1
Analysis and Design of Transmission Towers
Analysis and Design
Of
Transmission
Towers
A graduation project Submitted to the department of
civil engineering at The University of Baghdad
Baghdad - Iraq
In partial fulfillment of the requirement for the degree
of Bachelor of Science in civil engineering
By
Mohammed & Mustafa
Supervised by
Assistant lecturer, A. N. LAZEM
(M.Sc., in Structural Engineering)
July /2008
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Analysis and Design of Transmission Towers
Analysis and Design
Of
Transmission Towers
I certify that study entitled “Analysis and Design of Transmission Towers”, was
prepared by (
and
)
under
my
supervision at the civil engineering department in the University of Baghdad, in
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3
Analysis and Design of Transmission Towers
partial fulfillment of requirements for the degree of Bachelor of Science in civil
engineering.
Supervisor
Signature:
Name: A. N. LAZEM
Assistant lecturer
(M.Sc., in Structural
Engineering)
Date:
We certify that we have read this study “Analysis and Design of Transmission
Towers” and as examining committee examined the students in its content and in
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Analysis and Design of Transmission Towers
what is connected to with it, and that in our opinion it meets the standard of a study
for the degree of Bachelor of Science in civil engineering.
Committee Member
Committee Member
Signature:
Signature:
Name:
Name:
Date:
Date:
Signature:
Name:
Head of Civil Engineering Department
College of Engineering
Baghdad University
Date:
Abstract:
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Analysis and Design of Transmission Towers
The objective of this study is to develop a better understanding for the basic principles
of the Design and analysis of Transmission Towers so they can be efficiently
implemented on modern computers.
Demonstrate the effect of transverse loading on the Design of in-plane Truss
structures.
Develop an in-plane Stiffness Matrix that take into the effect of slenderness ration
limitations for each member during the design process inside computer program.
In addition a case study has been presented, that involve five different load
combinations that simulate a real Broken Wires Conditions, to inveterate the worst
loading case condition on the internal members stresses.
Project layout
The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject of transmission towers.
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the design of steel transmission
towers using I.S.(ASD) and the analysis process using stiffness matrix method.
Chapter four: presents a detailed procedure of the developed computer developed in
this project and their application for one case study with five different LoadCombinations.
Chapter five: discuses the results of analysis and design method and recommend
future steps.
Appendix I: program text
Contents:
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Analysis and Design of Transmission Towers
Title……………………………………………………………………………………2
Supervisor words……………………………………………………………….3
Committee words………………………………………………………………4
Thanks……………………………………………………………………………….5
Abstract…………………………………………………………………………….6
Project Layout………………………….……………………………………….6
Contents…….……..………………………………………………………………7
Chapter one; introduction…………..….…………………………………8
Chapter two; literature………………….………………………………..12
Chapter three; theory………………………………………………………17
Chapter four; computer program…………………………………….25
Chapter five; conclusions and recommendations…………….36
References…………………………………………………………………..……39
Appendix I…………………………………………………………………..……40
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Analysis and Design of Transmission Towers
CHAPTER ONE
INTRODUCTION
Transmission Towers
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Analysis and Design of Transmission Towers
5.1. GENERAL
Electricity is a major source of power for industries, agriculture, commercial and
residential use. because of its lesser cost, electricity is now being used for rail
transportation in place of fuel-powered engines. Electricity is generated from hydroelectric power plants, thermal generating stations and nuclear power plants. Hydroelectric plants are located on perennial rivers usually in remoted hilly gees and thermal
plants are situated near coal mines. Due to the diverse requirements of electricity
across the country and far-away locations of power plants, a grid of electric transmission
lines is required to cover the entire country. With the increase in transmission
distances, electricity is being transmitted at extra high voltages so as to minimize
transmission losses. This req uires greater ground clearance which means that taller
transmission towers are required. Various types of supporting structures are used
depending upon the type of electric transmission line or conductor. The supporting
structures are mainly of two types-poles and towers. Poles are normally used for supporting
conductors with lower voltages and requiring lesser ground clearance over smaller spans.
These poles can be made if timber, reinforced concrete, prestressed concrete, steel or
aluminum. The material and the shape of he pole depends upon location of transmission
line, required life span, initial cost, maintenance cost, voltage and availability of material.
Towers are provided where high voltage transmission conductors are to be supported
over longer ;pans and with greater ground clearances. These are designed as self supporting wide-based towers. Doles are designed for carrying loads in the transverse
direction only and depend upon conductors for longitudinal stability. In order to prevent the
simultaneous collapse of the whole line, self-supporting poles or towers are provided at
reasonable intervals.
2.2. TYPES OF TOWERS
The purpose of transmission line towers is to support conductors and one or two ground
wires at suitable distances above the ground level and from each other.
The selection of the most suitable type of tower for transmission lines depends upon
the actual terrain of the line and the number of circuits to be supported. Towers can be
broadly classified as follows
(i)
Tangent towers with suspension string (0° to 2°). These are used on
straight runs and for line deviation up to 2°. The conductor is
supported by a string of insulators hanging vertically from the tower
cross-arms.
(ii)
(ii) Small angle towers with tension strings (2° to 15°) These are used
for lines with deviation between 2° and 15°.
(iii)
Medium angle towers with tension strings (15° to 30°). These are used
for line deviation from 15° to 30°.
(iv)
Large angle (30° to 60°) and dead end towers with tension strings. These are
used for lines with deviation from 30° to 60° and for dead ends.
The angles of line deviation specified are for normal spans. The span may be increased
up to an optimum limit by reducing the angle of line deviation. Tangent towers are
designed for supporting the tensioned conductors. Angle towers, which are provided at
points of line deviation, are designed to resist the angular pull of the conductors. These
towers are positioned such that the axis of the cross arm bisects the angle in the line.
The height of the towers is fixed such that there is an adequate ground clearance (6 to
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Analysis and Design of Transmission Towers
10 m) at the point of greatest sag. The tower heights range from 10 to 45 m depending
upon the span, terrain and conductor voltage.
15.3. TOWER CONFIGURATION
Transmission towers are free-standing towers and are usually square in plan. These are
supported on ground by four legs and apt as cantilever trusses under horizontal loads.
Power transmission towers have horizontal arms called cross-arms for carrying the
conductors. The configuration of a tower depends upon the number of circuits, minimum
clearances of conductor from tower and ground, distance between conductors, terrain and
span. Various shapes of transmission towers are shown in Fig. 2.1.
Fig(2.1)
The most common type of tower for a single circuit is shown in Fig. 2.1 (a). For a double
circuit, tower shown in Fig. 2.1 (b) is used. The conductors in this case are hung one
above another from three horizontal cross-arms. Other forms of towers which are also used
are shown in Fig. 2.1 (c) and (d). A tower is subjected to horizontal loads due to wind on
tower and conductors and due to tension in conductors under broken-wire condition. These
forces tend to over-turn the tower. Where the overturning moments are large, as in the case
of tall towers, the base of the tower is widened and a pyramid shaped tower is provided.
The corner member of the tower, which are either vertical or nearly vertical, are called
`legs' or 'column members.' The main force is carried by these legs or column
members. The legs are interconnected by diagonal bracing members and sometimes
with horizontal members also. The bracing members carry very little force. Various types
of bracings are shown in Fig. 2.1.
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Analysis and Design of Transmission Towers
Fig.(2.2)
The tower outline diagram comprises:
(a) Tower height considered from ground level.
(b) Length of the cross arms and phase spacings.
(c) Tower widths at (i) base and (ii) top hamper and
(d) Bracing pattern adopted.
The various constituents are as shown in Fig. 2.2. Both electrical and mechanical
considerations determine these dimensions.
