Analytical calculation model used when dimensioning timber poles
used as overhead power line supports
Analytisk beräkningsmodell för att dimensionera trästolpar som används som luftledningsstolpar
Christopher Ekängen
Faculty: Faculty of Health, Science and Technology
Subject: CBAEM1 – Degree Project for Master of Science in Engineering, Mechanical Engineering Points: 30 credits
Supervisor: Mikael Grehk Examiner: Jens Bergström Date: 2018-06-13
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Abstract
Overhead power lines are more reliable and cost effective than underground cables when it comes to power transmission. The overhead power lines are often suspended above ground using timber pole supports.
As the placement of each support depends upon topography amongst other things the conditions for most of the supports will be different and each pole can be viewed as a unique object. This requires that each support is dimensioned individually. The purpose of the project is to clarify how the dimensioning of a support should be performed. The main objective for the project is to develop an analytical model and simulation tool which can be used to dimension timber pole supports. Octave, a scientific programming language, is used for the simulation tool and to visualize the analytical model for the timber pole sup- ports. The analytical model is validated by a finite element analysis, which is applied using the program ABAQUS/CAE. From the analytical model nomograms, which are diagrams that relate the parameters of the support, can be constructed. A nomogram is another tool that can be used when dimensioning timber supports. Many relevant design aspects are treated in the standard SS-EN 50341 and serves as the basis for the analytical calculation model.
The supports can be divided into three main types; tangential supports, where the conductors continue straight at the support, angle supports, where the conductors make an angle at the support, and terminal supports, which are used at the endpoint of the conductors. Loads taken into account are: self-weight of components, wind, ice, conductor tension and maintenance. The loads are combined using a design equation which takes into account the probability of certain events occurring simultaneously using com- bination factors. Partial safety factors are also used to take into account possible deviations in loads, material or geometry. The failure modes of interest are rupture due to excessive bending loads and buckling due to excessive axial loads in the support legs. In the analytical calculations the timber is considered to be isotropic in accordance with the standard while the timber is modeled both as isotropic and orthotropic in the finite element analysis. The analytical model assumes that the cross-arm, which is the horizontal part of the support, only acts to distribute the load between the legs of the support without providing any extra stability by linking them together. In the finite element analysis the legs of the support are considered both individually and linked together by a cross-arm to test the validity of the assumption.
To compare the results from the analytical model with finite element analysis one support of each type;
tangential, angular and terminal is selected. An Octave script is written and applied to the selected supports to calculate their necessary dimensions. Based on the calculated dimensions 3D-models are created using PTC Creo Parametric 3.0 which then are imported into ABAQUS/CAE where the supports are subjected to finite element analysis. In addition to using the analytical model directly expressions that can be used to create nomograms for timber supports are also derived.
Comparing results from FEM with the analytical model showed that the tangential and angular supports were over-dimensioned while the terminal supports were under-dimensioned in the analytical model.
This means that a smaller dimension could be used for the tangential and angular supports. For the terminal support the analytical model needs to be altered so that failure will not occur. Changing the buckling mode for the terminal support in the analytical solution fixed the under-dimensioning issue. The finite element calculations where a cross-arm was used to link the legs of the support together showed no increased stability for buckling but showed lower bending stresses in the support legs. Using an orthotropic material model for the timber did not effect the calculations greatly as the properties in the length direction of the pole were significantly more important than the properties in the perpendicular directions. Dimensioning using nomograms is less accurate, as it only narrows the results down to a timber pole grade rather than a specific diameter. This decrease in accuracy is not cause for concern as the timber poles are ordered by grade and not a specific diameter. Using nomograms can give quick results when it comes to dimensioning.
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Sammanfattning
Luftledningar ¨ar mer tillf¨orlitliga och kostnadseffektiva ¨an markkablar n¨ar det kommer till kraft¨overf¨oring.
F¨or att h˚alla luftledningar ovan mark anv¨ands ofta tr¨astolpar. Placeringen av varje stolpe beror bland annat p˚a topografin vilket g¨or att villkoren f¨or n¨astan varje stolpe skiljer sig ˚at. Detta g¨or att varje stolpe m˚aste dimensioneras individuellt. Projektets syfte ¨ar att klarg¨ora p˚a vilket s¨att dimensionen f¨or en stolpe ska best¨ammas. Det huvudsakliga m˚alet ¨ar att utveckla en analytisk modell som kan anv¨andas f¨or att dimensionera tr¨astolpar. Octave, ett vetenskapligt programspr˚ak, anv¨ands f¨or att till¨ampa den analytiska modellen p˚a tr¨astolparna. Den analytiska modellen valideras med finit elementanalys, som till¨ampas genom anv¨andandet av ABAQUS/CAE. Fr˚an den analytiska modellen kan ocks˚a nomogram konstrueras vilket ¨ar ett annat s¨att att dimensionera stolpar genom att anv¨ands diagram som relaterar stolparnas parametrar. M˚anga av de relevanta designaspekterna som ber¨or stolpar behandlas i standarden SS-EN 50341 och ligger till grund f¨or den analytiska modellen.
