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Analysis and design of single-pole transmission structures
Compurers& SrrucruresVol. 28, No. 4. pp. S-562. Printed in Great Bntam
WS-7949/88 $3.00+ 0.00 Pergamon Press plc
1988
COMPENDIUM ANALYSIS AND DESIGN OF SINGLE-POLE TRANSMISSION STRUCTURES D. VANDERBILT
M.
Department of Civil Engineering,
and M. E.
Colorado
State University,
(Receiwd
18 March
CRISWELL
Fort Collins,
CO 80523, USA
198’7)
Abstract-The unguyed single-pole transmission structure is widely used to support transmission lines of up to 345 kV capacity. Most such structures are wood poles, although prestressed concrete poles and tubular steel poles are sometimes used in high decay or high load situations. A number of features combine to make this type of structure an interesting structural analysis and design problem. Whether naturally grown or manufactured, the pole structure is almost always nonprismatic. Wind, ice and snow loads acting alone or in combination are variable and occasionally severe in nature. Poles are slender and the combination of transverse wind load with vertical dead and ice loads often leads to behavior which is geometrically highly nonlinear. The continuous exposure to weather, fungi, accidents, etc. results in structures with capacities that often deteriorate very significantly with time in service. The material properties (strength, stiffness), especially of wood, are highly variable, as are the soil conditions, The configurations of conductors, davit and cross arms, and attachments that are in use vary widely. This large variety of structural shapes plus the necessity of changing the configurations of in situ structures to meet new conductor and clearance requirements prevents the use of only a few designs. Thus rapid ways to assess existing structures and to design new structures are required. Since there are more than 130,000,000 poles currently in use in single-pole and H-frame transmission structures and several million new poles are put into service each year, the design effort related to poles is large. As part of an eight-year effort to develop reliability-based design methods for transmission structures, a correct procedure for analyzing and designing single-pole transmission structures was established. Implemented in program POLEDA (POLE Design and Analysis), this procedure enables unguyed planar single-pole structures of virtually any material and configuration to be rapidly analyzed or designed. The analysis is based on the Newmark numerical technique which, it is shown, in the limit provides an exact solution to the differential equation for large deflections of transversely loaded beams. The four design options provided in POLEDA use the load and resistance factor design (LRFD) format. This permits the user to use current, deterministic, design procedures or to use load and resistance factors based on reliability studies. The program provides the necessary high-quality analysis capability which forms an integral part of reliability-based design. It is being widely adopted by the electric utility industry.
1. INTRODUCTION
A
typical
structure
unguyed is shown
is typically in place, special
by and
aggregate,
wood in Fig.
auguring backfilling
single-pole
transmission
1. Foundation a
hole,
with
or manufactured
preparation
setting
tamped
the
natural
fillers.
The
pole soil, usual
setting depth is 10% of the pole length plus 2 ft for poles up to 90 ft long [I]. The structure supports current-carrying conductors located at the ends of cross arms or davit arms and usually supports a shield or ground wire, located at or above the pole top, to protect the conductors from lightning strikes. Conductor locations are described by their distance above the groundline and their eccentricity from the pole centerline. the ‘span’ relates to the conductor lengths supported by a structure. On flat terrain with uniformly spaced poles, the span is the distance between two adjacent poles. On sloping terrain, the wind (lateral) and weight (vertical) spans differ as shown in Fig. 2. Structures located in a straight line are tangent structures while structures located where a transmission line changes direction are angle structures. If the angle is small (a light angle structure), 551
then guy wires may not be required. If the angle is large, then guys wires are typically used with wood poles while heavier unguyed structures are used with steel. The transverse direction is the direction perpendicular to the span of a tangent structure. The most severe loading generally acts in this direction and most analysis and design is for transverse loads. Loads in the longitudinal direction are typically small unless a conductor breaks. Thus a planar analysis suffices for most design problems. In order to analyze an unguyed structure the engineer must define the geometry and material stiffness of the structure, the location, size and weight of each conductor and groundwire, the location and weight of each attachment (cross arm, insulator, transformer, etc.), the wind and weight spans, the ice/snow and wind loads, transverse loads due to components of conductor tension for angle structures and any known initial slope of the pole in the transverse direction. Deformations of the soil as load is applied are almost universally neglected and the resulting assumption that the structure is fixed at the groundline is also followed herein. The influence of soil
M. D. VANDERBILT and
552
M. E. CRISWELL
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deformations can be significant, however [2]. Design of a transmission structure typically means the determination of the span which can be supported by a structure of a given size. This practice arises from the method used to categorize wood poles by load carrying class [I]. In addition to information required for analysis, design requires that material strength and stiffness and the load and resistance factors be defined. While logically only one design procedure should exist, the current state-of-the-art encompasses several different procedures. Hence the designer must choose a design method as well as defining all the above data. The electric utility industry consists of more than 3000 public and private utilities including the REAs. Transmission structure design is sometimes performed in-house by engineers, in the past often electrical but now usually civil, employed by the larger utilities.
fir pole.
Smaller utilities and many of the REAs use consulting services provided by engineering firms and some of the larger utilities. The Electrical Power Research Institute (EPRI) headquartered in Palo Alto, CA is the research arm of the industry. Supported by several hundred of the U.S. utilities, EPRI conceives and supports research in many areas. Since 1978 EPRI has supported research at Colorado State University (CSU) on the development of reliability-based design techniques for transmission line structures. The first type of structure studied was the single-pole wood tangent transmission structure. About 130,000,000 wood poles are in use in the U.S.A. and several million new and replacement poles are placed in service each year. Since the in-place cost of one single-pole structure is currently $100&$2000, the nation-wide investment in wood pole structures is significant and improved design techniques can have
Analysis and design of single-pole transmission structures
Vertical
05L,
L,
tFor analysis,
, I
spa”
-I
3. CURRENT
05L,
I.
input span =
resistance and load effects on the same basis so they are directly comparable. Any error in the analysis will result in the as-built structure having a different reliability than that desired by the design engineer. If the method of analysis used in design is highly approximate, then the structures placed in service will have unknown and highly variable reliabilities. Using a method of analysis which introduces only small errors in converting factored loads to load effects is the sine qua non of reliability-based design.
span
Horlzontai
LZ
I
(H, + H, I
Fig. 2. Definition of horizontal (wind) and vertical (weight) spans.
an important economic payoff. Wood transmission poles are typically 55 to 125 ft long or about 48 to 113ft high above groundline. Their highly nonlinear behavior must be considered in any correct analysis and design procedure. 2. BASIC CONCEmS OF LOAD AND RESISTANCE FACTOR DESIGN
Of the many different ways to implement reliabilitybased design, the load and resistance factor design (LRFD) formulation is receiving the most widespread acceptance. The LRFD concept is expressed by b*R
2A(y*Q)>
553
(1)
where 4 is a capacity reduction (i.e. resistance) factor almost always less than 1, R is a measure of resistance such as the modulus of rupture of wood or yield stress of steel, y is a load related factor usually greater than one, Q is a load causing phenomenon such as wind pressure, ice thickness, etc. and A ( ) means that an analysis is performed of the structure loaded with the factored loads so that the factored resistance (4 *R) and the load effects can be compared on the same basis (e.g. normal stress). In words eqn (1) states that the factored resistance must equal or slightly exceed the effects of the factored loads. In reliabilitybased LRFD the four terms in eqn (1) are determined on the basis of studies of the statistical variations in R and Q and the level of reliability provided by various 4 and y factors taken in combination with various measures of the probability distributions of R and Q such as the mean values or upper and lower exclusion limits. Discussions of the studies required to establish LRFD equations for transmission structures are given in [3]. The role of structural analysis in reliability-based design, the A( ) in eqn (l), is to place factored
LINEAR
ANALYSIS
AND DESIGN
The first design document usually considered by the U.S. transmission structure designer is the National Electrical Safety Code [4] published as ANSI Standard C2. The NESC is primarily concerned with electrical requirements such as clearances between adjacent conductors and conductors and ground. The only structural provisions prescribe that structures be designed to carry the specified loads multiplied by overload capacity factors (OCFs). The NESC provisions can be expressed in LRFD form as (l.O)*R
