Journal of Constructional Steel Research (2001) Structural Engineering Research Centre, Chennai , India b

Transcription

1 Journal of Constructional Steel Research (00) 3 Non-linear behaviour of lattice panel of angle towers N. Prasad Rao a, V. Kalyanaraman b,* 7 a Structural Engineering Research Centre, Chennai 0003, India b Department of Civil Engineering, Indian Institute of Technology, Madras, Chennai 0003, India 8 Received 8 September 000; accepted 8 September Abstract Lattice microwave towers and transmission towers are frequently made of angles bolted 3 together directly or through gussets. Such towers are normally analysed to obtain design forces using the linear static methods, assuming the members to be subjected to only axial loads and the deformations to be small. The effects of the end restraints, eccentricity of connections and secondary bracings (redundants) on the strength of the compression members are usually accounted for in the codal recommendations by modifying the effective length of the members 7 and thus the design compressive strength. Hence, forces in the redundants are not known from 8 the analysis and their design is empirical. In this study, non-linear analysis of angle com- 9 pression members and the single panel of angle planar as well as three-dimensional lattice 0 frames, as in typical lattice towers, are carried out using MSC-NASTRAN software. Account is taken of member eccentricity, local deformation as well as rotational rigidity of joints, beam- column effects and material non-linearity. The analytical models are calibrated with test results. 3 Using this calibrated model, parametric studies are carried out to evaluate the forces in the redundants. The results are compared with codal provisions and recommendations for the design of redundants are presented. 00 Published by Elsevier Science Ltd. Keywords: Lattice towers; Non-linear analysis; Compression members; Secondary bracings * Corresponding author address: kalyan@civil.iitm.ernet.in (V. Kalyanaraman) X/0/$ - see front matter 00 Published by Elsevier Science Ltd. PII: S03-97X(0)000- JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P DTD v..0 / JCSR0

2 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 3 9. Introduction 70 Microwave and overhead electric transmission line towers are usually fabricated 7 using angles for the main legs and the bracing members. The members are bolted 7 together, either directly or through gusset plates. In order to reduce the unsupported 73 length and thus increase their buckling strength, the main legs and the bracing mem- 7 bers are laterally supported at intervals in between their end nodes, using secondary 7 bracings or redundants (Fig. ). 7 The lattice towers are usually analysed assuming the members to be concentrically 77 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 and 79 hence are not included in the linear analysis models. However, the main legs and 80 the bracing members are not axially loaded and the redundant forces are not negligi- 8 bly small, due to the following reasons: 8 83 The main legs are usually continuous through the joint Fig.. Tower configuration. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

3 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Usually more than one bolt is used in the connections and hence the joints are 8 semi-rigid The angle members are normally bolted through only one of their legs and hence 89 the force transfer in the members is eccentric The joints are flexible due to the local deformation of the leg of the angles under 9 the concentrated bolt forces The towers with high electric ratings tend to be flexible and hence equilibrium 9 in the deformed configuration has to be considered (large deformation effects) The compression member deformation increases the bending moments (P d 98 effect). 99 Therefore, the angle members of the tower experience both axial force and bending 00 moments, even well before the tower fails. This also produces forces in the redundant 0 members due to their participation in overall frame action, which are not negligible 0 as often assumed in designs. 03 Roy et al. [] studied the effects of joint rigidity and large deformation of tall 0 high-power electric transmission towers and concluded that these towers experienc- 0 ing heavier loads are more flexible and the secondary effects are more pronounced. 0 Al-Bermani and Kitipornchai [] evaluated the ultimate strength of towers consider- 07 ing the material (lumped plasticity) and geometric non-linearity, joint flexibility and 08 large deflection, using an equivalent tangent stiffness matrix for the members. They 09 concluded that the material and geometric non-linearity have a major effect on the 0 ultimate strength of towers. They attributed the larger difference between their analy- sis and experimental results to the bolt slippage, not modeled in the analysis. Hui et al. [3] presented details of geometric non-linear analysis of transmission towers 3 to trace the load deformation behaviour, treating the main legs as beam-columns and the bracings as truss members, using updated Lagrangian formulation. Chuenmei [] and Shan et al. [7] used rectangular plate elements to model the lattice tower members, which is impractical in the analysis of full towers. Rajmane 7 [8] used the beam-column element with seven degrees of freedom per node 8 (including the warping deformation) to analyse the braced frames including the 9 effects of eccentricity. Stoman [9] used minimisation of total potential energy to 0 study the plastic stability of X-braced systems and demonstrated the restraining effects of tension diagonals. Experimental studies have been conducted on concentrically and eccentrically 3 loaded single angles [,8,0 ], planar and three-dimensional lattice frames made of angles [9,3 ] and full-scale towers []. It is seen that the analytical studies reported have not considered all the important factors that may influence the behaviour of lattice towers before failure, particularly 7 the eccentricity of connections, and the flexibility of the joints due to the local defor- 8 mation of the bolted leg of the angles. Rao and Kalyanaraman [8] presented details 9 of a non-linear analysis of a panel of lattice towers, considering the effects listed 30 earlier, which affect the tower member forces. In their study, plate elements were 3 used at joints and at plastic hinge locations, and beam-column elements at the rest 3 of the locations of members, to model the angle members in the towers. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

