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34 Journal of Constructional Steel Research (2001) 37www.elsevier.com/locate/jcsr
38
42
43 Non-linear behaviour of lattice panel of angle44 towers
45
N. Prasad Rao
a
, V. Kalyanaraman
b,*
46 a Structural Engineering Research Centre, Chennai 600036, India
47 b 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 forces
53 using the linear static methods, assuming the members to be subjected to only axial loads and
54 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 usually
56 accounted for in the codal recommendations by modifying the effective length of the members
57 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 lattice
60 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 the
65 design of redundants are presented. 2001 Published by Elsevier Science Ltd.
66 Keywords: Lattice towers; Non-linear analysis; Compression members; Secondary bracings
67
68
1
27 *28 Corresponding author.
2930 E-mail address:[email protected] (V. Kalyanaraman).1
DTD v4.1.0 / JCSR2045
2
3 0143-974X/01/$ - see front matter 2001 Published by Elsevier Science Ltd.
4 PII: S0 1 4 3 - 9 7 4 X ( 0 1 ) 0 0 0 5 4 - 2
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69 1. Introduction
70 Microwave and overhead electric transmission line towers are usually fabricated
71 using angles for the main legs and the bracing members. The members are bolted
72 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-
74 bers are laterally supported at intervals in between their end nodes, using secondary
75 bracings or redundants (Fig. 1).
76 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-81 bly small, due to the following reasons:
82 83 The main legs are usually continuous through the joint.
516517
518
519520
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 hence
89 the force transfer in the members is eccentric.
90 91 The joints areflexible due to the local deformation of the leg of the angles under
92 the concentrated bolt forces.
93 94 The towers with high electric ratings tend to be flexible and hence equilibrium
95 in the deformed configuration has to be considered (large deformation effects).
96 97 The compression member deformation increases the bending moments (Pd98 effect).
99 Therefore, the angle members of the tower experience both axial force and bending
100 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 negligible
102 as often assumed in designs.
103 Roy et al. [1] studied the effects of joint rigidity and large deformation of tall
104 high-power electric transmission towers and concluded that these towers experienc-
105 ing heavier loads are moreflexible 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 and
108 large deflection, using an equivalent tangent stiffness matrix for the members. They
109 concluded that the material and geometric non-linearity have a major effect on the
110 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. Hui
112 et al. [3] presented details of geometric non-linear analysis of transmission towers
113 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 the
116 lattice tower members, which is impractical in the analysis of full towers. Rajmane
117 [8] used the beam-column element with seven degrees of freedom per node
118 (including the warping deformation) to analyse the braced frames including the
119 effects of eccentricity. Stoman [9] used minimisation of total potential energy to
120 study the plastic stability of X-braced systems and demonstrated the restraining
121 effects of tension diagonals.
122 Experimental studies have been conducted on concentrically and eccentrically
123 loaded single angles [6,8,1012], planar and three-dimensional lattice frames made
124 of angles [9,1315] 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, particularly
127 the eccentricity of connections, and theflexibility of the joints due to the local defor-
128 mation of the bolted leg of the angles. Rao and Kalyanaraman [18] presented details
129 of a non-linear analysis of a panel of lattice towers, considering the effects listed
130 earlier, which affect the tower member forces. In their study, plate elements were
131 used at joints and at plastic hinge locations, and beam-column elements at the rest
132 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 analysis
136 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 factors
138 that influence the strength of lattice towers and the design of redundant members.
139 Finally, the analysis results are compared with the empirical methods recommended
140 in codes of practice for the design of members. Based on this approach, a method
141 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 or
144 through one of the legs, are modeled and analysed. Subsequently, latticed plane
145 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 theoretically
149 should fail by bifurcation buckling about their weak axis, at the Euler buckling load.
150 However, due to imperfections they undergo a beamcolumn type of failure at loads151 below the Euler buckling load. At some stage, a part of the section subject to
152 maximum stress under combined bending and compression and residual stress yields.
153 Thefinal member failure may be by progressive yielding and plastic hinge formation
154 or partial yielding and local plate buckling, depending upon the width to thickness
155 ratio of legs and the overall slenderness ratio of the member.