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Analysis and Design of Transmission Towers
CHAPTER TWO
LITERATURE
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Analysis and Design of Transmission Towers
2.1
LINEAR
ANALYSIS
OF
IN-PLANE
STRUCTURES
USING
STIFFNESS
MATRIX METHOD
The theoretical foundation for matrix (stiffness) method of structural analysis was laid and
developed by many scientists:
James, C. Maxwell, [1864] who introduced the method of Consistent Deformations
(flexibility method).
Georg, A. Maney, [1915] who developed the Slope-Deflection method (stiffness method).
These classical methods are considered to be the precursors of the matrix (Flexibility and
Stiffness) method, respectively. In the pre-computer era, the main disadvantage of these
earlier methods was that they required direct solution of Simultaneous Equations (formidable
task by hand calculations in cases more than a few unknowns).
The invention of computers in the late-1940s revolutionized structural analysis. As computers
could solve large systems of Simultaneous Equations, the analysis methods yielding solutions
in that form were no longer at a disadvantage, but in fact were preferred, because
Simultaneous Equations could be expressed in matrix form and conveniently programmed for
solution on computers.
Levy, S., [1947] is generally considered to have been the first to introduce the flexibility
method, by generalizing the classical method of consistent deformations.
Falkenheimer, H., Langefors, B., and Denke, P. H., [1950], many subsequent researches
extended the flexibility method and expressed in matrix form are:
Livesley, R. K., [1954], is generally considered to have been the first to introduce the stiffness
matrix in 1954, by generalizing the classical method of slop-deflections.
Argyris, J. H., and Kelsey, S., [1954], the two subsequent researches presented a formulation
for stiffness matrices based on Energy Principles.
Turner, M. T., Clough, R. W., and Martin, H. C., [1956], derived stiffness matrices for truss
members and frame members using the finite element approach, and introduced the now
popular Direct Stiffness Method for generating the structure stiffness matrix.
Livesley, R. K., [1956], presented the Nonlinear Formulation of the stiffness method for
stability analysis of frames.
Since the mid-1950s, the development of Stiffness Method has been continued at a
tremendous pace, with research efforts in the recent years directed mainly toward formulating
procedures for Dynamic and Nonlinear analysis of structures, and developing efficient
Computational Techniques (load incremental procedures and Modified Newton-Raphson for
solving nonlinear Equations) for analyzing large structures and large displacements. Among
those researchers are: S. S. Archer, C. Birnstiel, R. H. Gallagher, J. Padlog, J. S.
przemieniecki, C. K. Wang, and E. L. Wilson and many others.
LIVESLEY, R. K. [1964] described the application of the Newton- Raphson procedure to
nonlinear structures. His analysis is general and no equations are presented for framed
structures. However, he did illustrate the analysis of a guyed tower.
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Analysis and Design of Transmission Towers
CHAPTER THREE
THEORY
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Analysis and Design of Transmission Towers
3.0 I.S. SPECIFICATION FOR ANALYSIS AND DESIGN OF
TRANSMISSIONS TOWERS
3.1. TOWER DEFINITIONS
Transmission towers are free-standing towers and are usually square in plan. These are
supported on ground by four legs and apt as cantilever trusses under horizontal loads.
Power transmission towers have horizontal arms called cross-arms for carrying the
conductors.
The definition of a tower depends upon the number of circuits, minimum clearances of
conductor from tower and ground, distance between conductors , terrain and span .
Various shapes of transmission towers are shown in Fig. 3.1.
Fig. (3.1)
The most common type of tower for a single circuit is shown in Fig. 3.1 (a). For a double
circuit, tower shown in Fig. 3.1 (b) is used. The conductors in this case are hung one
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Analysis and Design of Transmission Towers
above another from three horizontal cross-arms. Other forms of towers which are also used
are shown in Fig. 3.1 (c) and (d). A tower is subjected to horizontal loads due to wind on
tower and conductors and due to tension in conductors under broken-wire condition. These
forces tend to over-turn the tower. Where the overturning moments are large, as in the case
of tall towers, the base of the tower is widened and a pyramid shaped tower is provided.
The corner member of the tower, which are either vertical or nearly vertical, are called
`legs' or 'column members.' The main force is carried by these legs or column
members. The legs are interconnected by diagonal bracing members and sometimes
with horizontal members also. The bracing members carry very little force. Various types
of bracings are shown in Fig. 3.3.
The tower outline diagram comprises:
(a) Tower height considered from ground level.
(b) Length of the cross arms and phase spacings.
(c) Tower widths at (i) base and (ii) top hamper and
(d) Bracing pattern adopted.
The various constituents are as shown in Fig. 3.2. Both electrical and mechanical
considerations determine these dimensions.
(a) Tower Height
The height of a tower (H) in level country comprises the permissible ground clearance of
conductors required in accordance with the overhead line regulations (ht), max sag for the
lower most conductor (h2), vertical spacing between conductors including maximum
insulator string length (h3) and height of ground wire peak portion (h4).
(i) Minimum Ground Clearance
Power conductors, along the entire route of the transmission line should maintain requisite
clearance to ground over open country, national highways, important roads, electrified and
unelectrified railway
tracks, navigable and non-navigable rivers, telecommunication and power lines, etc. as laid
down in the National standards issued by the respective authorities. According to clause.
"77., o1;. The Indian-Electricity Rules- 1956 (incorporating the latest. amendment) stipulates the
following clearances above ground the lowest point of conductor For extra ,of the high
voltage lines, this, clause stipulates that, the- clearance above ground shall not be less.-than 5.1
in .plus 0.3 m for every 33,000 volts or part thereof by which the voltage of the line exceeds
33000 volts.
The permissible ground clearances for different voltages; therefore, work out as follows:
66 kV
5490 mm
132 kV
6100 mm
220 kV
7015 mm
400 kV
8840 mm
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Analysis and Design of Transmission Towers
The above minimum ground clearance are applicable for transmission lines running in the
open country.
The minimum clearance of conductors over rivet is specified as 3050 mm over maximum
flood level for rivers which are not navigable. For navigable rivers, clearances are fixed
in relation to the tallest mast in consultation with the concerned navigating authorities:
In case, the power lines crosses .over, a telephone line, the minimum clearances between the
conductors of the power, line and telecommunication wires are as specified as follows:
66 kV ‫ ـــــــــ‬2440 mm
132 kV ‫ــــــ‬2745 mm
220 kV‫ـــــــــ‬3050 mm
400 kV‫ـــــــــــ‬4880 mm
66 kV ‫ ـــــــــ‬2440 mm
132 kV ‫ــــــ‬2745 mm
220 kV‫ـــــــــ‬3050 mm
400 kV‫ـــــــــــ‬4880 mm
Between power line up to 220 kV crossing over another power line of any other voltage
up to 220 kV, the clearances shall not be less than 4550 mm, between 132 kV the clearance is
2750 mm and for 66 kV line the figure is 2440 mm. For 400 kV, the clearance may be
assumed as 6000 mm respectively.
The minimum height. above rail level of the lowest portion of any conductor under conditions
of maximum sag are as follows it accordance with the Regulation for electrical crossings of
railway tracks, 1963.
a) For un-electrified tracks or tracks electrified on 1500 Volts D.C.
b) Tracks Electrified On 25 Kv A.C.
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Analysis and Design of Transmission Towers
(II) Maximum Sag of Lowermost Conductor
The size and type of conductor, wind and climatic conditions of the region and span
length determine the conductor sag and tensions. Span length is fixed from economic
considerations: The maximum sag for conductor-span occurs at the maximum
temperature and still wind conditions: TVs maximum value of sag is taken into
consideration in fixing the overall height of the tower.
