- Email: [email protected]
Nonlinear analysis of transmission towers
Nonlinear analysis of transmission
towers
F. G. A. AI-Bermani and S. K i t i p o r n c h a i Department of Civil Engineering, The University of Queensland, St Lucia, Queensland, 4072 Australia (Received May 1991; revised August 1991)
A nonlinear analytical technique for predicting and simulating the ultimate structural behaviour of self-supporting transmission towers under static load conditions is presented. The method considers both the geometric and material nonlinear effects and treats the angle members in the tower as general asymmetrical thin-walled beamcolumn elements. Modelling of material nonlinearity for angle members is based on the assumption of lumped plasticity coupled with the concept of a yield surface in force space. A formex formulation is used for automatic generation of data necessary for the analysis. The developed software, AK TOWER, is used to predict the ultimate behaviour of two full-scale towers recently tested in Australia.
Keywords: elasto-plastic analysis, formex algebra, full-scale testing, steel structure, transmission tower, ultimate strength
Transmission tower structures are generally constructed using asymmetric thin-walled angle section members which are eccentrically connected. They are widely regarded as one of the most difficult forms of lattice structure to analyse. Proof-loading or full-scale testing of such structures has traditionally formed an integral part of the development of tower design. Stress calculations in the structure are normally obtained from a linear elastic analysis where members are assumed to be axially loaded and, for the majority of cases, pinconnected. In practice, such conditions do not exist and members are detailed to minimize bending stresses. Despite this, results from full-scale testing of transmission towers indicate that bending stresses in the members can be as significant as axial stresses 1. Design practices for transmission towers are different from those for other steel structures in that stresses are permitted to be higher because towers are tested to their ultimate design strength and designs incorporate modifications based on test results. A recent study by the Electric Power Research Institute 2 (EPRI) indicated that current design practices have, for the most part, served the industry well. However, data from full-scale tests show that the behaviour of transmission towers under complex load situations cannot be consistently predicted using the present techniques. The investigation by EPRI 2 also revealed that out of the 57 structure load cases conducted, 23% experienced premature failure. On average, failure occurred at 95.4% of the design load level but failure could occur at unexpected locations. Further, available data showed considerable discrepancies between member forces computed from linear
elastic truss analyses and those measured from full-scale tests. The EPRI report 2 indicated that the linear elastic truss analysis method for transmission towers should be used with caution. For these reasons, there is a need to develop a method of analysis capable of predicting the ultimate structural behaviour of transmission towers more accurately than the present 3D linear elastic truss approach. Such a refined technique would provide the designer with a better understanding of tower behaviour which undoubtedly will lead to a more economical structural design. Any saving in the design of one tower is magnified many times over because large numbers of towers of the same designs are usually constructed. For example, in a 250 km transmission line, there may be 500 towers of which up to 80% are of the same type. This paper presents a nonlinear analytical technique for simulating the ultimate structural behaviour of selfsupporting transmission towers under static loading. The proposed technique incorporates both the geometric and material nonlinear effects including large displacements, and treats the angle members in the tower as general asymmetric thin-walled beam-column elements 3. Based on an updated Lagrangian formulation, a deformation stiffness matrix" is used together with the linear and geometric stiffness matrices to enable accurate modelling of transmission tower structures with the least possible number of elements. Modelling of material nonlinearity for angle members is based on the assumption of lumped plasticity coupled with the concept of a yield surface in force space. A formex formulation is used for automatic generation of data necessary
0141-0296/92/030139-13 © 1992 Butterworth-Heinemann Ltd
Eng. Struct. 1992, Vol. 14, No 3
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Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
for the analysis of transmission towers. These include the topology, geometry, load and support conditions. The developed software, the AK TOWER program (AIBermani and Kitipornchai), is used to predict the ultimate behaviour of two full-scale transmission towers recently tested in Australia. The tests were conducted by the industry in conjunction with the electricity authority requirements for design verification. No measurements of member forces were reported but comparisons of test and predicted results are possible through the comparison of deflections, ultimate loads and the deflected shapes of the towers at ultimate load.
Brief review of current practice General There are many types of configurations of transmission towers, and a multitude of load combinations that can act on them. Self-supporting transmissions towers are the more conventional type and may be classified according to their function along the transmission line 5. Tower structures are subject to different kinds of loads. These include conductor weight, tower weight, ice load, transverse wind load, longitudinal or oblique wind loads, transverse load resulting from an angle in the line, longitudinal loads caused by some unbalanced forces in the conductor tensions, loads imposed during the stringing operation, torsional loads resulting from broken conductors and dynamic loading from galloping conductors and other effects. All of these loads must be considered to act in various combinations as specified by the electricity safety codes, statutory regulation and industry standards 6'7.
Design specifications The two most widely used design specifications for the design of axially loaded angle members in selfsupporting transmission towers are the ASCE manual 52: guide for design of steel transmission towers 5'8 and the ECCS recommendations for angles in lattice transmission towers 9. Tower structures are considered to consist of members supported by stress-carrying bracing and redundant members which are nominally unstressed. The design manuals specify limiting slenderness ratios for different member types to account for partial end restraint and joint eccentricity. In the ASCE manuals 5'8, the ultimate maximum stress is determined using the SSRC 1° basic column curves (curves 1 or 4 as appropriate), whereas in the ECCS manual 9 the ultimate maximum stress is based on the ECCS multiple column curves 11 (Curve a0).
failure is repeated until the tower is able to support the ultimate design load. Although the ultimate load testing will, to some extent, verify the adequacy of the tower in withstanding the specified static design loads, it cannot predict exactly how the structure will behave in practice under differing load conditions.
Methods of analysis and design Stress calculations in a transmission tower structure are generally based on a linear elastic analysis, normally assuming that members are axially loaded and pinconnected, with the stiffer main leg members considered as continuous beams. Forces or stresses in the members are usually determined using a computer-aided method of analysis. Two basic approaches have been used in developing computer programs for analysing transmission towers. The first approach translates the logic of conventional methods into routines to carry out the analysis of the structure. The second approach uses structural analysis methods such as the stiffness method. Most of the computer programs available are based on the linear 3D elastic truss approach using the stiffness method, for example the BPA TOWER program 13 and the TRANTOWER program TM. The BPA TOWER program is a linear elastic analysis program adjusted to handle, long, slender, tension only bracing members. The analysis requires a certain number of iterations to determine which bracing members are loaded beyond their compression capacity and to remove such members from the model, thus forcing the remaining bracing members to carry the tensile load. In the TRANTOWER program ~'~4, members are assumed to be fully active when in tension and are capable of sustaining only a certain compression. The compression members are characterized as having a bilinear force-displacement relationship where the member buckling load is obtained through the use of appropriate design formulae recommended by codes or design manuals 5. Secondary effects were incorporated by considering the geometric nonlinearity due to large tower displacements ~. When a truss type model is used to analyse a transmission tower, the structure should be free of planar joints which cause local instability. Significant effort on the part of the designer is required to remove planar joints, a process which requires the addition of stabilizing members. Identifying and correcting such instabilities generally requires a few additional computer runs.
Proposed nonlinear analytical technique
Procedure for full-scale load testing
General
Full-scale testing of transmission tower structures plays an important and integral role in the development of the designs. Guidelines for transmission tower testing are available 5'1z. The test is generally set up to simulate the most critical design conditions. Loads are normally incremented to 50%, 90%, 95% and 100% of the maximum specified loads. Typically, each load increment is held for one or two minutes. When a premature failure occurs, corrective measures are taken and all failed members are replaced. The load case which caused the
In the proposed nonlinear analytical technique, the tower is modelled as an assembly of general thin-walled beam-column elements. Since most of the tower connections are multiple-bolted end connections offering some degree of restraint, it is assumed that the restraint offered by a connection relative to the moments induced in the tower members is large enough to regard the connection as rigid. The effect of joint flexibility can also be inco,rporated in the technique provided information on joint flexibility is known ~5.
140
Eng. Struct. 1992, Vol. 14, No 3
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
(a) YX plane
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x Element generalized forces and displacements and reference axes
Figure 1
y\y Basic assumptions Figure l shows an element of general thin-walled open section. The right-hand orthogonal coordinate system x, y, z is chosen such that y and z pass through the end cross-section shear centres S and S' of the dement before deformation, and are parallel to the principal y and ~ axes of the cross-section. A parallel set of coordinates ~, y, ~ passes through the end cross-section centroids C and C' of the element. By neglecting the effect of warping, there are six possible actions (F,, F r, F z, M,, Mr and Mz) with corresponding displacement components (u, v, w, 0x, 0y and 0z) that can be applied at each end of the thin-walled element. The following assumptions are made: ¢ the element, but not necessarily the member, is prismatic and straight • cross-sections are rigid and do not distort • shear deformations are assumed to be negligible • the material is homogeneous, isotropic and elasticperfectly plastic • strains are small but displacements and rotations can be large • warping of the cross-section is neglected • loads are conservative Geometric nonlinearity
Figure 2 shows the deformation of an element in the projected YX and ZX planes of the global X, Y and Z coordinate system. The element deformation may be described using three different configurations, Co, C] and C2. These configurations represent the initial undeformed state, the current (known) deformed state and a neighbouring (desired) deformed state, respectively. The tensor notation and nomenclature used by Yang and McGuire ~6 is adopted. A left superscript denotes the configuration in which the quantity occurs.