2 A(OCF*Q).
(2)
The implied 4 of 1 and the prescribed OCF values are based on judgement and not on any formal type of reliability studies. In eqn (2), R is the ultimate resistance taken from a document specified by the NESC in the case of wood ]I] and left undefined for other materials, OCF is 4 for most new wood pole construction, and Q is a loading value such as 4 psf wind plus 0.5 in. radial ice thickness on conductors taken from a map. The entire U.S. is divided into only three NESC districts by this map. The method of analysis is left undefined as are other details of design. Since the public utilities commissions of most states require that NESC provisions be satisfied, the structural engineer is bound by eqn (2). For wood structures, the second design document of interest is the Specifications and Dimensions for Wood Poles, ANSI Standard 05.1 [ 11. This document categorizes poles by classes. The class of a pole of a given diameter and length is obtained by assuming the pole has a uniform taper from groundline to tip, that it is loaded by a single transverse load placed two feet from the tip, and that the maximum flexural stress occurs at the groundline. By assuming a value for the modulus of rupture, R, the value of the transverse load can be computed. Tables in ANSI 05.1 give the pole sizes as functions of class load and wood species. Thus a 70 ft class 2 pole can carry a class load of 3700 lb regardless of species. The only document containing examples of current design of wood structures is the Design Manual for High Voltage Transmission Lines published by the Rural Electrification Administration [5]. All of the examples in the REA manual use only linear analysis, and stresses in single wood pole structures are checked
M. D.
554
VANDERBILT and M. E. CRSWELL
only at groundline. This use of linear analysis coupled with the ANSI 05.1 practice of checking stresses only at ground line has produced a current design practice which is significantly in error. Thus a large and unknown part of the apparent ‘factor of safety’ of 4 implied by the OCF of 4 for wood poles is consumed by errors in analysis. Examples of design of steel poles are given in the Design of Steel Transmission Pole Structures publication by the American Society of Civil Engineers [6]. The ASCE steel pole manual uses linear analysis in all examples, checks stresses everywhere in the structure, and uses a moment magnifier to approximately account for P-delta effects. The fact that nonlinear behavior has not been adequately recognized by the writers of the NESC is shown by the change in basic design criteria introduced in the 7th (1977) edition. In the previous 6th edition, the design equation was (0.25)*R
> =A(I.O*Q).
(2a)
Thus the 4 factor was 0.25 and the y factors were 1.Oin the 6th edition. If the behavior was truly linear, then eqns (2) and (2a) would produce identical results. Since the behavior is nonlinear, multiplying the loads by four before making a nonlinear analysis will produce P-delta stresses on the order of at least 16 times greater (i.e. four times the load acting through at least four times the lateral deflection) than the P-delta stresses computed for (1.0) times the loads. Based on this brief review of the design and code documents now in use, it may be concluded that the role of correct analysis in design is not yet understood by many people involved with transmission structure design. 4. ANALYSIS
METHOD
A correct method of analysis defines the equilibrium position of a structure in terms of its final deflected shape. In linear analysis the change in geometry as load is applied is ignored which greatly simplifies analysis. If the change in geometry is considered, then the equilibrium position must be found through an iterative procedure which involves some type of searching technique to locate the equilibrium shape. A host of analytical techniques is available for use with structures that soften with increase in load. A major criterion in selecting a method for POLEDA was that it be computationally very efficient since it was planned that analyses of several thousand structures for multiple load cases would be made as part of the effort to establish reliability-based design methods. This criterion plus the observation that the unguyed single-pole structure is statically determinate led to the selection of the Newmark numerical technique [7-91 as the basic solution technique. The fundamental concept of the Newmark tech-
nique, as applied to transversely loaded beams, is that the real structure, which has a continuous variation in curvature over its length, is replaced by a series of rigid segments connected by rotationally flexible springs. The total curvature occurring over the length of the beam is concentrated at the springs. For small deflections the familiar differential equation. neglecting shear effects, is d2y
M(x)
dx’
El(x)
(3)
where x = axial coordinate, J’ = transverse coordinate (deflection), M(x) = moment caused by external loads and which varies only with s for small deflections, E = Young’s modulus, and I(s) = the moment of inertia which varies with s in nonprismatic members. The M/EI term in eqn (3) is the curvature. If the curvature is concentrated exactly. then the Newmark technique gives the exact solution at each of the nodes (flexural springs) and the chords between nodes closely approximate the true elastic curve. It is demonstrated in [7-91 that the concentration formulae exactly concentrate the curvature for some loading cases on prismatic beams. When applied to nonprismatic members, the formulae are approximate but the error is reduced to a small level by using reasonably short segments. For long slender members such as transmission structures, the effects of shear are negligible and omitting the shear term of eqn (3) has little effect. For large deflections of transversely loaded beams the governing differential equation is M(.r, $‘)
d2yjdx’
(1 + (dy/dx)2)3,2 = El(.u)’
(4)
where M(x, y) indicates that the moment is a function of the deflected shape. Figure 3 shows that curvature and the slope of the elastic curve are related by l/p = d0/ds. From
Fig. 3 it is also noted
(5)
that
dx = ds cos 0 0 = arctan (dy/dx). By substituting that [IO]
(6)
eqn (6) into eqn (5) it may be shown
1
d2y/dx2
p
(1 + (dy/dx)‘)3’2’
(7)
The usual substitutions may be made to use eqn (7) in developing eqn (4). With the Newmark technique, finite lengths A-X replace the infinitesimal lengths dx and thus eqn (6)
Analysis and design of single-pole transmission structures
555
ds = pd8
I=!!?! P dS Fig. 3. Elastic curve of benm. is rewritten
as Ax=Ascose 0 = arctan (By/Ax).
(8)
The computational procedure for computing the equilibrium shape of a transversely loaded slender beam using the Newmark method consists of the following steps: I. The beam is divided into segments and I(x) is computed at each node. 2. The moment M(x) and the curvature M/EZ are computed at each node. 3. The slope, 0, of each chord between adjacent nodes, and the deflection, y, of each node are computed. The deflected shape must satisfy kinematic boundary conditions. 4. The moment M(x, y) is now computed at each node. Because the beam is deflected, eqn (8) can be written for each segment. In general M(x, y) will be less than M(x) if only transverse loads act but will be greater than M(x) if both transverse and significant axial loads act. 5. The curvature, slope and deflection are computed. 6. The latest deflected shape is compared with the previous shape by dividing y,_ 1 by y, at each node, where y, is the deflection for the latest iteration and y, _ , is the deflection for the previous iteration. If the ratio equals unity plus or minus a small tolerance (say 0.01) at every node, then convergence is obtained. Otherwise the procedure returns to step 4 and continues until convergence or until examination of results after several cycles shows that the solution is oscillating or diverging. Where axial loads act at different points along the length of a beam-column the same basic procedure is used. The effects of the axial loads is null in steps 2 and 3. In step 4 the
P-delta moments caused by axial loads are computed and added to M(x, y). An example solved by hand is given in the Appendix to show the details of the technique. Note that the tabular format used with the Newmark technique enables the method to be implemented using a spread sheet program. Closed form solutions for large deflection problems are difficult to obtain. A large deflection solution for a cantilever beam transversely loaded with a single tip load was obtained by Bisshopp and Drucker using elliptic integrals [111.Their results are shown in Fig. 4 by solid lines. In this plot, A is the transverse deflection while S is the shortening of the moment span. The small deflection theory solution is shown by the broken straight line plot. For small deflections, the ratio of the tip deflection to the span is given by
A/L = WL2/3EI IO
Fig. 4. Large deflection
\
of tip-loaded
(9) I
cantilever
beam.
556
M.
D. VANDERBILT and
and the moment at the fixed end is II%. For large deflections the moment at the fixed end is W(L - S) and thus it is conservative to design transverselyloaded beams using linear theory. However, in beamcolumns the reduction in transverse moment (WS) is usually much smaller than the increase in PA moments. Thus the use of linear, small-deflection theory in designing beam-columns can be in error by as much as a factor of 2. The solution obtained using the Newmark procedure is shown by the circles in Fig. 4. From the closeness of the exact and numerical solutions it may be concluded that the Newmark procedure in the limit (as Ax approaches dx) gives an exact solution to this problem. Convergence studies may be made by analyzing a structure several times using successively shorter segments. Experience shows that dividing a structure into lo-20 segments gives good results. Closed form solutions for nonprismatic beam-columns have not been found in the literature. However, it is believed that the use of the Newmark method for their analysis gives good results. The Newmark method can also be used to compute the buckling capacity of straight members supporting only axial loads. For the case of a single axial load, the buckling capacity can be given in terms of the force which will caused bifurcation of the equilibrium position. For the case where two or more axial loads act, the buckling capacity is expressed as a multiplier applied equally to each load [12]. Thus if a tip load P and a midlength load Q act on the real structure, then the buckling capacity is given by i and the structure would buckle under the simultaneous application of U’ and AQ. If the axial components of the applied loads on a beam-column cause buckling, then a solution for the combined transverse and axial loads cannot be obtained. Hence a complete analysis begins with a buckling analysis to determine if a large-deflection beam-column analysis solution can exist. An example of a buckling analysis is given in the Appendix. Extensions of the Newmark technique to the analysis of beam-columns experiencing large deflections have not been found in the previous literature.
5.
DESIGN OPTIONS
The POLEDA program allows the user to select a different design option for each load case, where design consists of determining the allowable span for a given structure. This feature was included so that the effect of using different design criteria can be quickly examined. The design options provided are: Option Option
&only an analysis is performed. l-the current linear design method is used, thus only a linear analysis for transverse loads is performed, P-delta effects are ignored, and stresses are checked only at groundline.
