CHAPTER 4 STEEL LATTICE TOWERS

Transcription

1 70 CHAPTER 4 STEEL LATTICE TOWERS 4.1 TRANSMISSION LINE TOWERS In the power industry, steel lattice towers are commonly used for transmission of power through electrical conductors from the place of power generation to the place of distribution. The transmission line towers support electrical power conductors and ground-wires at suitable height above ground to satisfy certain functional requirements. It is reported that transmission line towers contribute to about 35-45% of the total cost of a transmission line. Hence optimisation of tower design can therefore result in substantial economy. Great responsibility thus rests on the design engineer who has to prepare not only economical, but also safe and reliable design. Structurally the tower should be adequate to resist loads such as wind load, snow load and self-weight Specification of Transmission Line Towers Transmission line towers are generally specified by voltage, number of circuits and type. Thus, these parameters become the basic parameters, which govern structural design of the tower.

2 71 The voltage classification of transmission line towers is according to the voltage of the line it carries. The common voltages used in India for power transmission are 110 kv, 220/230 kv and 440 kv. The configurations adopted are generally rectangular and square types. The square type broad based towers are the most commonly used. The number of circuits the tower can carry is either single, double or multi circuit. The number of earth wires, right of way, etc. also affect the configuration of the tower. Along the transmission line route, depending upon the profile along the centre line of the transmission line, towers are classified into three categories such as tangent tower, angle tower and dead end tower. Further, transmission line towers are also classified according to their shape as Barrel, Corset and Guyed towers. The Barrel type towers are considered in this study for optimisation as the generation and geometrical data are modular based. The functional requirements such as minimum ground clearance, and clearance between conductor and tower body, are governed by the electrical regulations and they mainly depend on the voltage carried by the conductor. The number of circuits decides the number of cross arms on the tower. Parameters such as number of cross arms, vertical spacing between cross arms, height of ground-wire peak, minimum ground clearance, maximum sag and other clearances decide the overall height of the tower. The staging of transmission line tower should be high enough to provide minimum ground clearance under maximum sag condition. As transmission line towers have components such as a number of cross arms and ground-wire peaks, the staging below the bottom cross arm is more useful for optimisation than the portion above.

3 Transmission Line Tower Configuration Typical barrel type and corset type transmission line tower configurations are shown in Figure 4.1. Choosing a preliminary configuration is pre-requisite for detailed analysis and design of a transmission line tower and this is to be chosen based on functional and structural requirements. The geometric parameters of transmission line tower configuration are height of the tower, base width of the tower, top-hamper width, length and depth of crossarm. Some of the parameters governing the geometry of a tower are shown in Figure 4.2. Approximate structural behaviour of the tower or conventional practice is taken as the basis for fixing these parameters of the tower. Sag tension and clearances also play an important role in deciding the configuration Tower Configuration Parameters For optimisation of transmission line towers, it is important to know various design parameters that control the design of the tower. Some of the parameters that dictate the configuration of the transmission line towers are briefly described below: Tower Height: The height of the tower is determined by parameters such as number of cross arms, vertical spacing between cross arms, height of ground-wire peak, minimum ground clearance, maximum sag and other clearances. The cost of the tower increases with the height of the tower. Hence, it is desirable to keep the tower height minimum to the extent possible without sacrificing the structural safety and functional requirement such as ground clearance and electrical clearance.

4 73 BARREL TYPE TOWERS NARROW BASE BARREL TYPE TOWER CORSET TOWER Figure 4.1 Typical Barrel and Corset Tower Configurations

5 74 n ~n E A -- Ground wire peak B Cross arm height C ~ Panel height D -Vertical spacing between conductors E - Staging F -- Base width G -- Panel width H1, H2 - Cross arm length I - Top hamper width Figure 4.2 Geometric Parameters of a Transmission Line Tower

6 75 Sag: The conductor wires and ground-wires sag due to self-weight. The size and type of the conductor, wind and climatic conditions of the region and span length determine the conductor s sag and tension. Span length is fixed from economic considerations. The maximum sag occurs at the maximum temperature and still wind conditions. Sagging of the conductor cables is considered in determining the height of the tower. It is essential to have minimum clearance between the bottom-most conductor and the ground, at the point where the sag is maximum. Sag tension is the force on the conductor, which in turn is transferred to the tower. Sag tension is maximum at the time of maximum temperature and when wind is at maximum. Loads such as selfweight and snow load on the conductors contribute to the sag tension. Spacing between the towers, ground level difference between tower locations, the mechanical properties of the conductors and ground-wires decide the sag distance and sag tension in the cables. The conductors assume catenary profile and the sag is calculated based on parabolic formulae or procedure given in codes of practices. 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 various national standards. The maximum sag for the normal span of the conductor should be added to the minimum ground clearance to get the staging height of the tower, i.e. the vertical distance from the ground level to the bottom of the lowest cross arm.