Stolparna kan delas in i tre huvudtyper; raklinjestolpar, d¨ar ledarna forts¨atter rakt vid stolpen, vinkel- stolpar, d¨ar ledarna avviker med en vinkel, och ¨andstolpar, som anv¨ands vid avslut av ledarna. De laster som beaktas ¨ar egenvikt hos komponenter, vind, is, sp¨anning i ledare och underh˚all. F¨or att kombinera lasterna anv¨ands en designekvation som tar h¨ansyn till sannolikheten att vissa h¨andelser sker samtidigt genom att applicera kombinationsfaktorer. Dessutom anv¨ands partials¨akerhetsfaktorer f¨or att kompensera f¨or eventuella avvikelser i last, material eller geometri. Stolparna dimensioneras mot brott p˚a grund av f¨or h¨og b¨ojande belastning och kn¨ackning p˚a grund av f¨or h¨og axialbelastning. I de analytiska ber¨akningarna antas tr¨a vara ett isotropt material i enlighet med standarden medan det modelleras b˚ade som isotropt och ortotropt vid analys med finita elementmetoden. Vidare antas att regeln, den horison- tella delen av stolpen, endast f¨ordelar lasten mellan stolpens ben utan att bidra med n˚agon extra stabilitet genom att l¨anka benen samman. F¨or att testa detta antagande modelleras stolpbenen b˚ade individuellt och sammanl¨ankade med en regel vid analys med finita elementmetoden.
F¨or att kunna j¨amf¨ora resultatet fr˚an den analytiska modellen med finit elementanalys v¨aljs en stolpe av varje typ; raklinje-, vinkel-, och ¨andstolpe, ut. Ett skript skrivs i Octave och till¨ampas p˚a de utvalda stolparna f¨or att best¨amma vilka dimensioner som kr¨avs. Baserat p˚a de ber¨aknade dimensionerna skapas 3D-modeller i PTC Creo Parametric 3.0 vilka sedan importeras till ABAQUS/CAE f¨or analys med finita elementmetoden. F¨orutom att till¨ampa den analytiska modellen direkt h¨arleds uttryck f¨or att kunna generera nomogram.
En j¨amf¨orelse av resultaten fr˚an FEM med den analytiska modellen visade att raklinje- och vinkelstolpar blev ¨overdimensionerade medan ¨andstolpar blev underdimensionerade. Detta inneb¨ar att en mindre dimension kan anv¨andas f¨or raklinje- och vinkelstolpar. F¨or ¨andstolpar m˚aste modellen korrigeras f¨or att stolpen ska h˚alla. Genom att ¨andra kn¨ackningsfall f¨or ¨andstolpen i den analytiska modellen l¨ostes problemet med underdimensionering. Ber¨akningarna i finita elementmetoden d¨ar en regel anv¨andes f¨or att sammanl¨anka stolpbenen visade ingen ¨okad stabilitet f¨or kn¨ackning men b¨ojsp¨anningarna i stolpbenen blev l¨agre. Att anv¨anda en ortotrop materialmodell p˚averkade inte ber¨akningarna n¨amnv¨art eftersom egenskaperna i l¨angsriktningen var mycket mer signifikanta ¨an egenskaperna i tv¨arriktningarna. Vid dimensionering med hj¨alp av nomogram f˚as en s¨amre noggrannhet, eftersom resultatet endast blir en viss stolpklass ist¨allet f¨or en specifik diameter. Denna minskade noggrannhet ¨ar ingen anledning till oro eftersom tr¨astolparna best¨alls utifr˚an klass och inte utifr˚an specifik diameter. Anv¨andandet av nomogram kan ge snabba resultat vid dimensionering.
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Acknowledgements
This master thesis has been carried out at NEKTAB and at Karlstads university and I would like to extend my gratitude to both organizations for making this project possible. I would like to thank my supervisors Mikael Grehk at Karlstads university, Emma Edvardsson at NEKTAB and Linnea Sj¨oholm at NEKTAB for continuously showing support, providing professional guidance and invaluable input throughout this project.