Figure 2
planes
Element deformations in projected global YX and ZX
The absence of such a superscript indicates that the quantity is an increment between C~ and C2. A left subscript denotes the configuration in which the quantity is measured. In an updated Lagrangian formulation, the principle of virtual displacements can be expressed as 16.17 '
I ~o6t~1 dV= ~W
l
(1)
where ~o is the second Piola-Kirchhoff stress tensor and dire is the variation of the Green-Lagrange strain tensor. Equation (2) represents the equilibrium of the element in the displaced configuration C2, in which the stresses 12oare corresponding to C2 but measured in Cl. The stresses 12o can be decomposed into
~0~--"1r"]"lO
(2)
where lz denotes the Cartesian component of the Cauchy stress tensor and ~o is the Cartesian component of the second Piola-Kirchhoff stress increment tensor referred to IC. Similarly the strain increments ~ can be
Eng. Struct. 1992, Vol. 14, No 3 141
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai decomposed into i~ =
(3)
]E L -F l eN
in which ~eL and I~N are the incremental linear and nonlinear strain components of the Green-Lagrange incremental strain tensor. The linearized constitutive relation between stress and strain increment is ,a = ID, eL
(4)
where ~D is the material matrix = diag [ E G G]. Substitution of equations (2)-(4) into equation (1) yields the incremental equilibrium equation of the element in C2
tl'
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IT616NIdV
ment. Plastic hinges are assumed to be elastic prior to the full plastification so that the initial stiffness of the complete element corresponds to that of the elastic beam. As the stress resultants at the ends of the element increase, the hinges yield resulting in a reduction in the element stiffness. For a steel section the hinges are assumed to become fully elastic again upon unloading. A solution method suitable for elasto-plastic nonlinear analysis of large scale structures has been presented by the authors 19. Since transmission towers are almost invariably constructed of angle sections, a single equation representing the stress-resultant yield surface for angle sections under a combination of axial force and biaxial moments 19'2° may be used. Approximate yield surfaces for angle sections are shown in Figure 3 for varying normalized axial (compressive and tensile) force values. A single equation describing the yield surfaces can be expressed as
+ ,i 'vlD16L61f:N ldV =
lw
4 d~(p, my, m:) = ~ ~b3(~ - l) + (9 +/z)3sign(1, p) (5)
-3([2 + #)
sign(l, p)
in which +ff(• + #)2 + ~b
~W= l'v ]r6teL'dV
(6)
The linear, geometric and deformation stiffness matrices, [KL], [Ka] and [Ko] can be determined, respectively, from the first, second and the third integrals on the left hand side of equation (5). Hence the tangent stiffness obtained will take into account not only the stress state but also the deformation state of the element. The linear stiffness matrix [Kt.] is available in the standard texts ~8. The geometric and deformation stiffness matrices for a general thin-walled beam-column element, [KG] and [Ko], have been presented by Kitipornchai and Chan 3 and by the authors 4, respectively. The deformation stiffness matrix [Ko] introduces the necessary coupling between the axial stretching and the flexural and torsional deformations thereby greatly reducing the number of elements needed to model the tower structure accurately in a nonlinear large displacement analysis.
Solution procedure The solution methods used in this paper are similar to those presented elsewhere 4']9. The analysis of large
-
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.
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.
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.
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For large scale structures such as transmission towers, modelling of material nonlinearity based on the assumption of lumped plasticity, coupled with the concept of yield surface in force space, provides a compact and practical method for modelling nonlinear global structural behaviour. The solution to the nonlinear response is obtained as a sequence of linearized solutions in which either the tangent stiffness is modified to reflect the extent of the development of plastic flow, or the load is modified by a residual force to maintain equilibrium. The stress resultants in the cross-section interact to produce yielding for the section. Any plastic behaviour is deemed to be concentrated at the familiar generalized plastic hinges located at the two extremities of an ele-
0.2
Eng. S t r u c t . 1 9 9 2 , Vol. 14, No 3
(7)
in which p, mr and mz are the normalized axial and bending moments about the centroidal axes parallel to the legs, and the coefficients or, /3, ~b, 7, ~b,/z, 9, and are expressed in terms of p, my and mz2°.
Material nonlinearity
142
= 0
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Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
scale structures such as transmission towers involves the solution of several hundred simultaneous equations and, for this purpose, an out-of-core solution scheme 2~ is used. For a given load increment {AR}, a corresponding displacement increment {Ar}, may be obtained by solving the incremental equilibrium equation tAR} = [Kr] {Ar}
(8)
in which [Kr] is the tangent stiffness matrix. Using the nodal displacements and the incremental constitutive law, the incremental resisting forces of the structure can be obtained. These are then compared with the externally applied forces to obtain the out-of-balance forces, {AR,}, which must be dissipated through an iterative procedure subjected to some imposed constraint conditions depending on the solution strategy selected. The element stiffness matrices are formed in the local principal axes of the element where the axial forces are referred to the centroidal axis, the shear forces and the torque to the shear centre axis and the bending moments to axes passing through the shear centre and parallel to the principal axes. In order to assemble these matrices to obtain the tangent stiffness matrix of the structure [Kr], the tangent stiffness matrix for each element has to be transformed to the global axes. Since general thinwalled sections are considered, this transformation process consists of three steps: rotational, translational and local to global transformations. The solution strategy chosen is the arc-length method 22. To avoid the case where the force-point for a certain element jumps from within the yield surface to a point outside the surface during a loading cycle, and so as to avoid the case where the force-point deviates excessively from the curved yield surface, a simple solution advancement control method is used in conjunction with the arc-length method. The solution advancement control is achieved by using the arc-distance from the previous cycle and the maximum value of the yield function in the last two cycles to extrapolate to maximum arc-distance for the present cycle. This brings the forcepoint gradually to the yield surface and guards against excessive deviation from the surface. The iteration process stops and the solution proceeds to the next load cycle when a certain norm criterion is satisfied and the force-point for any element which has entered the plastic stage returns to the yield surface. An out-of-balance force convergence criteria is adopted using the Euclidean norm measure with a convergence tolerance set to 5 %. When analysing a large scale structure such a s a transmission tower, the self-weight of the structure has to be considered. In the present analysis the tower's selfweight is generated automatically and applied incrementally on the tower prior to the incremental application of external loads.
Formex configuration processing In the analysis and design of transmission tower structures, the processes of data generation describing the topology and geometry of such large scale structures presents a real challenge. This task could be very time consuming, tedious and prone to error. The formex
algebra approach due to Nooshin 23 provides a very general and elegant avenue for handling this type of problem. Formex algebra is a mathematical system that consists of a set of abstract objects, known as formices, and a number of rules according to which of these objects may be manipulated. The formex is a mathematical entity that consists of an arrangement of integers. This approach has been used successfully for the data processing of double-layer grids and braced domes 23. A detailed description of the formex formulation of transmission towers is given elsewhere 24. A preprocessor which is capable of generating the topology, geometry, loading and supporting conditions of transmission towers has been implemented on a personal computer. Only a small amount of data is required for this program to generate details of the tower structure.
Practical application The developed software, the AK TOWER program, has been used to predict the ultimate structural behaviour of two transmission towers recently tested in Australia. They are (see Figure 4) the Nebo-Ross 275 kV tension tower 25 and the Ross-Chalumbin 275 kV double circuit heavy suspension tower 26. Descriptions of them are given in Figure 4.
Nebo-Ross 275 kV tension tower The Nebo-Ross 275 kV tension tower forms part of the Nebo- Ross transmission line constructed in Queensland, Australia. Details of test procedure and results have been reported by Transfield Pty Ltd 25, and some of the results are summarized below.
Test programme. The topological outline (front and side views) of the test tower is shown in Figure 5. The tower was tested for the load conditions shown in Figures 6(a)- (e). These are: Test no 1 temporary termination of both circuits (see Figure 6(a)) Test no 2 temporary termination of a single circuit (see Figure 6(b)) Test no 3 intact/broken maintenance (see Figure
6(c)) Test no 4 broken earth-wire and upper phase (see Figure 6 (d)) Test no 5 cross-arm rigging (see Figure 6 (e)) The test loads (in kN) shown in Figure 6 represent the design ultimate loads for each load condition. Loads indicated by Q represent static loads, while the remaining loads are applied incrementally to the tower. In the analysis, static loads Q together with the tower selfweight are applied prior to the application of the test loads. Deflections at selected points on the tower were recorded at different stages during and at the complexion of each test. When the full design load was reached, the loads were held for 2 to 3 minutes before they were finally released. The points at which deflections were recorded for the different tests are shown in Figure 609. A fixed notation
Eng. Struct. 1992, Vol. 14, No 3
143
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
Nebo-Ross 275 kV Tension Tower: Ross-Chalumbin
Tension tower, 15 degrees maximum line deviation Tower Structure: Square base, lattice steel, all members of hot-rolled angle section Height: 50500 mm Self-Weight: 146.2 kN Conductor Configuration: 275 kV double circuit, pyramid type Tower Type:
Nebo-Ross
0 0 to
Ross-Chalumbin 275 kV Double Circuit Heavy Suspension Tower: o
tO
Tower Type: Tower Structure:
Heavy suspension tower Rectangular base, lattice steel, all members of hot-roiled angle section Height: 59600 mm Self-Weight: 99.27 kN (original tower) 100.43 kN (upgraded tower) Conductor Configuration: 275 kV Double circuit Figure 4
T o w e r s used in case studies
Earthwire Peak Top c r o s s - a r m Super structure
Middle c r o s s - a r m Bottom c r o s s - a r m
Common body
!