7 76 Ground-wire peak: Ground-wire peaks are provided to support the ground-wires, which shield the tower from lightning and provide earthing to the tower. The height of the ground-wire peak is chosen in such a way that the cross arm falls within the shield angle. The bottom width of the ground-wire peak is assumed equal to the top hamper width and is normally 0.75m to lm. Cross-arm spacing: Cross arms are provided to support the transmission line power conductors. The number of circuits carried by the tower determines the number of cross arms. In general three cross arms for single circuit towers and six cross arms for double circuit towers are required. The vertical spacing between the cross arms must satisfy the minimum clearance between circuit lines and other electrical requirements. The minimum horizontal clearance required between the conductors and the tower steel is based on the swing conditions, and it determines the length of the cross arm. The depth of the cross arm is assumed in general such that the angle at the tip of the arm is in the range of 15 to 20 degrees. Base Width: The base width of the tower is determined heuristically. For example, the ratio of base width to total height may vary from one-tenth for tangent towers to one-fifth for large angle tower. Also, there are formulae for preliminary determination of economical base width. The widths may be varied to satisfy other constraints like foundation design and land availability. Top Hamper Width: Top hamper width is the width of the tower at lower cross-arm level. The top hamper width is also determined heuristically and is generally about one third of the base width. Other parameters like horizontal spacing between conductors and slope of the leg may also be considered while determining the top hamper width.

8 MICROWAVE TOWERS Steel lattice towers are also used in electronic and communication industries for communication of microwave signals through different types of antennas. Several antennae are fixed on the tower in different directions at different heights as per the requirement and usage. The antenna positions decide the height of the tower. Symmetrical cross sections are preferred for microwave towers due to reversal of wind direction. Generally steel lattice towers with square or triangular plan are used for microwave towers. Angle sections and tubes are commonly used for the fabrication of these towers. Microwave towers are generally self-supporting steel lattice towers. Guyed towers are also used for microwave communication, but are least preferred for supporting heavy disc antennae. Wind load on the tower body and antennae is the major load on the structure besides the self-weight of the tower. Microwave towers are generally supported either at ground or at rooftop of some buildings. The tip deflection of the tower is a governing parameter for the functional requirement. Typical configuration of a 102-m high microwave tower is shown in Figure 4.3. The tower is triangular in plan. The two-dimensional and three-dimensional views of the tower are shown. 4.3 GEOMETRICAL MODELLING OF STEEL LATTICE TOWERS The mathematical modelling is an important step in the design of steel lattice towers. Steel lattice towers are treated as a pin jointed skeletal system. The influence of dimension in modelling is presented below:

9 78 N> Is) N> 3 m 3 m 3,rff~ JnT 3 m 7^ 3 m 3 ni 7^ 3 m 7 ~ ~?C 3 m 3 it? 3 n? 3 n? 6 in 6 m 6 m 7^ 6 m 71~ 6 m <- -7<- 6 mn 6 m "7 ~ 7 6 m 7 / ~ 6 m 7^ 6 m * 16.5 m 2-D VIEW 3-D VIEW Figure 4.3 Typical Configuration of a 102 m Triangular Microwave Tower

10 One-Dimensional Modelling In one-dimensional modelling, the tower is treated as a cantilever beam-column with varying inertia along the height. The properties of the beam are calculated assuming that the legs are the flanges of an equivalent I-beam so that the moment of inertia and area can be used in calculating the forces due to bending and axial loads. But the effect of bracing and torsion are very difficult to include mainly because the contribution of each of these in resisting the load as a beam can be done only by approximations. Further, as the beam is nonprismatic and skeletal, the approximations completely neglect the integral action of the members. This brings in disproportionate distribution of loads on to the various components resulting in either over-design or under-design Two-Dimensional Modelling * Even though one dimensional beam model of a square tower gives an idea of the general behaviour of the tower, the spatial nature of the system calls for better modelling, reflecting the response more realistically. Here again, basic engineering knowledge can be extended to visualise in two-dimension so that axial, lateral and torsional loads are distributed in an appropriate way on to the four faces of the tower. Typically lateral and axial loads are transferred equally on to the two faces of the tower. For torsion, the resisting couple for the torsional moment is assumed to be provided by shear on four faces. Using this distribution of the loads on the faces, member forces are determined using method of joints. Here again conservation in stiffness and absence of integrated action prevent realistic estimation of forces in all members leading to overdesign.