M. E.
CRISWELL
Option
2-basically the same as option 1 except that moments from vertical dead and ice loads acting at the ends of the cross arms are also considered. This option was included since the 1980 edition of the REA manual [5], which replaced the 1972 edition, adopted this one improvement over option 1. Option 3-all nonlinear behavior is correctly included in the analysis but the stresses are checked only at the groundline. Option &-all nonlinear behavior is correctly included in the analysis and the maximum stress wherever it occurs in the structure is considered. No matter which design option is chosen by the user, POLEDA always concludes the design of each load case by performing a correct analysis and outputting the stresses and deflections at each node point, Design options 1 and 2 require that only one analysis be made. The program begins with an assumed span and performs an analysis. Part of the total capacity at groundline is used to support wind that acts on the pole. The remaining capacity is used to support wind which acts on the iced conductors and, for option 2, to support moments due to vertical ice and conductor weights. For either option the loads on the conductors are linear functions of span and thus the allowable span, which causes eqn (1) to be satisfied as an equality, is easily found. Design options 3 and 4 require that an iterative method be used to compute the span [13]. This requirement arises because the stresses in the structure are a nonlinear function of the span. Figure 5 shows a plot of trial span, TS, versus V, where V equals the maximum normal stress divided by the factored resistance, Cp*R. The correct span is the one for which V = 1. POLEDA computes TS 1 using the groundline moment computed ignoring P-delta and large geometry effects (i.e. using option I). A correct analysis is performed and the first value of VI is computed. The TSl, VI point plots as point 1 in Fig. 5. For design options 1 or 2 the procedure stops here. For options 3 and 4 a straight line from the origin through point 1 gives point A and TS2. A correct analysis gives point 2. The line through points 2 and 1 gives point B and TS3. A third nonlinear analysis gives point 3. A parabola with equation TS = A V2 + BV + C can now be fitted through points 1, 2 and 3. Three simultaneous equations are solved to determine A, B and C. This parabola, plotted as a broken line in Fig. 5, is used to find TS4. Analysis gives V4 and finally point 4 is plotted. If point four plots as V equal to one plus or minus a small tolerance (say 0.01) the design is complete. If it plots outside the tolerance limits, the procedure continues with the coefficients of the parabola always based on the latest three trial spans, until either convergence or a prescribed limit on number of trials is reached. Experience shows that more than four
Analysis
and design
of single-pole
transmission
structures
I-
557
I+T
TSI
TS:
TS;
Parabola
through
3
points
Curve of span versus V
Acceptable
V = (maximum
Fig. 5. Interpolation
normal
procedure
trial spans is rarely required. Note that an iterative buckling analysis followed by an iterative beamcolumn analysis is required for each trial span. 6. PROGRAM
POLEDA
Program POLEDA is written in FORTRAN 77 and can execute on an IBM PC or AT. It contains about 1200 source statements and the executable file on an AT is about 63,000 bytes in length when compiled with the Microsoft 4.00 FORTRAN compiler. The program operates in a batch mode with the input data file being easily constructed with any text editor. The program is available through the Electric Power Software Center, 1930 Hi Line Drive, Dallas TX 75207, (214)-655-8883. It is distributed at no charge to EPRI members and is for sale to nonmembers. The source code is available to users who wish to install the program on a mainframe. A package of programs for transmission line design is
stress)
/(Phi
range
*R)
for span computation.
available from EPRI as the TL-Workstation (tm). This package, which requires an IBM AT as part of the workstation, includes a compiled copy of POLEDA, an interactive front-end program to generate the input data file, and a post-processor to provide graphics displays of stresses, deflected shapes, etc. Data are input in blocks with the length of each block dependent upon the complexity and type of structure. The blocks are briefly summarized to give an idea of the capabilities of the program. Block 1. Control
(a) The type of pole, either wood, steel or general geometry, is defined. (b) The common values of material unit weight, modulus, and material strength are defined and become the default values in each load case. (c) The tolerances for accepting the deflected shape and the design span are each input or default to 0.0 1.
558
M. D. VANDERBILT and M. E.
The program contains default values for many of the data and uses the default value whenever the user inputs a 0 value. Block 2. Structure geometry
(a) For a wood pole the length, and the tip and butt circumferences are input. The program uses the ANSI 05.1 assumption of uniform taper to compute the area, A, section modulus, S, and moment of inertia, I, at each node point. (b) For steel or aluminum poles the length of each section (six maximum), the height from groundline to bottom of section, plate thickness, and section diameter at bottom and top are input. Metal poles may be circular or polygonal in cross-sectional shape. The program computes A, S and I values. (c) For general geometry poles, the cross-sectional dimensions and properties A, S and I must be given at each node. Block 3. Load level dejinition
(a) Control data to define the number of load cases, number of conductors, number of levels at which loads may be applied and number of segments to use in the Newmark analysis are input. (b) A table of conductor properties is input which includes the name, weight per foot, and diameter of each conductor (including groundwires) to be used. (c) The location of each level at which loads may be input is defined. At any load level, conductors may be attached or concentrated actions, such as from an attached transformer, may be applied, or both. (d) The number of each type of conductors acting at each load level is input. Block 4. Load and design data
For each load case the user selects the design option, the load and resistance factor data, the wind model, the ice model, other load data such as extra concentrated actions, and the resistance and stiffness data. This flexibility was provided so that comparisons of the effects of choosing different design options could be quickly made. The data include: (a) Control data = design option, No. of extra concentrated actions, and span (required for option 0 only); (b) Extra concentrated actions, if any; (c) 4 and y factors for every load cause. All values default to 1.0; (d) Wind model. Either a uniform wind pressure corresponding to the NESC district loadings or a velocity profile is chosen. If a velocity profile is selected, then wind velocity, reference height, power law exponent, and air density are input. Drag factors for conductors and pole are input for either model; (e) Ice thickness and density; (f) Young’s modulus, material density, and material strength. These default to the common values if input as 0.
CRISWELL
7. EXAMPLERESULTS Figures 6 and 7 show analysis results for the pole of Fig. 1 for an NESC heavy loading consisting of 4 psf wind, a radial ice thickness of 0.5 in. on each conductor, an ice density of 57 pcf, drag factors of unity, a load factor of 4 on the wind and ice weight and a load factor of 1 on conductor and pole weight. The NESC specifies OCFs of 4, but it is clear that the code writers assumed a linear analysis would be performed for which dead load has negligible effect. Hence selection of 1 for dead load seems reasonable. The NESC also specifies that stresses in wood poles are to be checked at the groundline. Using design option 2 POLEDA computed a design span of 390 ft. Figure 6 shows deflected shapes for two cases. The first is for a linear analysis which includes moments from unbalanced conductors. The second is for the correct analysis which is automatically output by POLEDA as the last part of each load case. The undeflected and deflected shapes of Fig. 6 are drawn to scale. Figure 7 shows that the linear analysis gives a stress of 8000 psi at groundline which is the designated fiber stress from ANSI 05.1. The maximum stress of 8340psi occurs 20 ft above groundline. The maximum stress in wood transmission structures almost always occurs above groundline, a behavior ignored by both the NESC and ANSI 05.1. In Fig. 7 the correct nonlinear analysis gives a maximum stress of 13,980 psi at 29 ft above groundline. Thus the current code sanctioned design of this structure is in error, on the nonconservative side, by 75% which is not untypical. Additional examples showing design results for various wood and steel structures are given in [3, 10, 12, 131. 8. CLOSURE The role of analysis in the design process is to enable resistances and loads to be compared on the same basis such as normal stress, axial compressive force, etc. An incorrect analysis introduces an unknown error and makes it impossible to design structures to have the level of reliability desired by the structural engineer. The behavior of the unguyed single-pole transmission structure, especially those of wood, is geometrically nonlinear to a significant degree and a correct method of analysis must properly consider this nonlinearity. Program POLEDAprovides a complete analysis and design procedure for unguyed single-pole transmission structures of wood or steel. The analysis method is based on extensions of the Newmark numerical method which, for the cases tested, provides a very accurate analysis for both combined axial and transverse loading response and buckling capacities. The design provisions allow spans to be determined for either currently specified deterministic loadings or for loads, resistances, and load and resistance factors chosen [3] on
Analysis and design of single-pole transmission structures
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M. D. VANDERBILT and M. E. CRISWELL
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the basis of reliability studies. The analysis capability of POLEDA provides the necessary high quality analysis which, if lacking, prevents design from being placed on a rational basis.
4. 5.