While working out tension in arriving at the maximum sag the following stipulations
laid down in Indian Electricity Rules (1956) are to be satisfied
(a) The minimum factor of safety for conductors shall be based on their ultimate
strength (tensile).
(b) The conductor tension at 32 °C without external load shall not exceed the following
percentages of the ultimate tensile strength of the conductor.
Initial unloaded tension 35 percent Final unloaded tension 25 percent
(III) Height and Location of Ground Wires. Earthwire provides protection
against direct stroke of lightening. It intercepts the direct lightning strokes and conducts
the charge to the nearest ground connections. The location of the ground wire/ground
wires determines the. height of the ground wire peak portion. The height and location of
the overhead ground wires shall be such that the line joining the ground wire to the
outermost conductor shall make angles of approximately 20 to 30 degrees with the
vertical. This angle is called the shield angle. The smaller the angle, the better is the
shielding provided. The practice is to specify 30 degrees for 66 kV and 25-30 degrees
for 220 kV.
(IV) Minimum mid span clearance. In case of direct lightning stroke on the mid
span of overhead earthwires, the potential of the mid span is built up during the
propagation of the surge current, and the midspan flashover may occur from ground wire to
conductor before the current is discharged through the tower. The midspan clearance
between the earthwires and conductor is, therefore, kept more than the clearance at the
tower. The usual practice in this regard is to maintain the sag of the groundwire at least
10 percent less than that of the power conductor under all temperature conditions in still
wind at the normal spans so as to give a midspan separation greater than that at the
supports. It is however ensured that under minimum temperature and maximum wind
conditions, the sag of the ground wire does not exceed the sag of the power conductor.
(b) Length of X-arm and conductor spacings The length and composition of
insulator strings, jumpers, their swings and the corresponding safe electrical clearances to
earthed parts in the deflected position of strings and jumpers and tower width at the
cross arms level determine the length of the cross-arms and the horizontal and vertical
spacings between the phases. The width of tower at cross-arms level is generally
determined from the torsional forces it has to resist under broken conductor conditions. The
larger width reduces the torsional forces transmitted to the bracing below that level and helps
in reducing the forces in bracings of the tower body. The optimum cross-arms length evolve
t he most economical tower outline.
(c) Tower widths at base. Spacing between the tower footings, that is, the base
width at the concrete level (or at the foot of bottom panel) is the distance from the
centre of gravity of one corner leg to the centre of gravity of the adjacent corner leg
angle. This width depends upon the magnitude of the physical loads imposed upon the
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Analysis and Design of Transmission Towers
towers, calculated from the size, 'type of the conductors and wind loads, and also depends
upon heights of application of external loads from ground level. Towers with larger basewidth result in lour footing, costs and lighter main leg members at the expense of long
bracing members. There is a particular base width which gives the best compromise and for
which total cost of the tower and foundations is minimum.
It is observed that the relation between total height of tower up to the lower cross-arm
and base width is generally 2.4 to 4.0. As per an American practice, the ratio of base width to
height which is the height of the intersection of the slope of the legs from ground wires is I : 3
for single circuits and is 1 : 4 for double circuits. There is a formula, which gives the
economical base width of lattice tower,
Where;
B = Base width of tower at ground level in cm
M = overturning moment in kg-m
.
K = a constant
The value of K lies between 1.35 and 2.5, and 1.93 is a good average figure.
In medium and heavy angle towers for the bracings to carry minimum possible loads it is
suggested that the base width and the slopes of the leg members may be adjusted in
such a manner that the legs when extended may preferably meet at the line of action of
the resultant loads. This reduces the forces in bracings to a large extent and a stronger
and more stable tower emerges.
(d) Tower widths at top hamper. Top hamper width is the width of the tower at
the level of the tower cross-arm in the case of barrel type of tower (in double circuits
it may be at the middle cross arm level). The width of top hamper is mainly decided by
torsion loading. The torsional stresses are evenly distributed on the four faces of the
square tower. The top hamper width is also decided in a manner that the angle
between the lower main member and the tie member of the same cross arm is not less
than 20 degrees as angle less than 20 degrees may introduce bending stresses in the
members. The top hamper width is found to be generally about 1/3 to 1/3.5 of the base
width.
(e) Type of bracing pattern
Several bracing patterns are adopted for towers. A few of them are shown in Fig. 3.3 below
and are as discussed under
(i) Single web system [Fig. 15 3 (a) and 3.3 (b)]
It comprises either diagonals and struts [Fig. 3.3 (a)] or all diagonals [ 3.3 (b)]. In diagonal
and strut system, struts are designed in compression while diagonals in tension, whereas in
system with all diagonals, the members are designed both for tension and compressive load
to permit reversal of the applied external shear. This system is particularly used for narrow
base towers.
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Analysis and Design of Transmission Towers
(ii) Double web or Warren system [Fig. 3.3 (c)]
This system is made up with diagonal cross bracings. Shear is equally distributed between
the two diagonals, one in compression and other in tension. Both diagonals are
designed for tension and compressive loads in order to permit reversal of externally
applied shears. The diagonal bracings are connected at their cross point. The tension
diagonal gives an effective support to the compression diagonal at the point of their
connections, and reduces the unsupported length of the bracings which results in lighter sizes
of bracing members. This system is used for both large and small towers.
(iii) Pratt system [Fig. 3.3 (d)]
Shear is carried entirely by one of the diagonal members under tension, the other
member is assumed to be redundant carrying no stress. Struts, i.e.. horizontal members in
compression are necessary at every panel to provide continuity to the bracing system.
Advantage of this system is that the sizes of diagonal members would be small because
these are designed for high slenderness ratio in order to make them act in tension. This
type of bracing results in larger deflection of tower under heavy loadings, because the
tension members are smaller in X-section than compression members would be for
similar loading. If such a tower is over loaded, the inactive diagonal will fail in
compression due to large deflection in the panel, although the active tension member
can very well take the tension loads. This system of bracing imparts torsional stresses in
leg members of the square based tower and also result in unequal shears at the top of
four
stubs
for
design
of
foundation.
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Analysis and Design of Transmission Towers
(iv) Portal system (Shear divided 50 : 50 between diagonals K system) Fig. 3.3 (e)
and (f)
The diagonals are designed for both tension and compression. It is stiffer than Pratt system
and has the advantage that the horizontal struts are supported at mid-length by the diagonals
and the same are exceedingly smaller than that in Pratt system. It is used when it is
desirable to provide clearance between the bottom legs of a tower. It has been found
advantageous to use the portal system for bottom panels, extensions and heavy river
crossing towers when rigidity is a prime consideration. If side-hill or corner extensions are
anticipated, the portal panel is particularly attractive due to versatility of its application.
(v) Modified System of Bracings (Fig. 3.3 (h) and 3.3 (g)]
In EHV towers, where torsional loads are of high magnitude, the top hamper width is kept
large to resist the torsional loads. Standard Warren system if used gives longer
unsupported length which increases the weight of the tower disproportionately. For such
system, modified bracing system is used. The advantage of this system is that the
unsupported lengths of leg members and bracings are reduced substantially thereby
increasing their strength and reducing their member sizes. Although there is an increase
in the number of bolts, fabrication and erection cost yet the above system gives overall
reduction in weight and cost of steel.
Bracings of type (a), (b) and (c) are used for small towers and type (ei) and (e2) are used
for tall towers. Type (e,) and (e2) provide greater head-room and are, therefore, provided
in panels next to the ground. A combination of various types of bracings is normally
used. Bracings (a), (b) or (c) may be used in the top portions of the tower and (e1) or (e2)
in the lower portions.