156 In practice, the angle members in towers are usually loaded eccentrically through
157 only one leg, which is connected to gussets or directly to a leg of adjacent angle
158 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 and
160 fails by the formation of a plastic hinge under the combined action of axial load
161 and magnified biaxial bending. Further, the bolted leg of the angle undergoes local
162 deformation under the bearing force of the bolts, causingflexibility 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 of
166 angles loaded through end gussets, covering a range of slenderness ratio, size and
167 yield strength. Natarajan et al. [22] tested angles as part of a plane lattice. These
168 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 for
173 evaluating the compressive strength of angle members in lattice towers, accounting
174 for the effects of residual stresses, imperfections and end conditions. This method
175 involves modifying the effective slenderness ratio of the member, depending upon
176 the location of the member in the tower and the eccentricity of connection. The
177 strength of the angle members tested is compared with the results based on the code
178 recommendations, in Tables 1 and 2 under the column Code.
179 It is seen that the theoretical strengths evaluated based on code provisions are
180 conservative compared to concentric compression test results, which is under-
181 standable, since these code provisions are for the design of angle members with
182 eccentric end connections through one leg. However, the angle strengths based on183 code provisions are highly unconservative compared to eccentric compression test
184 results. It is also seen that the extent of the unsafe nature of code provisions decreases
185 with increases in the slenderness ratio. This comparison indicates that code pro-
186 visions do not seem to adequately account for eccentricity, imperfection and residual
187 stress effects, which have a major influence on the strength of compression members
188 in the intermediate slenderness ratio ranges (60l/r120).
189 2.1.2. Numerical method
190 The non-linear finite element analysis methods are effective for evaluating the
191 behaviour and strength of compression members and space frames, considering vari-
192 ous effects discussed earlier. The angles under compression were analysed with the
193 help of MSC-NASTRAN. The non-linear analysis capability of the software was
194 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 was
196 assumed in the analysis, to trace the non-linear large deformation behaviour. In
197 eccentrically loaded members the effect of member imperfection was neglected, since
198 the eccentric load caused much larger lateral deflection of members. Three different
199 models, with increasing elaboration, as given below, were used for angle members
200 under compression.
201202 Model 1 (Fig. 2(a)). A number of beam-column line elements (six in total) along
203 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 of
205 the member and the point of application of the load. The limit load in this model
206 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 at
213 the center of the member (0.2 times the length), where the member plastification
214 is expected to occur, the two legs of the angles were modeled using flat-shell
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3573574575
576
577
578Table1
579Singleanglesunderconc
entriccompression[8]
580
587594601 Angle
Length
L/r
Fy
Failureload(kN)
%
Differen
cewithtestresults:
section
(mm)
(N/mm2)
100(Theo
ryTest)/Test
614
Test
Code
Analysismodel
Analysism
odel
623
626629632
M1
M2
M3
Code
M1
M2
M3
646
660674688 50506
576
60
330
162
159.2
153
152
152
1.7
5.7
6.1
6.1
702 50506
814
85
330
139
131.1
145
140
140
5.9
4.0
0.4
0.4
716 50506
960
100
330
132
109
140
130
133
17.3
6.1
1.4
0.8
730 Mean
8.3
1.5
2.3
1.6
736 Standarddeviation
8.0
6.3
3.4
3.9
742
748754
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3757758759
760
761
762Table2
763Singleanglesunderecce
ntriccompressiona
764
771778785 Angelsection
Len
gth
L/r
Fy
Failureload(kN)
%
Differencewithtestresults:
(mm
)
(N/mm2)
100(TheoryTest)/Test
797
Test
Code
Analysismodel
Analysismodel
806
809812815
M1
M2
M3
Code
M1
M2
M3
829
843857871 50506
57
6
60
330
87.3
107
92.0
94.0
90.0
22.5
5.4
7.7
3.1
885 50506
96
0
100
330
78.5
92
75.0
79.0
78.0
17.2
4.5
0.6
0.6
899 50506
144
0
150
330
50.5
58
47.0
56.9
58.6
14.8
6.9
12.7
16.0
913 45453
62
5
72
255
47.3
62.2
42.4
45.0
49.3
31.5
10.4
4.9
4.2
927 90906
169
0
97
255
133.4
165.9
122.0
131.6
135.9
24.4
8.5
1.3
1.9
941 90906
200
0
114
255
122.6
148.7
111.6
122.9
121.9
21.3
9.0
0.2
4
0.6
955 40404
39
5
51
270
52.2
59.7
45.0
45.9
51.1
14.4
13.8
12.0
2.1
969 40404
63
2
82
270
39.3
52.4
37.5
39.6
43.1
33.3
4.6
0.8
9.7
983 40404
79
0
103
270
39.1
47.0
35.9
39.2
41.2
20.2
8.1
0.2
3
5.3
997 40404
94
8
123
270
37.4
40.7
32.8
34.9
39.4
8.8
12.3
6.7
5.3
1011
40404
110
4
143
270
33.8
33.6
29.3
32.3
33.8
0.6
13.3
4.4
0.0
1025
Mean
18.9
7.8
0.6
4
3.8
4
1031
Standarddeviation
9.7
5.4
6.7
5.3
1037
1043
1049
1055aM1:model-1;M2:m
odel-2;M3:model-3.