AcknowIedgemenl.~-The research, of which developing was a small part, was supported by EPRI as RP 1352, ‘Reliability-Based Design of Transmission Line Structures’. Mr Richard E. Kennon of the Transmission and Distribution Division is the Manager of the EPRI Electrical Systems Division which funded the project and Mr Paul Lyons of the EPRI Transmission Line Mechanical Research Center was project manager. Their continued support has been instrumental to the successful development of a reliability-based design methodology for transmission line structures. Colorado State University Professors J. R. Goodman and J. Bodig have been key researchers in various aspects of the project since its inception. Numerous graduate students have contributed to the project. In particular M. D. False developed the first version of POLEDA and E. M. Hendrickson extended it to include metal and general geometry poles. POLEDA
6.
I.
8. 9.
10.
II.
REFERENCES Specifications and dimensions for wood poles. ANSI 05.1- 198 I, American National Standards Institute, 1430 Broadway, NY 10018 (1981). K. H. Kim, M. E. Criswell and M. D. Vanderbilt, Inclusion of foundation effects in analysis of pole structures, Structural Research Report 61, Colorado State University (1986). M. E. Criswell and M. D. Vanderbilt, Reliability-Bused
APPENDIX:
EXAMPLES
12.
13.
OF ANALYSIS
Design of Transmission Line Structures: Final Report. Vols 1 and 2. EPRI (1987). National Electric Safety Code. American National Standard C2- 1984 (1984). Design manual for high voltage transmission lines. United States Department of Agriculture, Rural Electrification Administration, REA Bulletin 62-l. revised August 1980. Task Committee on Steel Transmission Poles of the Committee on Analysis and Design of Structures of the Structural Division. In Design of Sfeel Transmis.won Pole Strucrures. American Society of Civil Engineers. New York (1978). N. M. Newmark, Numerical procedure for computing deflections, moments and buckling loads. Paper No. 2202, Trans. AXE 108 (1943). W. G. Godden, Numerical Analysis of Beam und Column Sfruclures. Prentice-Hall, Englewood Cliffs, NJ (1965). N. C. Lind, Newmark’s numerical method. Department of Civil Engineering, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada (1975). E. M. Hendrickson, M. D. Vanderbilt, M. E. Criswell and J. R. Goodman, Reliability design of steel transmission poles. Structural Research Report 41, Colorado State University, CO (1982). K. E. Bisshopp and D. C. Drucker, Large deflection of cantilever beams. Q. appl. Math. 3 (1945). M. D. False, M. D. Vanderbilt, J. R. Goodman and M. E. Criswell, Probability-based design of wood transmission structures. Structural Research Report 35. Colorado State University, CO (1981). M. D. False, E. H. Hendrickson, M. D. Vanderbilt and J. R. Goodman, Program POLEDA-80: A computer program for the analysis and design of single pole transmission structures. EPRI EL-2040 (1981).
USING NEWMARK
METHOD
1. Concenlration formulae Assume
a = -M/EI
The concentrated
varies parabolically
curvature,
as shown.
a, at each node is given by [7-91 g, = h (7a, + 6a, - a,)/24 ZI,= h(a, + IOU,+ a,.)/12.
The equation
for ac is found
from the equation
a, and a,
for g, by interchanging
2. Beam analysis A cantilever prismatic beam is analyzed for simplicity. Assume Young’s modulus, E, =2000 ksi; moment of inertia, I, = 500 in’; a 40 ft span, and a transverse tip load of 5 kips. The beam is divided into four segments each of length h, = 120in. The computations are conveniently performed using the tabular arrangement shown. In the following A4 = moment, Icr. y = increment in deflection y between two successive nodes, and do. = ditto. Cycle 1. Linear analysis
1 5 kips
t
Computation
I lG b
II
1
-4
-3
:
22;
Icr. y = Bh, ? JI
0 0
H
II
;
22 22
36I1
I 1
58 58
II
1
-I
24I1
I I
82 82
80 57.60 19.94
I
1
-2
22 15.84 7.56
II
1
I1
0
I I I
12I
94 94
162 116.64 28.19
I#
II
32.31
I, I
1 256 184.32
II
Common 120 in. 600 in. kips 0.006 rad. do. 0.72 in. do. in. deg.
Analysis and design of single-pole transmission structures
561
The boundary conditions of zero slope and zero deflection at the left end have been used in computing 0 and Y. For subsequent cycles use h = h, cos tl and Icr. Y = h, sin 0. Cycle 2 h
M
-2195
a fJ Icr. Y Y YIIY, Cycle 3
I I
I I
118.96
6.86
I
6.86 14.33
, I 0
I , , , 0
0 Y YJY,
-
Cycle 4 Y Y,lY,
-
119.14 7.00
17.88 36.84
0
I I I I
1 - 103:
105.77
7.14
I I I I
14.33 1.11
-
h
112.81
-1600 11.02
25.02 so.75
0
I I !
28.52 57.30
ld1.92 1.14
I I I
108.74 25.68
1
101.42
3.50
I I I I
51.17 1.13 114.20 18.30
; - 507
I I
1 I 159.22 1.16
105.44 29.31
; ;
14.62 0.98
52.30 0.98
l&.30 0.98
163.04 0.98
14.56 1.004
52.08 1.004
103.84
162.31 1.004
1.004
in. in. kips deg. deg. in. in
say
o.k.
Note that faster convergence would be obtained by using Icr. y = h, sin 0 in Cycle 1, but this would prevent the linear analysis from being obtained. The linear analysis gives the starting shape for the combined analysis.
3. Beam-column analysis The same structure with a 5 kips transverse and 2 kips axial force is analyzed. The P-delta moment at node j is the axial force times Y,,~- Y,
Cycle 1 Y, (linear)
MO
P-delta Total M a
e
Icr. y Y Y,lY,
15.84
0
I I
h
118.96
-2195 - 369 -2564 8.10 I
I
I I
-1600 -337 - 1937 13.31 8.10
16.91 b
1
I
I
184.32 101.42
-50; -135 -642 30.27 60.49
0 0 0
60.72 0.95
; I
34.69 68.30
121.21 0.96
in.
/
4.42
1 1
21.41 43.81
I
105.77
- 103: -253 - 1289 8.86
16.91 0.94
-
116.64
57.60 112.81
; ;
in. kips deg. deg. in.
189.51 0.97
Cycle 2 h
MQ
P-delta YZ YIIYZ
-2164 -379
I1 0
-
181.80
II
-1570 -345 16.76 1.009
I
111.72
103.64
- 1012 -258 60.14 1.010
Comparison of values Linear M -2195 Y 0 Nonlinear M -2543 N:‘L
1 -493 -137 119.99 1.010
98.67
1
0 0 187.54 1.011
say o.k.
0 184.32 0
01.16
187.54 1.02
4. Buckling analysis
The procedure for computing the buckling multiplier i, is:
1, Assume a buckled shape which approximates the first mode. POLEDA uses a parabola with a 10.0 in. tip deflection. Call this starting deflection pattern y, _ , 2. Compute the deflected shape y,. 3. Compute SUM1 = summation of nodal deflections for shape
Y,_ ,
and SUM2 = summation for shape y,.
If y, _ , /y, divided by SUM 1/SUM2 = 1 k tolerance (say + 0.01) at each node, then 1 = SUM 1/SUMZ. Otherwise return to step 2 and use current Y, as the starting deflected shape.
562
M. D. VANDERBILT and M. E. CRISWELL
The same structure
is used with the following
concentrically
-2 b
a Cycle 1 Th:st DY Icr. M M : Y Ye/Y,
0
I1 1 II \
0.62 -4 0.62 -2.48
- 25.00 0.148 I1 0
-4
I, I I 1 1
1.88
5.62 3.12
15.00 0.181 1 0.67 3.73
0.413
2 k d-
C
-6.24 -
pattern:
kips
-2
-1.52
0.265 \I 0.18 4.56
loading
2.50
-22.52 0.148
applied
-2
I1 I I II
4.38 -8.76
0.594
in. kips
II I I ,I
in. kips
0
-8.76 0.103 I 1.39 4.04
Dy = difference in deflections of adjacent nodes, e.g. for segment cd. Dy = yd - .r, Thrust = total axial force in segment. Icr. A4 = increment in M over length of one segment Cycle 2
10.0
0.697
= Thrust
II 2.22 4.50
0.01 0.01 rad. rad. in.
times Dy.
I’! I‘, /.1’2
0
0.0407 4.42
0.153 4.38
0.310 4.48
0.49 4.52
I
in
?‘?
0
0.00906 4.49
0.0340 4.50
0.0688 4.Sl
0.1088 4.51
in
1.ooo
1.000
Cycle 3 !‘z /!‘I
SUM I/SUM2 yz/yJ4.S08
= 0.9947/0.2207
0.996
Thus the value of i. = 4.51 and the pole would
0.998 buckle
= 4.508
if each load was increased
to 4.51(2) = 9.02 kips.
WS-7949/88 $3.00+ 0.00 Pergamon Press plc
1988
COMPENDIUM ANALYSIS AND DESIGN OF SINGLE-POLE TRANSMISSION STRUCTURES D. VANDERBILT
M.