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Analysis and Design of Transmission Towers
3.2. LOADS
The various loads acting on transmission towers are
(i)
Vertical loads
(ii)
Transverse loads
(iii)
Longitudinal loads
(iv)
Thermal loads.
( i ) Vertical Loads; the vertical loads acting on a transmission tower are due to
(a) Dead weight of tower structure
(b) Weight of conductors, insulators, fittings
(c) Weight of linesman with tools (d) weight of ice coating.
The dead weight of the tower is assumed and then checked after the completion of
design. The weight may be assumed by comparison with similar existing towers or
from some empirical formulae, as given below
Where;
W = weight of tower in kN
C = Constant, with value ranging between 0.05 and 0.046
H = height of tower in meter
M = overturning moment at base in kN m
or
Where;
W = weight of tower in kN
C1= constant, varying from 0.043 to 0.065
1= maximum torque arm for longitudinal Load, m
H1 = height of centre of gravity of conductor loads above ground in meter.
Lt = total conductor transverse loads in kN
Ll = total conductor longitudinal loads in kN
Lr = total conductor vertical loads in kN
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Analysis and Design of Transmission Towers
The vertical load -due to conductors and ground wires is calculated on the weight span. The
weight span is the horizontal distance between the lowest points of the conductor on the
two spans adjacent to the tower. The lowest point is defined as the point at which the
tangent to the sag curve is horizontal. (Fig. 3.4)
A load of 1500 N is - taken as the weight of linesman with tools. An additional load of
3500 N is taken for the design of conductor(s) cross-arm only.
If the transmission line is subjected to snow load, ice loadings for conductors and ground
wires shall be calculated corresponding to a radial thickness of ice of 12 mm. No ice
loading is assumed for the tower body. The wind pressure is taken as 392 N/m2 on the
increased projected area of conductors and ground wires due to ice at the minimum
temperature.
(ii) Transverse Loads ; these are due to
(a) Wind or seismic load on conductors and ground wires
(b) Wind or seismic load on tower body
(c) Transverse components of cable tensions in case of angle
towers.
Wind load is more critical and most often controls the design of towers. The
seismic load is not critical as the mass of the structure is not heavy and it is near
the base. As wind pressure is the chief criterion for the design of transmission line
towers, IS : 802-1977 (Part I) has specified design pressures which are different
from those for general structures. The revised code on loading [i.e.. IS : 875 (Part
3)-1978] specifies that for the design of overhead transmission line towers, the
specific requirements of IS 802-1977 (Part I) should be used in conjunction with
the provisions of this code, as far as applicable. The transmission line towers are
designed to withstand maximum wind pressure including winds of short duration
as in squalls.
On the basis of measured maximum wind velocities for different parts of the country
including winds of short duration as in squall, the country. has been divided into
three zones of low, medium and heavy wind pressure, as shown in Fig. 3.5.
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Analysis and Design of Transmission Towers
(a) Wind Pressure Loads
The wind pressures on towers and conductors shall he as given in Table 3.1 and
3.2 and shall be assumed as acting horizontally.
In the case of towers the wind pressures shall be calculated on 1.5 times the
projected area of the members on the windward face. In the case of conductors and
ground wires the pressures given in "fable 3.2 shall be assumed as acting on the
full projected area.
Table 3.1. Wind pressures on Towers
Pressure in N/m2 (kgf / m2) on Towers and Supports at a Height
Intensity of
Pressure
Up to 30 m
above mean
30-35 m
35-40 m
4045 m surface
Light •
retarding
1270(130)
1320(135)
1340(137)
1370(140)
Medium
1910(195)
1990(203)
2020(206)
2060(210)
Heavy
2550(260)
2640)270)
2680(274)
2740(280)
Table 3.2. Wind Pressure on Conductors and Ground Wires
Intensity of Pressure
Maximum Wind Pressure N/m 2 (kgf/m2)
Light
420(43)
Medium
Heavy
440(45)
510(52)
The wind pressure values given in Fig. 3.5 and Table 3.1 and 3.2 are based on maximum
wind pressure likely to be experienced over different parts of the country, within a
height of about 30 m above mean retarding surface, irrespective of the height of the
place above the mean sea level. The altitude of the country traversed may, therefore,
be ignored in so far as the maximum wind pressure on towers, conductors, and ground
wires are concerned.
For the purpose of computing the wind load on bundle conductors (more than one
conductor per phase) wind pressure given in Table 3.2 shall be assumed as acting on
full projected area of each conductor in a bundle.
For the purpose of' computating the wind pressure on insulator strings, the effective
projected area of the string shall be assumed as 50 percent of the projected area of the
cylinder with a diameter equal to that of the insulators skirt. The pressure shall be
calculated as for tower members.
23
24
Analysis and Design of Transmission Towers
The transverse load due to wind on conductors and ground wires is calculated on the wind
span. The wind span is the sum of the two half spans adjacent to the support under
consideration (Fig. 3.4). Under broken-wire conditions, 50 percent of the intact span and 10
percent of the broken span shall be assumed as the wind span.
In angle towers, the transverse components of cable tension produce transverse loading on
the tower. This loading is greater for bigger line deviations i.e. for large angle towers.
(iii) Longitudinal loads. The longitudinal loads on a tower are due to,
(a) unbalanced pull due to broken wire condition
(b) seismic load on wires and tower
(c) Pull of conductors and ground wires in case of dead-end tower.
Longitudinal loads are caused by broken wire conditions. The unbalanced pull due to broken
conductors in the case of supports with suspension strings. may he assumed as equal to 50
percent of the maximum working tension of the conductor.
For bundle conductors, the pull due to broken conductors in the case of supports with
suspension strings, may be assumed as equal to 25 percent of the maximum working tension
of all the sub-conductors in one bundle. For the ground wire broken condition, 100 percent
of the maximum working tension shall be considered for the purpose of design of tower.
The unbalanced pull due to broken conductor or ground wire in the base of tension
strings, shall be equal to the component of the maximum working tension of the
conductor or the ground wire as the case may be, in the longitudinal direction along
with its components in the transverse direction. This will be taken for the maximum
as well as the minimum angle of the deviation for which the tower is designed and the
condition which is most stringent for a member shall be adopted.
When there is a possibility of the tower being used with a longer span by reducing
the angle of line deviation, the tower member shall also be checked for longitudinal
and transverse components arising out of the reduced angle of line deviations. The
broken-wire conditions (15-3) may be assumed in the design of towers
Dead-end towers are designed for longitudinal loads due to tension in all conductors and
ground wires.
The seismic loads may be considered in the design of towers in regions where earth quakes are experienced frequently. Specific provisions of earthquake forces have not
been specified in IS : 802, (Part I)-1977. The general code on earthquake IS 1893-1984
may be followed for the design of transmission towers.
(IV) Thermal Loads. These loads are due to temperature variations suns radiation and
heating due to current in the conductor.
The temperature range varies for different regions and under different diurnal and
seasonal conditions. The absolute maximum and minimum temperature which may be
expected in different localities in the country are indicated in National Climate Charts of
Temperature Variations. These figures may be used for guidance in assessing the
maximum variations of temperature.
The temperatures indicated in Fig. 3.6 and 3.7 are the air temperatures in the shade. The
range of variation in temperature of the building materials may be appreciably greater or
24
25
Analysis and Design of Transmission Towers
Table (3.3) Broken-Wire Conditions
For Lines With Single Conductor
For Lines With Bundle Conductor
(a) Single Circuit Towers
I. Tangent towers with
suspension string (0° to 2°)
Any one power conductor broken
or one ground-wire broken
whichever is more stringent for a
particular member.