1056
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3524525
526
527528
529 Fig. 2. Eccentrically compressed single angle model.530
215 elements. This enabled modeling of the progressive yielding at the point of plastic
216 hinge formation and the subsequent failure by local buckling of the elements. At
217 the transition between the beam-column element and the flat-shell elements, rigid
218 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, the
220 gussets and the legs of the angle over 0.2 times the length at the ends were
221 modeled using theflat-shell elements. Beam elements were used to represent the
222 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 is
225 modeled using a number offlat-shell elements. Whenever the load is transferred
226 through gussets at the ends, the gusset plates also are modeled using theflat-shell
227 elements and the connections between the gussets and the angles are modeled
228 using the gap elements available in MSC-NASTRAN. The bolts are modeled using
229 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 their
232 pre-ultimate behaviour and the limit loads. The elasticplastic material property of
233 steel was represented by a bi-linear model, having modulus of elasticity up to a yield
234 stress equal to 2.0105 MPa and 2000 MPa beyond yield stress. The incremental
235 load and predictorcorrector iteration under each load increment were used in the
236 non-linear range. The Von-Mises criterion was used to define yielding. The isotropic
237 hardening model was used in the post-yield range. The load increments were carried
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238 out in 2030 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 based
241 on the three MSC-NASTRAN models. The percentage error
242 ((TheoryTest)100/Test), the mean and standard deviation of the errors also are
243 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 with
245 the test results. The model M3 comparison with the test results is the best of the
246 three, although model M2 is quite adequate. The model M1 has the largest mean
247 error among the three models, particularly in the eccentrically compressed cases. For
248 the further study of lattice frame models M1 and M2 are used, since the model M3
249 consumes a large amount of time and memory due to the large number of degrees
250 of freedom.
251 2.2. Behaviour of lattice frames
252 Rajmane [8] tested planar angle lattice frames, and Natarajan [13] tested planar
253 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 35. The
255 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 equations
258 These lattice frame test results are compared with the strengths based on code
259 provisions by the following procedure. The member forces are obtained from a linear
260 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 obtained
262 from the code provisions and linear analysis member forces are used to calculate
263 the frame strength. It is seen (Tables 35) that the code provisions either under- or
264 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 in
266 the code based design of members, for forces obtained from the linear elastic analysis
267 of a concentrically connected truss model, could be high, particularly in the case of
268 slender bracing members.
269 2.2.2. Numerical analysis
270 The conventional assumption of hinged joints does not represent the real joint
271 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, theflexibility of the bolt and the legs of the angle at the
275 joint were disregarded and the joints were assumed to be rigid by enforcing the
276 compatibility of translations and rotations in all the members meeting at the joint.
277 However, the effect of an eccentric bolted connection between members was
278 accounted for by using rigid elements between the bolt lines and the centroid of the
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3532533
534
535536
537 Fig. 3. Joint models. (a) Rigid bolted joint model. (b) Flexible bolted joint model.538
1059
1060 Table 3
1061 X-braced plane framesa10621063106810731078
Reference Bracing L/r Panel failure load (kN) % Difference with test results:
100(TheoryTest)/Test
1089
Test Code Model Model Code Element Element
M1 M2 model M1 model M2110311131123
1133
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 5.0117311781183
1188 a Model 1: beam-column model; model 2: beam-column and flat-shell model.