11 Three-Dimensional Modelling In three-dimensional modelling, the tower is considered as it is, so that all the loads can be accounted for simultaneously. The member participation in the response of the tower for axial, bending and torsional effects is taken into account. But amenability of three-dimensional model to hand calculation is very difficult. Hence, the tower has to be modelled and analysed using one of the numerical techniques and computer programming. The mathematical model of a steel lattice tower is preferably a space truss for computer analysis. Hence three-dimensional geometric models are used to represent the mathematical model of the tower. Geometric modelling and configuration processing are prerequisites for any lattice tower analysis. Configuration processing refers to the phase in structural analysis where a mathematical model of a structure is formulated. For example, mesh generation in finite element analysis is a special case of configuration processing. Preparation of input data in tower analysis in three-dimension is a tedious and time-consuming process. Therefore, it is a general practice to write configuration generator programs to minimise input. The generator, which reads the input for the tower geometry, generates the data for the analysis program. The main features of the configuration generator are: ease and minimisation of input data, automatic generation of joint co-ordinates, numbering of nodes and members, and members connectivity. Modification can be done easily and the chances of making errors in input are minimised. These types of programs are useful to the structural engineers and designers for configuration optimisation and to evolve various alternatives for analysis and design. It will considerably reduce the drudgery of data preparation for the analysis.

12 MODULAR GENERATION TECHNIQUE While writing configuration generator, it is preferable to be more general rather than specific to a particular configuration. For generating tower configurations, the methods are not generalised and not many attempts have been made to establish any technique. In an attempt to do so, a knowledge based modular generation technique has been developed, as this method helps in the process of optimisation. The tower configuration is decomposed into various generic modules as shown in Figure Using various combinations of these modules, it is possible to generate various transmission line tower configurations and microwave tower configurations. Various modules for different panels of tower are encoded in the knowledge and can be assembled to the required configuration of the tower as shown in Figure 4.5. The modular generator produces the geometry of the tower from simple input and the function of the generator is shown in the flow chart given in Figure 4.6. A 220 kv double circuit tangent transmission line tower generated using this technique is shown in Figure 4.7. The purpose of considering modules is to simplify the task of generating the tower configuration. Also, this will enable to generalise the development of configuration of a similar nature. The modules can be repeatedly used in different parts to get the desired configuration. The entire height of the transmission line tower is divided into different segments, such that the leg members of the tower in that portion have the same slope along the height. Hence, the width of the tower at any location along its height can be calculated by linear variation, when the top width and bottom width of the segment are known. These segments are considered to have one or more panels and each panel is represented by some module.

13 Figure 4.4 Typical Modules for Tower Modelling 82

14 83 Figure 4.5 Tower Assembly EXPLODED VIEW

15 Figure 4.6. Flow Chart of Modular Generator 84

16 Figure 4.7 Geometric Model OF A 220 kv Double Circuit Transmission Line Tower

17 86 In a way, modules are the representation of any interchangeable portion or unit of the tower having a particular type of bracing system. Only the type of module has to be chosen and depending upon the context, appropriate data for a module are automatically generated. Different modules can be used for generating different panels, cross arms, and ground-wire peaks. For each type, varieties of modules may be built in and stored in the knowledge base. By this way, numerous combinations can be explored to experiment with radically new structural configurations without much difficulty. This is the major advantage of decomposing the tower into modules. The system automatically generates nodes, spatial co-ordinates, member numbers, member groups and members connectivity for the module locally. They are independent of the full tower configuration. Then these data are converted to global format for the required tower configuration. When connecting one module to the other, the common variables between modules are properly taken into account to eliminate repetition. Different types of modules with different patterns can be generated for a particular section and this adds more flexibility to the choice Member Grouping Member grouping has practical significance in tower design as well as optimisation. Tower members are grouped into different types such as leg members, diagonal bracing members, tie bracing members and plan bracing members. Group numbers are assigned to each member to identify the type and cross section of the member. Freestanding microwave towers and transmission line towers are subjected to wind load. Reversal in the direction of wind is always considered, and hence the members are kept identical in each face of the tower generally. For design purposes, members of similar type at a particular height or a panel, at a different face of the tower are grouped together with

18 87 same area of cross section. For example, the leg members in a panel are considered to be of the same group. Similarly, the diagonal bracing members in all the faces of a panel are likely to have same section and considered as a single group entity. The members are designed for the maximum forces in the group rather than the force in the member. These group numbers are also useful in identifying the members for applying the design rules such as slenderness ratio limitations Tower Analysis Data The geometrical data of the tower are generated by assembling the modules from top to bottom of the tower. While assembling the geometric model with modules, some parameters like last node, last member, last group number in each group, are passed internally from module to module for data continuity. When the generation is done modularly, the input required for assembling the tower is minimal. The configuration can be changed easily by changing the modules or the geometric parameters to get a new configuration. This not only facilitates the analysis and design, but also enables design revision and optimisation. The top width, number of slopes, bottom width of each segment, number of the modules, bracing pattern and height of each module only are specified as input. A 60m high square microwave tower model having two slopes assembled with modules using this method is shown in Figure 4.8. The output of the generator contains all the geometrical co-ordinates of the nodes, member numbers, members connectivity, member group numbers and restrained nodal data. Bottom-most nodes are assumed to be restrained against all the degrees of freedom.