Body
+o
Bo.,y Leg 12
(a)
(b)
Figure 5
Elevation views of N e b o - R o s s test tower: (a), front view; (b), side view
is used for deflections: deflection in the transverse (T) direction at point AL is TAL, in the longitudinal (L) direction at point CR is LCR and in the vertical (V) direction at point EL is VEL.
Test loading procedure. During the test, loads were applied in an incremental-cycle manner 25. That is, a certain load increment was given to one load and then followed by the same percentage increment to another 144
Eng. Struct. 1992, Vol. 14, No 3
load so as to bring them to the same percentage level of the design load. This procedure was repeated for every load increment of every load point in the tower in order to achieve the specified load level. When the load increment is applied to a particular point, however, loads at other locations are relaxed and they in turn must be re-adjusted to the same percentage level. The sequence of loading and re-adjustment every time a certain load increment level is applied to the structure cannot be simulated easily in the analysis where all the loads are incremented simultaneously. This effect becomes more significant as the structure begins to yield. Hence, some discrepancies in the ultimate loads and the deflections from the analysis and the test read-outs can be expected.
Test results. At the commencement of test no. 1, deflection read-outs assigned to LER, LAR and LBR (see Figure 609) were malfunctioning and therefore ignored. Loads were incremented in stages up to 100% of the design ultimate loads and held for 2 min. No permanent deformation or over-stressing was reported. At 90 % of the design ultimate load during test no. 2, the compression leg of Body :v0 (see Figure 5(a)) collapsed due to the failure of the diagonals in the body. The design was checked and as a result it was decided that the diagonal members of Body :F0, Body - 6 . 5 , and Body +7.5 would be upgraded. Larger angle sections were chosen with a higher grade steel of 350 MPa used instead of 250 MPa as used in the original design. During test no. 3, at 50% of the design ultimate load, malfunction of the deflection read-out assigned to LAR
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
18.5Q
~
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T (f) Nomenclature
Figure 6 Load conditions for N e b o - R o s s test t o w e r
was reported. Adjustments were attempted but were unsuccessful and LAR deflection readings were subsequently disregarded. The test tower was able to hold loads at 100% of the design ultimate load condition for three minutes with no permanent deformation or overstressing observed. Test no. 4 passed without incident. The tower held 100% of the design ultimate load for two minutes with no permanent deformation or over-stressing observed. Cross-arm AR was selected for test no. 5 (cross-arm rigging). At 70% of the design ultimate load, the bottom chord adjacent to point of load application failed due to local plate buckling. Since the present analysis does not account for local plate buckling, this test load condition is not considered in the study.
Predicted results. The N e b o - R o s s test tower under the various test load conditions has been analysed using the developed AK TOWER program. The tower topology, geometry, load and support conditions were generated automatically using the formex algebra approach 23. A total of 1238 elements and 487 nodes has been used to model the tower. It should be noted that although the developed software can handle any joint flexibility, the present analysis assumes that no joint slippage in the tower occurred. Test no. 1: This test represents a load condition for temporary termination of both circuits as shown in Figure 6 (a)). The tower passed this test without incident and this is confirmed by the present analysis. Further, the
Eng. Struct. 1992, Vol. 14, No 3 145
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
l
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Figure 7 T h e o r e t i c a l l o a d - d e f l e c t i o n c u r v e s f o r N e b o - R o s s
/
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test tower:
Test no. 2. This test represents a load condition for temporary termination of a single circuit as shown in Figure
.
. (d)
Test no.4 Transverse deflection at Xl
/
/ [
150
f -T_est_.Lup - - - - - to X= 1.0
Theory 0.4
AK TOWER program is able to predict an ultimate load factor ), = 1.22 for this load condition. The theoretical. load-deflection curve at TER (see Figure 6(/9) are shown m Figure 7(a). The tower deflected shape at failure (X = 1.22) is shown in Figure 8(a). The first plastic hinge starts in the compression leg of the common body at a load factor ), = 0.92. This is followed by gradual spread of plastic hinges in the compression legs of the common body until a maximum load of ~, = 1.22 is reached. After this stage, plastic hinges start to spread in the compression leg of the lower body (Leg + 12) causing a drop in the load and a rapid spread of plastic hinges in the diagonals of the common body. As reported by Transfield 25, the test was passed successfully. It can be seen from Figure 7(a) that the derived load-deflection curve is still quite linear up to the design ultimate load (i.e. at ~, = 1.0). A comparison between the test measurements and the numerical results for the deflections of different nodes at the design ultimate load (), = 1.0) is presented in Table 1. The predicted deflections agree reasonably with the test measurements considering the effect of bolt slippage which must have occurred at joints and the manner in which loads were applied during the test.
146 Eng. Struct. 1992, Vol. 14, No 3
- -- ..- _ ~ ~ _
0 ..J
Test no.3
II/
II1
"o
(c)
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at X1 I 45
Deflection, TX1 (mm) I
[ l h = 1.62
~
failed at ;~= 0.9
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i
50
i
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100 150 Deflection, TX 1 (mrn)
200
(a), t e s t no. 1; (b), t e s t no. 2; (c), t e s t no. 3; (d), t e s t no. 4
6(b). During the test, the tower failed at a load factor X = 0.90 due to the collapse of the compression leg in Body ~:0 which was initiated by the buckling of the diagonal members in the body. The load-deflection curves obtained from the AK TOWER program are shown in Figure 7(b) for the deflection TX1. The deflected shape of the tower is shown in Figure 8(b). The first plastic hinge forms at X = 0.52 in the common body and spreads down to the diagonals of Body - 6 . 5 and Body ± 0 . However, when the applied load reaches X = 0.75, the numerical method starts to diverge and the analysis breaks down as the tangent stiffness matrix becomes ill-conditioned. Results predicted by the AK TOWER program indicate a weakness in the tower under this load condition and a nonlinear behaviour at a relatively low load level (i.e. at X ~- 0.5, see Figure 7(b)). The reason for the higher test load (h = 0.90 compared to the predicted ), = 0.75) may be attributed to the different loading (incremental-cyclic) method used during the test.
Test no. 3. This test represents a load condition for intact/broken maintenance as shown in Figure 6(c). The tower passed this test without incident z5 and this is confirmed by the present analysis. The load-deflection curve for the deflection VAR is shown in Figure 7(c). The first plastic hinge forms at X = 0.95 and hinges
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai Test
Figure 8
Test
no. 1
no.2
Test
no.3
Test
no.4
)~=1.22
h=0.75
~=1.62
;k= 1 . 5
(a)
(b)
(c)
(d)
=neoreticai failure deflected shapes of N e b o - R o s s test tower: (a), test no. 1; (b), test no. 2; (c), test no. 3; (d), test no. 4
spread but remain confined to the cross-arms AR and CL. A maximum load factor ~, = 1.62 is reached as indicated by the plateau in the load-deflection curve in Figure 7(c). The predicted deflected shape of the tower at the maximum load value is shown in Figure 8(c). A comparison between the test read-outs and the numerical results for various nodal deflections at the design ultimate load (X = 1.0) is presented in Table 1. It can be seen that the predicted deflections agree reasonably with the test measurements.
Table 1 Comparison
Test no 4. This test represents a loading for a broken earth-wire and upper phase as shown in Figure 6(d). The tower passed this test without incident and this is confirmed by the present analysis. The load-deflection curve for the deflection TX1 is shown in Figure 7(d). The first plastic hinge forms at X = 0.75 in the diagonals of the common body. More hinges start to spread to the other diagonals but remain confined to the common body until a maximum load factor ~, --- 1.5 is reached. After this point, plastic hinges start to form in the leg
of predicted nodal deflections with full-scale test measurements for Nebo-Ross 2 7 5 kV tension t o w e r
Measured Deflections (turn) Test no. 1
Test no. 3
Test no. 4
Measurement locations
Test
Predicted
Test
Predicted
Test
Predicted
TER LER LEL TAR LAR LAL TBR LBR TCR LCR VCR VCL
-54 ** -87 ** -74 ** -44 -148 -
-88 -513 -68 -392 -47 -263 -29 -159 -
-86 -218 -93 -86 ** -19 . . -24 . +16 -77
-115 -161 -105 -93 -151 -44 . . -39 . + 5 -54
-114 -453 +13 -150 -225 +44
-215 -251 -82 -159 -230 -14
-48
-65
-
-
* * T h e s e malfunctioned during the test and were
. . .
. . .
-
disregarded
Eng. Struct.
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Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
Body
-6
(a) Figure 9
Front and side views of R o s s - C h a l u m b i n test towers: (a), original tower; (b), upgraded t o w e r
members of the common body causing a drop in the load followed by a failure of the numerical method indicating that a collapse condition has been reached. The predicted deflected shape of the tower at the end of the analysis for this test is shown in Figure 8(d). A comparison between the test read-outs and the numerical results for different nodal deflections at the design ultimate load (X = 1.0) is presented in Table 1. It can be seen that some of the predictions are in reasonable agreement with the test read-outs while others are not so good. The differences between the predicted results and the test read-outs for LER, LEL and LAL can be attributed to the effects of bolt slippage, damage to some of the members from earlier tests and the incrementalcyclic loading procedure used in the test.
Ross-Chalumbin 275 kV double circuit heavy suspension tower This tower (see Figure 4) is part of the R o s s Chalumbin transmission line recently constructed in Queensland, Australia. Details of test procedure and results have been reported by Electric Power Transmis2~ sion (EPT) Pry Ltd . It is worth noting that the loading procedure used in this test is different from that employed in the tension (Nebo-Ross) test tower. In the Ross-Chalumbin tower, all loads were applied simultaneously in an incremental manner which is similar to that used in the analysis hence better predictions are expected.