4 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 3 33 This paper initially presents details of non-linear analyses of angle members and 3 lattice towers made of angles, using MSC-NASTRAN. These analysis models con- 3 sider all the factors listed earlier, which affect the tower behaviour, and the analysis 3 results are calibrated against test results. Using the model thus developed, a para- 37 metric study has been done in order to understand the effects of the various factors 38 that influence the strength of lattice towers and the design of redundant members. 39 Finally, the analysis results are compared with the empirical methods recommended 0 in codes of practice for the design of members. Based on this approach, a method for designing redundants in lattice towers is recommended.. Calibration of non-linear analysis model 3 Initially, single angle compression test specimens, loaded through the centroid or through one of the legs, are modeled and analysed. Subsequently, latticed plane frame and space frame tests using angle members are also modeled and analysed. These analyses help to calibrate the models used in the subsequent parametric studies. 7.. Single angles under compression 8 Concentrically loaded, ideal single angle compression members theoretically 9 should fail by bifurcation buckling about their weak axis, at the Euler buckling load. 0 However, due to imperfections they undergo a beam column type of failure at loads below the Euler buckling load. At some stage, a part of the section subject to maximum stress under combined bending and compression and residual stress yields. 3 The final member failure may be by progressive yielding and plastic hinge formation or partial yielding and local plate buckling, depending upon the width to thickness ratio of legs and the overall slenderness ratio of the member. In practice, the angle members in towers are usually loaded eccentrically through 7 only one leg, which is connected to gussets or directly to a leg of adjacent angle 8 members. Consequently, they undergo bi-axial bending in addition to axial com- 9 pression. Under this loading, the cross section of the angle progressively yields and 0 fails by the formation of a plastic hinge under the combined action of axial load and magnified biaxial bending. Further, the bolted leg of the angle undergoes local deformation under the bearing force of the bolts, causing flexibility in the connection, 3 and shear lag in the member. Rajmane [8] tested single angles under concentric compression and eccentric com- pression by loading through end gussets. Chuenmei [] presented test results of angles loaded through end gussets, covering a range of slenderness ratio, size and 7 yield strength. Natarajan et al. [] tested angles as part of a plane lattice. These 8 test results are compared with strengths obtained from design equations and numeri- 9 cal analysis in the following sections. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