19 H 3-D VIEW ALL DIMENSIONS IN mm Figure 4.8 Geometric Model of a Square Microwave Communication Tower

20 LOADS Transmission line towers are subjected to loads acting in all the three mutually perpendicular directions namely vertical, normal to the direction of line, and parallel to the direction of line Transverse Loads The load perpendicular to the direction of the line is known as transverse load. It acts at the points of conductor, i.e. at the tip of the cross arms and ground-wire support. In addition to that, a load distributed over the transverse face of the tower due to wind also acts on the tower. Wind load is applied horizontally, acting in the direction normal to the transmission line. Angle towers are used where the line makes a horizontal angle greater than 2. They must resist a transverse load from components of line tension induced by this angle in addition to the usual wind, ice and broken conductor loads. They are located such that the axis of the cross-arms bisects the angle formed by the conductors. The governing wind direction on conductors for the angle condition is assumed to be parallel to the cross-arms. Wind load over the wind span on bare or ice-covered conductors, ground-wire and wind load on insulator strings contribute transverse load on transmission line tower. The two half spans adjacent to the tower under consideration is known as wind span Longitudinal Loads Longitudinal load on transmission line tower is due to the unbalanced conductor tensions. It acts on the tower in a direction parallel to the line. The unbalanced conductor tension may be due to broken wire condition, unequal

21 90 adjacent spans of the tower, dead-ending of the tower, etc. The unbalanced pull due to a broken conductor or ground-wire in the case of tension strings is assumed 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 component in the transverse direction Vertical Loads Vertical load on transmission line tower is due to the weight of bare or ice-covered conductor over the governing weight span, weight of insulator, hardware, etc., covered with ice, if applicable, and a load equal to the weight of a line man with tools. This vertical load is applied at the ends of the cross-arms and on the ground-wire peak. The self-weight of the tower acts vertically and is calculated approximately as this is unknown until the actual design is complete. This may be revised, if required, before the final design. 4.6 WIND LOAD CALCULATIONS Wind load is the major load on all freestanding lattice towers. Wind load on transmission line tower body has to be calculated and transferred to all panel points to get more realistic effect. As this involves a number of laborious and complex calculations, it is the general practice to consider the equivalent loads. These loads are applied on the conductor and ground wire supports which are already subjected to certain other transverse, longitudinal and vertical loads. The projected area is an unknown quantity, but is required for the wind load calculation. This can be calculated exactly only after the design

22 91 process is over and actual sections are known. Therefore, it is necessary to assume certain area to arrive at the wind load on the structure. Depending on the spread and size of the structure, 15 to 25% of the gross area is generally taken as the net area. Gross area is the area bounded by the outside perimeter of single face of the tower. For accounting the wind force on the leeward side, a factor of 1.5 is used. The wind load on the tower is assumed to act at selected nodes, generally at the tip of the cross-arms. Some methods suggested by Murthy (1990) to determine the magnitude of wind load are given below, In one method, the wind loads on various parts of the tower or the members are calculated first. Then the moments about the tower base for all these loads are added and an equivalent load at selected points for that moment is calculated. The loads applied on the bottom cross-arms are increased with corresponding reduction in the loads applied on the upper cross-arms in another method. The second method is to divide the tower into a number of parts corresponding to the ground-wire and conductor support points. Based on solidity ratio, the wind load on each point is then calculated and the moment due to this wind load about the base is divided by the corresponding height, which gives the wind load on two points of support. In another method, the equivalent loads are applied at a number of points such as ground-wire peak, cross-arm points and waist level. An approximate solidity ratio is assumed and wind load on different parts of the tower are determined. An equivalent load, which can produce the same amount of moment at the base, is transferred to the upper loading point and the

