CHAPTER 5 STRUCTURAL ANALYSIS OF TRANSMISSION LINE TOWER STRUCTURES

Transcription

1 58 CHAPTER 5 STRUCTURAL ANALYSIS OF TRANSMISSION LINE TOWER STRUCTURES 5.1 INTRODUCTION Transmission tower structures are generally constructed using symmetric thin-walled angle section members which are eccentrically connected. They are widely regarded as one of the most difficult forms of lattice structures for analysis. Proof loading or full-scale, testing of such structures has traditionally formed an integral part of the development of tower design. Stress calculations in the structure are normally obtained from a linear elastic analysis where members are assumed to be axially loaded and, for the majority of cases, pin connected. In practice, such conditions do not exist and members are detailed to minimize bending stress. Despite this, results from full-scale testing of transmission towers indicate that bending stress in the members can be as significant as axial stresses (Roy et al. 1984). Design practices for transmission towers are different from those for other steel structures in that stresses are permitted to be higher because towers are tested to their ultimate design strength and designs incorporate modifications based on test results. A recent study by the Electric Power Research Institute (EPRI project-4756,1986) indicated that current design practices have, for the most part, served industry well. However, data from full-scale tests show that tire behaviour of transmission towers under complex load situations cannot be consistently predicted using the present techniques. The investigation by (EPRI Project-4756, 1986) also revealed that out of the 57 structure load cases conducted, 23%

2 59 experienced premature failure. On an average, failure occurred at 95.4% of the design load level but failure could occur at unexpected locations. Further, available data showed considerable discrepancies between member forces computed from linear elastic truss analyses and those measured from full-scale tests. The (EPRI project-4756,1986) indicated that the linear elastic truss analysis method for transmission towers should be used with caution. The influences of geometric and material nonlinearities play a very important role in determining the ultimate behaviour of the structure. In the analysis of transmission line structures, nonlinear effects, both material and geometric, are commonly neglected. Neglecting geometric nonlinearities when the axial loads are large fractions of the buckling loads causes error in the analysis. In this Chapter, a comparison is initially made between the linear elastic and nonlinear analysis of the transmission tower structures with respect to the member forces considering large deflection using Updated Geometry Method to show the effect of geometric nonlinearity on structural analysis. Secondly, the member level nonlinearity is introduced by modelling the compression member using, multi-level force-deformation characteristic behaviour. The member replacement technique is implemented and is shown that the ultimate capacity of the system is affected due to member nonlinearity. 5.2 FIRST ORDER LINEAR ELASTIC ANALYSIS OF TRANSMISSION TOWER In structural analysis, the actual complex structure and loading are modelled mathematically, using several simplifying assumptions. In the first order linear elastic analysis of transmission tower, nonlinear effects at member and system level (geometric) are not taken into consideration and the tower is analyzed as a space truss. Moments produced by the continuity of members are generally not considered since each leg member is assumed pinned between two joints. Redundant members are not included in the analysis since they have very little effect on the forces in the load-carrying members.

3 FIGURE 5.1 FLOW CHART FOR COMPUTER MODULE TLTOWER

4 61 Table 5.1 : Transmission Tower Loads and Load Combinations IEC load cases Wind Pressure Applied Circuits Load Description Vertical Transverse Longitudinal i A D.C Normal 90 wind _ 2 A S.C. Normal 90 wind - 3 B D.C. Normal 45 wind 45 wind 4 B S.C. Normal 45 wind 45 wind 5 A D.C. Minimum 90 wind - 6 A S.C Minimum 90 wind - 7 B D.C Minimum 45 wind 45 wind 8 B S.C Minimum 45 wind 45 wind D.C 2xNormal No wind - S.C. 2xNormal No wind 11 C D.C Normal 1 and 2 Unbalanced 90 wind 12 C D.C Normal 1 and 5 Unbalanced 90 wind 13 c D.C Normal 1 and 3 Unbalanced 90 wind 14 c D.C Normal 1 and 6 Unbalanced 90 wind 15 c D.C Normal 1 and 4 Unbalanced 90 wind 16 c D.C Normal land 7 Unbalanced 90 wind 17 D D.C Normal 1 and 2 Unbalanced 45 wind 18 D D.C Normal 1 and 5 Unbalanced 45 wind 19 D D.C Normal 1 and 3 Unbalanced 45 wind 20 D D.C Normal 1 and 6 Unbalanced 45 wind 21 D D.C Normal 1 and 4 Unbalanced 45 wind 22 D D.C Normal land 7 Unbalanced 45 wind 23 C D.C Minimum 1 and 2 Unbalanced 90 wind 24 C D.C Minimum 1 and 5 Unbalanced 90 wind 25 C D.C Minimum 1 and 3 Unbalanced 90 wind 26 C D.C Minimum 1 and 6 Unbalanced 90 wind 27 C D.C Minimum 1 and 4 Unbalanced 90 wind 28 C D.C. Minimum 1 and 7 Unbalanced 90 wind 29 D D.C Minimum 1 and 2 Unbalanced 45 wind 30 D D.C. Minimum 1 and 5 Unbalanced 45 wind 31 D D.C Minimum 1 and 3 Unbalanced 45 wind 32 D D.C. Minimum 1 and 6 Unbalanced 45 wind 33 D D.C. Minimum 1 and 4 Unbalanced 45 wind 34 D D.C. Minimum 1 and 7 Unbalanced 45 wind 35 - D.C. Normal Cascade Collapsed. No wind 36 - D.C. Minimum Cascade Collapsed. No wind -