Department of Civil Engineering,
and M. E.
Colorado
State University,
(Receiwd
18 March
CRISWELL
Fort Collins,
CO 80523, USA
198’7)
Abstract-The unguyed single-pole transmission structure is widely used to support transmission lines of up to 345 kV capacity. Most such structures are wood poles, although prestressed concrete poles and tubular steel poles are sometimes used in high decay or high load situations. A number of features combine to make this type of structure an interesting structural analysis and design problem. Whether naturally grown or manufactured, the pole structure is almost always nonprismatic. Wind, ice and snow loads acting alone or in combination are variable and occasionally severe in nature. Poles are slender and the combination of transverse wind load with vertical dead and ice loads often leads to behavior which is geometrically highly nonlinear. The continuous exposure to weather, fungi, accidents, etc. results in structures with capacities that often deteriorate very significantly with time in service. The material properties (strength, stiffness), especially of wood, are highly variable, as are the soil conditions, The configurations of conductors, davit and cross arms, and attachments that are in use vary widely. This large variety of structural shapes plus the necessity of changing the configurations of in situ structures to meet new conductor and clearance requirements prevents the use of only a few designs. Thus rapid ways to assess existing structures and to design new structures are required. Since there are more than 130,000,000 poles currently in use in single-pole and H-frame transmission structures and several million new poles are put into service each year, the design effort related to poles is large. As part of an eight-year effort to develop reliability-based design methods for transmission structures, a correct procedure for analyzing and designing single-pole transmission structures was established. Implemented in program POLEDA (POLE Design and Analysis), this procedure enables unguyed planar single-pole structures of virtually any material and configuration to be rapidly analyzed or designed. The analysis is based on the Newmark numerical technique which, it is shown, in the limit provides an exact solution to the differential equation for large deflections of transversely loaded beams. The four design options provided in POLEDA use the load and resistance factor design (LRFD) format. This permits the user to use current, deterministic, design procedures or to use load and resistance factors based on reliability studies. The program provides the necessary high-quality analysis capability which forms an integral part of reliability-based design. It is being widely adopted by the electric utility industry.
1. INTRODUCTION
A
typical
structure
unguyed is shown
is typically in place, special
by and
aggregate,
wood in Fig.
auguring backfilling
single-pole
transmission
1. Foundation a
hole,
with
or manufactured
preparation
setting
tamped
the
natural
fillers.
The
pole soil, usual
setting depth is 10% of the pole length plus 2 ft for poles up to 90 ft long [I]. The structure supports current-carrying conductors located at the ends of cross arms or davit arms and usually supports a shield or ground wire, located at or above the pole top, to protect the conductors from lightning strikes. Conductor locations are described by their distance above the groundline and their eccentricity from the pole centerline. the ‘span’ relates to the conductor lengths supported by a structure. On flat terrain with uniformly spaced poles, the span is the distance between two adjacent poles. On sloping terrain, the wind (lateral) and weight (vertical) spans differ as shown in Fig. 2. Structures located in a straight line are tangent structures while structures located where a transmission line changes direction are angle structures. If the angle is small (a light angle structure), 551
then guy wires may not be required. If the angle is large, then guys wires are typically used with wood poles while heavier unguyed structures are used with steel. The transverse direction is the direction perpendicular to the span of a tangent structure. The most severe loading generally acts in this direction and most analysis and design is for transverse loads. Loads in the longitudinal direction are typically small unless a conductor breaks. Thus a planar analysis suffices for most design problems. In order to analyze an unguyed structure the engineer must define the geometry and material stiffness of the structure, the location, size and weight of each conductor and groundwire, the location and weight of each attachment (cross arm, insulator, transformer, etc.), the wind and weight spans, the ice/snow and wind loads, transverse loads due to components of conductor tension for angle structures and any known initial slope of the pole in the transverse direction. Deformations of the soil as load is applied are almost universally neglected and the resulting assumption that the structure is fixed at the groundline is also followed herein. The influence of soil
M. D. VANDERBILT and
552
M. E. CRISWELL
_ l-l
0 75
I-
_
f
GW
dC
17 c-
-1
GW
C
_
7
6 --I
7
=
Ground 0 546 0 5429
= Drake I 108 IO94
w,re,
m diameter, Ib/ft
ASCR conductor, in diameter, Ib/ft
-I
85 45 7.5
63
68 5 ft
1 1 ‘0 5
i Fig.
1. Class 1 Douglas
deformations can be significant, however [2]. Design of a transmission structure typically means the determination of the span which can be supported by a structure of a given size. This practice arises from the method used to categorize wood poles by load carrying class [I]. In addition to information required for analysis, design requires that material strength and stiffness and the load and resistance factors be defined. While logically only one design procedure should exist, the current state-of-the-art encompasses several different procedures. Hence the designer must choose a design method as well as defining all the above data. The electric utility industry consists of more than 3000 public and private utilities including the REAs. Transmission structure design is sometimes performed in-house by engineers, in the past often electrical but now usually civil, employed by the larger utilities.
fir pole.
Smaller utilities and many of the REAs use consulting services provided by engineering firms and some of the larger utilities. The Electrical Power Research Institute (EPRI) headquartered in Palo Alto, CA is the research arm of the industry. Supported by several hundred of the U.S. utilities, EPRI conceives and supports research in many areas. Since 1978 EPRI has supported research at Colorado State University (CSU) on the development of reliability-based design techniques for transmission line structures. The first type of structure studied was the single-pole wood tangent transmission structure. About 130,000,000 wood poles are in use in the U.S.A. and several million new and replacement poles are placed in service each year. Since the in-place cost of one single-pole structure is currently $100&$2000, the nation-wide investment in wood pole structures is significant and improved design techniques can have
Analysis and design of single-pole transmission structures
Vertical
05L,
L,
tFor analysis,
, I
spa”
-I
3. CURRENT
05L,
I.
input span =
resistance and load effects on the same basis so they are directly comparable. Any error in the analysis will result in the as-built structure having a different reliability than that desired by the design engineer. If the method of analysis used in design is highly approximate, then the structures placed in service will have unknown and highly variable reliabilities. Using a method of analysis which introduces only small errors in converting factored loads to load effects is the sine qua non of reliability-based design.
span
Horlzontai
LZ
I
(H, + H, I
Fig. 2. Definition of horizontal (wind) and vertical (weight) spans.
an important economic payoff. Wood transmission poles are typically 55 to 125 ft long or about 48 to 113ft high above groundline. Their highly nonlinear behavior must be considered in any correct analysis and design procedure. 2. BASIC CONCEmS OF LOAD AND RESISTANCE FACTOR DESIGN
Of the many different ways to implement reliabilitybased design, the load and resistance factor design (LRFD) formulation is receiving the most widespread acceptance. The LRFD concept is expressed by b*R
2A(y*Q)>
553
(1)
where 4 is a capacity reduction (i.e. resistance) factor almost always less than 1, R is a measure of resistance such as the modulus of rupture of wood or yield stress of steel, y is a load related factor usually greater than one, Q is a load causing phenomenon such as wind pressure, ice thickness, etc. and A ( ) means that an analysis is performed of the structure loaded with the factored loads so that the factored resistance (4 *R) and the load effects can be compared on the same basis (e.g. normal stress). In words eqn (1) states that the factored resistance must equal or slightly exceed the effects of the factored loads. In reliabilitybased LRFD the four terms in eqn (1) are determined on the basis of studies of the statistical variations in R and Q and the level of reliability provided by various 4 and y factors taken in combination with various measures of the probability distributions of R and Q such as the mean values or upper and lower exclusion limits. Discussions of the studies required to establish LRFD equations for transmission structures are given in [3]. The role of structural analysis in reliability-based design, the A( ) in eqn (l), is to place factored
LINEAR
ANALYSIS
AND DESIGN
The first design document usually considered by the U.S. transmission structure designer is the National Electrical Safety Code [4] published as ANSI Standard C2. The NESC is primarily concerned with electrical requirements such as clearances between adjacent conductors and conductors and ground. The only structural provisions prescribe that structures be designed to carry the specified loads multiplied by overload capacity factors (OCFs). The NESC provisions can be expressed in LRFD form as (l.O)*R
