF, Astsuta T. Theory of beam-columns, behaviour and design, vol. 2. New York: McGraw-478 Hill, Inc., 1977.479 [6] Chuenmei G. Elasto plastic buckling of single angle columns. J Struct Div ASCE 1984;98(6):1391–480 5 [Pro. paper 18888].481 [7] Shan L, Peyrot AH. Plate element modeling of steel angle members. J Struct Engng 1988;114(4).482 [8] Rajmane Sanatkumar P. An investigation on the behaviour of X and K-bracings of hot rolled single483 angles. Thesis submitted in partial fulfillment of the requirements for a PhD, Indian Institute of484 Technology, Madras, Chennai, India; 1992.485 [9] Stoman SH. A stability criteria for X bracing system. J Struct Div ASCE 1988;114(ST8):1426–34.486 [10] Usami T, Galambos TV. Eccentrically loaded single angle columns. Publication of the International487 Association for Bridge and Structural Engineering, vol. 31-II. Zurich (Switzerland). p. 153–184.488 [11] Elgaaly M, Davids W, Dagher H. Non slender single angle struts. Engineering Journal, American489 Institute of Steel Construction, second quarter 1992:49–58.490 [12] Natarajan PR, Muralidharan K, Mohan SJ, Raghunathan MD. Buckling of eccentrically loaded single491 angle struts. Report no. RD-30/8. Madras (India): Structural Engineering Research Centre; 1992.492 [13] Natarajan PR, Muralidharan K, Mohan SJ. Buckling of K-bracing. Report no. RD-30/2. Madras493 (India): Structural Engineering Research Centre; 1991.494 [14] Natarajan PR, Muralidharan K, Mohan SJ, Raman NV. Studies on X-braced panels. Report no. RD-495 1418. Madras (India): Structural Engineering Research Centre; 1992.496 [15] Natarajan PR, Muralidharan K. Behaviour of a 6 m extension portion of a 220 kV transmission497 line tower with K-bracing. Report no. RD-14/6. Madras (India): Structural Engineering Research498 Centre; 1992.499 [16] BS Code of Practice for strength assessment of members of lattice towers and masts. London: British500 Standards Institute.501 [17] ECCS 1985. Recommendation for angles in lattice transmission towers. European Convention for502 Constructional Steel Work; Jan 1985.503 [18] Prasad Rao N, Kalyanaraman. Non-linear analysis of lattice panels in transmission line towers. In:504 Trans Tower 96: International Seminar on Modern Trends in Design of EHV Transmission Towers,505 Nagpur (India): Institute of Engineers; 1997:89–95.506 [19] ASCE manuals and reports on engineering practice no. 52, Guide for design of steel transmission507 towers. 2nd ed. New York: American Society of Civil Engineers.508 [20] Use of structural steel in over head transmission line towers—code of practice IS:802 (part 1/set509 2): 1992 third revision. New Delhi (India): Bureau of Indian Standards.

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510 [21] White DW, Chen WF, editors. Plastic hinge based methods for advanced analysis and design of511 steel frames, Bethlaham (USA): Structural Stability Research Council; 1993.512 [22] Natarajan PR, Muralidharan K, Mohan SJ, Raghunathan MD. Buckling of eccentrically loaded equal513 angle struts. In: International Conference on Stability of Structures, ICSS 95; Coimbatore (India),514 PSG College of Technology; 1995.51552353153954755556457275610581189144915361602168620812305

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