5 62 The loadings on the transmission towers are considered probabilistically under extreme value distribution of wind as described in Chapter 2 and 4. The loading criteria which governs the critical member load effects are fully based on the IEC-826 (1991), IS:802 (1989) and ASCE Manual-52 (1989). Thus, the transfer model for wind speed to wind load in addition to other loads has been automated in the computer program TLTOWER by Alam et al. (1990a). TLTOWER performs stiffness analyses of three dimensional transmission towers modeled using stiffness matrix method. The detail of the program is elaborated in the form of flow chart in Figure 5.1. The load and load combinations that have been incorporated in the TLTOWER is shown in Table UNCERTAINTY IN STRUCTURAL ANALYSIS AND CODE INTERPRETATION Based on the results of a recent experiment using 20 engineering firms, EPRI (EPRI project-5488, 1987) reported that there is significant variation in structural analysis results and interpretation of the ASCE Manual-52 (1988) procedures. In this EPRI experiment, all 20 firms analyzed the same tower which was tested to failure by EPRI. Results obtained from all 20 firms showed that the coefficients of variation of the calculated force and strength for the same component ranged from and respectively. The result of the full scale test yielded member force values within 10% of the mean value calculated by all the 20 firms. In structural analysis the actual complex structure and loading are modelled mathematically, using several simplifying assumptions. As a result several errors are introduced in the estimate of the real response. They are described below. In the analysis of transmission line structures, nonlinear effects (both material and geometric), as mentioned earlier are commonly neglected. In many cases this assumption introduces negligible errors. Neglecting geometric

6 63 nonlinearities when the axial loads are large fractions of the buckling loads also causes errors which can not be neglected. In modelling a lattice tower structure, joints are assumed as points with no finite size. The forces acting on all the members meeting at the joint are assumed to pass through this point considering the joint as a point rather than giving a finite size usually overestimates the stresses and deflections (EPRI Report-5488, 1987). In addition, the joints are assumed either pinned or fixed. In reality, the joints are partially fixed, which is somewhat in between the fully fixed and fully pinned condition. If necessary, bounds for this condition can be obtained by conducting the analysis twice with ends pinned and ends fixed. Alternatively, the stiffness matrix can be modified to properly model finite joint details. The assumption commonly made when designing the superstructure is that the soil is rigid. In reality, soil-structure interaction principles must be considered to evaluate the degree of fixity. Lattice towers are commonly analyzed as space trusses, and eccentricity at connections which introduce moments is considered as a secondary effect. Moments produced by the continuity of members (e.g. leg members) are generally not considered, since each leg segment is assumed pinned between two joints. The ASCE Manual-52 (1988) and IS code (IS:802, 1977) specified design equations approximately account for load eccentricity and end restraint. The Indian standards IS:802 (1977, 1989) specifications are silent on the analytical model to be adopted for the design of towers. The analytical models that are available to designer contain many variations and options for application of model such as linear, nonlinear, elastic, inelastic, large displacements, etc. The modelling technique shall also include additional testing and modelling capabilities for bracing members to improve the over all accuracy of modelling technique.