Test results. The topological outline of the test tower is shown in Figure 9. The tower was tested under six dif-
148
(b)
Eng. Struct. 1992, Vol. 14, No 3
ferent load conditions and passed the first five tests successfully. In the sixth test, which was under the intact maximum wind load condition, the tower collapsed at 90 % of the design ultimate, load. The failure described in the test report 26 as an unusual failure which involved a general collapse of the compressed face of the tower'. Figure 10 shows a photograph of the failed original tower after the test. The ultimate design load condition for the sixth test is shown in Figure 11(a). Figure 11(b) shows the notation adopted for different nodes and displacements. Following the collapse of the tower in the sixth test, the tower was strengthened by adding a number of horizontal members in an attempt to prevent the global failure experienced in the sixth test. The initial upgraded tower shown in Figure 9go) (compared with the original tower in Figure 9(a)) was tested and it again failed at a load only slightly higher than that encountered in the original tower. The tower finally passed the test after further strengthening but details of the final configuration are not given in the test report 26.
Results predicted by AK TOWER program. Both the original and the initial upgraded towers have been analysed using the proposed nonlinear analysis AK TOWER program. The tower topology, geometry, load and support conditions were generated automatically using the formex algebra approach 23. A total of 1732 elements and 768 nodes have been used to model the original tower. The initial upgraded tower has been modelled using 1761 elements and 772 nodes. In the analysis of the original tower, the first plastic
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
)0
1.95
Figure 12 Theoretical failure deflected shapes of R o s s Chalumbin test towers: (a), original tower; (b), initial upgraded tower
at collapse is shown in Figure 12(a). The shape closely resembles the actual tower failure shape obtained during the test as depicted by the photograph taken after failure
(Figure 10). Figure 10 Photograph of failed (original) Ross-Chalumbin test tower
hmge is formed at a load factor h = 0.7 in Body - 6 (see
Figure 9(a)). Subsequent plastic hinges spread rapidly to other parts of the tower below Body - 6 accompanied by a rapid increase in displacement. A maximum load factor )~ = 0.91 is reached followed by a break down of the numerical method indicating collapse of the tower. This compares extremely well with the 90% failure load reported in the test :6. The deflected shape of the tower
10.83
10.83
6.83--I 23.86 --145.39 23.86 --145.39 23.86 --145.39
In the analysis of the initial upgraded tower, the first plastic hinge is formed at a load factor )~ = 0.8. Plastic hinges then spread in a similar fashion as before but less rapidly and with less displacement. A maximum load factor X = 0.95 is obtained. This load is only a little higher than that of the original tower as was confirmed by the test report 26. The predicted deflected shape of the initial upgraded tower is shown in Figure 12(b). Load-deflection curves for vertical deflection VER at node ER of the original and the initial upgraded towers are shown in Figure 13. It can be seen that the addition of a number of horizontal members to the tower has, to some extent, improved the tower's behaviour.
I. 11.37 45.39 I - 3 2 - 3 2 45.391 - 35"78 45.391-40"66 ~20.5 --15.4 --35 ~35
= ER
EL =
BL=
=AR -- BR
eL--
-- CR
AL'-
Xl X2 X3
v
X4 7__ L
-
(a)
T
(b)
Figure 1 1 Critical load condition for Ross - Chalumbin test tower
Eng. S t r u c t 1992, Vol. 14, No 3
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Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai 1.2
,<
I
I
1.0 -
X=0.95
'
X=O.9
I
profound effect on tower deflection but should only have little influence on tower ultimate capacity 27.
I
Acknowledgements 7
0.8
o 0
0.6
"0 0
-~ 0 . 4 -
/
-Original tower
/ 0.2 -
-
/
/I
0
50 Vertical
Initial upgraded tower
I
I
;
100
150
200
deflection,
-
-
250
This project has been supported by funds from the Australian Research Council (ARC) under Project Grant No. 834 and from the Australian Electricity Supply Industry Research Board (AESIRB). The authors wish to thank the Queensland Electricity Commission (QEC) for making the full-scale test data of the two tested transmission towers available. In particular, the authors wish to acknowledge the assistance from Mr Henry Hawes of the QEC for his continued encouragement, advice and technical support throughout the course of this work. The authors wish also to thank Dr S. L. Chan of the Department of Civil and Structural Engineering (Hong Kong Polytechnic) for the initial work on developing the geometric stiffness for asymmetric thin-walled beamcolumn elements and Mr Warren Traves, (Gutteridge Haskins and Davey Pty Ltd) for proof-reading the manuscript.
VER (mm)
Figure 13 Theoretical l o a d - d e f l e c t i o n curves of original and
References
initial upgraded R o s s - C h a l u m b i n test t o w e r s
Conclusions Accurate structural analysis of transmission towers is complicated because the structure is three-dimensional and comprised of asymmetric angle section members eccentrically connected. The influences of geometric and material nonlinearities play a very important role in determining the ultimate behaviour of the structure. The paper describes a nonlinear analytical method in which most factors affecting the ultimate behaviour of the tower structure can be incorporated. These include geometric and material nonlinearities, joint flexibility and the effects of large deflection. The developed software, the AK TOWER program, was used to predict the ultimate structural behaviour of two electric transmission towers recently tested in Australia. No comparison of the calculated and the actual member forces was made due to the lack of such field data in these tests, but predictions of the ultimate loads, the nodal deflections at various points and the tower failure deflected shapes have been made. The developed software was able to predict accurately the collapse load (predominantly flexural load) for the Ross-Chalumbin heavy suspension tower. As for the N e b o - R o s s tension tower, the predicted collapse load for test no. 2 (torsional load) was 16.7% lower than the test collapse load. This may be attributed to the different loading procedures used. In the test, loads were applied in an incremental-cyclic manner whereas in the analysis all loads were incremented simultaneously. Moreover, a torsional load case is generally regarded as more difficult to predict than a flexural load case. Comparison of measured and predicted deflections have not all been good. This may be attributed to the effect of bolt slippage which must have occurred at most joints during test, particularly at high load level, but was totally ignored in the analysis. Bolt slippage will have a
150
Eng. Struct. 1992, Vol. 14, No 3
1 Roy, S., Fang, S. and Rossow, E. 'Secondary stresses on transmission tower structures', J. Energy Engng, ASCE 1984, 110, (2), 157 - 172 2 Electric Power Research Institute. 'Structural development studies at the EPRI transmission line mechanical research facility', Interim Report No. 1: EPRI EL-4756, August 1986, Sverdrup Technology. Inc., Tullahoma, Tennessee. 3 Kitipornchai, S. and Chan, S. L. 'Nonlinear finite element analysis of angle and tee beam-columns', J. Struct. Engng, ASCE 1987, 113, 4, 721-739 4 AI-Bermani, F. G. A. and Kitipornchai, S. 'Nonlinear analysis of thin-walled structures using least element/member', J. Struct. Engng, ASCE 1990, 116, (1), 215-234 5 American Society of Civil Engineers. 'Guide for design of steel transmission towers', Manuals and Reports on Engineering Practice, No. 52, ASCE 1988 6 American Society of Civil Engineers Committee on electric transmission structures. 'Loading of electrical transmission structures', J. Struct. Div., ASCE 1982, 108, (ST5.), 1088-1105 7 American Society of Civil Engineers Committee on electric transmission structures. 'Guidelines for transmission line structural loading', ASCE, New York, 1984 8 American Society of Civil Engineers 'Guide for design of steel transmission towers', Manuals and Reports on Engineering Practice, No. 52, ASCE, New York, 1971 9 European Convention for Constructional Steelwork (ECCS). 'Recommendations for angles in lattice transmission towers', January 1985 10 Structural Stability Research Council: SSRC. "Guide to stability criteria for metal structures', (4th edn) T. V. Galambos (Ed.) John Wiley New York, 1988 11 Eurocode, 'Common Unified Code of Practice for Steel Structures'. Eurocode no. 3, 1984, Commission of the European Communities, Directorate-General, Brussels 12 International Electrotechnical Commission (IEC). 'Loading tests on overhead line towers', Publication No. 652, 1979 13 Bonneville Power Administration. 'Elastic design program', Portland, Oregon, 1987 14 Lo, D., Morcos, A. and Goel, S. 'Use of computer in transmission tower design', J. Struct. Div., ASCEI975, 101, (ST7), 1443-1453 15 AI-Bermani, F. G. A. and Kitipornchai, S. 'Elasto-plastic analysis of flexibly-jointed space frames, J. Struct. Engng, ASCE 1992, 118, (1), 108- 127 16 Yang, Y. B. and McGuire, W. 'Stiffness matrix for geometric nonlinear analysis', J. Struct. Engng, ASCE 1986, 112, (4), 853 -877 17 Bathe, K. J. and Bolourchi, S. 'Large displacement analysis of threedimensional beam structures', Int. J. Num. Meth. Engng 1979, 14, 961 - 986
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai 8 Przemieniecki, J. S. "Theory of matrix structural analysis ', McGrawHill, New York 1968 9 AI-Bermani, F. G. A. and Kitipornchai, S. 'Elasto-plastic large deformation analysis of thin-walled structures', Engng Struct. 1990, 12, 28-36 ~) Kitipornchai, S., Zhu, K., Xiang, Y. and AI-Bermani, F. G. A. 'Single-equation yield surfaces for monosymmetric and asymmetric sections', Engng Struct 1991, 13, (4), 366-370 1 Wilson, E. L. and Dovey, H. H. 'Solution or reduction of equilibrium equations for large complex structural systems', Adv. Engng Software 1978, 1, (1), 19-25 2 Crisfield, M. A. 'A fast incremental/iterative solution procedure that handles snap-through', Comp. Struct. 1981, 13, (1), 55-62
23 Nooshin, H. Formex configuration processing in structural engineering', Elsevier Applied Science Publishers, 1984 24 Al-Bermani, F. G. A., Kitipornchai, S. and Chan, S. L. 'Formex formulation of transmission towers', Int. J. Space Struct. 1992, 7 (1) 25 Transfield Pty Ltd 'Test Report, The Queensland Electricity Generating Board, Northern Region, Nebo-Ross Transmission Line, Tower D2T15 + 12M Queensland, Australia', November 1983 26 Electric Power Transmission (EPt) Pry Ltd, Test Report No. 497 275 kV T/L Ross-Chalumbin D. C. Heavy Suspension Tower D2S2D' February 1988, NSW, Australia 27 Kitipornchai, S., AI-Bermani, F. G. A. and Peyrot, A. H. 'Effect of bolt slippage on the ultimate behaviour of lattice structures', (to be published)
Eng. Struct. 1992, Vol. 14, No 3
151
towers
F. G. A. AI-Bermani and S. K i t i p o r n c h a i Department of Civil Engineering, The University of Queensland, St Lucia, Queensland, 4072 Australia (Received May 1991; revised August 1991)
A nonlinear analytical technique for predicting and simulating the ultimate structural behaviour of self-supporting transmission towers under static load conditions is presented. The method considers both the geometric and material nonlinear effects and treats the angle members in the tower as general asymmetrical thin-walled beamcolumn elements. Modelling of material nonlinearity for angle members is based on the assumption of lumped plasticity coupled with the concept of a yield surface in force space. A formex formulation is used for automatic generation of data necessary for the analysis. The developed software, AK TOWER, is used to predict the ultimate behaviour of two full-scale towers recently tested in Australia.