23 remaining part to the base. This process is repeated for various parts of the tower. 92 Even though the design wind load based on the last method is more logical than the others, it is reported that these loads are lower than the actual. In microwave towers, which are square in plan, the wind load acting in the diagonal direction of the tower is generally considered to be critical. Similarly, face wind is critical in microwave towers that are triangular in plan. In some cases the wind load on antennae may decide the critical wind direction. An assumed configuration and bracing pattern are used in the preliminary design. Based on the preliminary design, approximate member sizes are arrived at for calculating wind load on towers. Recalculation of wind load, if required, is done before arriving at the final designs. A realistic approach is to apply the wind load at each node of the tower and this method is possible with computer programs. A generic load generator based on the modular approach is developed to calculate the wind load on the tower. The details are given below: 4.7 SELF-WEIGHT AND WIND LOAD GENERATOR In the calculation of wind load, the projected area is an unknown quantity and. in the calculation of self-weight, the section weights are unknown. Hence, the actual quantities of loads can be calculated only after the design process is complete. But for design, the analysis needs to be completed based on the load. Hence, it becomes necessary to do an approximate calculation of load and revise the load after final design in an iterative manner. Generally, preliminary design calculations are carried out to find the force in members approximately and the sections are decided based on the experience. Once the

24 93 sections are assumed, the wind loads as per codal provisions are calculated. This process is laborious if it is done manually. A computer based method is necessary to do the process as it may be required to perform calculations repeatedly whenever there is a change. The modular generation technique used for tower geometry is also found suitable for this process. The load generator program is generic and isolated from other programs so that it can work independently to give wind load and self-weight on each panel. The basic tower geometry has to be same as the analytical model. But the modules developed by this program need not be same as the analytical model. This may include additional members or bracing patterns. The secondary bracing can also be included in this program for the purpose of calculating loads. The tower can have the same input as in the case of geometric model. In addition to that, the input should include the member sizes for each group of members. The tower is divided into modules. Each module may represent a panel. The geometric data of the panel is generated first. The wind load on each panel and self-weight can be calculated from the geometry of the tower. The wind loads are calculated according to IS:875 (Part 3) , Indian Standard Code of Practice for Design Loads (other than Earthquake) for Buildings and Structures and are incorporated in the program. The panel wise wind load generated by this program for a tower shown in Figure 4.9 is given in Table 4.1. The self-weight of the panels are calculated using the length of the members and weight per unit length obtained from the database. This self-weight is assumed to act at all bottom nodes connecting the panel to the next panel and distributed equally.

25 94 Table 4.1 Typical output of load generator Basic Wind Speed Vb: 50 m/s Design Life in Years : 100 Terrain Category: 2 Class C kl Factor : 1.08 k3 Factor : 1.00 Force Coeff. for Square Tower with Angles considered Wind Pressure Pz = 0.6Vz**2 = 0.6{Vb.kl.k2.k3)**2 Design Wind Pressure 0. 6(Vb.kl k3.)**2(k2**2) (k2**2) Kg/m2 Pan. Ht k2 PZ SR Cf Pz.Cf Exp.Ar. T.D.L. T.W.L. DL si.2 W.L. Ho. cm. Kg/m2 Kg/m2 cm2 Kg Kg YSall Sail Total Weight : Kg. APPLIED LOAD FACTOR «1.00

26 95 Figure 4.9 Details of a 60 m Tower for Wind Load Calculations

27 96 A general database of all angle sections given in IS: Dimensions for Hot Rolled Steel Beam, Column, Channel and Angle Sections, and tubular sections from IS: Specification for Steel Tubes for Structural purposes are linked to the program. This database contains the designation, the size, mass, sectional area, moment of Inertia and the radius of gyration, which are often used in the design. The variation of k2 factor and design wind pressure along the height of the tower is given in Figure The variation of wind load and dead load for panels from top to bottom of the tower is shown in Figure 4.11 Since the height of the bottom most panel is much greater than the rest of the panels the self weight drastically increases. Also since heavier sections are provided in the bottom-most panel, it attracts more wind pressure and hence, there is a drastic increase in the wind pressure in that panel. 4.8 ANALYTICAL PROCEDURE Steel lattice towers are highly indeterminate space frames with semirigid joints and assumptions are made to simplify the complexities involved in the analysis of the actual tower. It is reported in literature that the comparison of space frame analysis with space truss analysis showed insignificant difference (less than 10%) and also different analyses (plane/space/truss/frame) with and without secondary bracings ultimately gave same member forces. Hence, the members are considered as three dimensional truss elements for modelling of the tower. Secondary members are not considered in the analytical modelling, as these members are assumed to carry less than 2.5% of the main leg / bracing members.