7 64 Agreement on the proper modelling for lattice towers does not exist. Some designers use program which models the tower as a space truss. Since no continuity, member weight or wind force moments are computed, the design is typically completed by considering axial forces plus eccentricity moments. Other designers use a space frame program and consider all moments. The ASCE Manual-52 (1988) agrees on the presence of moments in members of a tower, because of framing eccentricities for slightly eccentric loads, lateral wind load on members etc., but usually they are not significant. The Manual emphasizes an exclusive analysis of the tower as an ideal space truss. It also argues that while not considered in analysis, moments from normal framing eccentricities of angles are accounted for in the design of the member by derating the angles load capacity. 5.4 GEOMETRICALLY NONLINEAR ELASTIC ANALYSIS OF TRANSMISSION TOWERS The power industry is moving toward the use of high-voltage transmission lines to meet the demand for greater transmission capacity. This results in taller, more slender transmission towers that are subjected to heavier loads and undergo larger displacements. These displacements of a deformed transmission tower create forces in addition to those calculated in a first order analysis. A second-order or nonlinear, in the geometric sense, analysis is one that produces forces that are in equilibrium in the deformed geometry. A nonlinear analysis is normally performed as a succession of first-order analyses, the geometry of the tower is updated at the end of each iteration. The analysis of a structure in which the geometry changes due to the loading leads to a set of nonlinear equations. Various techniques exist for treating this type of analysis. A general principle of all techniques is that the final deflected position of the tower structure must be used when the equilibrium equations are evaluated. Once such method is the Updated Geometry Method.

8 Updated geometry method For a three-dimensional truss, the equilibrium equations at a typical joint can be written as n T\ M S Cos (plx Px (5.1) n Z i=l n Z i=l Sj COS (P'yy = Py (5.2) Sj Cos <pa = Pz (5.3) in which S} = the force in the ith member of the n members at the Joint; <pix, 0iy and <f>a = the angle between the axis of that member and the x, y and z coordinate axis and Px, Py and Pz = the component of the applied joint force in the x, y and z direction respectively. When Equations (5.1), (5.2) and (5.3) are nonlinear, they must be solved in an iterative method, using Updated Geometry Method, the changed geometry is used to rewrite and resolve the equations. The procedure used is shown in the flow chart in Figure 5.2. If the structure is stable under the action of the loads, this method will generally converge in about two to four iterations. The mathematical representation of the scheme can be written simply for any joint of the tower as follows: n Z Sjk+1 Cos 0iak = Pa; a = x, y, z (5.4) i=l in which the Cos (p-la represents the direction cosines for the geometry of the member at the end of the kth iteration; and the Sjk+1 = the unknown member

9 FIGURE 5.2 FLOW CHART FOR UPDATED GEOMETRY METHOD

10 67 forces that are to be determined from the solution of Equation (5.4). After the member forces, Sjk+1, and the joint displacements have been determined, the equilibrium equations for the iteration cycle k+1 become n 2 Sik+1 Cos <piak+1 - Pa = Rak+1; a = x,y,z (5.5) i=l The forces Ra represent the difference between the internal and the external forces at the joint. If they are sufficiently small, the structure is considered to have reached the final equilibrium position. The above procedure has been implemented in the TLTOWER program (Alam et ai 1990a). This program uses the stiffness matrix method of analysis for three-dimensional trusses; the computations are based on using the displacements as unknown in the equilibrium equation. For a tower structure, the equations are written in the form K0 = P (5.6) in which 0 represents the unknown displacements of the joints of the tower, and K = the stiffness matrix based on individual member properties and the initial geometry of the structure. Equation (5.6) is solved for the displacements Uj; these displacements are added to the coordinates of the joints of the structure, and a new stiffness matrix Kj, is formed. Then, the equation Kj U2 = P (5.7) is solved for the U2, which is compared with the U obtained previously. If the two sets of displacements are sufficiently close, the solution is complete. Otherwise, the process continues. The general form of the equation is KkUU+1 = P (5.8)

11 68 The process ends when Uk and Uk+1 are close. The Updated Geometry Method Is easy to implement on any existing elastic tower analysis program. The solution process usually averages 2-4 cycles. It can only treat a structure under a single loading condition. 5.5 COMPARISON OF MEMBER FORCES WITH LINEAR AND GEOMETRIC NONLINEAR ANALYSIS FOR TRANSMISSION TOWERS This section presents the geometric nonlinear effects on two towers which were studied using the methodology described above. The first tower is a 220 kv double circuit with an overall height of 38.92m is shown in Figure 5.3 and the second tower is a 132 kv single circuit with an overall height of 21.98m with a 6m extension is shown in Figure 5.4. The probabilistic extreme wind loading governs the analysis both for the linear and nonlinear cases. The analysis results reveal that the 132 kv single circuit lower is less critical than 220 kv double circuit tower with respect to the nonlinear member forces. The maximum increase in member forces due to displacement effects is generally small and a maximum of 2.53% in the cross-arm member under broken wire condition and the leg members are less prone to the secondary effects in comparison to cross-arm members. There are negligible increase in member forces in the bracing due to secondary effects. On the other hand, the maximum increase in member force for cross-arm members is about 7.2% and that of leg members is 4.8% in the case of 220 kv double circuit tower. The preceding results clearly indicate that secondary effects on a tower increase with an increase in the tower flexibility and the applied loads. The 220 kv tower is more flexible than the 132 kv tower due to its greater height and longer cross arms and is subjected to heavier loading. So the geometric nonlinear effects