Keywords: elasto-plastic analysis, formex algebra, full-scale testing, steel structure, transmission tower, ultimate strength
Transmission tower structures are generally constructed using asymmetric thin-walled angle section members which are eccentrically connected. They are widely regarded as one of the most difficult forms of lattice structure to analyse. Proof-loading or full-scale testing of such structures has traditionally formed an integral part of the development of tower design. Stress calculations in the structure are normally obtained from a linear elastic analysis where members are assumed to be axially loaded and, for the majority of cases, pinconnected. In practice, such conditions do not exist and members are detailed to minimize bending stresses. Despite this, results from full-scale testing of transmission towers indicate that bending stresses in the members can be as significant as axial stresses 1. Design practices for transmission towers are different from those for other steel structures in that stresses are permitted to be higher because towers are tested to their ultimate design strength and designs incorporate modifications based on test results. A recent study by the Electric Power Research Institute 2 (EPRI) indicated that current design practices have, for the most part, served the industry well. However, data from full-scale tests show that the behaviour of transmission towers under complex load situations cannot be consistently predicted using the present techniques. The investigation by EPRI 2 also revealed that out of the 57 structure load cases conducted, 23% experienced premature failure. On average, failure occurred at 95.4% of the design load level but failure could occur at unexpected locations. Further, available data showed considerable discrepancies between member forces computed from linear
elastic truss analyses and those measured from full-scale tests. The EPRI report 2 indicated that the linear elastic truss analysis method for transmission towers should be used with caution. For these reasons, there is a need to develop a method of analysis capable of predicting the ultimate structural behaviour of transmission towers more accurately than the present 3D linear elastic truss approach. Such a refined technique would provide the designer with a better understanding of tower behaviour which undoubtedly will lead to a more economical structural design. Any saving in the design of one tower is magnified many times over because large numbers of towers of the same designs are usually constructed. For example, in a 250 km transmission line, there may be 500 towers of which up to 80% are of the same type. This paper presents a nonlinear analytical technique for simulating the ultimate structural behaviour of selfsupporting transmission towers under static loading. The proposed technique incorporates both the geometric and material nonlinear effects including large displacements, and treats the angle members in the tower as general asymmetric thin-walled beam-column elements 3. Based on an updated Lagrangian formulation, a deformation stiffness matrix" is used together with the linear and geometric stiffness matrices to enable accurate modelling of transmission tower structures with the least possible number of elements. Modelling of material nonlinearity for angle members is based on the assumption of lumped plasticity coupled with the concept of a yield surface in force space. A formex formulation is used for automatic generation of data necessary
0141-0296/92/030139-13 © 1992 Butterworth-Heinemann Ltd
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Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
for the analysis of transmission towers. These include the topology, geometry, load and support conditions. The developed software, the AK TOWER program (AIBermani and Kitipornchai), is used to predict the ultimate behaviour of two full-scale transmission towers recently tested in Australia. The tests were conducted by the industry in conjunction with the electricity authority requirements for design verification. No measurements of member forces were reported but comparisons of test and predicted results are possible through the comparison of deflections, ultimate loads and the deflected shapes of the towers at ultimate load.
Brief review of current practice General There are many types of configurations of transmission towers, and a multitude of load combinations that can act on them. Self-supporting transmissions towers are the more conventional type and may be classified according to their function along the transmission line 5. Tower structures are subject to different kinds of loads. These include conductor weight, tower weight, ice load, transverse wind load, longitudinal or oblique wind loads, transverse load resulting from an angle in the line, longitudinal loads caused by some unbalanced forces in the conductor tensions, loads imposed during the stringing operation, torsional loads resulting from broken conductors and dynamic loading from galloping conductors and other effects. All of these loads must be considered to act in various combinations as specified by the electricity safety codes, statutory regulation and industry standards 6'7.
Design specifications The two most widely used design specifications for the design of axially loaded angle members in selfsupporting transmission towers are the ASCE manual 52: guide for design of steel transmission towers 5'8 and the ECCS recommendations for angles in lattice transmission towers 9. Tower structures are considered to consist of members supported by stress-carrying bracing and redundant members which are nominally unstressed. The design manuals specify limiting slenderness ratios for different member types to account for partial end restraint and joint eccentricity. In the ASCE manuals 5'8, the ultimate maximum stress is determined using the SSRC 1° basic column curves (curves 1 or 4 as appropriate), whereas in the ECCS manual 9 the ultimate maximum stress is based on the ECCS multiple column curves 11 (Curve a0).
failure is repeated until the tower is able to support the ultimate design load. Although the ultimate load testing will, to some extent, verify the adequacy of the tower in withstanding the specified static design loads, it cannot predict exactly how the structure will behave in practice under differing load conditions.
Methods of analysis and design Stress calculations in a transmission tower structure are generally based on a linear elastic analysis, normally assuming that members are axially loaded and pinconnected, with the stiffer main leg members considered as continuous beams. Forces or stresses in the members are usually determined using a computer-aided method of analysis. Two basic approaches have been used in developing computer programs for analysing transmission towers. The first approach translates the logic of conventional methods into routines to carry out the analysis of the structure. The second approach uses structural analysis methods such as the stiffness method. Most of the computer programs available are based on the linear 3D elastic truss approach using the stiffness method, for example the BPA TOWER program 13 and the TRANTOWER program TM. The BPA TOWER program is a linear elastic analysis program adjusted to handle, long, slender, tension only bracing members. The analysis requires a certain number of iterations to determine which bracing members are loaded beyond their compression capacity and to remove such members from the model, thus forcing the remaining bracing members to carry the tensile load. In the TRANTOWER program ~'~4, members are assumed to be fully active when in tension and are capable of sustaining only a certain compression. The compression members are characterized as having a bilinear force-displacement relationship where the member buckling load is obtained through the use of appropriate design formulae recommended by codes or design manuals 5. Secondary effects were incorporated by considering the geometric nonlinearity due to large tower displacements ~. When a truss type model is used to analyse a transmission tower, the structure should be free of planar joints which cause local instability. Significant effort on the part of the designer is required to remove planar joints, a process which requires the addition of stabilizing members. Identifying and correcting such instabilities generally requires a few additional computer runs.
Proposed nonlinear analytical technique
Procedure for full-scale load testing
General
Full-scale testing of transmission tower structures plays an important and integral role in the development of the designs. Guidelines for transmission tower testing are available 5'1z. The test is generally set up to simulate the most critical design conditions. Loads are normally incremented to 50%, 90%, 95% and 100% of the maximum specified loads. Typically, each load increment is held for one or two minutes. When a premature failure occurs, corrective measures are taken and all failed members are replaced. The load case which caused the
In the proposed nonlinear analytical technique, the tower is modelled as an assembly of general thin-walled beam-column elements. Since most of the tower connections are multiple-bolted end connections offering some degree of restraint, it is assumed that the restraint offered by a connection relative to the moments induced in the tower members is large enough to regard the connection as rigid. The effect of joint flexibility can also be inco,rporated in the technique provided information on joint flexibility is known ~5.