2 A(OCF*Q).
(2)
The implied 4 of 1 and the prescribed OCF values are based on judgement and not on any formal type of reliability studies. In eqn (2), R is the ultimate resistance taken from a document specified by the NESC in the case of wood ]I] and left undefined for other materials, OCF is 4 for most new wood pole construction, and Q is a loading value such as 4 psf wind plus 0.5 in. radial ice thickness on conductors taken from a map. The entire U.S. is divided into only three NESC districts by this map. The method of analysis is left undefined as are other details of design. Since the public utilities commissions of most states require that NESC provisions be satisfied, the structural engineer is bound by eqn (2). For wood structures, the second design document of interest is the Specifications and Dimensions for Wood Poles, ANSI Standard 05.1 [ 11. This document categorizes poles by classes. The class of a pole of a given diameter and length is obtained by assuming the pole has a uniform taper from groundline to tip, that it is loaded by a single transverse load placed two feet from the tip, and that the maximum flexural stress occurs at the groundline. By assuming a value for the modulus of rupture, R, the value of the transverse load can be computed. Tables in ANSI 05.1 give the pole sizes as functions of class load and wood species. Thus a 70 ft class 2 pole can carry a class load of 3700 lb regardless of species. The only document containing examples of current design of wood structures is the Design Manual for High Voltage Transmission Lines published by the Rural Electrification Administration [5]. All of the examples in the REA manual use only linear analysis, and stresses in single wood pole structures are checked
M. D.
554
VANDERBILT and M. E. CRSWELL
only at groundline. This use of linear analysis coupled with the ANSI 05.1 practice of checking stresses only at ground line has produced a current design practice which is significantly in error. Thus a large and unknown part of the apparent ‘factor of safety’ of 4 implied by the OCF of 4 for wood poles is consumed by errors in analysis. Examples of design of steel poles are given in the Design of Steel Transmission Pole Structures publication by the American Society of Civil Engineers [6]. The ASCE steel pole manual uses linear analysis in all examples, checks stresses everywhere in the structure, and uses a moment magnifier to approximately account for P-delta effects. The fact that nonlinear behavior has not been adequately recognized by the writers of the NESC is shown by the change in basic design criteria introduced in the 7th (1977) edition. In the previous 6th edition, the design equation was (0.25)*R
> =A(I.O*Q).
(2a)
Thus the 4 factor was 0.25 and the y factors were 1.Oin the 6th edition. If the behavior was truly linear, then eqns (2) and (2a) would produce identical results. Since the behavior is nonlinear, multiplying the loads by four before making a nonlinear analysis will produce P-delta stresses on the order of at least 16 times greater (i.e. four times the load acting through at least four times the lateral deflection) than the P-delta stresses computed for (1.0) times the loads. Based on this brief review of the design and code documents now in use, it may be concluded that the role of correct analysis in design is not yet understood by many people involved with transmission structure design. 4. ANALYSIS
METHOD
A correct method of analysis defines the equilibrium position of a structure in terms of its final deflected shape. In linear analysis the change in geometry as load is applied is ignored which greatly simplifies analysis. If the change in geometry is considered, then the equilibrium position must be found through an iterative procedure which involves some type of searching technique to locate the equilibrium shape. A host of analytical techniques is available for use with structures that soften with increase in load. A major criterion in selecting a method for POLEDA was that it be computationally very efficient since it was planned that analyses of several thousand structures for multiple load cases would be made as part of the effort to establish reliability-based design methods. This criterion plus the observation that the unguyed single-pole structure is statically determinate led to the selection of the Newmark numerical technique [7-91 as the basic solution technique. The fundamental concept of the Newmark tech-
nique, as applied to transversely loaded beams, is that the real structure, which has a continuous variation in curvature over its length, is replaced by a series of rigid segments connected by rotationally flexible springs. The total curvature occurring over the length of the beam is concentrated at the springs. For small deflections the familiar differential equation. neglecting shear effects, is d2y
M(x)
dx’
El(x)
(3)
where x = axial coordinate, J’ = transverse coordinate (deflection), M(x) = moment caused by external loads and which varies only with s for small deflections, E = Young’s modulus, and I(s) = the moment of inertia which varies with s in nonprismatic members. The M/EI term in eqn (3) is the curvature. If the curvature is concentrated exactly. then the Newmark technique gives the exact solution at each of the nodes (flexural springs) and the chords between nodes closely approximate the true elastic curve. It is demonstrated in [7-91 that the concentration formulae exactly concentrate the curvature for some loading cases on prismatic beams. When applied to nonprismatic members, the formulae are approximate but the error is reduced to a small level by using reasonably short segments. For long slender members such as transmission structures, the effects of shear are negligible and omitting the shear term of eqn (3) has little effect. For large deflections of transversely loaded beams the governing differential equation is M(.r, $‘)
d2yjdx’
(1 + (dy/dx)2)3,2 = El(.u)’
(4)
where M(x, y) indicates that the moment is a function of the deflected shape. Figure 3 shows that curvature and the slope of the elastic curve are related by l/p = d0/ds. From
Fig. 3 it is also noted
(5)
that
dx = ds cos 0 0 = arctan (dy/dx). By substituting that [IO]
(6)
eqn (6) into eqn (5) it may be shown
1
d2y/dx2
p
(1 + (dy/dx)‘)3’2’
(7)
The usual substitutions may be made to use eqn (7) in developing eqn (4). With the Newmark technique, finite lengths A-X replace the infinitesimal lengths dx and thus eqn (6)
Analysis and design of single-pole transmission structures
555
ds = pd8
I=!!?! P dS Fig. 3. Elastic curve of benm. is rewritten
as Ax=Ascose 0 = arctan (By/Ax).
(8)
The computational procedure for computing the equilibrium shape of a transversely loaded slender beam using the Newmark method consists of the following steps: I. The beam is divided into segments and I(x) is computed at each node. 2. The moment M(x) and the curvature M/EZ are computed at each node. 3. The slope, 0, of each chord between adjacent nodes, and the deflection, y, of each node are computed. The deflected shape must satisfy kinematic boundary conditions. 4. The moment M(x, y) is now computed at each node. Because the beam is deflected, eqn (8) can be written for each segment. In general M(x, y) will be less than M(x) if only transverse loads act but will be greater than M(x) if both transverse and significant axial loads act. 5. The curvature, slope and deflection are computed. 6. The latest deflected shape is compared with the previous shape by dividing y,_ 1 by y, at each node, where y, is the deflection for the latest iteration and y, _ , is the deflection for the previous iteration. If the ratio equals unity plus or minus a small tolerance (say 0.01) at every node, then convergence is obtained. Otherwise the procedure returns to step 4 and continues until convergence or until examination of results after several cycles shows that the solution is oscillating or diverging. Where axial loads act at different points along the length of a beam-column the same basic procedure is used. The effects of the axial loads is null in steps 2 and 3. In step 4 the
P-delta moments caused by axial loads are computed and added to M(x, y). An example solved by hand is given in the Appendix to show the details of the technique. Note that the tabular format used with the Newmark technique enables the method to be implemented using a spread sheet program. Closed form solutions for large deflection problems are difficult to obtain. A large deflection solution for a cantilever beam transversely loaded with a single tip load was obtained by Bisshopp and Drucker using elliptic integrals [111.Their results are shown in Fig. 4 by solid lines. In this plot, A is the transverse deflection while S is the shortening of the moment span. The small deflection theory solution is shown by the broken straight line plot. For small deflections, the ratio of the tip deflection to the span is given by
A/L = WL2/3EI IO
Fig. 4. Large deflection
\
of tip-loaded
(9) I
cantilever
beam.
556
M.
D. VANDERBILT and
and the moment at the fixed end is II%. For large deflections the moment at the fixed end is W(L - S) and thus it is conservative to design transverselyloaded beams using linear theory. However, in beamcolumns the reduction in transverse moment (WS) is usually much smaller than the increase in PA moments. Thus the use of linear, small-deflection theory in designing beam-columns can be in error by as much as a factor of 2. The solution obtained using the Newmark procedure is shown by the circles in Fig. 4. From the closeness of the exact and numerical solutions it may be concluded that the Newmark procedure in the limit (as Ax approaches dx) gives an exact solution to this problem. Convergence studies may be made by analyzing a structure several times using successively shorter segments. Experience shows that dividing a structure into lo-20 segments gives good results. Closed form solutions for nonprismatic beam-columns have not been found in the literature. However, it is believed that the use of the Newmark method for their analysis gives good results. The Newmark method can also be used to compute the buckling capacity of straight members supporting only axial loads. For the case of a single axial load, the buckling capacity can be given in terms of the force which will caused bifurcation of the equilibrium position. For the case where two or more axial loads act, the buckling capacity is expressed as a multiplier applied equally to each load [12]. Thus if a tip load P and a midlength load Q act on the real structure, then the buckling capacity is given by i and the structure would buckle under the simultaneous application of U’ and AQ. If the axial components of the applied loads on a beam-column cause buckling, then a solution for the combined transverse and axial loads cannot be obtained. Hence a complete analysis begins with a buckling analysis to determine if a large-deflection beam-column analysis solution can exist. An example of a buckling analysis is given in the Appendix. Extensions of the Newmark technique to the analysis of beam-columns experiencing large deflections have not been found in the previous literature.
5.