Christopher Ek¨angen Karlstad, 2018-06-13
Contents
List of Figures . . . . x
List of Tables . . . . xi
Nomenclature . . . . xii
1 Introduction . . . . 1
2 Theory . . . . 3
2.1 Relevant dimensions and parts of a support for an overhead line . . . . 3
2.2 Support types . . . . 4
2.3 Loads . . . . 6
2.3.1 Permanent loads . . . . 6
2.3.2 Wind load . . . . 6
2.3.3 Ice load . . . . 7
2.3.4 Loads related to conductor tension . . . . 7
2.3.5 Construction and maintenance loads . . . . 8
2.3.6 Security loads . . . . 9
2.3.7 Partial factor design . . . . 9
2.3.8 Limit states . . . . 10
2.4 Load cases . . . . 10
2.4.1 Tangential and angle supports . . . . 11
2.4.2 Terminal support . . . . 11
2.5 Analytical structural analysis . . . . 12
2.5.1 Analysis of bending . . . . 12
2.5.2 Analysis of buckling . . . . 12
2.6 Timber poles . . . . 13
2.7 Finite element analysis . . . . 14
3 Method . . . . 16
3.1 Selected supports for comparison of analysis methods . . . . 16
3.1.1 Selected tangential support . . . . 16
3.1.2 Selected angular support . . . . 16
3.1.3 Selected terminal support . . . . 17
3.2 Analytical calculations . . . . 17
3.2.1 Bending of tangential supports . . . . 18
3.2.2 Bending of angle supports . . . . 20
3.2.3 Bending of terminal supports . . . . 21
3.2.4 Buckling of supports . . . . 21
3.2.5 Nomograms . . . . 23
3.2.6 Octave . . . . 24
3.3 Comparison with FEM-software . . . . 26
3.3.1 Bending . . . . 28
3.3.2 Buckling . . . . 28
4 Results . . . . 29
4.1 Bending . . . . 29
4.2 Buckling . . . . 34
4.3 Nomograms . . . . 37
5 Discussion . . . . 40
6 Conclusions . . . . 43
7 References . . . . 44
Appendices . . . . 45
A Timber pole grades . . . . 45
B Conductor data . . . . 46
C Fraction of pole self-weight in Euler buckling analysis . . . . 47
D Detrimental load case for a two-legged terminal support . . . . 48
List of Figures
1 Hierarchical diagram of the three network levels that make up the power grid. . . . 1
2 Relevant dimensions and parts of a support. . . . 3
3 The horizontal span or wind span, ah, and the vertical span or weight span, avfor the i:th power line support dictated by the two adjacent spans. . . . 4
4 Different types of supports used for overhead lines seen from above. . . . . 5
5 Insulators used to attach the overhead line to the supports. . . . 5
6 Load on the support caused by conductor tension. . . . 8
7 The bending situation for different support types with different boundary conditions. . . . 12
8 Effect of boundary conditions on buckling behavior of columns. . . . 13
9 A tangential support buried in the ground stabilized by a guy-wire exposed to bending loads. 19 10 Angles necessary for calculating the vertical component of the force caused by the guy acting on the support. . . . 22
11 Buckling of a support with different boundary conditions. . . . 23
12 Window where the user selects the phase conductor. . . . 24
13 Input windows for the octave program where the user specifies the parameters for the support being analyzed. . . . 25
14 Output windows from the Octave program showing the calculated diameters, loads and recommended pole grade. . . . 26
15 Rendered models created in CAD-software. . . . 27
16 Bending analysis of a two legged tangential support where the legs are considered individually. 30 17 Bending analysis of a two legged tangential support where the legs are considered linked together by a cross-arm. . . . 30
18 Bending analysis of a two legged tangential support where the legs are considered linked together by a cross-arm. . . . 31
19 Bending analysis of a three legged angular support where the legs are considered individually. 32 20 Bending analysis of a three legged angular support where the legs are considered linked together by a cross-arm. . . . 32
21 Bending analysis of a two legged terminal support where the legs are considered individually. 33 22 Bending analysis of a two legged terminal support where the legs are considered linked together by a cross-arm. . . . 34
23 Buckling analysis of a two legged tangential support. . . . 35
24 Buckling analysis of a three legged angular support. . . . . 36
25 Buckling analysis of a two legged terminal support. . . . 37
26 Bending and buckling nomograms for the selected tangential support . . . . 38
27 Bending and buckling nomograms for the selected angular support. . . . 38
28 Bending and buckling nomograms for the selected terminal support. . . . 39
29 Output window from the Octave program showing the calculated diameters, loads and recommended pole grade. . . . 41
A.1 Standard measurements and tolerances for timber poles . . . . 45
D.1 Free body diagram of the cross-arm from a two-legged terminal support with three con- ductors attached. . . . 48
List of Tables
1 Dynamic wind pressure under normal and extreme wind conditions in Sweden . . . . 6
2 Partial factors for actions . . . . 10
3 Standard load cases for analysis of supports . . . . 10
4 Load cases for analysis of tangential and angle supports . . . . 11
5 Load cases for analysis of terminal supports . . . . 11
6 Design values for timber poles . . . . 14
7 Average values for elastic constants for Scots pine (Pinus Sylvestris) . . . . 15
8 Parameters for the selected tangential support . . . . 16
9 Parameters for the selected angular support . . . . 17
10 Parameters for the selected terminal support . . . . 17
11 Necessary top diameters and largest loads for the selected supports . . . . 26
12 Design values for elastic constants for Scots pine (Pinus Sylvestris) . . . . 27
13 Results from bending calculations using both the analytical expressions and FEM for the tangential support . . . . 29
14 Results from bending calculations using both the analytical expressions and FEM for the angular support . . . . 31
15 Results from bending calculations using both the analytical expressions and FEM for the terminal support . . . . 33
16 Results from buckling calculations using both the analytical expressions and FEM for the tangential support . . . . 34
17 Results from buckling calculations using both the analytical expressions and FEM for the angular support . . . . 35
18 Results from buckling calculations using both the analytical expressions and FEM for the terminal support . . . . 36