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3119011911192
1193
1194
1195
Table4
1196
K-bracedplaneandspac
eframes[13]a
1197
1202
1207
1212
Panel
L/r
Pan
elfailureload(kN)
%
Differencewithrespecttotestresults:
type
bracing
100(TheoryTest)/Test
1225
Tes
t
Code
Rigidjointmodel
Flexiblejointmodel
Code
Rigidjointmodel
Flexiblejointmodel
1235
1237
1239
1241
MembermodelM1MembermodelM1Member
Model
M1
ModelM1
Model
model
M2
M2
1256
1263
1270
1277
PF
SF
PF
SF
SF
PF
SF
PF
SF
SF
1293
1309
1325
1341
A*
144
36.8
30.1
37.8
36.4
18
2.7
1.0
1357
B
105
51.7
52.1
56.0
55.7
54.7
52.8
55.6
0.8
8.0
7.7
5.8
2.0
7.5
1373
C
105
52.0
52.1
57.9
57.8
55.1
54.3
55.6
0.2
11.0
11.0
6.0
4.0
6.9
1389
D
96
59.1
57.0
59.4
55.5
58.3
3.5
0.5
6.0
1.3
1405
Mean
5.1
7.2
4.5
5.9
0.0
5.2
1413
Standarddeviation
8.8
4.2
5.7
0.1
5.3
3.4
1421
1429
1437
1445aPF,planeframemod
el;SF,spaceframemodel.
Paneltype:A*:nosecondarybracingandwithsinglebolt
connection;B:onelevelsecondarybracing
;
1446
C:twolevelsecondaryb
racings;D:twolevelsecondarybracingsan
dcornerstays.
1447
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31450
1451 Table 51452 K-braced panel of a 6 m extension of a 220 kV tower [15]14531454145814621466
BracingL/r Panel failure load (kN) % Difference with respect to test
result: 100(TheoryTest)/Test
1476
Test Code Analysis Code Member model
11486
Member model
11494150115081515
191 51.6 66.4 54.3 28.6 5.2315221529
279 angle members (Fig. 3, elements 1 and 3 for members A and B, respectively) and
280 a beam element (element 2) joining these rigid elements was used to represent the
281 bolts. Freedom of relative rotation of the members about the axis of the single bolt
282 was modeled by keeping the torsional stiffness of the beam element very low (60
283 mm4).
284 The above rigid joint model does not account for the flexibility and the local
285 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 model
287 shown in Fig. 3(a). In this flexible joint model (FJM) a short segment of angles
288 joining at the node along with the bolts were studied. The angles were modeled
289 using flat-shell elements. The contact force transfer between the legs of the angles
290 was modeled using the gap elements, available in MSC-NASTRAN. The bolts in
291 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 two
294 different sets of member sizes to obtain the joint stiffness values in the practical
295 range of member sizes. These joint analyses results were used to evolve a beam
296 element connecting the centroidal lines of the two angles, with an equivalent stiff-
297 nesses. Theflexural stiffnesses of the connecting equivalent beam elements are given
298 in Table 6. The equivalent link elements were used in the full frame model, to
299 represent the jointflexibility and eccentricity. Such full frames with equivalent beam
300 elements corresponding to the flexible joint model are referred to as FJM. The FJM
301 has been used in the analyses of K-braced frames only.
1537
1538 Table 6
1539 Joint flexibility model results15401541154515491553
Angle member Joint rotation/unit moment Equivalent moment of
(rad/N mm) inertia of joint member
(mm4)1562
Leg Bracing1567157215771582
90908 45453 1.78108 10,2571587
45453 45453 1.93107 62115921597
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3540541
542
543544
545 Fig. 4. X-braced plane frame.546
302 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
304 of beam-column elements (model M1). In the second model 20% of the central
305 length of the angle compression members and 20% of the length closer to the joints
306 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 member
309 length/1000 in a few cases, to study their effects. The eccentricity of connections
310 had greater influence than the initial bow in these frames. The K-braced latticed
311 space frames were tested for different patterns of secondary bracings and in the
312
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-braced
315 frames, tested as three-dimensional lattices, were analysed as both plane and space
316 frames. The angle member models (M1) and (M2) were used in the case of the
317 flexible joint model of space frames and only the angle member model (M1) was
318 used in the case of the rigid model. Typical analytical models are shown in Figs. 4
319 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 different
321 nonlinear analysis models are presented in Tables 35. These non-linear analysis
322 results when compared with the test results indicate the following:
548549
550
551552
553 Fig. 5. K-braced frame [13].554
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558
559560
561 Fig. 6. Three-dimensional model of 6 m extension of a 220 kV tower [15]. (a) Secondary bracing
562 pattern (I); (b) secondary bracing pattern (II).563
323 324 The numerical analyses results for X- and K-braced lattice frames compare well
325 with the test results. The maximum error is 8%.