Any ground-wire or one
sub-conductor from any bundle
conductor broken, whichever is
more stringent for a particular
member. The unbalanced pull due
to sub-conductor broken may be
taken as specified above.
2. Small angle tension towers
Any ground-wire broken or all
(2° to 15°)
sub-conductors in the bundle
broken whichever is more
Any one power conductor broken
stringent for a particular
member 3. Medium angle tension towers
or one ground-wire broken;
(15° to 30°)
4. Large angle tension (30° to 60°)
and dead end towers
whichever is more stringent for a
particular member
(b) Double Circuit Towers
I . Tangent tower with suspension
strings (0° to 2°)
Any one power-conductor broken
or one ground-wire broken
whichever is more stringent for a
particular member
2. Small angle towers with tension Any two of the power-conductors
strings (2° to 15°)
3. Medium angle towers with
tension strings (IS° to 30°)
4. Large angle (30° to 60°) and
-
broken on the same side and on the
same span or any one of the
power-conductors and any one
ground-wire broken on the same span
whichever combination is more
stringent for a particular member
Three power-conductors broken
dead end towers with tension
on the same side and on the same
strings
span or any two of the powerconductors and any one ground wire
broken on the same span,
whichever
combination
constitutes the most stringent
condition for a particular member
(c) Cross Arms
-
In all types of towers, the powerconductor supports and ground wire supports
-
-
shall be designed for the broken-wire conditions also.
less than the variation of air temperature and is influenced by the condition of exposure and
the rate at which the materials composing the structure absorb or radiate heat. This
difference in temperature variations of the material and air should be given due
consideration. The absolute maximum temperatures given in Fig. 3.6 shall be increased by
17 °C to allow for Sun's radiation, heating effect of current etc in the conductor.
25
26
Analysis and Design of Transmission Towers
3.3. ANALYSIS OF TOWERS
Overhead transmission line tower is a high order indeterminate cantilever space truss and its
analysis as a space truss is possible only with the help of a computer. The conventional
method of analyzing a tower is by resolving it into plane trusses. While analyzing the tower
as plane trusses, the loads are applied at joints and the members are designed as ties or
struts. As the legs of most towers are sloping to the vertical, the sides of the tower are not
in one plane. Thus, the solution of a tower by resolving it into planar trusses is no more than a
good approximation.
The tower can be subjected to forces acting in three different directions.
(a) Force is parallel to two trusses
(b) Force is inclined to the trusses
(c) Force is acting at a distance from the tower axis.
The various loading cases are shown in Fig. 3.8. In Fig. 3.8 ( a ) , the load is parallel to two
trusses and passes through the axis of the tower. This load is normally inclined to the
horizontal. Let the horizontal component of the load P be P,, and vertical component be P,..
The vertical load P,, is shared equally by all the four legs A, B, C and D, Where as the
horizontal load P,, is shared by trusses 1 and 3.
In the second case, Fig. 3.8 (b), the load P is acting at an angle 8 to the plane of
trusses 1 and 3. The load P is also inclined to the horizontal. Let its vertical
component be P Z and horizontal component be P,,. The vertical component P, will be
distributed equally among the four legs A, B, C and D.
The horizontal component P h is further resolved into two components i.e. P,,, parallel to
trusses I and 3, and P,,2 parallel to trusses 2 and 4. Load Ph1 is shared equally by
trusses 1 and 3 and load Ph2 is shared equally by trusses 1 and 3 and load P,,2 is shared
equally by, trusses 2 and 4.
In Fig. 3.8 (c), the load is acting at a distance `e' from the tower axis. The load may be
inclined at an angle 0 to the plane of trusses I and 3. It is also inclined to the
horizontal. Let the vertical component be P, and horizontal component be P h. The
horizontal component is further resolved into P,,, and Ph2 parallel to trusses I and 3, and
2 and 4. The vertical load P,, causes a moment, M = P,, • e which will be resisted
equally by trusses 2 and 4. Besides, the vertical load P,, will be shared equally by all the
four legs. The load P,, 2 is shared equally by the trusses 2 and 4. The force P,,,
produces torsion in the tower, equal to M, = P hI • e. If the cross-bracing, which is provided
26
27
Analysis and Design of Transmission Towers
at this level, is rigid, then the torsion M, will be resisted equally by the two sets of
trusses I and 3, and 2 and 4. The torsion will produce forces Pt1 and P t2 in the trusses I
and 3, and 2 and 4 respectively, or
Trusses 1 and 3 also resist the load P h1, equally. The trusses are analyzed separately for
various loading conditions after resolving the forces. The forces are then tabulated and
members are designed for the worst conditions. In case the trusses are statically
determinate, the stresses are found by analytical or graphical methods. In case crossed
diagonals and horizontals are provided, the tension brace is designed to take all the force or
it is divided equally between the two diagonals.
3.4. DESIGN OF TOWERS
The Indian Code IS : 802 (Part T)-1977 has specified the factor of safety to be adopted for
design permissible stresses and the slenderness ratios. The factor of safety in the design of
structural member of steel transmission line towers may be assumed as 2.0 under normal
conditions and 3.5 under broken wire conditions.
(a) Permissible Axial Stresses in Tension.
The estimated tensile stresses on the net effective sections area in various members,
multiplied by the appropriate factor of safety shall not exceed minimum guaranteed yield
stress of the material.
For steel conforming to IS : 226-1975, the permissible axial stress shall not exceed 255 N/m2
(2600 kgf/cm2).
(b) Permissible Axial Stresses in Compression
The estimated compressive stresses in various members multiplied by the appropriate factor of
safety shall not exceed the value give by the formulae below.
The allowable unit stress on the gross section of the axially loaded compression member shall
be:
Where;
Fa = allowable unit stress in compression,
F,. = minimum guaranteed yield stress of the material,
27
28
Analysis and Design of Transmission Towers
K =restraint factor,
L = length of the compression member
E = modulus of elasticity of steel that is 200000 N/mm2 (2047000 kgf/cm2), and
KL = largest effective slenderness ratio of any unbraced segment of the member.
These formulae are applicable provided the largest width-thickness ratio bit is not more
than the limiting value given by
Or
where;
b = distance from edge of fillet to the extreme fibre, and
t = thickness of material
For steel conforming to IS .: 226-1975 the formulae given above will reduce to the
following provided the width-thickness ratio does not exceed 13
Where the width-thickness ratio exceeds
formulae given in (a) and (b) shall be used
28
29
Analysis and Design of Transmission Towers
substituting for Fy the value Fcr, given by
For steel conforming to IS : 226-1975 the formulae given in (e) and (f) above will reduce to
the following:
Stress in Bolts. The estimated stresses in the bolts multiplied by the appropriate factor
of
safety
shall not exceed the value given in Table 3.4.
Table (3.4) Permissible stress in bolts
Nature of Stress (1)
1. Shear
2. Bearing
Shear stress on gross area
of bolts shear the area to be
assumed shall be twice
the area defined
Bearing stress on gross
diameter of bolts
Permissible
Stress (2)
N/mm2
(kgf/cm2)
218
(2220)
436
(4440)
Remarks (3)
For gross area of
bolts. For bolts in
double
For the bolt area in
bearing 3. Tension
29
30
Analysis and Design of Transmission Towers
3.