19 Results from buckling calculations using both the analytical expressions and FEM for the terminal support using a mean value of Euler 2 and 3 . . . . 41
20 Material properties and results from FEM-simulations where the material parameters were altered to simulate a varying degree of orthotropy. . . . 42
B.1 Data on conductors used for overhead power lines . . . . 46
Nomenclature
α Angle between guy-wires in a plane per- pendicular to the length of the support β Deviation angle of conductor at an angle
support
γ Angle between guy-wire and support γG Partial factor for permanent actions γQ Partial factor for variable actions ρ Density of the support material σ0 Pre-tension in conductor A Cross-sectional area of support A Cross-sectional area of the pole Ac Cross-sectional area of conductor ah Horizontal span, wind span an Ruling span
av Vertical span, weight span
c Factor used to describe the conicity of timber poles
D Diameter of pole dc Diameter of conductor
Dg Diameter of pole at ground level Dt Diameter of pole at the top of the pole dcI Diameter of conductor covered by ice E Modulus of elasticity of support Ec Modulus of elasticity of conductor F0 Load in conductor due to pre-tension Fhg Horizontal component of the load acting
on the pole from the guy-wire
Fiw Load in conductor due to ice and wind Fvg Vertical component of the load acting on
the pole from the guy-wire G Permanent action
g Standard acceleration due to gravity Gc Self-weight of conductor
gc Mass of conductor per meter Gs Self-weight of support Gc−a Self-weigh of cross-arm
Hc Height from ground level up to cross-arm mount
Hg Height from ground level up to guy-wire mount
Hs Height from ground level up to shield con- ductor mount
Ht Distance between shield conductor mount and top of the pole or cross-arm mount and top of the pole
Htot Total height of support above ground I Load transferred to support due to ice
load on conductor affected by wind Ig Area moment of inertia of the pole at
ground level
IK Load transferred to support due to ice load on conductor in still air
It Area moment of inertia of the pole at the top
Iy Area moment of inertia
K Factor used in buckling calculations de- pendent on boundary conditions
L Buckling length of support
Li Distance between two adjacent supports, span
M Moment, bending moment mc−a Mass of cross-arm
P Axial load
Pcr Critical axial load at which buckling oc- curs
Q Variable action
qh Dynamic wind pressure
QK Load from a linesman with tools, mainte- nance load
Qwc Load transferred to support due to wind load on conductor
qwi Wind load on ice covered conductor per meter
T Load on support from tension in conduc- tor
V Volume of pole
w Combined load (wind, ice and self-weight) per meter of conductor
1 Introduction
The power grid is an hierarchical structure built up by sub networks with different operating voltages and geographical distribution. Structurally it is divided into transmission networks, sub-transmission networks (known as regional network in Sweden) and distribution networks or local networks (Figure 1).
The transmission networks and sub-transmission networks collect and transfer the produced energy while the local networks distribute the energy to the consumers [1].
In Sweden the transmission network is owned by Svenska Kraftn¨at and has operating voltages in the range of 200 to 400 kV. The sub-transmission network is owned by network companies which in turn are owned by larger electricity companies and has operating voltages in the range of 40 to 200 kV. The power lines used in transmission and sub-transmission networks are almost exclusively overhead lines. In the local networks the lines are either overhead lines or underground cables. In urban areas underground cables is the most common solution [1].
Geographical distribution Transmission network (200-400 kV)
Regional/sub-transmission network (40-200 kV)
Local/distribution network (230 v - 40 kV)
Consumers
Figure 1: Hierarchical diagram of the three network levels that make up the power grid. A wider box indicates a greater geographical distribution.
Underground power cables are seen by many as the modern alternative when it comes to power transmis- sion. However an underground cable is eight to twelve times more expensive compared to an overhead line when it comes to construction [2]. In addition to this overhead lines have a life span that is twice as long, 70 years compared to 35 years, and a greater reliability [2]. For high voltage cables large phase shifts between current and voltage occur when they are buried in the ground [2]. In order to correct this phase shift, stations are built every 20 to 40 kilometers. The cable also contains joints every 700 metres.
Every station and joint introduces a source for a potential error [2]. Another important aspect is the time it takes to troubleshoot and repair the line where underground cables take longer to troubleshoot and longer to repair [3].
To suspend the power line above ground supports are used. The main structural components of the overhead power line support are the legs and the cross-arm. In the regional network the most common material for support legs is timber poles, more specifically Scots pine. The cross-arm, which is mainly made of steel, attaches the timber poles together and is usually where the power line is attached using insulators. As the topography is seldom in level and not all locations are suitable for placement of a support, e.g. in the proximity of roads, railways or bodies of water, height differences and irregular distances between supports occur. This results in different conditions for almost every support of the overhead power line. Each support along the stretch of the conductor is dimensioned individually to assure reliability throughout the life-time of the overhead power line.
Many relevant design aspects related to overhead power line supports and conductors are treated in the standard SS-EN 50341 and it will be the basis for many of the calculations in this thesis. The advantages of having common technical regulations regarding aspects such as safety, performance and maintenance of electrical products and power plants are many. By formulating standards the requirements on safety become clear and the cost of development will be kept at a reasonable level [4].