5 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Design equations 7 The British Standards Institute [], the American Society of Civil Engineers [9] 7 and the Bureau of Indian Standards [0] specify essentially the same method for 73 evaluating the compressive strength of angle members in lattice towers, accounting 7 for the effects of residual stresses, imperfections and end conditions. This method 7 involves modifying the effective slenderness ratio of the member, depending upon 7 the location of the member in the tower and the eccentricity of connection. The 77 strength of the angle members tested is compared with the results based on the code 78 recommendations, in Tables and under the column Code. 79 It is seen that the theoretical strengths evaluated based on code provisions are 80 conservative compared to concentric compression test results, which is under- 8 standable, since these code provisions are for the design of angle members with 8 eccentric end connections through one leg. However, the angle strengths based on 83 code provisions are highly unconservative compared to eccentric compression test 8 results. It is also seen that the extent of the unsafe nature of code provisions decreases 8 with increases in the slenderness ratio. This comparison indicates that code pro- 8 visions do not seem to adequately account for eccentricity, imperfection and residual 87 stress effects, which have a major influence on the strength of compression members 88 in the intermediate slenderness ratio ranges (0 l/r 0) Numerical method 90 The non-linear finite element analysis methods are effective for evaluating the 9 behaviour and strength of compression members and space frames, considering vari- 9 ous effects discussed earlier. The angles under compression were analysed with the 93 help of MSC-NASTRAN. The non-linear analysis capability of the software was 9 used for the strength evaluation. In the case of concentrically loaded members, sinus- 9 oidal initial imperfection amplitude of /000 of the length of the member was 9 assumed in the analysis, to trace the non-linear large deformation behaviour. In 97 eccentrically loaded members the effect of member imperfection was neglected, since 98 the eccentric load caused much larger lateral deflection of members. Three different 99 models, with increasing elaboration, as given below, were used for angle members 00 under compression. 0 0 Model (Fig. (a)). A number of beam-column line elements (six in total) along 03 the centroid of the section were used to model each angle in this model (M). 0 The eccentric loading was applied through a rigid link between the centroid of 0 the member and the point of application of the load. The limit load in this model 0 is reached in the MSC-NASTRAN analysis when the stress at the maximum 07 stressed point in the member reaches the yield stress. This is obviously conserva- 08 tive, especially in slender members, since it does not account for the post-first- 09 yield plastification of the maximum stressed section before failure. 0 Model (Fig. (b)). In this model (M) a major segment of the member is mod- elled using the beam-column elements as before. However, over a short length at 3 the center of the member (0. times the length), where the member plastification is expected to occur, the two legs of the angles were modeled using flat-shell JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

6 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Table Single angles under concentric compression [8] Angle Length L/r F y Failure load (kn) % Difference with test results: section (mm) (N/mm ) 00 (Theory Test)/Test Test Code Analysis model Analysis model M M M3 Code M M M Mean Standard deviation JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

7 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Table Single angles under eccentric compression a Angel section Length L/r F y Failure load (kn) % Difference with test results: 797 (mm) (N/mm ) 00 (Theory Test)/Test Test Code Analysis model Analysis model M M M3 Code M M M Mean Standard deviation a M: model-; M: model-; M3: model-3. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

8 8 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Fig.. Eccentrically compressed single angle model. elements. This enabled modeling of the progressive yielding at the point of plastic hinge formation and the subsequent failure by local buckling of the elements. At 7 the transition between the beam-column element and the flat-shell elements, rigid 8 elements were used to connect the beam column node to the nodes of the flat- 9 shell elements. Whenever the load is transferred through gussets at the ends, the 0 gussets and the legs of the angle over 0. times the length at the ends were modeled using the flat-shell elements. Beam elements were used to represent the bolts, connecting the gussets and the beam-columns/flat-shell elements. 3 Model 3 (Fig. (c)). In this case (M3), the entire length of the angle member is modeled using a number of flat-shell elements. Whenever the load is transferred through gussets at the ends, the gusset plates also are modeled using the flat-shell 7 elements and the connections between the gussets and the angles are modeled 8 using the gap elements available in MSC-NASTRAN. The bolts are modeled using 9 beam elements. 30 The non-linear analysis capability of MSC-NASTRAN, accounting for the geo- 3 metric and material non-linearity, was used to analyse the models and obtain their 3 pre-ultimate behaviour and the limit loads. The elastic plastic material property of 33 steel was represented by a bi-linear model, having modulus of elasticity up to a yield 3 stress equal to.0 0 MPa and 000 MPa beyond yield stress. The incremental 3 load and predictor corrector iteration under each load increment were used in the 3 non-linear range. The Von-Mises criterion was used to define yielding. The isotropic 37 hardening model was used in the post-yield range. The load increments were carried JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