DESIGN OPTIONS
The POLEDA program allows the user to select a different design option for each load case, where design consists of determining the allowable span for a given structure. This feature was included so that the effect of using different design criteria can be quickly examined. The design options provided are: Option Option
&only an analysis is performed. l-the current linear design method is used, thus only a linear analysis for transverse loads is performed, P-delta effects are ignored, and stresses are checked only at groundline.
M. E.
CRISWELL
Option
2-basically the same as option 1 except that moments from vertical dead and ice loads acting at the ends of the cross arms are also considered. This option was included since the 1980 edition of the REA manual [5], which replaced the 1972 edition, adopted this one improvement over option 1. Option 3-all nonlinear behavior is correctly included in the analysis but the stresses are checked only at the groundline. Option &-all nonlinear behavior is correctly included in the analysis and the maximum stress wherever it occurs in the structure is considered. No matter which design option is chosen by the user, POLEDA always concludes the design of each load case by performing a correct analysis and outputting the stresses and deflections at each node point, Design options 1 and 2 require that only one analysis be made. The program begins with an assumed span and performs an analysis. Part of the total capacity at groundline is used to support wind that acts on the pole. The remaining capacity is used to support wind which acts on the iced conductors and, for option 2, to support moments due to vertical ice and conductor weights. For either option the loads on the conductors are linear functions of span and thus the allowable span, which causes eqn (1) to be satisfied as an equality, is easily found. Design options 3 and 4 require that an iterative method be used to compute the span [13]. This requirement arises because the stresses in the structure are a nonlinear function of the span. Figure 5 shows a plot of trial span, TS, versus V, where V equals the maximum normal stress divided by the factored resistance, Cp*R. The correct span is the one for which V = 1. POLEDA computes TS 1 using the groundline moment computed ignoring P-delta and large geometry effects (i.e. using option I). A correct analysis is performed and the first value of VI is computed. The TSl, VI point plots as point 1 in Fig. 5. For design options 1 or 2 the procedure stops here. For options 3 and 4 a straight line from the origin through point 1 gives point A and TS2. A correct analysis gives point 2. The line through points 2 and 1 gives point B and TS3. A third nonlinear analysis gives point 3. A parabola with equation TS = A V2 + BV + C can now be fitted through points 1, 2 and 3. Three simultaneous equations are solved to determine A, B and C. This parabola, plotted as a broken line in Fig. 5, is used to find TS4. Analysis gives V4 and finally point 4 is plotted. If point four plots as V equal to one plus or minus a small tolerance (say 0.01) the design is complete. If it plots outside the tolerance limits, the procedure continues with the coefficients of the parabola always based on the latest three trial spans, until either convergence or a prescribed limit on number of trials is reached. Experience shows that more than four
Analysis
and design
of single-pole
transmission
structures
I-
557
I+T
TSI
TS:
TS;
Parabola
through
3
points
Curve of span versus V
Acceptable
V = (maximum
Fig. 5. Interpolation
normal
procedure
trial spans is rarely required. Note that an iterative buckling analysis followed by an iterative beamcolumn analysis is required for each trial span. 6. PROGRAM
POLEDA
Program POLEDA is written in FORTRAN 77 and can execute on an IBM PC or AT. It contains about 1200 source statements and the executable file on an AT is about 63,000 bytes in length when compiled with the Microsoft 4.00 FORTRAN compiler. The program operates in a batch mode with the input data file being easily constructed with any text editor. The program is available through the Electric Power Software Center, 1930 Hi Line Drive, Dallas TX 75207, (214)-655-8883. It is distributed at no charge to EPRI members and is for sale to nonmembers. The source code is available to users who wish to install the program on a mainframe. A package of programs for transmission line design is
stress)
/(Phi
range
*R)
for span computation.
available from EPRI as the TL-Workstation (tm). This package, which requires an IBM AT as part of the workstation, includes a compiled copy of POLEDA, an interactive front-end program to generate the input data file, and a post-processor to provide graphics displays of stresses, deflected shapes, etc. Data are input in blocks with the length of each block dependent upon the complexity and type of structure. The blocks are briefly summarized to give an idea of the capabilities of the program. Block 1. Control
(a) The type of pole, either wood, steel or general geometry, is defined. (b) The common values of material unit weight, modulus, and material strength are defined and become the default values in each load case. (c) The tolerances for accepting the deflected shape and the design span are each input or default to 0.0 1.
558
M. D. VANDERBILT and M. E.
The program contains default values for many of the data and uses the default value whenever the user inputs a 0 value. Block 2. Structure geometry
(a) For a wood pole the length, and the tip and butt circumferences are input. The program uses the ANSI 05.1 assumption of uniform taper to compute the area, A, section modulus, S, and moment of inertia, I, at each node point. (b) For steel or aluminum poles the length of each section (six maximum), the height from groundline to bottom of section, plate thickness, and section diameter at bottom and top are input. Metal poles may be circular or polygonal in cross-sectional shape. The program computes A, S and I values. (c) For general geometry poles, the cross-sectional dimensions and properties A, S and I must be given at each node. Block 3. Load level dejinition
(a) Control data to define the number of load cases, number of conductors, number of levels at which loads may be applied and number of segments to use in the Newmark analysis are input. (b) A table of conductor properties is input which includes the name, weight per foot, and diameter of each conductor (including groundwires) to be used. (c) The location of each level at which loads may be input is defined. At any load level, conductors may be attached or concentrated actions, such as from an attached transformer, may be applied, or both. (d) The number of each type of conductors acting at each load level is input. Block 4. Load and design data
For each load case the user selects the design option, the load and resistance factor data, the wind model, the ice model, other load data such as extra concentrated actions, and the resistance and stiffness data. This flexibility was provided so that comparisons of the effects of choosing different design options could be quickly made. The data include: (a) Control data = design option, No. of extra concentrated actions, and span (required for option 0 only); (b) Extra concentrated actions, if any; (c) 4 and y factors for every load cause. All values default to 1.0; (d) Wind model. Either a uniform wind pressure corresponding to the NESC district loadings or a velocity profile is chosen. If a velocity profile is selected, then wind velocity, reference height, power law exponent, and air density are input. Drag factors for conductors and pole are input for either model; (e) Ice thickness and density; (f) Young’s modulus, material density, and material strength. These default to the common values if input as 0.
CRISWELL
7. EXAMPLERESULTS Figures 6 and 7 show analysis results for the pole of Fig. 1 for an NESC heavy loading consisting of 4 psf wind, a radial ice thickness of 0.5 in. on each conductor, an ice density of 57 pcf, drag factors of unity, a load factor of 4 on the wind and ice weight and a load factor of 1 on conductor and pole weight. The NESC specifies OCFs of 4, but it is clear that the code writers assumed a linear analysis would be performed for which dead load has negligible effect. Hence selection of 1 for dead load seems reasonable. The NESC also specifies that stresses in wood poles are to be checked at the groundline. Using design option 2 POLEDA computed a design span of 390 ft. Figure 6 shows deflected shapes for two cases. The first is for a linear analysis which includes moments from unbalanced conductors. The second is for the correct analysis which is automatically output by POLEDA as the last part of each load case. The undeflected and deflected shapes of Fig. 6 are drawn to scale. Figure 7 shows that the linear analysis gives a stress of 8000 psi at groundline which is the designated fiber stress from ANSI 05.1. The maximum stress of 8340psi occurs 20 ft above groundline. The maximum stress in wood transmission structures almost always occurs above groundline, a behavior ignored by both the NESC and ANSI 05.1. In Fig. 7 the correct nonlinear analysis gives a maximum stress of 13,980 psi at 29 ft above groundline. Thus the current code sanctioned design of this structure is in error, on the nonconservative side, by 75% which is not untypical. Additional examples showing design results for various wood and steel structures are given in [3, 10, 12, 131. 8. CLOSURE The role of analysis in the design process is to enable resistances and loads to be compared on the same basis such as normal stress, axial compressive force, etc. An incorrect analysis introduces an unknown error and makes it impossible to design structures to have the level of reliability desired by the structural engineer. The behavior of the unguyed single-pole transmission structure, especially those of wood, is geometrically nonlinear to a significant degree and a correct method of analysis must properly consider this nonlinearity. Program POLEDAprovides a complete analysis and design procedure for unguyed single-pole transmission structures of wood or steel. The analysis method is based on extensions of the Newmark numerical method which, for the cases tested, provides a very accurate analysis for both combined axial and transverse loading response and buckling capacities. The design provisions allow spans to be determined for either currently specified deterministic loadings or for loads, resistances, and load and resistance factors chosen [3] on
Analysis and design of single-pole transmission structures
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559
M. D. VANDERBILT and M. E. CRISWELL
560
the basis of reliability studies. The analysis capability of POLEDA provides the necessary high quality analysis which, if lacking, prevents design from being placed on a rational basis.
4. 5.