NEKTAB, a consulting company in the energy sector, is currently undergoing an expansive phase due to
reconstruction of old lines and new lines connecting wind power plants to the grid. In the construction of overhead power lines in the regional network NEKTAB uses a number of timber pole support variations and wishes to construct an analytical model which can be used to determine the necessary dimension for all timber pole supports. Another request is that nomograms based on the analytical model are constructed.
Nomograms are diagrams that relate three dependent parameters where one can be determined if the other two are known and will be used to relate height of the support, diameter of the support and the distance between two adjacent supports. The purpose of the project is to clarify how the necessary dimension for a support should be determined.
Objectives for the project are:
– Analytical model which can be used to determine the necessary diameter for the different timber support types with respect to type of conductor, angle at which the conductors of adjacent spans meet, height difference in the terrain and distance between supports.
– Constructing a simulation tool using the scientific programming language Octave.
– A comparison of calculations using the analytical model with calculations using finite element analysis software.
– Nomograms based on the analytical model for each support variation which can be used to easily determine what diameter is necessary for a timber pole support with a specified set of boundary conditions.
The approach consists mainly of analytical calculations but is also compared with calculations using a finite element method (FEM) software. Loads are calculated in accordance with the standards that apply in Sweden. The analytical calculations are performed using a program script written in Octave in which the analytical equations are applied to the parameters of the timber support. Finite element analysis using the software Abaqus/CAE is applied to a timber support of dimensions corresponding to the results from the analytical calculations to validate the results.
2 Theory
To perform the analytical and FEM calculations it is necessary to investigate structural parts, dimensions, shape, loads, load combinations and material parameters relevant for the supports. Parameters such as support height, distance between supports and height difference between support placements are determined by the placement of the poles onto the current ground profile. The placement is performed by power line engineers using a software that imports a profile of the ground. The software will calculate how much the conductor will sag. Based on the parameters calculated in the software in combination with the loads, load combinations and material properties from the following sections the required support dimension can be determined.
2.1 Relevant dimensions and parts of a support for an overhead line
In general, a support consists of one to three poles and a cross-arm (Figure 2). The cross-arm is a horizontal member of the support that holds the insulator, which in turn holds the phase conductor, at a height, Hc, from the ground. Hs− Hc is the distance from where the phase conductor is mounted to where the shield conductor is mounted, if there is one. If the tip of the pole is above all the mounted components Htis the distance from the top of the pole to the closest attached component. If the support needs further stability a guy can be used to help keep it upright. The height at which the guy acts on the pole is here denoted Hg. The pole, if it does not need a foundation, is buried at a depth, Hb, into the ground. The diameter of the pole at ground level is usually denoted Dg. The components and their position relative to each other can vary. For example the guy-wire could be attached above the cross-arm and the shield conductor could be mounted below the phase conductor.
HcHs−HcHtHb Hg
Dg Guy
Ground level Cross-arm
Pole
Shield conductor
Phase conductor Insulator
Figure 2: Relevant dimensions and parts of a support.
The horizontal span or wind span, denoted ah, is measured between the mid points for two adjacent spans and is thus the mean value of the two adjacent span lengths, i.e. ah = L1+L2 2 (Figure 3). The vertical span or weight span, denoted av, is measured between the lowest points of the sagging conductor of the adjacent spans, i.e. av= LW 1+ LW 2(Figure 3). The width of the support is negligible compared to the horizontal and vertical spans. The vertical span is dependent upon the height difference between the poles while the horizontal span is always distributed equally.
i - 1
i
i + 1
L1 L2
LW 1 LW 2
ah av
Ground line
Figure 3: The horizontal span or wind span, ah, and the vertical span or weight span, av for the i:th power line support dictated by the two adjacent spans.
2.2 Support types
Three different support types are used when constructing an overhead line. When the conductor makes an angle above the support instead of continuing straight, an angle support (Figure 4a) is used. A tangential support (Figure 4b) is used when the conductor continues straight at the support. At the end of an overhead line, e.g. at a transformation station, a terminal support (Figure 4c) is used. A terminal support can also be used in place of a tangential support to section of parts of the conductor.
Guys
Conductors
a) Angle support b) Tangential support
Cross-arm Pole
c) Terminal support
Figure 4: Different types of supports used for overhead lines seen from above.
Different types of supports used for overhead lines seen from above. a) shows an angle support with guys, b) shows a tangential support without guys and c) shows a terminal support.
For each of the three support types different insulators can be used. Tension supports (Figure 5a) are used to make sure that the tension in the conductor is constant for each span. Tension supports can also section off parts of the line preventing cascading failure of the entire line. The conductors are attached by tension insulators to the support and are connected to each other by jumpers. When suspension insulators are used (Figure 5b) the conductor is merely hanging from the insulator below the cross-arm.
Jumper
Tension insulator
Conductor
a) Tension support
Suspension insulator
Conductor
b) Suspension support
Figure 5: Insulators used to attach the overhead line to the supports. a) shows a support with tension insulators seen from the side (perpendicular to the stretch of the conductor) and b) shows a support with suspension insulators seen at an angle from the side.