9 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) out in 0 30 steps, until the limit point was reached in the load deformation behav- 39 iour. 0 The test results are compared in Tables and with the strength evaluated based on the three MSC-NASTRAN models. The percentage error ((Theory Test) 00/Test), the mean and standard deviation of the errors also are 3 presented in Tables and, corresponding to the three models (M, M and M3, respectively). It is seen that the MSC-NASTRAN model results compare well with the test results. The model M3 comparison with the test results is the best of the three, although model M is quite adequate. The model M has the largest mean 7 error among the three models, particularly in the eccentrically compressed cases. For 8 the further study of lattice frame models M and M are used, since the model M3 9 consumes a large amount of time and memory due to the large number of degrees 0 of freedom... Behaviour of lattice frames Rajmane [8] tested planar angle lattice frames, and Natarajan [3] tested planar 3 and three-dimensional angle lattice frames, consisting of X bracings and K bracings. Details of the test specimens and results are presented in Fig. 3 and Tables 3. The experimental strengths of these frames are compared with the code based and numeri- cal analysis based strengths as discussed below Code equations 8 These lattice frame test results are compared with the strengths based on code 9 provisions by the following procedure. The member forces are obtained from a linear 0 elastic analysis of concentrically connected lattice truss models of the frame, as com- monly done in practice. The design strength of the critical angle member as obtained from the code provisions and linear analysis member forces are used to calculate 3 the frame strength. It is seen (Tables 3 ) that the code provisions either under- or overestimate the actual strength of the lattice frame by as much as 8% (conservative) to +9% (unconservative). It is clear from this study that the error in the code based design of members, for forces obtained from the linear elastic analysis 7 of a concentrically connected truss model, could be high, particularly in the case of 8 slender bracing members Numerical analysis 70 The conventional assumption of hinged joints does not represent the real joint 7 behaviour in latticed towers. Two types of joint models given below, to represent 7 the bolted connections between angles in the frames, were evaluated in the numeri- 73 cal study. 7 In the rigid joint model, the flexibility of the bolt and the legs of the angle at the 7 joint were disregarded and the joints were assumed to be rigid by enforcing the 7 compatibility of translations and rotations in all the members meeting at the joint. 77 However, the effect of an eccentric bolted connection between members was 78 accounted for by using rigid elements between the bolt lines and the centroid of the JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

10 0 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Fig. 3. Joint models. (a) Rigid bolted joint model. (b) Flexible bolted joint model Table X -braced plane frames a Reference Bracing L/r Panel failure load (kn) % Difference with test results: 00 (Theory Test)/Test Test Code Model Model Code Element Element M M model M model M Mean Standard deviation a Model : beam-column model; model : beam-column and flat-shell model. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

11 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Table K-braced plane and space frames [3] a Panel L/r Panel failure load (kn) % Difference with respect to test results: type bracing 00 (Theory Test)/Test Test Code Rigid joint model Flexible joint model Code Rigid joint model Flexible joint model Member model M Member model M Member Model M Model M Model model M M PF SF PF SF SF PF SF PF SF SF A* B C D Mean Standard deviation a PF, plane frame model; SF, space frame model. Panel type: A*: no secondary bracing and with single bolt connection; B: one level secondary bracing; C: 7 two level secondary bracings; D:two level secondary bracings and corner stays. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