140
Eng. Struct. 1992, Vol. 14, No 3
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
(a) YX plane
z /
Y'lYn /~= "C) IMY1'0~yl ~ o/ Mzl,Ozl /1" /;~S~_~// "~iS'~'-- Fzl'W
Centroida, =/" ax,s / ,,p"
/'/
\
~.~ ,?"/"/M,x~,'OxlFyl,Vl ~
,, 4~'/'~'/
/~// )/
Fx.,U,
z/ r .4hea
x
/
s\
1~
/
Lo
Co ,o 2
X
ax,s,,/
.Mz2,ez2 ~ tT2My2'OY2 .... #I
\z
/ / ~ . center
S
_*" . x ,0
C2
/
(b) ZX plane
/
l~y ...~-C2
F,;,v=
x Element generalized forces and displacements and reference axes
Figure 1
y\y Basic assumptions Figure l shows an element of general thin-walled open section. The right-hand orthogonal coordinate system x, y, z is chosen such that y and z pass through the end cross-section shear centres S and S' of the dement before deformation, and are parallel to the principal y and ~ axes of the cross-section. A parallel set of coordinates ~, y, ~ passes through the end cross-section centroids C and C' of the element. By neglecting the effect of warping, there are six possible actions (F,, F r, F z, M,, Mr and Mz) with corresponding displacement components (u, v, w, 0x, 0y and 0z) that can be applied at each end of the thin-walled element. The following assumptions are made: ¢ the element, but not necessarily the member, is prismatic and straight • cross-sections are rigid and do not distort • shear deformations are assumed to be negligible • the material is homogeneous, isotropic and elasticperfectly plastic • strains are small but displacements and rotations can be large • warping of the cross-section is neglected • loads are conservative Geometric nonlinearity
Figure 2 shows the deformation of an element in the projected YX and ZX planes of the global X, Y and Z coordinate system. The element deformation may be described using three different configurations, Co, C] and C2. These configurations represent the initial undeformed state, the current (known) deformed state and a neighbouring (desired) deformed state, respectively. The tensor notation and nomenclature used by Yang and McGuire ~6 is adopted. A left superscript denotes the configuration in which the quantity occurs.
Figure 2
planes
Element deformations in projected global YX and ZX
The absence of such a superscript indicates that the quantity is an increment between C~ and C2. A left subscript denotes the configuration in which the quantity is measured. In an updated Lagrangian formulation, the principle of virtual displacements can be expressed as 16.17 '
I ~o6t~1 dV= ~W
l
(1)
where ~o is the second Piola-Kirchhoff stress tensor and dire is the variation of the Green-Lagrange strain tensor. Equation (2) represents the equilibrium of the element in the displaced configuration C2, in which the stresses 12oare corresponding to C2 but measured in Cl. The stresses 12o can be decomposed into
~0~--"1r"]"lO
(2)
where lz denotes the Cartesian component of the Cauchy stress tensor and ~o is the Cartesian component of the second Piola-Kirchhoff stress increment tensor referred to IC. Similarly the strain increments ~ can be
Eng. Struct. 1992, Vol. 14, No 3 141
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai decomposed into i~ =
(3)
]E L -F l eN
in which ~eL and I~N are the incremental linear and nonlinear strain components of the Green-Lagrange incremental strain tensor. The linearized constitutive relation between stress and strain increment is ,a = ID, eL
(4)
where ~D is the material matrix = diag [ E G G]. Substitution of equations (2)-(4) into equation (1) yields the incremental equilibrium equation of the element in C2
tl'
'v1DIeL61eLI dV + t"v
IT616NIdV
ment. Plastic hinges are assumed to be elastic prior to the full plastification so that the initial stiffness of the complete element corresponds to that of the elastic beam. As the stress resultants at the ends of the element increase, the hinges yield resulting in a reduction in the element stiffness. For a steel section the hinges are assumed to become fully elastic again upon unloading. A solution method suitable for elasto-plastic nonlinear analysis of large scale structures has been presented by the authors 19. Since transmission towers are almost invariably constructed of angle sections, a single equation representing the stress-resultant yield surface for angle sections under a combination of axial force and biaxial moments 19'2° may be used. Approximate yield surfaces for angle sections are shown in Figure 3 for varying normalized axial (compressive and tensile) force values. A single equation describing the yield surfaces can be expressed as
+ ,i 'vlD16L61f:N ldV =
lw
4 d~(p, my, m:) = ~ ~b3(~ - l) + (9 +/z)3sign(1, p) (5)
-3([2 + #)
sign(l, p)
in which +ff(• + #)2 + ~b
~W= l'v ]r6teL'dV
(6)
The linear, geometric and deformation stiffness matrices, [KL], [Ka] and [Ko] can be determined, respectively, from the first, second and the third integrals on the left hand side of equation (5). Hence the tangent stiffness obtained will take into account not only the stress state but also the deformation state of the element. The linear stiffness matrix [Kt.] is available in the standard texts ~8. The geometric and deformation stiffness matrices for a general thin-walled beam-column element, [KG] and [Ko], have been presented by Kitipornchai and Chan 3 and by the authors 4, respectively. The deformation stiffness matrix [Ko] introduces the necessary coupling between the axial stretching and the flexural and torsional deformations thereby greatly reducing the number of elements needed to model the tower structure accurately in a nonlinear large displacement analysis.
Solution procedure The solution methods used in this paper are similar to those presented elsewhere 4']9. The analysis of large
-
-
Compression p -ve Tension p +ve
- x
,
1.0
p=O
.
/1-~
~...
.
.
.
my -
0.8 0.6 0.4
For large scale structures such as transmission towers, modelling of material nonlinearity based on the assumption of lumped plasticity, coupled with the concept of yield surface in force space, provides a compact and practical method for modelling nonlinear global structural behaviour. The solution to the nonlinear response is obtained as a sequence of linearized solutions in which either the tangent stiffness is modified to reflect the extent of the development of plastic flow, or the load is modified by a residual force to maintain equilibrium. The stress resultants in the cross-section interact to produce yielding for the section. Any plastic behaviour is deemed to be concentrated at the familiar generalized plastic hinges located at the two extremities of an ele-
0.2
Eng. S t r u c t . 1 9 9 2 , Vol. 14, No 3
(7)
in which p, mr and mz are the normalized axial and bending moments about the centroidal axes parallel to the legs, and the coefficients or, /3, ~b, 7, ~b,/z, 9, and are expressed in terms of p, my and mz2°.
Material nonlinearity
142
= 0
2 L J - ~ - - - - - ~ >, " \ ~ \
I/J
0 -0.2
-
/ , /
-0.4 -0.6
-
-0.8
-
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I
I
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Figure 3
0
~
0.2
Yield surfaces for angle section
0.4
0.6
I
0.8
I
1.0
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
scale structures such as transmission towers involves the solution of several hundred simultaneous equations and, for this purpose, an out-of-core solution scheme 2~ is used. For a given load increment {AR}, a corresponding displacement increment {Ar}, may be obtained by solving the incremental equilibrium equation tAR} = [Kr] {Ar}
(8)
in which [Kr] is the tangent stiffness matrix. Using the nodal displacements and the incremental constitutive law, the incremental resisting forces of the structure can be obtained. These are then compared with the externally applied forces to obtain the out-of-balance forces, {AR,}, which must be dissipated through an iterative procedure subjected to some imposed constraint conditions depending on the solution strategy selected. The element stiffness matrices are formed in the local principal axes of the element where the axial forces are referred to the centroidal axis, the shear forces and the torque to the shear centre axis and the bending moments to axes passing through the shear centre and parallel to the principal axes. In order to assemble these matrices to obtain the tangent stiffness matrix of the structure [Kr], the tangent stiffness matrix for each element has to be transformed to the global axes. Since general thinwalled sections are considered, this transformation process consists of three steps: rotational, translational and local to global transformations. The solution strategy chosen is the arc-length method 22. To avoid the case where the force-point for a certain element jumps from within the yield surface to a point outside the surface during a loading cycle, and so as to avoid the case where the force-point deviates excessively from the curved yield surface, a simple solution advancement control method is used in conjunction with the arc-length method. The solution advancement control is achieved by using the arc-distance from the previous cycle and the maximum value of the yield function in the last two cycles to extrapolate to maximum arc-distance for the present cycle. This brings the forcepoint gradually to the yield surface and guards against excessive deviation from the surface. The iteration process stops and the solution proceeds to the next load cycle when a certain norm criterion is satisfied and the force-point for any element which has entered the plastic stage returns to the yield surface. An out-of-balance force convergence criteria is adopted using the Euclidean norm measure with a convergence tolerance set to 5 %. When analysing a large scale structure such a s a transmission tower, the self-weight of the structure has to be considered. In the present analysis the tower's selfweight is generated automatically and applied incrementally on the tower prior to the incremental application of external loads.
Formex configuration processing In the analysis and design of transmission tower structures, the processes of data generation describing the topology and geometry of such large scale structures presents a real challenge. This task could be very time consuming, tedious and prone to error. The formex
algebra approach due to Nooshin 23 provides a very general and elegant avenue for handling this type of problem. Formex algebra is a mathematical system that consists of a set of abstract objects, known as formices, and a number of rules according to which of these objects may be manipulated. The formex is a mathematical entity that consists of an arrangement of integers. This approach has been used successfully for the data processing of double-layer grids and braced domes 23. A detailed description of the formex formulation of transmission towers is given elsewhere 24. A preprocessor which is capable of generating the topology, geometry, loading and supporting conditions of transmission towers has been implemented on a personal computer. Only a small amount of data is required for this program to generate details of the tower structure.
Practical application The developed software, the AK TOWER program, has been used to predict the ultimate structural behaviour of two transmission towers recently tested in Australia. They are (see Figure 4) the Nebo-Ross 275 kV tension tower 25 and the Ross-Chalumbin 275 kV double circuit heavy suspension tower 26. Descriptions of them are given in Figure 4.