2.3 Loads
Loads acting on the support, conductors and other components are known as actions. The actions are classified by their variation in time as permanent actions, variable actions or accidental actions[4, 5].
The actions which are classified as permanent, denoted G, are not expected to vary under the design life of the structure [5]. Examples of permanent loads are self-weight of the support, conductor or other structural parts [4]. Variable actions, denoted Q, are expected to vary over time such as wind load or ice load [5]. The accidental actions, denoted A, include loads that deviate from normality such as avalanches or earthquakes [4].
Actions can also be divided into static and dynamic actions. Static actions do not cause significant acceleration of the elements they act upon while dynamic actions cause significant acceleration. Usually it is sufficient to assume an equivalent static response of quasi-static actions such as wind loads [4].
When evaluating actions characteristic values are assumed since the value of an action typically lies within a range of possible values. Characteristic values are values with a certain level of probability of not being exceeded [5].
2.3.1 Permanent loads
To permanent loads the self-weight of supports, conductors and other fix equipments are included. The self-weight of the support is calculated as
Gs= ρV g [N] (1)
where ρ is the density of the support material, V is the volume of the support and g is the standard acceleration due to gravity. The self-weight of the cross-arm with insulators to which the conductor is attached is
Gc−a= mc−ag [N] (2)
where mc−a is the mass of the cross-arm with insulators and other components and g is the standard acceleration due to gravity. The self-weight of each conductor is calculated as
Gc= gcgav [N] (3)
where gc is the mass of the conductor per meter, g is the standard acceleration due to gravity and av is the weight span.
2.3.2 Wind load
The wind load per meter for timber poles is:
QW p= 0.16qh [N/m] (4)
where qh is the characteristic dynamic wind pressure, given in Table 1, and is dependent upon height and gust wind speed. In general the values given in Table 1 are valid for Sweden [6].
Table 1: Dynamic wind pressure under normal and extreme wind conditions in Sweden[6]
For normal wind conditions
h [m] qh[N/m2]
≤ 25 500
> 25 500 + 6(h - 25)
For extreme wind conditions
h [m] qh[N/m2]
≤ 10 800
> 10 800 + 6(h - 10)
The wind load transferred to the support due to wind pressure on the conductor is
QW c= qhGqGcCcdcahcos2(φ) [N] (5)
where qh is the characteristic dynamic wind pressure, given in Table 1, Gq is the gust response factor and should be taken as 1.0, Gc is the structural resonance factor and should be taken to be Gc= 1.0 for jumpers and Gc= 0.5 for spans, Cc is the drag coefficient for the conductor which for round conductors under normal design wind speeds can be taken to be 1.0, dc is the diameter of the bare conductor, ah is the horizontal or wind span and φ is the angle of incidence (related to the normal direction) for the critical wind direction, where a normal incidence (φ = 0°) is the most severe [6].
For winds acting on an ice covered conductor the diameter should be compensated by an ice thickness of 18 mm giving the diameter as
dcI= dc+ 2 × 0.018 [m] (6)
where dc is the bare conductor diameter in meters. For ice covered conductors the drag coefficient is CcI= 1.0.
2.3.3 Ice load
The ice load on conductors under normal wind conditions in Sweden is
I = 9.2 + 0.51 × 104d [N/m] (7)
where d is the bare conductor diameter in meters.
Under still air conditions the ice load on conductors is IK =
q
(I + gcg)2+ qwi2 − gcg [N/m] (8)
where IK is the uniform ice load at no wind, minimum 20 N/m, I is the uniform ice load at normal wind, g is the standard acceleration due to gravity, gc is the mass of the conductor per meter and qwi is the normal wind load per meter on conductor covered by uniform ice load calculated from (5) with φ = 0, d = dcI and without including ah as
qwi= qhGqGcCcdcI [N/m] (9)
The ice load acting on the conductor causes vertical forces, acting on the support, as well as increased tensions in the conductors. The ice force caused by the two adjacent spans acting vertically on the support from each sub-conductor is
QI = Iav [N] (10)
where I is the ice load per unit length of the conductor according to (7) or (8) depending on wind conditions and av is the contribution of the weight spans of the two adjacent spans.
The load from ice placed directly on the support can in general be neglected [6].
2.3.4 Loads related to conductor tension
Angle and terminal supports will be affected by a load resulting from the tension in the conductors. Part of the load is considered permanent, the load from pre-tension, and part of the load is considered variable, the increase in conductor tension due to load from ice and wind on the conductor. When calculating sag and tension in the conductor a length parameter known as ruling span is usually used. The ruling span is a theoretical length which can be used to estimate the sag and tension behavior of all spans between two tension supports [7]. The ruling span is defined as
an=
sP L3i P Li
[m] (11)
where Li are the individual span lengths between two terminal supports. Calculation of the ruling span quickly becomes tedious and is usually carried out in the support placement software. The conductor tension can be calculated from equation (12) which is based on the ruling span method [8].