12 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 30 Table 3 K-braced panel of a m extension of a 0 kv tower [] Bracing L/r Panel failure load (kn) % Difference with respect to test result: 00 (Theory Test)/Test Test Code Analysis Code Member model Member model angle members (Fig. 3, elements and 3 for members A and B, respectively) and 80 a beam element (element ) joining these rigid elements was used to represent the 8 bolts. Freedom of relative rotation of the members about the axis of the single bolt 8 was modeled by keeping the torsional stiffness of the beam element very low (0 83 mm ). 8 The above rigid joint model does not account for the flexibility and the local 8 deformation of the legs of the angles at the bolted joint. For evaluating these effects, 8 a finite element analysis of the joint region alone was carried out using the model 87 shown in Fig. 3(a). In this flexible joint model (FJM) a short segment of angles 88 joining at the node along with the bolts were studied. The angles were modeled 89 using flat-shell elements. The contact force transfer between the legs of the angles 90 was modeled using the gap elements, available in MSC-NASTRAN. The bolts in 9 the joint were modeled using a rod element. 9 Static analyses of the joint model were carried out to obtain the joint stiffness 93 considering the local deformation effects. The analyses were carried out for two 9 different sets of member sizes to obtain the joint stiffness values in the practical 9 range of member sizes. These joint analyses results were used to evolve a beam 9 element connecting the centroidal lines of the two angles, with an equivalent stiff- 97 nesses. The flexural stiffnesses of the connecting equivalent beam elements are given 98 in Table. The equivalent link elements were used in the full frame model, to 99 represent the joint flexibility and eccentricity. Such full frames with equivalent beam 300 elements corresponding to the flexible joint model are referred to as FJM. The FJM 30 has been used in the analyses of K-braced frames only Table 39 0 Joint flexibility model results Angle member Leg Bracing Joint rotation/unit moment Equivalent moment of (rad/n mm) inertia of joint member (mm ) , JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

13 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Fig.. X-braced plane frame. 30 Two types of models were used to represent the angle members in the frame, as 303 discussed earlier. In one, the entire length of the angle is represented by a number 30 of beam-column elements (model M). In the second model 0% of the central 30 length of the angle compression members and 0% of the length closer to the joints 30 in the angle tension members were modeled using flat-shell elements (model M), 307 as discussed earlier. 308 The non-linear analyses were carried out assuming an initial bow of member 309 length/000 in a few cases, to study their effects. The eccentricity of connections 30 had greater influence than the initial bow in these frames. The K-braced latticed 3 space frames were tested for different patterns of secondary bracings and in the 3 analytical model of these frames, the different secondary bracing patterns were rep- 33 resented. 3 Only plane frame analyses were carried out in X-braced frames, whereas K-braced 3 frames, tested as three-dimensional lattices, were analysed as both plane and space 3 frames. The angle member models (M) and (M) were used in the case of the 37 flexible joint model of space frames and only the angle member model (M) was 38 used in the case of the rigid model. Typical analytical models are shown in Figs. 39 and. Some of the failure mode shapes are shown in Fig.. 30 The strength of the frames as obtained for different frames tested and different 3 nonlinear analysis models are presented in Tables 3. These non-linear analysis 3 results when compared with the test results indicate the following: Fig.. K-braced frame [3]. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

14 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Fig.. Three-dimensional model of m extension of a 0 kv tower []. (a) Secondary bracing 3 pattern (I); (b) secondary bracing pattern (II) The numerical analyses results for X- and K-braced lattice frames compare well 3 with the test results. The maximum error is 8% The flexible joint frame models generally compare better with test results in the 38 case of K-braced frames There is not much of a difference in the results obtained using the two angle 33 member element models, M and M The mean and the standard deviation of the error in numerical analysis results 33 are less than %. 33 The results of the finite element analysis using member model M, considering 33 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 for 338 parametric studies in the following sections without incurring the high expenses of 339 experimental studies Behaviour of secondary bracings 3 The secondary bracing members are provided to reduce the unsupported length 3 and thus increase the buckling strength of the main compression members. Linear 33 elastic analysis of lattice towers with secondary bracings, assuming the member con- 3 nectivity to be concentric and hinged, would normally indicate zero or near zero 3 force in the secondary members. Hence no force for the design of secondary bracings 3 can be obtained from such analyses. However, secondary bracings should have some 37 minimum strength and stiffness to perform intended functions Code methods 39 Codes of practice suggest provisions for the design of the secondary bracings as 30 given below. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