Nebo-Ross 275 kV tension tower The Nebo-Ross 275 kV tension tower forms part of the Nebo- Ross transmission line constructed in Queensland, Australia. Details of test procedure and results have been reported by Transfield Pty Ltd 25, and some of the results are summarized below.
Test programme. The topological outline (front and side views) of the test tower is shown in Figure 5. The tower was tested for the load conditions shown in Figures 6(a)- (e). These are: Test no 1 temporary termination of both circuits (see Figure 6(a)) Test no 2 temporary termination of a single circuit (see Figure 6(b)) Test no 3 intact/broken maintenance (see Figure
6(c)) Test no 4 broken earth-wire and upper phase (see Figure 6 (d)) Test no 5 cross-arm rigging (see Figure 6 (e)) The test loads (in kN) shown in Figure 6 represent the design ultimate loads for each load condition. Loads indicated by Q represent static loads, while the remaining loads are applied incrementally to the tower. In the analysis, static loads Q together with the tower selfweight are applied prior to the application of the test loads. Deflections at selected points on the tower were recorded at different stages during and at the complexion of each test. When the full design load was reached, the loads were held for 2 to 3 minutes before they were finally released. The points at which deflections were recorded for the different tests are shown in Figure 609. A fixed notation
Eng. Struct. 1992, Vol. 14, No 3
143
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
Nebo-Ross 275 kV Tension Tower: Ross-Chalumbin
Tension tower, 15 degrees maximum line deviation Tower Structure: Square base, lattice steel, all members of hot-rolled angle section Height: 50500 mm Self-Weight: 146.2 kN Conductor Configuration: 275 kV double circuit, pyramid type Tower Type:
Nebo-Ross
0 0 to
Ross-Chalumbin 275 kV Double Circuit Heavy Suspension Tower: o
tO
Tower Type: Tower Structure:
Heavy suspension tower Rectangular base, lattice steel, all members of hot-roiled angle section Height: 59600 mm Self-Weight: 99.27 kN (original tower) 100.43 kN (upgraded tower) Conductor Configuration: 275 kV Double circuit Figure 4
T o w e r s used in case studies
Earthwire Peak Top c r o s s - a r m Super structure
Middle c r o s s - a r m Bottom c r o s s - a r m
Common body
!
Body
+o
Bo.,y Leg 12
(a)
(b)
Figure 5
Elevation views of N e b o - R o s s test tower: (a), front view; (b), side view
is used for deflections: deflection in the transverse (T) direction at point AL is TAL, in the longitudinal (L) direction at point CR is LCR and in the vertical (V) direction at point EL is VEL.
Test loading procedure. During the test, loads were applied in an incremental-cycle manner 25. That is, a certain load increment was given to one load and then followed by the same percentage increment to another 144
Eng. Struct. 1992, Vol. 14, No 3
load so as to bring them to the same percentage level of the design load. This procedure was repeated for every load increment of every load point in the tower in order to achieve the specified load level. When the load increment is applied to a particular point, however, loads at other locations are relaxed and they in turn must be re-adjusted to the same percentage level. The sequence of loading and re-adjustment every time a certain load increment level is applied to the structure cannot be simulated easily in the analysis where all the loads are incremented simultaneously. This effect becomes more significant as the structure begins to yield. Hence, some discrepancies in the ultimate loads and the deflections from the analysis and the test read-outs can be expected.
Test results. At the commencement of test no. 1, deflection read-outs assigned to LER, LAR and LBR (see Figure 609) were malfunctioning and therefore ignored. Loads were incremented in stages up to 100% of the design ultimate loads and held for 2 min. No permanent deformation or over-stressing was reported. At 90 % of the design ultimate load during test no. 2, the compression leg of Body :v0 (see Figure 5(a)) collapsed due to the failure of the diagonals in the body. The design was checked and as a result it was decided that the diagonal members of Body :F0, Body - 6 . 5 , and Body +7.5 would be upgraded. Larger angle sections were chosen with a higher grade steel of 350 MPa used instead of 250 MPa as used in the original design. During test no. 3, at 50% of the design ultimate load, malfunction of the deflection read-out assigned to LAR
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
18.5Q
~
8.50
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(d) Test no.4
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- 8.8
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: AR
AL =
BL CL
BR X1
CR
X2 X3
(e) Test no.5
T (f) Nomenclature
Figure 6 Load conditions for N e b o - R o s s test t o w e r
was reported. Adjustments were attempted but were unsuccessful and LAR deflection readings were subsequently disregarded. The test tower was able to hold loads at 100% of the design ultimate load condition for three minutes with no permanent deformation or overstressing observed. Test no. 4 passed without incident. The tower held 100% of the design ultimate load for two minutes with no permanent deformation or over-stressing observed. Cross-arm AR was selected for test no. 5 (cross-arm rigging). At 70% of the design ultimate load, the bottom chord adjacent to point of load application failed due to local plate buckling. Since the present analysis does not account for local plate buckling, this test load condition is not considered in the study.
Predicted results. The N e b o - R o s s test tower under the various test load conditions has been analysed using the developed AK TOWER program. The tower topology, geometry, load and support conditions were generated automatically using the formex algebra approach 23. A total of 1238 elements and 487 nodes has been used to model the tower. It should be noted that although the developed software can handle any joint flexibility, the present analysis assumes that no joint slippage in the tower occurred. Test no. 1: This test represents a load condition for temporary termination of both circuits as shown in Figure 6 (a)). The tower passed this test without incident and this is confirmed by the present analysis. Further, the
Eng. Struct. 1992, Vol. 14, No 3 145
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
l
1.4
l
,
1.2
I
1
I
X=1.22 1.0
1.2 .
1
.
0
~ -
-
-
~
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tO
-
Test
0.8
0
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/--Theory
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/
0.2
/
!
I
I
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I
I
(b)
eory
I
200 300 400 Deflection. TER (ram)
Test no.2 Transverse deflection
"o 0.4
Test no. 1 Transverse deflection at ER
/
0
"-" 0.6
(a)
/
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0.2
I 15
0
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I 30
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'
-~
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°0"811 / - ' T h e ° r y --
Vertical deflection I
50
I
100 Deflection, VAR (ram)
Figure 7 T h e o r e t i c a l l o a d - d e f l e c t i o n c u r v e s f o r N e b o - R o s s
/
0.8
test tower:
Test no. 2. This test represents a load condition for temporary termination of a single circuit as shown in Figure
.
. (d)
Test no.4 Transverse deflection at Xl
/
/ [
150
f -T_est_.Lup - - - - - to X= 1.0
Theory 0.4
AK TOWER program is able to predict an ultimate load factor ), = 1.22 for this load condition. The theoretical. load-deflection curve at TER (see Figure 6(/9) are shown m Figure 7(a). The tower deflected shape at failure (X = 1.22) is shown in Figure 8(a). The first plastic hinge starts in the compression leg of the common body at a load factor ), = 0.92. This is followed by gradual spread of plastic hinges in the compression legs of the common body until a maximum load of ~, = 1.22 is reached. After this stage, plastic hinges start to spread in the compression leg of the lower body (Leg + 12) causing a drop in the load and a rapid spread of plastic hinges in the diagonals of the common body. As reported by Transfield 25, the test was passed successfully. It can be seen from Figure 7(a) that the derived load-deflection curve is still quite linear up to the design ultimate load (i.e. at ~, = 1.0). A comparison between the test measurements and the numerical results for the deflections of different nodes at the design ultimate load (), = 1.0) is presented in Table 1. The predicted deflections agree reasonably with the test measurements considering the effect of bolt slippage which must have occurred at joints and the manner in which loads were applied during the test.
146 Eng. Struct. 1992, Vol. 14, No 3
- -- ..- _ ~ ~ _
0 ..J
Test no.3
II/
II1
"o
(c)
0.4 I-l/ 0
at X1 I 45
Deflection, TX1 (mm) I
[ l h = 1.62
~
failed at ;~= 0.9
.~ 0.8 -___h=0.75
-
i
50
i
I
100 150 Deflection, TX 1 (mrn)
200
(a), t e s t no. 1; (b), t e s t no. 2; (c), t e s t no. 3; (d), t e s t no. 4
6(b). During the test, the tower failed at a load factor X = 0.90 due to the collapse of the compression leg in Body ~:0 which was initiated by the buckling of the diagonal members in the body. The load-deflection curves obtained from the AK TOWER program are shown in Figure 7(b) for the deflection TX1. The deflected shape of the tower is shown in Figure 8(b). The first plastic hinge forms at X = 0.52 in the common body and spreads down to the diagonals of Body - 6 . 5 and Body ± 0 . However, when the applied load reaches X = 0.75, the numerical method starts to diverge and the analysis breaks down as the tangent stiffness matrix becomes ill-conditioned. Results predicted by the AK TOWER program indicate a weakness in the tower under this load condition and a nonlinear behaviour at a relatively low load level (i.e. at X ~- 0.5, see Figure 7(b)). The reason for the higher test load (h = 0.90 compared to the predicted ), = 0.75) may be attributed to the different loading (incremental-cyclic) method used during the test.