28 97 k2 Factor Height Vs k2 Design Wind Pressure in kn/sq.m Height Vs Pz Figure 4.10 Variation of k2 Factor and Design Wind Pressure

29 98 Panel Number Panel Number Height Vs Dead load Height Vs Wind load Figure 4.11 Comparison of Dead Load and Wind Load on Panels

30 99 Self-supporting towers act basically as a-cantilever structure and the bending moments at different heights of the tower are useful in the preliminary calculation of width of the tower for the given load. Matrix method of structural analysis has been used, as large structural systems like lattice towers can be analysed using matrix method with highspeed computers. Stiffness method, which is also known as Displacement method or Equilibrium method, which is commonly employed in the analysis of lattice towers, is adopted in the analysis program. The equations of equilibrium is given by [K] [8] = [P] (4.7) where [K] = Global stiffness matrix [8] = Displacement vector [P] = Force vector The analysis program solves the set of linear algebraic equations formed by the above equation using Choleskey s Decomposition method. 4.9 DESIGN PROCEDURE Members of lattice towers are designed for either compression or tension as axial force is the only force in the members. Reversal of loads may induce alternate nature of force and hence, these members are designed for both compression and tension. The calculated tensile or compressive stress in various members shall not exceed the permissible stress limit as prescribed in the code. For the design of transmission line tower members IS:

31 100 (part 1 / Section 2) - Use of Structural Steel in Overhead Transmission Line Towers - Material, Loads and Permissible Stresses is used. For the design of microwave towers, IS: Indian Standard Code of Practice for General Construction in Steel is adopted SENSITIVITY STUDIES WITH RESPECT TO DESIGN PARAMETERS IN TOWER DESIGN In engineering design optimisation problems, most of the proposed methodologies call for assessment of response for any change in design and configuration variables. For optimal design problems, the direction and the step length in the complex design space of different tower configurations are required in some methods. The influence of design variables on selected performance requirement can identify the most efficient way of modifying a design concept to obtain the desired structural performance. Studying the influence of a selected design parameter on the performance of a group, enables to determine (a) which performance characteristics will benefit most and (b) which performance characteristics will be most adversely affected. Structural optimisation codes require the designer to identify all important constraints, the constraint bounds and the objective function. Design variables that are most critical to performance and the performance requirements that are most critically affected by a particular design variable are valuable information in optimisation. This can help in selecting the most critical parameters as design variables in optimisation using genetic algorithm. Hence, sensitivity studies were carried out to estimate the response for various design parameter changes.

32 101 In the design of transmission line towers, optimisation possibilities are more between waist and ground portion and hence, the change in performance of the design parameters for panels below the waist height alone are taken. Sensitivity methods for design changes were developed for member design parameters. Sensitivity studies are needed for the evaluation of response of transmission line systems when design parameters are varied. So these studies were carried out for leg and bracing members at different levels below the waist level to find out the response in terms of stress. The cross sectional areas of leg and bracing members are taken as design parameter in this study. They are varied independently keeping the other variables constant. Necessary relations are derived for evaluating the response Sensitivity of Deflection for Leg Members A typical elevation of a 132 kv tower with numbering scheme and member groups is shown in Figure A three-dimensional truss model is considered for analysis. Three typical load cases viz., Load case 1- Normal condition, Load case 2 - Ground wire broken and Load case - 3 Top conductor broken condition, are considered in the analysis and these load cases are shown in Figure.4.13 (a), (b) and (c). The leg members in the panel are grouped from top to bottom with code numbers 1006 to The code numbers above 1000 and below 2000 are used to identify leg member group in the generation. Cantilever moment (M) at the bottom of the tower for normal load case - 1 is calculated and the initial area of bottom leg member group is assumed based on this moment.

33 Figure 4.12 Typical Elevation of a 132 kv Single Circuit Tower 102

34 Figure (a) Load case -1 (Normal Load Condition) Figure (b) Load Case -2 (Ground-wire Broken Condition)

35 Figure 4.13 (c) Load Case -3 (Top Conductor Broken Condition) The approximate area is calculated using the equation given below: Area required M*FOS W * 0.67 * Fy (4.8) Where Yield Stress of Steel FOS W Factor of Safety Width at the top of the bottom panel The nearest area of the equal angle section, from IS Handbook is chosen as the initial area for all the leg members. The ratio of actual stress to allowable stress is known as the criticality ratio. The criticality ratio is high in Load Case - 2 and this may govern the design for the given set of load

36 105 conditions and initial sections chosen. Hence, the Load Case - 2 is considered for the deflection sensitivity study. The changes in deflection due to the change in area of leg members at different panels are tabulated. The variations of areas and deflections are given as ratios to the initial section and initial deflections in order to have nondimensional quantities for comparison. The change in area of the leg members at the bottom panel, which is grouped as 1009 and the corresponding change in maximum deflection of the tower is given in Table 4.2. The deflection vs. the area behaviour for leg member group 1009 is shown in Figure Table 4.2 Deflection Variation for Leg member group 1009 Area Deflection (8z) (mm2) (mm) x = A/Ao a = 8/5o The leg members at the panel immediately above the bottom are grouped as 1008 and same initial area is chosen. The change in area of the leg members in this group and the corresponding change in maximum deflection is given in Table 4.3.