15 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) British code. The British code prescribes the application of a fictitious load acting 33 transverse to the main member being stabilized by the secondary member, at the 3 node of attachment of the secondary member to the main member. This force to 3 be applied is prescribed as a percentage of the main leg or other main bracing 3 member force, depending upon the slenderness ratio of the member (Table 7). 37 This force should be applied in the plane of the bracings in turn at each node 38 where the secondary members meet the main member. The secondary bracing 39 forces should also be analysed separately, by applying.% of the force in the 30 main leg distributed equally at all the interior nodal points along the length of 3 the leg excluding the first and the last node. The nodal forces should be applied 3 transverse to the leg member in the plane of the bracing ASCE Manual. The maximum slenderness ratio of the secondary bracing mem- 3 bers is restricted to be below 330. This manual does not require calculation of 3 forces for which the secondary bracing members have to be designed. However, 37 it suggests that the magnitude of the load in the redundant members can vary 38 from 0. to.% of the force in the supported member IS: 80 (99). This standard specifies the maximum limit on the slenderness 37 ratio of the redundants to be equal to Thus, it is seen that some variations in the design requirements of the secondary 373 bracings exist in codes. The non-linear finite element analysis method, discussed in 37 the earlier section, can be used to evaluate the forces in the secondary bracings prior 37 to failure. The forces in the secondary bracings so evaluated could serve as a guide- 37 line for the design of secondary bracing members Numerical parametric study 378 For this purpose a parametric study was carried out to evaluate the forces in the 379 secondary bracings in a typical bottom panel of a K-braced three-dimensional latticed 380 frame (Fig. ). In a typical tower the force resultants in the form of vertical force 38 V, the shear force H and the over turning moment M vary over the height of the 38 tower. In the parametric study of the single panel of the tower, the force resultants 383 at the top of the panel were applied corresponding to different values of V/H and 38 M/bH ratios in the practical range, where V, H and M are vertical force, shear force 03 0 Table Secondary member forces calculation BSI DD Applied force as percentage of leg load, F Slenderness 0 to ratio (L/r) 0 Applied force (percentage of F L ) JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

16 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 3 38 and over turning moment resultants acting at the centre at the top of the panel. The 38 forces in the four corner nodes in the model were evaluated corresponding to these 387 force resultant values and were applied in the three orthogonal directions at the four 388 top nodes, so as to obtain the desired ratio of the force resultants as given in Table The sections of the main bracing, secondary bracing and leg members were kept 390 constant in most cases. Changes in the size of these members were made in a few 39 of the analysis cases (Sl. nos., 3,, and 7), to understand the impact of such 39 changes. In all cases secondary ties joining the main bracings on two adjacent faces 393 of the three-dimensional latticed tower were provided. A typical displacement con- 39 figuration prior to failure is shown in Fig. 7(b). The shear H, corresponding to the 39 failure of the structure as obtained from the non-linear analysis is given in Table 8, 39 in addition to the corresponding maximum compressive forces in the main leg, F L, 397 main bracing, F b, and the secondary bracings F sb. Further, the maximum values of 398 equivalent panel shear, t max, corresponding to the secondary bracing forces, F sb, from 399 the non-linear analysis at limit load, are also presented in Table The parametric study results in Table 8 indicate the following: 0 0 The leg forces, F L, obtained from linear and non-linear analyses are nearly the 03 same in all the cases, the maximum difference being %. 0 0 The non-linear analysis results indicate appreciable increase in the maximum axial 0 force in the bracing. The increase can be as high as 38%. This is usually more 07 in cases where secondary bracings are very light or type secondary bracings 08 are used. It is therefore essential to design the bracing members conservatively 09 for the force obtained from the linear analysis. 0 As the size of the secondary bracings decreases from the standard value ( 3 having l/r 0) to a lesser value ( having l/r 330), the strength of the 3 panel is appreciably decreased (Sl. no. versus Sl. nos. and versus Sl. no.7 in Table 8). However, increases in the size of the secondary bracing above the standard value do not seem to improve the strength of the panel appreciably (Sl. no. versus Sl. no. 3 in Table 8). This indicates the importance of the minimum 7 stiffness requirement of secondary bracings. 8 9 It is seen from the results of Sl. no. in Table 8 that the same secondary bracings 0 ( 3 ) are able to sustain even a larger panel force without initiating failure when the other (leg and main bracing) member sizes are increased. Similarly, reduction in the main leg size (Sl. no. in Table 8) causes reduction in the strength 3 of the panel, due to the strength being governed by the leg buckling. The design recommendations of various codes are compared with the parametric study results in Table 9. The following conclusions can be drawn based on this com- parison: 7 8 The secondary bracing forces calculated based on BS recommendations, F sb, are 9 compared with the secondary bracing forces obtained from the non-linear analysis 30 results, F sb,nla, in terms of their ratios in Table 9. It is seen that the correlation 3 is very poor, with the mean value of the ratio equal to 0.8 and the coefficient 3 of variation equal to JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