Test no. 3. This test represents a load condition for intact/broken maintenance as shown in Figure 6(c). The tower passed this test without incident z5 and this is confirmed by the present analysis. The load-deflection curve for the deflection VAR is shown in Figure 7(c). The first plastic hinge forms at X = 0.95 and hinges
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai Test
Figure 8
Test
no. 1
no.2
Test
no.3
Test
no.4
)~=1.22
h=0.75
~=1.62
;k= 1 . 5
(a)
(b)
(c)
(d)
=neoreticai failure deflected shapes of N e b o - R o s s test tower: (a), test no. 1; (b), test no. 2; (c), test no. 3; (d), test no. 4
spread but remain confined to the cross-arms AR and CL. A maximum load factor ~, = 1.62 is reached as indicated by the plateau in the load-deflection curve in Figure 7(c). The predicted deflected shape of the tower at the maximum load value is shown in Figure 8(c). A comparison between the test read-outs and the numerical results for various nodal deflections at the design ultimate load (X = 1.0) is presented in Table 1. It can be seen that the predicted deflections agree reasonably with the test measurements.
Table 1 Comparison
Test no 4. This test represents a loading for a broken earth-wire and upper phase as shown in Figure 6(d). The tower passed this test without incident and this is confirmed by the present analysis. The load-deflection curve for the deflection TX1 is shown in Figure 7(d). The first plastic hinge forms at X = 0.75 in the diagonals of the common body. More hinges start to spread to the other diagonals but remain confined to the common body until a maximum load factor ~, --- 1.5 is reached. After this point, plastic hinges start to form in the leg
of predicted nodal deflections with full-scale test measurements for Nebo-Ross 2 7 5 kV tension t o w e r
Measured Deflections (turn) Test no. 1
Test no. 3
Test no. 4
Measurement locations
Test
Predicted
Test
Predicted
Test
Predicted
TER LER LEL TAR LAR LAL TBR LBR TCR LCR VCR VCL
-54 ** -87 ** -74 ** -44 -148 -
-88 -513 -68 -392 -47 -263 -29 -159 -
-86 -218 -93 -86 ** -19 . . -24 . +16 -77
-115 -161 -105 -93 -151 -44 . . -39 . + 5 -54
-114 -453 +13 -150 -225 +44
-215 -251 -82 -159 -230 -14
-48
-65
-
-
* * T h e s e malfunctioned during the test and were
. . .
. . .
-
disregarded
Eng. Struct.
1992,
Vol.
14, No 3
147
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
Body
-6
(a) Figure 9
Front and side views of R o s s - C h a l u m b i n test towers: (a), original tower; (b), upgraded t o w e r
members of the common body causing a drop in the load followed by a failure of the numerical method indicating that a collapse condition has been reached. The predicted deflected shape of the tower at the end of the analysis for this test is shown in Figure 8(d). A comparison between the test read-outs and the numerical results for different nodal deflections at the design ultimate load (X = 1.0) is presented in Table 1. It can be seen that some of the predictions are in reasonable agreement with the test read-outs while others are not so good. The differences between the predicted results and the test read-outs for LER, LEL and LAL can be attributed to the effects of bolt slippage, damage to some of the members from earlier tests and the incrementalcyclic loading procedure used in the test.
Ross-Chalumbin 275 kV double circuit heavy suspension tower This tower (see Figure 4) is part of the R o s s Chalumbin transmission line recently constructed in Queensland, Australia. Details of test procedure and results have been reported by Electric Power Transmis2~ sion (EPT) Pry Ltd . It is worth noting that the loading procedure used in this test is different from that employed in the tension (Nebo-Ross) test tower. In the Ross-Chalumbin tower, all loads were applied simultaneously in an incremental manner which is similar to that used in the analysis hence better predictions are expected.
Test results. The topological outline of the test tower is shown in Figure 9. The tower was tested under six dif-
148
(b)
Eng. Struct. 1992, Vol. 14, No 3
ferent load conditions and passed the first five tests successfully. In the sixth test, which was under the intact maximum wind load condition, the tower collapsed at 90 % of the design ultimate, load. The failure described in the test report 26 as an unusual failure which involved a general collapse of the compressed face of the tower'. Figure 10 shows a photograph of the failed original tower after the test. The ultimate design load condition for the sixth test is shown in Figure 11(a). Figure 11(b) shows the notation adopted for different nodes and displacements. Following the collapse of the tower in the sixth test, the tower was strengthened by adding a number of horizontal members in an attempt to prevent the global failure experienced in the sixth test. The initial upgraded tower shown in Figure 9go) (compared with the original tower in Figure 9(a)) was tested and it again failed at a load only slightly higher than that encountered in the original tower. The tower finally passed the test after further strengthening but details of the final configuration are not given in the test report 26.
Results predicted by AK TOWER program. Both the original and the initial upgraded towers have been analysed using the proposed nonlinear analysis AK TOWER program. The tower topology, geometry, load and support conditions were generated automatically using the formex algebra approach 23. A total of 1732 elements and 768 nodes have been used to model the original tower. The initial upgraded tower has been modelled using 1761 elements and 772 nodes. In the analysis of the original tower, the first plastic
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai
)0
1.95
Figure 12 Theoretical failure deflected shapes of R o s s Chalumbin test towers: (a), original tower; (b), initial upgraded tower
at collapse is shown in Figure 12(a). The shape closely resembles the actual tower failure shape obtained during the test as depicted by the photograph taken after failure
(Figure 10). Figure 10 Photograph of failed (original) Ross-Chalumbin test tower
hmge is formed at a load factor h = 0.7 in Body - 6 (see
Figure 9(a)). Subsequent plastic hinges spread rapidly to other parts of the tower below Body - 6 accompanied by a rapid increase in displacement. A maximum load factor )~ = 0.91 is reached followed by a break down of the numerical method indicating collapse of the tower. This compares extremely well with the 90% failure load reported in the test :6. The deflected shape of the tower
10.83
10.83
6.83--I 23.86 --145.39 23.86 --145.39 23.86 --145.39
In the analysis of the initial upgraded tower, the first plastic hinge is formed at a load factor )~ = 0.8. Plastic hinges then spread in a similar fashion as before but less rapidly and with less displacement. A maximum load factor X = 0.95 is obtained. This load is only a little higher than that of the original tower as was confirmed by the test report 26. The predicted deflected shape of the initial upgraded tower is shown in Figure 12(b). Load-deflection curves for vertical deflection VER at node ER of the original and the initial upgraded towers are shown in Figure 13. It can be seen that the addition of a number of horizontal members to the tower has, to some extent, improved the tower's behaviour.
I. 11.37 45.39 I - 3 2 - 3 2 45.391 - 35"78 45.391-40"66 ~20.5 --15.4 --35 ~35
= ER
EL =
BL=
=AR -- BR
eL--
-- CR
AL'-
Xl X2 X3
v
X4 7__ L
-
(a)
T
(b)
Figure 1 1 Critical load condition for Ross - Chalumbin test tower
Eng. S t r u c t 1992, Vol. 14, No 3
149
Nonlinear analysis of transmission towers: F. G. A. AI-Bermani and S. Kitipornchai 1.2
,<
I
I
1.0 -
X=0.95
'
X=O.9
I
profound effect on tower deflection but should only have little influence on tower ultimate capacity 27.
I
Acknowledgements 7
0.8
o 0
0.6
"0 0
-~ 0 . 4 -
/
-Original tower
/ 0.2 -
-
/
/I
0
50 Vertical
Initial upgraded tower
I
I
;
100
150
200
deflection,
-
-
250
This project has been supported by funds from the Australian Research Council (ARC) under Project Grant No. 834 and from the Australian Electricity Supply Industry Research Board (AESIRB). The authors wish to thank the Queensland Electricity Commission (QEC) for making the full-scale test data of the two tested transmission towers available. In particular, the authors wish to acknowledge the assistance from Mr Henry Hawes of the QEC for his continued encouragement, advice and technical support throughout the course of this work. The authors wish also to thank Dr S. L. Chan of the Department of Civil and Structural Engineering (Hong Kong Polytechnic) for the initial work on developing the geometric stiffness for asymmetric thin-walled beamcolumn elements and Mr Warren Traves, (Gutteridge Haskins and Davey Pty Ltd) for proof-reading the manuscript.
VER (mm)
Figure 13 Theoretical l o a d - d e f l e c t i o n curves of original and
References
initial upgraded R o s s - C h a l u m b i n test t o w e r s
Conclusions Accurate structural analysis of transmission towers is complicated because the structure is three-dimensional and comprised of asymmetric angle section members eccentrically connected. The influences of geometric and material nonlinearities play a very important role in determining the ultimate behaviour of the structure. The paper describes a nonlinear analytical method in which most factors affecting the ultimate behaviour of the tower structure can be incorporated. These include geometric and material nonlinearities, joint flexibility and the effects of large deflection. The developed software, the AK TOWER program, was used to predict the ultimate structural behaviour of two electric transmission towers recently tested in Australia. No comparison of the calculated and the actual member forces was made due to the lack of such field data in these tests, but predictions of the ultimate loads, the nodal deflections at various points and the tower failure deflected shapes have been made. The developed software was able to predict accurately the collapse load (predominantly flexural load) for the Ross-Chalumbin heavy suspension tower. As for the N e b o - R o s s tension tower, the predicted collapse load for test no. 2 (torsional load) was 16.7% lower than the test collapse load. This may be attributed to the different loading procedures used. In the test, loads were applied in an incremental-cyclic manner whereas in the analysis all loads were incremented simultaneously. Moreover, a torsional load case is generally regarded as more difficult to predict than a flexural load case. Comparison of measured and predicted deflections have not all been good. This may be attributed to the effect of bolt slippage which must have occurred at most joints during test, particularly at high load level, but was totally ignored in the analysis. Bolt slippage will have a
150
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