Conditions of interest
z }| {
Fiw Ac
−w2a2nEc 24Fiw2 =
Initial conditions
z }| {
F0
Ac−(gcg)2a2nEc
24F02 [N/mm2] (12)
where Fiw is the load in the conductor due to ice and wind on the conductor, Ac is the conductor area, w is the combined load from self-weight, ice and wind acting on the conductor per meter, an is the ruling span, Ec is the elastic modulus for the conductor, F0= σ0Ac is the load in the conductor due to pre-tension, σ0, in the conductor with cross-sectional area, Acand gcg is the self-weight of the conductor.
In the case of combined self-weight, ice and wind loads w is w =
q
q2wi+ (gcg + I)2 [N/m] (13)
where qwiis the wind load per meter, gcg is the self-weight per meter and I is the ice load per meter.
For angle supports (Figure 6a) the loads acting on the support from conductor tension from each conductor is
T =
P ermanent
z }| {
F0sin(β/2) +
V ariable
z }| {
(Fiw− F0) sin(β/2) [N] (14)
where F0 = σ0Ac is the load in the conductor related to the pre-tension, σ0, in the conductor with cross-sectional area, Ac, Fiw is the load in the conductor due to ice and wind on the conductor and β is the deviation angle (Figure 6a). For a terminal support (Figure 6b) the load acting on the support from conductor tension from each conductor is
T =
P ermanent
z}|{F0 +
V ariable
z }| {
(Fiw− F0) [N] (15)
where F0= σ0Ac is the load in the conductor related to the pre-tension, σ0, in the conductor with cross- sectional area, Ac and Fiw is the load in the conductor due to ice and wind on the conductor.
β
T T
a)
T
b)
Figure 6: Load on the support caused by conductor tension. a) shows the loads on an angle support and b) shows the load on a terminal support.
2.3.5 Construction and maintenance loads
Construction loads include working procedures, temporary guying, lifting arrangement and any other loads which are likely to occur. The supports shall be able to withstand a vertical load QK = 1000N , corresponding to the weight of a linesman with tools, acting in the most unfavorable position.
2.3.6 Security loads
Security loads are unbalanced loading situations occurring due to loss of tension in one or more of the conductors. When the security loads are evaluated two methods are available, basic assumption and alternative assumption where the most favorable can be chosen. In the basic assumption a reduction factor is applied to the conductor tension on one side of the support. A reduction factor of 0.4 is applied for fixed attached conductors, i.e. tension insulator set, and a reduction factor of 0.7 is applied for non- fixed conductors, i.e. suspension insulator set. The basic assumption assumes: uniform ice, a conductor temperature of 0°C, in still air and at initial conductor tension [6].
In the alternative assumption the conductor tension is fully reduced (100 %) for a conductor placed at the most unfavourable position. The alternative assumption assumes: no ice, a conductor temperature of 0 °C, in still air and at initial conductor tension [6].
2.3.7 Partial factor design
Reliability in the design of overhead power lines is achieved by using partial factors for actions and material properties. The partial factor method ensures that the limit states (Section 2.3.8) are not reached [4].
There are two types of partial factors commonly used; partial safety factors and combination factors [5]. The partial safety factors, denoted by γ, takes into consideration possible unfavorable deviations of actions, material properties or geometry. Combination factors are used when combining actions and taking into account the reduced probability that the most unfavorable values of several actions occur simultaneously [4].
By combining characteristic values with partial factors, design values are formed [4]. The design values for actions, denoted Fd, are calculated according to (16) [4].
Fd= γfFK (16)
The design values for permanent actions (Gd), variable actions (Qd) and accidental actions (Ad) are calculated in analogue with Fd. For material properties the design values are calculated according to (17) [4].
Xd= XK/γM (17)
The actions are combined using the basic design equation [6]
Ed=
∞
X
i=1
γGGiK+ ψ
∞
X
n=1
γQQnK [N/m] (18)
where ψ is the combination factor and shall be taken as 1.0. The partial factors for actions are taken according to Table 2.
Table 2: Partial factors for actions [6]
Action Partial factor Load combination
1 2 3
Permanent actions
Dead weight of supports, foundations and conductors γG 1.1 0.94 1.0
Dead weight of soil and ground water γG 1.1 1.1 1.0
Conductor tension for bare conductors at 0°C γG 1.1 1.1 1.0 Dynamic maintenance and construction loads γG 1.8 1.8 1.3 Variable actions
Wind and ice loads, additional loads, residual static load at one-sided conductor tension reduction, al- ternative assumption, difference in actual conductor tension and tension at 0°C on bare conductor
γQ 1.43 1.43 1.0
Load combination 1) is normally determinant for supports and guys.
Load combination 2) might be determinant for foundations.
Load combination 3) is valid for conductors and insulators. Also valid when checking defor- mations, electrical clearances and concrete cracks (Serviceability limit state).