17 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Table 8 93 Parametric 9 study results a Sl. no. V/H M/bH Section Failure H Forces in members at failure H t max 730 Linear analysis Non-linear analysis 73 Leg Main brace Belt Redundant F L F b F L F b F sb Secondary bracing pattern I Secondary bracing pattern II a F L =force in the leg; F b =force in the main bracing; H=total shear in the structure, F sb =force in the secondary bracing; t max =horizontal component of shear 078 in 079the secondary bracings. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

18 8 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Fig. 7. Failure modes of X- and K-braced frames The ratio of the maximum value of secondary bracing forces obtained from non- 3 linear analyses to the maximum leg forces, (F sb /F L ) is also presented in Table 9. 3 The ratio is in the range of 0.9.7%, comparable to the ASCE recommended 37 range of 0..%. The mean value of the ratio is equal to.% and the coefficient 38 of variation is equal to The ratio of t max to leg force, F L, as a percentage is also given in Table 9. Usual design practice has been to use a value of.%. It is seen that the mean value of t max /F expressed as a percentage is equal to.0% with a coefficient of vari- 3 ation of 0.. It is seen that designing the secondary bracings for a characteristic panel shear of.3% of the leg force is the most consistent method for designing secondary bracings in addition to prescribing a limiting slenderness ratio in the range of JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

19 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) Table Comparison of non-linear analysis results with code provisions Sl. V/H M/bH Maximum force in Comparison no. redundants, F Sb (kn) F Sb,NAL F Sb,Code F Sb,Code /F Sb,NAL t max /F L 00 F sb,nla /F L 00 Secondary bracing pattern I Secondary bracing pattern II Summary and conclusions 8 Non-linear FEM models were developed for the analysis of panels of latticed angle 9 towers by calibration with test results. It is found that the current methods of design 0 of main leg members based on the forces obtained from a linear analysis are not consistent with test results. The results obtained using non-linear analyses compare well with test results. Using such a model, full tower analysis can be done to obtain 3 more accurate values of member forces including secondary bracing forces prior to failure and the strength of a tower. This analysis model was used to perform a parametric study to obtain forces in the secondary bracing members prior to failure. Based on this study it is rec- 7 ommended that the secondary bracing member designs should meet both strength 8 requirements (t max.30f L /00) and stiffness requirements (l/r 0 330) to per- 9 form their functions adequately. 0. Uncited references [,,7,]. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