37 106 Deflection Variation Figure 4.14 Variation of Deflection Vs Area for Leg Members Table 4.3 Deflection Variation for Leg member group 1008 Area (mm2) Deflection (8z) (mm) x = A/A0 a = 8/6o

38 107 Similarly the change in area and corresponding change in deflection for the leg members at the bottom panel, which is grouped as 1007, is given in Table 4.4. Table 4.4 Deflection Variation for Leg Member Group 1007 Area Deflection (8z) x = A/A0 a = S/5o (mm2) (mm) I Leg members in fourth panel from the bottom are grouped as 1006 and the change in area and corresponding change in deflection for these leg members are given in Table 4.5. Table 4.5 Deflection Variation for Leg Member Group 1006 Area Deflection (8z) 1 (mm2) (mm) x = AJA0 a = 5/5o

39 Sensitivity Curves for Deflection The behaviour is asymptotic and can be divided into three regions. The first region where the area variation revision is above 10% and below 50% the deflection can be linearly interpolated. The second region is the centre portion close to the initial section area, i.e. where the variation of area is between 50 % and 200% and can be fitted with a curve. A quadratic equation fitting the curve can be arrived by solving a set of equations satisfying the three points. The third region where the area revision is above 2.0 times and below 10 times can also be linearly interpolated. For area revisions above 10 times and below 0.1 time (10%), this fitting is not suitable and it is suggested that the initial section itself is revised. The quadratic equation for curve fitting is arrived for values in Table 4.2. ax2+ bx + c = a (4.9) 0.25a b + c =1.176 (4.10) a + b + c =1.0 (4.11) 4 a + 2 b + c = (4.12) Solving the above equations, the curve fitting equation is Where x x =a (4.13) x = A/A0 a = S/80 8 = Revised Deflection 80 = Initial Deflection A = Revised Area A0 = Initial Area

40 109 Similar equations for other leg member groups in subsequent panels from bottom to top are also arrived. The quadratic equations for the curves fitted for the centre three points for all the leg member groups are given in Table 4.6. Table 4.6 Equations for Deflection Estimation Leg Member Equation Group x x x x x x x x Validation of Sensitivity Equations The validation of the above study can be carried out by finding the deflection of the tower for an area revision using the above formulation and compared with the actual deflection by complete analysis with revised area. For example, the maximum criticality ratio is 1.65 for the bottom panel leg members. If the area is revised as per this stress ratio, (i.e.) 1.65 times the initial area, the deflection due to the change in area is calculated without doing a reanalysis. The new area for this leg member shall be revised as 1.65* A0 = 1.65 * 1167 = 1926 mm2 The deflection using the curve fitted with equation (4.12) is (1.65) (1.65) = a = 8/60 = = * 60 = mm

41 no The actual deflection calculated by a separate analysis is noun. The sensitivity equation underestimated the actual value of deflection by 3 %. The estimated value and actual value of deflection for revision of leg member areas as per stress ratio or criticality ratio are tabulated in Table 4.7. In all cases the sensitivity equation underestimated the actual value of deflection. Table 4.7 Comparison of Estimated and Actual Deflection Leg Member Group Area Revision A/A0 Estimated Deflection (mm) Actual 1 Deflection (mm) Difference % Sensitivity of Deflection for Bracing Members The bracing members in the panels are grouped from top to bottom with code numbers. An initial area of 52.7 mm2 is chosen for the member groups. The analysis is carried out for three typical load cases, viz. Normal condition, Ground wire broken and Top conductor broken condition. The criticality ratio is high in the top conductor broken for the given set of load conditions and the initial sections chosen. Hence this load case is considered throughout for sensitivity on bracing. Table 4.8 gives the change in deflection for corresponding change in area of the bottom panel bracing member group. The deflection and area values are tabulated with variation ratios, which are normalised with initial section.

42 Ill Table 4.8 Deflection Variation for Bottom Bracing Member Group Area 1 Deflection (5z) 1 (mm2) 8 (mm) x = A/A0 a = 8/8o From the above values, the variation of deflection to the variation of area is not significant and hence, it can be observed that the sensitivity of deflection for area change in bracing member is insignificant Design Sensitivity The factors controlling the design of compression members are the force coming on the member and the allowable stress for the member. But, the allowable stress depends on the 1/r ratio of the member. The force in the leg member also changes when the area of the member is changed. Hence, when the area of the member is revised, both the allowable stress and actual stress on the member are changed. In order to optimise the design of members, the actual stress shall be equated to the allowable stress. The allowable stress (aali) is calculated from the Euler bucking load as below: crall = n2 El (Ur)2 (4.14)