326 327 The flexible joint frame models generally compare better with test results in the
328 case of K-braced frames.
329 330 There is not much of a difference in the results obtained using the two angle
331 member element models, M1 and M2.
332 333 The mean and the standard deviation of the error in numerical analysis results
334 are less than 5%.
335 The results of the finite element analysis using member model M1, considering
336 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.
340 3. Behaviour of secondary bracings
341 The secondary bracing members are provided to reduce the unsupported length
342 and thus increase the buckling strength of the main compression members. Linear
343 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 zero
345 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 some
347 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 as
350 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 the
354 node of attachment of the secondary member to the main member. This force to
355 be applied is prescribed as a percentage of the main leg or other main bracing
356 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 node
358 where the secondary members meet the main member. The secondary bracing
359 forces should also be analysed separately, by applying 2.5% of the force in the
360 main leg distributed equally at all the interior nodal points along the length of
361 the leg excluding the first and the last node. The nodal forces should be applied
362 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 vary
368 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 slenderness
371 ratio of the redundants to be equal to 250.
372 Thus, it is seen that some variations in the design requirements of the secondary
373 bracings exist in codes. The non-linearfinite element analysis method, discussed in
374 the earlier section, can be used to evaluate the forces in the secondary bracings prior
375 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 the
379 secondary bracings in a typical bottom panel of a K-braced three-dimensional latticed
380 frame (Fig. 6). In a typical tower the force resultants in the form of vertical force
381 V, the shear force H and the over turning moment M vary over the height of the
382 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
384 M/bHratios in the practical range, where V, Hand Mare vertical force, shear force
1603
1604 Table 7
1605 Secondary member forces calculation BSI DD 13316061607160916111613
Applied force as percentage of leg load, F1615161716191621
Slenderness 0 to 45 50 55 60 65 70 75 80 85 90 95 100
ratio (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 2.0
force
(percentage
ofFL)16561671
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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 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 8
389 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
391 of the analysis cases (Sl. nos. 2, 3, 4, 5 and 7), to understand the impact of such
392 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-
394 figuration prior to failure is shown in Fig. 7(b). The shear H, corresponding to the
395 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, from
399 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 the
403 same in all the cases, the maximum difference being 4%.
404 405 The non-linear analysis results indicate appreciable increase in the maximum axial
406 force in the bracing. The increase can be as high as 38%. This is usually more
407 in cases where secondary bracings are very light or type 2 secondary bracings
408 are used. It is therefore essential to design the bracing members conservatively
409 for the force obtained from the linear analysis.
410 411 As the size of the secondary bracings decreases from the standard value (45355
412 having l/r250) to a lesser value (25254 having l/r330), the strength of the413 panel is appreciably decreased (Sl. no. 1 versus Sl. nos. 2 and 6 versus Sl. no.7
414 in Table 8). However, increases in the size of the secondary bracing above the
415 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 minimum
417 stiffness requirement of secondary bracings.
418 419 It is seen from the results of Sl. no.4 in Table 8 that the same secondary bracings
420 (45355) are able to sustain even a larger panel force without initiating failure
421 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 strength
423 of the panel, due to the strength being governed by the leg buckling.
424 The design recommendations of various codes are compared with the parametric
425 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, are
429 compared with the secondary bracing forces obtained from the non-linear analysis
430 results, Fsb,NLA, in terms of their ratios in Table 9. It is seen that the correlation
431 is very poor, with the mean value of the ratio equal to 0.86 and the coefficient
432 of variation equal to 0.38.