Tension
Axial tension stress on the
root area of the thread of
bolt
194
(1980)
-
3.5. SLENDERNESS RATIOS
(a) Compression Members
The slenderness ratios of compression members shall be determined as follows
Type of Members
Table (15-5)
Value of KL/r
(a) Leg sections or joint members L/r
bolted at connections in both faces (curves 1 and 4 of Fig. (3.9)
(b) Members with concentric loading L/r
at both ends of the unsupported
panel with value of (L/r) up to and
including 120 (curve I of Fig. 3.9)
(c) Member with concentric loading at 30 +0.75 L/r
one end and normal eccentricities
at the other end of the
unsupported panel with values of
L/r up to and including 120 (curve
2 of Fig 3.9)
(d) Members with normal framing 60 + 0.50 L/r
30
31
Analysis and Design of Transmission Towers
eccentricities at both ends of the
unsupported panel for value of L,
upto and including 120 (curve 3 of
Fig. 3.9)
(e) Member
unrestrained
against L/r
rotation at both ends of the
unsupported panel for values of L/r
from 120 to 200 (curve 4 of Fig.
3.9)
(f) Members partially restrained 28.6 + 0.762 L/r
against rotation at one end of the
unsupported panel for Values of
L/r over 120 up to and including
225 (curve 5 of Fig. 3.9)
(g) Members partially restrained 46.2 + 0.615 L/r
against rotation at both ends of
the unsupported panel for values
of L over 120 up to and including
250 (curve 6 of Fig. 3.9)
A single bolt connection shall not be considered as offering restraint against rotation. A
multiple bolt connection properly detailed to minimize eccentricities shall be considered to
offer partial restraint if connection is made to a member having adequate flexural strength
to resist rotation of the joint. Points of intermediate support shall not be considered as
offering restraint to rotation unless they meet the criteria outlined above.
In the design of members, the length L shall be from centre to centre of intersection at each
end of the member.
Table (15-6 )
limiting values of L/r
31
32
Analysis and Design of Transmission Towers
Leg members and lower members of the
cross-arms in compression
Other members carrying computed stresses
Redundant members and those carrying
nominal stresses
150
200
250
Table (3.7) gives for ready reference, the values of allowable unit stresses in N/mm2 (kg
f/cm2) for
L/r ratios of compression members of the types listed above for steel conforming to IS :
226-1975.
3.6. Connections
The angle between any two members common to a joint of a trussed frame shall preferably
be greater than 20° and never less than l5°, due to uncertainty of stress distribution between
two closely spaced members.
(b) (B)Tension Member
The slenderness ratio of a member carrying axial tension only, shall not exceed 375.
Table (15-7)
Allowable unit stresses Fac in N/mm2 (kgf/cm2)
for measured slenderness ratios (L/r) of steel with yield stress Fy= 255 N/mm2 (2600
kgf/cm2)
L/r
fac (N/mm2)
n=2
0
127.5
10
127.1
20
125.9
30
123.8
40
120.9
50
117.2
60
112.7
70
107.3
80
101.1
90
94.1
100
86.3
where n = Factor of safety
L/r
n=1.5
170.0
169.5
167.8
165.0
161.2
156.3
150.2
143.1
134.8
125.5
13.1
fac (N/mm 2 )
n=2
110
120
130
140
150
160
170
180
190
200
77.7
68.2
58.0
50.0 _
43.6
38.3
33.9
30.2
27.1
24.5
n=1.5
103.5
90.7
77.3
66.7
58.1
51.0
45.2
40.3
36.2
32.7
3.7. MINIMUM THICKNESS
Minimum Thickness of galvanized and painted tower members shall be as follows :
32
33
Analysis and Design of Transmission Towers
Minimum Thickness, mm
Galvanized
Painted
Leg members and lower members of cross-arms in compression
5
6
Other members
4
5
3.8. BOLTING
(a) Minimum Diameter of Bolts. The diameter of bolts shall not be less than 12 mm.
(b) Preferred Sizes of Bolts. Bolts used for the erection of transmission line towers shall
be of 3 diameters 12, 16 and 20 mm.
(c) The length of bolts shall be such that the threaded portion does not lie in the plane of
contact of 3 members.
(d) Gross area of Bolts for purposes of calculating the shear stress the gross area of bolts
shall be taken as the nominal area of the bolt.
(e) The bolt area for bearing shall be taken as d x t where d is the diameter of bolt and t
the thickness of the thinner of the parts joined.
(f) The net area of a bolt in tension shall be taken as the area at the root of the thread.
(g) Holes for Bolting. The diameter of the hole drilled or punched shall not be more than
the nominal diameter of the bolt plus 1.5 mm.
3.9. GENERAL INTRODUCTION TO STIFFNESS METHOD
This method of analyzing structures is probably(14) used more widely than the flexibility
method, especially for large and complex structures (with multiple nodes). Such structures
require the use of electronic computers for carrying out the extensive numerical calculations,
and the stiffness method is much more suitable for computer programming than the flexibility
method!
The reason is that the stiffness method can be put into the form of a standardized procedure
which dose not requires any engineering decisions during the calculation process. And also
the unknown quantities in the stiffness method are prescribed more clearly than the flexibility
method.
When analyzing a structure by the stiffness method, normally we use the concepts of
kinematic indeterminacy, fixed-end reactions, and stiffnesses. These definitions will be
explained as follows:
3.9.1 KINEMATIC INDETERMINACY
In stiffness method the unknown quantities in the analysis are the joint displacements of the
structure, rather than the redundant reactions and stress resultants as is the case of flexibility
method. The Joints in any structure will be define as points where two or more members
intersect, the points of support, and the free ends of any projecting members.
When the structure is subjected to loads, all or some of the joints will undergo displacements
in the form of translations and rotations. Of course, some of the joints displacements will be
33
34
Analysis and Design of Transmission Towers
zero because of the restraint conditions; for instance, at a fixed support there will be no
displacements of any kind.
The unknown joint displacements are called kinematic unknowns and their number is called
either the degree of kinematic indeterminacy or the number of degrees of freedom (DOF) for
joint displacements.
3.9.2 FIXED-END ACTIONS
In stiffness method we regulatory encounter fixed-end beam, because one of the first steps in
this method is to restrain all of the unknown joint displacements. The imposition of such
restrains causes a continuous beam or plane frame to become an assemblage of fixed-end
beams. Therefore, we need to have readily available a collection of formulas for the reactions
of fixed-end beams for multiple case. These reactions which consist of both; forces and
couples (moments), are known collectively as Fixed-End actions. Values of fixed-end actions
for multiple cases are shown in Appendix I.
3.9.3 STIFFNESSES
In the stiffness method we make use of actions caused by unit displacement. These
displacement may be either unit translation (or unit rotation for in-plane frame), and the
resulting actions are either forces of couples (moments). These actions caused by unit
displacement are known as stiffness influence coefficients, or stiffnesses. These coefficients
called also member stiffnesses which they are frequently used in this method. Here by two of
the most useful cases as shown in fig. (1.2).
Note: all basic relations of stiffness matrix will be presented in chapter four as part of
computer program development.
3.9.4. GENERAL EQUATION OF STIFFNESS METHOD
Now most of the preliminary ideas and definitions have been set fourth, and the problem of
analyzing a structure can be established. Interpreting of Equilibrium Equations, and making
use of the Principles of Superposition, for the case of a structure having (n x n) Degrees of
Kinematic Indeterminacy will lead to the following sets of linear equations are obtained:
𝑆11 𝐷1
𝑆21 𝐷1
:
𝑆𝑛1 𝐷1
+ 𝑆12 𝐷2
+ 𝑆22 𝐷2
:
+ 𝑆𝑛2 𝐷2
+ 𝑆13 𝐷3 … … . + 𝑆1𝑛 𝐷𝑛
+ 𝑆23 𝐷3 … … . + 𝑆2𝑛 𝐷𝑛
:
:
:
+ 𝑆𝑛3 𝐷3… … . + 𝑆𝑛𝑛 𝐷𝑛
+ 𝐴1
+ 𝐴2
:
+ 𝐴𝑛
= 𝑃1
= 𝑃2
: ……………….Eq. (1.1)
= 𝑃𝑛
This can be reduced to General Equation form:
[𝑘]|∆| = |𝑝|…………..Eq. (1.2)
Hence, the principles of superposition are used in developing fixed-end actions (forces),
therefore, this method is limited to linearly elastic structures with small displacements. The n
equations can be solved for the n unknown joint displacement of the structure.