2.3.8 Limit states
Limit states are used to describe conditions under which the design performance of the overhead line is no longer fulfilled. In general these limit states are divided into ultimate limit states and serviceability limit states [4]. Ultimate limit states are related to collapse or other forms of structural failure due to excessive deformation, loss of stability, overturning, rupture, buckling etc. Serviceability limit states correspond to conditions beyond which specified service requirements for an overhead line and parts of its supporting structure are no longer met.
2.4 Load cases
When analyzing the supports different load cases (Table 3) need to be evaluated. Many of the load cases can be ignored in Sweden and two of the load cases are not applicable for naturally grown timber poles [4]. The load cases describe different combinations of loads that occur in different scenarios.
Table 3: Standard load cases for analysis of supports[6]
Load case
Conditions Remark
1a Extreme wind load Not applicable for naturally grown timber poles 1b Wind load at a minimum temperature This load can normally be ignored in Sweden 2a Uniform ice loads on all spans
2b Uniform ice loads, transversal bending This load can normally be ignored in Sweden 2c Unbalanced ice loads, longitudinal bending This load can normally be ignored in Sweden 2d Unbalanced ice loads, torsional bending This load can normally be ignored in Sweden 3 Combined wind and ice loads
4 Construction and maintenance loads
5a Security loads, torsional loads Not applicable for naturally grown timber poles used as tangential or angle support
Suspension insulator sets and line-post insulators do not have to be designed for this load case
5b Security loads, longitudinal loads This load can normally be ignored in Sweden for tan- gential, angle and terminal supports
2.4.1 Tangential and angle supports
Table 4 shows the load cases and their corresponding loads which are relevant when analyzing tangential and angle supports.
Table 4: Load cases for analysis of tangential and angle supports[6]
Load case
Conditions Included loads
1a Extreme wind load - Self weight
- Extreme wind load on the conductors and support 2a Uniform ice loads on all spans - Self weight
- Uniform ice load at no wind on the conductors - Load of a linesman with tools
3 Combined wind and ice loads - Self weight
- Uniform ice load at normal wind on the conductors - Normal wind load on the conductors and support 4 Construction and maintenance loads - Self weight
- Construction and maintenance loads - Load of a linesman with tools 5a Security loads, torsional loads - Self weight
- Security loads: Basic or alternative assumption
2.4.2 Terminal support
Table 5 shows the load cases and their corresponding loads which are relevant when analyzing terminal supports.
Table 5: Load cases for analysis of terminal supports[6]
Load case
Conditions Included loads
1a Extreme wind load - Self weight
- Extreme wind load on the conductors and support - Conductor tensions from all conductors or conductors on one side of the support, whichever is most severe 2a Uniform ice loads on all spans - Self weight
- Uniform ice load at no wind on the conductors - Load of a linesman with tools
- Conductor tensions from all conductors on one side of the support
3 Combined wind and ice loads - Self weight
- Uniform ice load at normal wind on the conductors - Normal wind load on the conductors and support - Conductor tensions from all conductors or conductors on one side of the support, whichever is most severe 4 Construction and maintenance loads - Self weight
- Construction and maintenance loads - Load of a linesman with tools 5a Security loads, torsional loads - Self weight
- Conductor tensions from all conductors, with reduced tension for some conductors using the basic assumption for security loads
2.5 Analytical structural analysis
The structural analysis is divided into bending and buckling analysis. Large bending loads are applied to the supports as a result of wind load and conductor tension and large axial loads are applied due to self-weight, ice load, maintenance loads and the load caused by the guy-wire. The different support types have different boundary conditions and therefor need to be treated separately when it comes to structural analysis. If the supports are placed on foundations or stabilized by guy-wires the boundary conditions will also be altered and require special consideration.
2.5.1 Analysis of bending
The stress that occurs in the support due to vertical forces in combination with horizontal forces giving a bending moment are calculated as
σ(x) = Nx(x)
A(x) +My(x)
Iy(x)z [N/m2] (19)
where Nx(x) is the normal force, A(x) is the area of the cross-section, My(x) is the bending moment and Iy(x) is the area moment of inertia at position x calculated according to (20) for a circular cross- section.
Iy(x) = πa4(x)
4 =πD4(x)
64 [m4] (20)
where a(x) is the radius and D(x) is the diameter of the circular cross-section at position x.
The bending moment varies along the length of the support and depending on the type of support being analyzed and the boundary conditions of the support the bending moment will have its maximum at different positions. A support without guys will be subjected to horizontal loads from wind acting on the conductors and pole (Figure 7a). A guyed support (Figure 7b) will, in addition to the wind load, be affected by the horizontal component of the load from the guy-wire(s), denoted FGH.
x
QW c
QW p
a)
QW c
FGH QW p
b)
Figure 7: The bending situation for different support types with different boundary conditions. a) shows a support buried in the ground without guys, b) shows a support buried in the ground with guys,
2.5.2 Analysis of buckling
When a long initially straight pole is subjected to an axial compressive force, P, it will maintain a straight equilibrium shape for lower values of P while for higher values of P both a straight and bent equilibrium shape is possible [9]. The critical axial load according to Euler, denoted Pcr after which the support can no longer maintain its straight shape is calculated as
Pcr= π2EI
(KL)2 [N ] (21)