43 112 Where E - Young s modulus 1 - Length of the member r - Minimum radius of gyration relationship The radius of gyration can be approximately defined with a simple Where r = h (4.15) h - size of the angle leg in mm (A) is given by For equal angle mild steel sections, the approximate area of the angle where A = 2 t h h = A / 2t A - Area of the angle section in mm2 t - thickness of the angle in mm (4.16) (4.17) For equal angle sections of thickness 5 mm, h = 0.1A and hence, r = A aall = 7T2 E/ (1/r)2 = tc2e/( 1/0.0197A)2 (4.18) (4.19)

44 113 Substituting E Gall 2.0 x 105 N/mm *A2 /12 (4.20) Gact = F/A (4.21) Equating actual stress to the allowable stress F/A = * A2 /12 (4.22) V (4.23) Where aac, - Actual stress in N/mm2 A - Area of the member in mm2 F - Force on the member in N 1 - Length of the member in mm The forces obtained in the analysis with assumed initial areas can be used in this expression to arrive at a set of minimum theoretical values of areas. Using the relationship in equation (4.23), the area required for all the leg member groups are calculated and given in Table 4.9. Table 4.9 Estimated Minimum Areas for Leg Members Leg Member Group Force in Member Area (mm2)

45 114 For bottom bracing member group, the maximum force in the member is N. Using the relationship in equation (4.23) the area required for bracing member group is calculated as below: Using these areas as the revised area of the member groups, reanalysis is carried out and it gives a criticality ratio of 0.98, which is very close to 1. This means the members have attained 98% of their load capacity, which can lead to optimality. The equation (4.23) is useful for obtaining the minimum theoretical area required for the members of tower, which in turn helps in fixing the variable bound for area of cross section of members in the optimisation process. This can form the knowledge base required for optimisation. 4,10.6 Sensitivity for Base Width The forces in the leg members are mostly governed by the base width as it resists the moment due to load. Hence, the variations of forces in the leg members are studied by varying the base width. The same tower configuration and loadings are assumed. The base width is increased by 10% and then by 20%. Similarly the base width is reduced by 10% and then by 20%. The comparison of maximum force in compression and tension on top leg member group for different load cases are tabulated in Table (a) and (b) respectively. Similarly for other leg member groups, the values are tabulated in Tables to 4.13.

46 Table 4.10 (a) Comparison of Forces (Compression) in Leg Member Group SI. No. 0 1 Base Width 1 Initial Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation Force Variation Force Variation (%) (%) 1 (%) % % % % Table 4.10 (b) Comparison of Forces (Tension) in Leg Member Group 1006 SI. No. 0 Base Width Initial Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation Force Variation Force Variation (%) (%) (%) % % % %

47 116 Table 4.11 (a) Comparison of Forces (Compression) in Leg Member Group 1007 SI No Base Width Initial Width Load case -1 Load Case - 2 Load Case - 3 Force Variation Force Variation Force (%) (%) Variation (%) % % % I % I Table 4.11 (b) Comparison of Forces (Tension) in Leg Member Group SL Base No. Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation Force Variation Force Variation (%) (%) (%) 0 Initial Width % % % %

48 117 Table 4.12 (a) Comparison of Forces (Compression) in Leg Member Group 1008 Load Case -1 Load Case - 2 Load Case - 3 SI. Base Force Variation Force Variation Force Variation N. Width (%) (%) (%) Initial Width % % % % Table (b) Comparison of Forces (Tension) in Leg Member Group 1008 S1. No. 0 I 4 1 Base Width Initial Width Load Case -1 Load Case - 2 Load Case - 3 j Force Variation (%) Force Variation (%) Force Variation (%) % % % %

49 118 Table 4.13 (a) Comparison of Forces (Compression) in Leg Member Group 1009 SI. No. 0 Base Width Initial Width Load Case -1 Load Case - 2 Load Case - 3 Force (m Variation 1 (%) 1 Force Variation (%) Force Variation (%) % % % % Table 4.13 (b) Comparison of Forces (Tension) in Leg Member Group 1009 SI. No. 0 Base Width Initial Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation Force Variation Force Variation (%) (%) (%) % % % %

50 119 The comparison of maximum force in compression and tension on top bracing members group for different load cases are tabulated in Tables 4.14 (a) and (b) respectively. Similarly for other bracing member groups, the values are tabulated in Tables 4.15 to Table 4.14 (a) Comparison of Forces (Compression) in Bracing Member Group 2008 SI. No. Base Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation (%) Force Variation (%) Force Variation (%) 0 Initial Width % % % % Table 4.14 (b) Comparison of Forces (Tension) in Bracing Member Group 2008 SI. No. Base Width Load Case -1 Load Case - 2 Load Case - 3 Force Variation (%) Force Variation (%) Force Variation (%) 0 Initial Width % % % %