The important fact which need to be established: that Equilibrium Equations of the Stiffness
Method express the superposition of actions (forces) corresponding to unknown
displacements. While the compatibility equations of the Flexibility Method express the
superposition of displacements corresponding unknown actions (forces).
34
35
Analysis and Design of Transmission Towers
Also; it should be noticed that above equilibrium equations (1.1) are written in a form which
takes into account only the effects of applied loads on the structure, but the equation can be
readily modified to include the effects of temperature changes, restrains, and support
settlements. It is only necessary to include these effects in the determination of the actions
(forces) A1, A2,…, An. Furthermore, Eq. (1.2) apply to many types of structures, including
trusses and space frames, although in this project is limited to in-plane structure (beams), and
hence the stiffness method is applicable only to linearly elastic structures.
3.9.5. STIFFNESS METHOD VERSUS FINITE ELEMENT METHOD
(FEM)
Stiffness method can be used to analyze structures only, finite element analysis, which
originated as an extension of matrix (stiffness and flexibility), it is detected to analyze surface
structures (e. g. plates and shells). FEM has now developed to the extent that it can be applied
to structures and solids of practically any shape or form. From theoretical viewpoint, the basic
difference between the two is that, in stiffness method, the member force-displacement
relationships are based on the exact solutions of the underlying differential equations, whereas
in FEM, such relations are generally derived by Work-Energy Principles from assumed
displacement or stress functions.
Because of the approximate nature of its force-displacements relations, FEM analysis yield
approximate results for small node numbers. However, FEM is always more accurate than
stiffness matrix especially in nonlinear analysis.
35
36
Analysis and Design of Transmission Towers
U11= 1.0
K31= - EA/L
K11 = +EA/L
K21 = 0.0
K41 = 0.0
L
L = L’
U44= 1.0
K14= 0.0
K34 = 0.0
K24 = 0.0
K44 = 0.0
L
U33= 1.0
K13= -EA/L
K33= +EA/L
K23= 0
K43= 0
L
L = L’
U22= 1.0
K12= 0.0
K32= 0.0
K22 = 0
K42 = 0.0
L
Fig.(1.2) Axial Member Stiffnesses.
36
37
Analysis and Design of Transmission Towers
Chapter Four
Computer Program
37
38
Analysis and Design of Transmission Towers
4.1 INTRODUCTION
This chapter presents a brief description of the computer program developed in this study
which governs the problem of the Analysis and Design of Transmission Towers.
4.1 PROGRAM PROCEDURE
Based on theoretical equations presented in chapter three, the following step-by-step
procedure for the analysis and design of In-plane structures (Truss) using Stiffness Matrix
Method will be presented;
The sign convention used in this analysis is as follow: the joint translations are considered
positive when they act in positive direction of Y-axis, and joint rotations are considered
positive when they rotate in counterclockwise direction:
Prepare the analytical model of in-plane structure, as follows:
1. Draw a line diagram of the in-plane structure (beam), and identify each joint member
by a number.
2. Determine the origin of the global (X-Y) coordinate system (G.C.S.). It is usually
located to the lower left joint, with the X and Y axes oriented in the horizontal
(positive to the right) and vertical (positive upward) directions, respectively.
3. For each member, establish a local (x-y) coordinate system (L.C.S), with the left end
(beginning) of the member, and the x and y axes oriented in the horizontal (positive
to the right) and vertical (positive upward) directions, respectively.
4. Number the degrees of freedom and restrained coordinates of the beam elements and
nodes.
5. Assume an initial section properties; such like (Ag, Ix, Ex,…)
6. Evaluate the Overall Stiffness Matrix [k]. The number of rows & columns of [S] must
be equal to the number of DOF of the structure. For each element of the in-plane
structure, perform the following operations:
a) Compute the Element stiffness matrix [ke] in (L.C.S) by apply the basic stiffness
equation, as follow:
{𝑓} = [𝑘 𝑒 ]{𝑒}.
b) Transform the force vector {𝑓} form (L.C.S) to {𝑃} in (G.C.S.) using
transformation matrix [A], as follow:
{𝑃} = [𝐴]{𝑓}.
c) Transform the deformation vector {𝑒}form (L.C.S) to
transformation matrix [B], as follow:
{𝑋} (G.C.S.) using
{𝑒} = [𝐵]{𝑋}.
d) It is evident that matrix [B] is the transpose of matrix [A], therefore ;
{𝑒} = [𝐴]𝑇 {𝑋}.
38
39
Analysis and Design of Transmission Towers
e) Substituting step (d) in step (a), resulting in:
{𝑓} = [𝑘 𝑒 ][𝐴]𝑇 {𝑋}.
f) Substituting step (e) in step (b), resulting in:
{𝑃} = [𝐴][𝑘 𝑒 ][𝐴]𝑇 {𝑋}.
g) Inverting equation in step (f), resulting in:
{𝑋} = [[𝐴][𝑘 𝑒 ][𝐴]𝑇 ]−1 {𝑃}.
h) Store the element stiffness matrix, in (G.C.S.), [𝑘 𝑒 ] = [[𝐴][𝑘 𝑒 ][𝐴]𝑇 ]−1 , for each
element.
7. Assemble Overall Stiffness Matrix [K] for the System of in-plane structure. By
assembling the element stiffness matrices for each element in the in-plane structure,
using their proper positions in the in-plane structure Stiffness Matrix [K], and it must
be symmetric.
8. Compute the Joint load vector {Pj} for each joint of the in-plane structure.
9. Added the Fixed-Ends (lateral loads) forces Vector {Pf} to their corresponding Joint
load vector {P} using their proper positions in the in-plane structure Stiffness Matrix
[K].
10. Determine the structure joint displacements {X}. Substitute {Pj} and [K] into the
structure stiffness relations, {𝑃𝑗 } = [𝐾]{𝑋} and solve the resulting system of
simultaneous equations for the unknown joint displacements {X}.
11. Compute Element end displacement {e} and end forces {f}, and support reactions for
each Element of the beam, as following:
12. Obtain Element end displacements {e} form the joint displacements {X}, using the
Element code numbers.
13. Compute Element end forces {f}, using the following relationship:
{𝑓} = [𝑘 𝑒 ]{𝑒}
14.
15.
16.
17.
18.
Compute element internal Stresses
Compare the computed internal stresses with allowable stresses given by AISC-ASD.
If its check then ok.
Otherwise; select a larger (but economical) from AISC design manual
Repeat steps form (6) to step (15), until all members check according to ASD design
criteria.
19. Using the Element code numbers, store the pertinent elements of {f}, in their proper
position in the Support Reaction Vector {R}
20. Check the calculation of the member end-forces and support reactions by applying
the Equation of Equilibrium to the free body of the entire in-plane structure;
∑𝑛𝑖=0 Fy = 0,
∑𝑛𝑖=0 Fx = 0.
39
40
Analysis and Design of Transmission Towers
4.2 Computer program application;
4.2.1. Case Study:
Analysis and design a 220 kV transmission tower having a height of 94.0ft (28.65 m) and
with a base width of 22.0ft (6.7 m), as shown in Fig. (4.1).