AcknowIedgemenl.~-The research, of which developing was a small part, was supported by EPRI as RP 1352, ‘Reliability-Based Design of Transmission Line Structures’. Mr Richard E. Kennon of the Transmission and Distribution Division is the Manager of the EPRI Electrical Systems Division which funded the project and Mr Paul Lyons of the EPRI Transmission Line Mechanical Research Center was project manager. Their continued support has been instrumental to the successful development of a reliability-based design methodology for transmission line structures. Colorado State University Professors J. R. Goodman and J. Bodig have been key researchers in various aspects of the project since its inception. Numerous graduate students have contributed to the project. In particular M. D. False developed the first version of POLEDA and E. M. Hendrickson extended it to include metal and general geometry poles. POLEDA
6.
I.
8. 9.
10.
II.
REFERENCES Specifications and dimensions for wood poles. ANSI 05.1- 198 I, American National Standards Institute, 1430 Broadway, NY 10018 (1981). K. H. Kim, M. E. Criswell and M. D. Vanderbilt, Inclusion of foundation effects in analysis of pole structures, Structural Research Report 61, Colorado State University (1986). M. E. Criswell and M. D. Vanderbilt, Reliability-Bused
APPENDIX:
EXAMPLES
12.
13.
OF ANALYSIS
Design of Transmission Line Structures: Final Report. Vols 1 and 2. EPRI (1987). National Electric Safety Code. American National Standard C2- 1984 (1984). Design manual for high voltage transmission lines. United States Department of Agriculture, Rural Electrification Administration, REA Bulletin 62-l. revised August 1980. Task Committee on Steel Transmission Poles of the Committee on Analysis and Design of Structures of the Structural Division. In Design of Sfeel Transmis.won Pole Strucrures. American Society of Civil Engineers. New York (1978). N. M. Newmark, Numerical procedure for computing deflections, moments and buckling loads. Paper No. 2202, Trans. AXE 108 (1943). W. G. Godden, Numerical Analysis of Beam und Column Sfruclures. Prentice-Hall, Englewood Cliffs, NJ (1965). N. C. Lind, Newmark’s numerical method. Department of Civil Engineering, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada (1975). E. M. Hendrickson, M. D. Vanderbilt, M. E. Criswell and J. R. Goodman, Reliability design of steel transmission poles. Structural Research Report 41, Colorado State University, CO (1982). K. E. Bisshopp and D. C. Drucker, Large deflection of cantilever beams. Q. appl. Math. 3 (1945). M. D. False, M. D. Vanderbilt, J. R. Goodman and M. E. Criswell, Probability-based design of wood transmission structures. Structural Research Report 35. Colorado State University, CO (1981). M. D. False, E. H. Hendrickson, M. D. Vanderbilt and J. R. Goodman, Program POLEDA-80: A computer program for the analysis and design of single pole transmission structures. EPRI EL-2040 (1981).
USING NEWMARK
METHOD
1. Concenlration formulae Assume
a = -M/EI
The concentrated
varies parabolically
curvature,
as shown.
a, at each node is given by [7-91 g, = h (7a, + 6a, - a,)/24 ZI,= h(a, + IOU,+ a,.)/12.
The equation
for ac is found
from the equation
a, and a,
for g, by interchanging
2. Beam analysis A cantilever prismatic beam is analyzed for simplicity. Assume Young’s modulus, E, =2000 ksi; moment of inertia, I, = 500 in’; a 40 ft span, and a transverse tip load of 5 kips. The beam is divided into four segments each of length h, = 120in. The computations are conveniently performed using the tabular arrangement shown. In the following A4 = moment, Icr. y = increment in deflection y between two successive nodes, and do. = ditto. Cycle 1. Linear analysis
1 5 kips
t
Computation
I lG b
II
1
-4
-3
:
22;
Icr. y = Bh, ? JI
0 0
H
II
;
22 22
36I1
I 1
58 58
II
1
-I
24I1
I I
82 82
80 57.60 19.94
I
1
-2
22 15.84 7.56
II
1
I1
0
I I I
12I
94 94
162 116.64 28.19
I#
II
32.31
I, I
1 256 184.32
II
Common 120 in. 600 in. kips 0.006 rad. do. 0.72 in. do. in. deg.
Analysis and design of single-pole transmission structures
561
The boundary conditions of zero slope and zero deflection at the left end have been used in computing 0 and Y. For subsequent cycles use h = h, cos tl and Icr. Y = h, sin 0. Cycle 2 h
M
-2195
a fJ Icr. Y Y YIIY, Cycle 3
I I
I I
118.96
6.86
I
6.86 14.33
, I 0
I , , , 0
0 Y YJY,
-
Cycle 4 Y Y,lY,
-
119.14 7.00
17.88 36.84
0
I I I I
1 - 103:
105.77
7.14
I I I I
14.33 1.11
-
h
112.81
-1600 11.02
25.02 so.75
0
I I !
28.52 57.30
ld1.92 1.14
I I I
108.74 25.68
1
101.42
3.50
I I I I
51.17 1.13 114.20 18.30
; - 507
I I
1 I 159.22 1.16
105.44 29.31
; ;
14.62 0.98
52.30 0.98
l&.30 0.98
163.04 0.98
14.56 1.004
52.08 1.004
103.84
162.31 1.004
1.004
in. in. kips deg. deg. in. in
say
o.k.
Note that faster convergence would be obtained by using Icr. y = h, sin 0 in Cycle 1, but this would prevent the linear analysis from being obtained. The linear analysis gives the starting shape for the combined analysis.
3. Beam-column analysis The same structure with a 5 kips transverse and 2 kips axial force is analyzed. The P-delta moment at node j is the axial force times Y,,~- Y,
Cycle 1 Y, (linear)
MO
P-delta Total M a
e
Icr. y Y Y,lY,
15.84
0
I I
h
118.96
-2195 - 369 -2564 8.10 I
I
I I
-1600 -337 - 1937 13.31 8.10
16.91 b
1
I
I
184.32 101.42
-50; -135 -642 30.27 60.49
0 0 0
60.72 0.95
; I
34.69 68.30
121.21 0.96
in.
/
4.42
1 1
21.41 43.81
I
105.77
- 103: -253 - 1289 8.86
16.91 0.94
-
116.64
57.60 112.81
; ;
in. kips deg. deg. in.
189.51 0.97
Cycle 2 h
MQ
P-delta YZ YIIYZ
-2164 -379
I1 0
-
181.80
II
-1570 -345 16.76 1.009
I
111.72
103.64
- 1012 -258 60.14 1.010
Comparison of values Linear M -2195 Y 0 Nonlinear M -2543 N:‘L
1 -493 -137 119.99 1.010
98.67
1
0 0 187.54 1.011
say o.k.
0 184.32 0
01.16
187.54 1.02
4. Buckling analysis
The procedure for computing the buckling multiplier i, is:
1, Assume a buckled shape which approximates the first mode. POLEDA uses a parabola with a 10.0 in. tip deflection. Call this starting deflection pattern y, _ , 2. Compute the deflected shape y,. 3. Compute SUM1 = summation of nodal deflections for shape
Y,_ ,
and SUM2 = summation for shape y,.
If y, _ , /y, divided by SUM 1/SUM2 = 1 k tolerance (say + 0.01) at each node, then 1 = SUM 1/SUMZ. Otherwise return to step 2 and use current Y, as the starting deflected shape.
562
M. D. VANDERBILT and M. E. CRISWELL
The same structure
is used with the following
concentrically
-2 b
a Cycle 1 Th:st DY Icr. M M : Y Ye/Y,
0
I1 1 II \
0.62 -4 0.62 -2.48
- 25.00 0.148 I1 0
-4
I, I I 1 1
1.88
5.62 3.12
15.00 0.181 1 0.67 3.73
0.413
2 k d-
C
-6.24 -
pattern:
kips
-2
-1.52
0.265 \I 0.18 4.56
loading
2.50
-22.52 0.148
applied
-2
I1 I I II
4.38 -8.76
0.594
in. kips
II I I ,I
in. kips
0
-8.76 0.103 I 1.39 4.04
Dy = difference in deflections of adjacent nodes, e.g. for segment cd. Dy = yd - .r, Thrust = total axial force in segment. Icr. A4 = increment in M over length of one segment Cycle 2
10.0
0.697
= Thrust
II 2.22 4.50
0.01 0.01 rad. rad. in.
times Dy.
I’! I‘, /.1’2
0
0.0407 4.42
0.153 4.38
0.310 4.48
0.49 4.52
I
in
?‘?
0
0.00906 4.49
0.0340 4.50
0.0688 4.Sl
0.1088 4.51
in
1.ooo
1.000
Cycle 3 !‘z /!‘I
SUM I/SUM2 yz/yJ4.S08
= 0.9947/0.2207
0.996
Thus the value of i. = 4.51 and the pole would
0.998 buckle
= 4.508
if each load was increased
to 4.51(2) = 9.02 kips.