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3168716881689
1690
1691
1692
Table8
1693
Parametricstudyresultsa
1694
1702
1710
1718
Sl.no.
V/H
M/bH
Section
FailureHForcesinmembersatfailureH
tmax
1730
Linearanalysis
Non-linearanalysis
1743
Leg
Mainbrace
Belt
Redundant
FL
Fb
FL
Fb
Fsb
1758
1773
1788
1803Secondarybracingpatte
rnI
1805
1
0.5
3
1001008
75756
75756
45355
258
435
20.9
433
21.1
9.0
4.0
0
1820
2
0.5
3
1001008
75756
75756
25254
204.2
345
16.6
343
19.4
4.7
3.5
5
1835
3
0.5
3
1001008
75756
75756
60605
259.2
438
21.0
436
20.6
11.7
4.6
6
1850
4
0.5
3
1501501275756
1001008
45355
448
757
36.3
754
35.1
17.0
9.7
5
1865
5
0.5
3
656510
75756
75756
45355
194.5
328
15.8
327
15.8
5.3
3.6
0
1880
6
1.3
3
1001008
75756
75756
45355
229
433
18.6
431
18.5
6.8
3.4
0
1895
7
1.3
3
1001008
75756
75756
25254
153.6
290
12.5
278
17.3
3.7
2.1
7
1910
8
0.5
4
1001008
75756
75756
45355
195.2
409
7.3
407
9.3
10.0
4.6
0
1925
9
1.3
4
1001008
75756
75756
45355
178.8
410
6.7
408
8.8
8.0
4.5
5
1940
10
4.5
3
1001008
75756
75756
45355
133.2
426
12.8
428
14.3
3.7
3.6
0
1955
11
6
3
1001008
75756
75756
45355
139.3
427
11.3
426
10.9
3.7
3.6
5
1970Secondarybracingpatte
rnII
1972
1
0.5
3
1001008
75756
75756
45355
259.2
438
16.6
436
21.4
4.2
3.5
5
1987
2
1.3
3
1001008
75756
75756
45355
229.8
433
15.8
433
20.9
4.1
3.4
7
2002
3
1.3
4
1001008
75756
75756
45355
178.4
410
6.7
408
6.9
4.6
4.5
5
2017
4
0.5
4
1001008
75756
75756
45355
195.3
409
7.3
407
10.7
4.8
4.5
5
2032
2047
2062
2077aFL=forceintheleg;Fb=forceinthemainbracing;H=totalshear
inthestructure,
Fsb=forceinthesecondary
bracing;tmax=horizontalcomponentofshea
r
2078
inthesecondarybracings.
2079
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567
568569
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 0.92.7%, comparable to the ASCE recommended
437 range of 0.52.5%. The mean value of the ratio is equal to 1.6% and the coefficient
438 of variation is equal to 0.375.
439 440 The ratio oftmax to leg force, FL, as a percentage is also given in Table 9. Usual
441 design practice has been to use a value of 2.5%. It is seen that the mean value442 oftmax/Fexpressed 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 characteristic
444 panel shear of 1.3% of the leg force is the most consistent method for designing
445 secondary bracings in addition to prescribing a limiting slenderness ratio in the
446 range of 250330.
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32082
2083 Table 92084 Comparison of non-linear analysis results with code provisions20852086209220982104
Sl. V/H M/bH Maximum force in Comparison
no. redundants, FSb (kN)2112211521182121
FSb,NAL FSb,Code FSb,Code/FSb,NAL tmax/FL100 Fsb,NLA/FL1002130213921482157
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 1.1822872296
447 4. Summary and conclusions
448 Non-linear FEM models were developed for the analysis of panels of latticed angle
449 towers by calibration with test results. It is found that the current methods of design
450 of main leg members based on the forces obtained from a linear analysis are not
451 consistent with test results. The results obtained using non-linear analyses compare
452 well with test results. Using such a model, full tower analysis can be done to obtain
453 more accurate values of member forces including secondary bracing forces prior to
454 failure and the strength of a tower.
455 This analysis model was used to perform a parametric study to obtain forces in
456 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 strength
458 requirements (tmax1.30FL/100) and stiffness requirements (l/r250330) 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
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