20 0 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 3 Acknowledgements 3 The authors acknowledge the constant support given by Dr. T.V.S.R. Appa Rao, Director, Dr. R. Narayanan, DGS, Structural Engineering Research Centre, Madras. The authors also wish to thank Mr. P.R. Natarajan, former Head, Tower Testing & Research Station, SERC, Madras for the technical support during the work. 7 References 8 [] Roy S, Fang S-J, Rossow EC. Secondary stresses on transmission tower structures. J Energy 9 Engng 98;0(). 70 [] Al-Bermani FGA, Kitipornchai S. Nonlinear analysis of transmission towers. J Engng Struct 7 99;(3):39. 7 [3] Yan H, Liu Y, Zhao D. Geometric nonlinear analysis of transmission tower with continuous legs. 73 In: Advances in steel structures volume I, Proceedings of International Conference on Advances 7 in Steel Structures, Dec ; Hong Kong, 99: [] Kitipornchai S, Lee HW. Inelastic buckling of single-angle, tee and double angle struts. J Construct 7 Steel Res 98;():3. 77 [] Chen WF, Astsuta T. Theory of beam-columns, behaviour and design, vol.. New York: McGraw- 78 Hill, Inc., [] Chuenmei G. Elasto plastic buckling of single angle columns. J Struct Div ASCE 98;98():39 80 [Pro. paper 8888]. 8 [7] Shan L, Peyrot AH. Plate element modeling of steel angle members. J Struct Engng 988;(). 8 [8] Rajmane Sanatkumar P. An investigation on the behaviour of X and K-bracings of hot rolled single 83 angles. Thesis submitted in partial fulfillment of the requirements for a PhD, Indian Institute of 8 Technology, Madras, Chennai, India; [9] Stoman SH. A stability criteria for X bracing system. J Struct Div ASCE 988;(ST8): 3. 8 [0] Usami T, Galambos TV. Eccentrically loaded single angle columns. Publication of the International 87 Association for Bridge and Structural Engineering, vol. 3-II. Zurich (Switzerland). p [] Elgaaly M, Davids W, Dagher H. Non slender single angle struts. Engineering Journal, American 89 Institute of Steel Construction, second quarter 99: [] Natarajan PR, Muralidharan K, Mohan SJ, Raghunathan MD. Buckling of eccentrically loaded single 9 angle struts. Report no. RD-30/8. Madras (India): Structural Engineering Research Centre; [3] Natarajan PR, Muralidharan K, Mohan SJ. Buckling of K-bracing. Report no. RD-30/. Madras 93 (India): Structural Engineering Research Centre; [] Natarajan PR, Muralidharan K, Mohan SJ, Raman NV. Studies on X-braced panels. Report no. RD Madras (India): Structural Engineering Research Centre; [] Natarajan PR, Muralidharan K. Behaviour of a m extension portion of a 0 kv transmission 97 line tower with K-bracing. Report no. RD-/. Madras (India): Structural Engineering Research 98 Centre; [] BS Code of Practice for strength assessment of members of lattice towers and masts. London: British 00 Standards Institute. 0 [7] ECCS 98. Recommendation for angles in lattice transmission towers. European Convention for 0 Constructional Steel Work; Jan [8] Prasad Rao N, Kalyanaraman. Non-linear analysis of lattice panels in transmission line towers. In: 0 Trans Tower 9: International Seminar on Modern Trends in Design of EHV Transmission Towers, 0 Nagpur (India): Institute of Engineers; 997: [9] ASCE manuals and reports on engineering practice no., Guide for design of steel transmission 07 towers. nd ed. New York: American Society of Civil Engineers. 08 [0] Use of structural steel in over head transmission line towers code of practice IS:80 (part /set 09 ): 99 third revision. New Delhi (India): Bureau of Indian Standards. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P

21 N. Prasad Rao, V. Kalyanaraman / Journal of Constructional Steel Research (00) 3 0 [] White DW, Chen WF, editors. Plastic hinge based methods for advanced analysis and design of steel frames, Bethlaham (USA): Structural Stability Research Council; 993. [] Natarajan PR, Muralidharan K, Mohan SJ, Raghunathan MD. Buckling of eccentrically loaded equal 3 angle struts. In: International Conference on Stability of Structures, ICSS 9; Coimbatore (India), PSG College of Technology; 99. JCSR: j. of constructional steel research - ELSEVIER :00:3 Rev.03x JCSR$$0P