40
41
Analysis and Design of Transmission Towers
8’
6’
6’
6’
6’
6’
4’
12’
6’
10’
10’
6’
8’
4’.6”
3’.6”
3’
3’
3’.6”
4’.6”
Fig.(4.1), Analytical Model of Transmission Tower
Solution;
The structure has been analyzed as a space truss using stiffness method. A computer
program has been used for analyzing the following loading conditions;
(a) Normal condition (NCL)
(b) Top conductor broken (TCB)
41
42
Analysis and Design of Transmission Towers
(c) Lower conductor broken (LCB)
(d) Middle wire broken (MWB)
(e) Ground wire broken (GWB)
For the analysis of the structure, the joints and the members have been numbered, as shown
in fig.(4.1). The loading diagrams for the presented case study are shown in Fig. (4.1).
Fig.(4.2), Loading Conditions of Transmission Tower
The details of analysis have not been given as these are available in reference on analysis of
structures. Only the resultant forces for the worst condition have been tabulated for each
group of members. For case in designing, the members have been grouped as follows
Grouping of Members
Members locations
Major members (Legs)
Minor members (Legs)
Major diagonals (Lacing)
Minor diagonals (Lacing)
Other members
Section Designation (HRS)
L 5 x 5 x 8/16
L 3,1/2 x 3.1/2 x 8/16
L 2,1/2 x 2,1/2 x 8/16
L 2 x 2 x 6/16
L 1,1/2 x 1,1/2, x 4/16
Location Symbol
B1, B2, and B3
TS1, TS2, and TS3
B1, B2, and B3
TS1, TS2, and TS3
Everywhere.
42
Analysis and Design of Transmission Towers
96
88
80
72
64
56
Tower height (ft)
43
48
40
32
24
load case one
16
load case two
load case three
load case four
8
load case five
0
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
Vertical Displacements (in)
Fig.(4.2) Vertical Displacement Diagram
43
Analysis and Design of Transmission Towers
96
88
80
72
64
56
Tower height (ft)
44
48
40
32
24
load case one
load case two
16
load case three
load case four
8
load case five
0
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Horizontal Displacements (in)
Fig.(4.3) Horizontal Displacement Diagram
44
4.5
Analysis and Design of Transmission Towers
96
88
80
72
64
56
Tower height (ft)
45
48
40
32
24
load case one
16
load case two
load case three
load case four
8
load case five
0
0
10
20
30
40
50
60
70
Compressive Forces (kip)
Fig.(4.4) Compressive Forces Distribution
45
80
Analysis and Design of Transmission Towers
96
88
80
72
64
56
Tower height (ft)
46
48
40
32
24
16
load case one
load case two
load case three
8
load case four
load case five
0
0
-10
-20
-30
-40
-50
-60
Tensile Forces (kip)
Fig.(4.5) Tensile Forces Distribution
46
Analysis and Design of Transmission Towers
96
88
load case one
80
load case two
load case three
load case four
72
load case five
Allowable Compressive Stress
Allowable Tensile Stress
64
56
Tower height (ft)
47
48
40
32
24
16
8
0
25
20
15
10
5
0
-5
-10
-15
-20
Axial Stresses (ksi)
Fig.(4.6) Actual Axial Stresses Versus Allowable Stress
47
-25
48
Analysis and Design of Transmission Towers
CHAPTER FIVE
CONCLUSIONS AND RECOMMENDATIONS
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49
Analysis and Design of Transmission Towers
CONCLUSIONS
Depending on the results obtained from the present study, several conclusions could be
summarized as follows:
Results indicate that in-plane structures (Steel Transmission Towers) can be can be dealt with
successfully by the Stiffness Matrix Method.
Developed Program in this study is quite efficient and reliable for both analysis and design.
Design process developed in this study is quit forward and easy to implement which depends
on the design criteria given by AISC-89 design manual (Allowable Stress Design) and for I.S.
1977(part I) for wind loading specifications.
Five load-combinations (broken wires conditions) are investigated and results indicate the
following:
1. Fig.(4.6), indicate that compressive stress is inversely proportional to slenderness
ratio (KL/rx, rx=√(Ix/Ax)), Therefore; in order to reduce members internal axial stress
a larger radius of gyration and lower member length should be used.
2. Fig.(4.6), indicate that the second load-combination represent the worst case scenario
where the largest values of tensile and compressive stresses could be seen. This might
be justified because of higher arm of the horizontal forces which will produce a
higher rotational overturning moment over the tower base supports (tension and
compression reactions).
3. Fig.(4.4 and 4.5), indicate that compression and tension forces are gradually increased
form top to down of tower length, as it expected, therefore; an increased members
cross-sectional area of major members (Legs and arms) will be an efficient method to
reduce internal stresses.
4. Fig.(4.2 and 4.3), indicate that Horizontal displacement are directly proportional to
the horizontal wind loads while the vertical displacement are not directly proportional
to vertical cable (gravity) loads.
Presented results indicate also:
In order to overcome member Critical Compressive Stress Case a lower member length
(more refined tower geometry) could be used which will reduce the slenderness ratio (KL/rx)
and eventually the compressive stress.
A second solution is to increased member cross-sectional area which will increase the radius
of gyration and eventually the compressive stress. But it is not recommended since it is not
economical solution.
RECOMMENDATIONS
The analysis method, presented in this study for in-plane structures, could be extended to
include the following factors:






More revised space (three-dimensional) analysis to include torsional forces.
Differential settlement of tower base-foundation.
Member’s joint-connections (bolted, welded, or spliced).
Shear calculation (resistance) especially at tower main members.
Using of gust-plate at members-joints.
Optimization process for selecting economical members.
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50
Analysis and Design of Transmission Towers

Making a parametric study for comparing towers with different geometry
(shapes) to find best geometry for certain loading-case.
REFERENCES
1. Syal, I. C., and Satinder S., "Design of steel structures.", Standard Publishers
Distributers, Delhi, 2000.
2. Dayaramtnam. P., "Design of steel structures.", Chand S. Company ltd. for
publishing , New-Delhi, 2003.
3. Manual of Steel Construction (AISC-1989, Allowable Stress Design), ninth edition.
4. AMERICAN NATIONAL STANDARD SJI-JG–1.1, SECTION 1001. Adopted by
the Steel Joist Institute November 4, 1985 ( Revised to November 10, 2003 Effective March 01, 2005).
5. Asalam Kassimali, “Matrix Analysis of Structures”, Brooks/ Cole Publishing
Company, 1999.
6. Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks."
Manchester University Press, Manchester, 1956.
7. Livesley, R.K., "The Application of an Electronic Digital Computer to Some Problem
of Structural Analysis." The Structural Engineer, Vol. 34, no.1, London, 1956, PP. 112.
8. Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis."
Pergamon Press, London, 1964, PP. 115-145.
9. Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London,
1964. PP. 241-252.
10. Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering."
McGraw-Hill Book Co., New York, 1974, pp. 190-210.
11. Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York,
1986, Fourth Edition, pp. 380-230.
12. Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No. 6, Nov.-Dec. 1986, pp. 10101017.
13. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition,
McGraw-Hill Book Company, New York, 1961, pp. 1-17.
14. Timoshenko, S.P. and Gere, J.M., "Mechanics of Materials." 2nd Edition, Von
Nostrand Reinhold Book Company, England, 1978.
15. KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of
Structural Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
16. LAZEM, A. N., "Large Displacement Elastic Stability of Elastic Framed Structures
Resting On Elastic Foundation" M.Sc. Thesis, University of Technology, Baghdad,
2003, pp. 42-123.
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