12 ! FIGURE kv DOUBLE CIRCUIT TANGENT TOWER WITH PRIMARY MEMBER TAGS

13 FIGURE kv SINGLE CIRCUIT TANGENT TOWER 70

14 71 on the 220 kv double circuit tower are larger than 132 kv single circuit for these particular configurations of the example towers considered for this study. 5.6 MEMBER-LEVEL NONLINEARITY OF TRANSMISSION TOWER In the theoretical analysis performed it has been assumed that all members are elastic and that the deflections are small or large (with geometric nonlinearity). However, these assumptions are valid only up to a particular load limit. Near ultimate load the members which are subjected to axial load are likely to behave nonlinearly. Though it is possible to take into account by an exact computer analysis the nonlinearity, for a practical design of a transmission line tower, the ultimate capacity is the only parameter which is important. The ultimate capacity of the overall tower depends mostly on the ultimate capacity of the leg members. It can be stated explicitly that the leg subjected to compression will fail first before other main members. When one member yields or buckles, other members carry the load and insures the integrity of the tower. This is the concept of force redistribution. One can see how force redistribution would develop in a truss configuration by studying Figure 5.5. It is an indeterminate truss system. For this discussion it will be assumed that members 1-2, 1-4, 2-3 and 3-4 are large members capable of carrying any force required of them. Members 1-3 and 2-4, the diagonals, are the members of interest in this illustration. Their compression and tension capacities are given in Figure 5.6. They are expressed in terms of member load vs. axial deflection. These are the member performance curves. The first case to be considered is when the system of Figure 5.5 is designed as a tension/compression system. This system has been designed such that the compression member will withstand 17.8 kn. Assuming an equal distribution of force to the diagonals, P would be P = ( ) * Cos 45 = kn (5.9)

15 FIGURE 5.5 SIMPLE INDETERMINATE TRUSS SYSTEM C (COMPRESSIVE FORCE) T (TENSILE FORCE) FIGURE 5.6 MEMBER PERFORMANCE CURVES FOR MEMBERS OF THE TRUSS SYSTEM IN FIGURE 5.5

16 73 when (T) and (C) of Figure 5.6 are 17.8 kn each. This is the elastic capacity of the system. But (T), in this example, can be 44.5 kn at which time the compression member unloads but sustain 8.9 kn. P at this point is P = ( ) * cos 45 = kn (5.10) This is a nonlinear analysis (considering member nonlinearity) in its fundamental form. One can see from this simple example how an elastic analysis could under estimate the true ultimate or limit capacity of a structural system. Now let us assume that the system of Figure 5.5 is a tension only system. That is, the diagonal member in compression is ignored when calculating the capacity of the system. One of the easiest technique to ignore this compression member effect is to set to a very small value of area so that with such a small area no significant compression load could flow to the member. The results of the analysis are, within usable mathematical significance, the same as if the members were removed from the model. Now, in this case the system capacity calculated using an elastic analysis would be P = 44.5 * Cos 45 = kn (5.11) Equation (5.11) ignores the 8.9 kn capacity of the compression diagonal. Including this 8.9 kn in the analysis results in a ultimate or limit value of kn as given by Equation (5.10). In other words, the linear elastic method under estimates the ultimate capacity or limit capacity by 16.7 per cent. The above two examples were chosen as the simplest forms of indeterminacy to illustrate how the residual strength is developed in a truss system. In a practical example, such as a transmission tower, the indeterminacy will be of a higher degree in 3-D environment. It is seen from the two simple examples that to account for the residual strength or perform a limit analysis two things are

17 74 necessary. The first is a nonlinear or limit structural analysis method capable of handling 3-D structures; second is the performance of the members in terms of load Vs. axial deflection i.e. the member performance curves. From the above simple examples it is seen that to perform a limit analysis, complete member performance is needed. This includes the loading portion and the unloading or post buckling region of the member performance curve as shown in Figure 3.2. To quantify member performance, a series of experimental investigations have been carried out by Mueller et al. (1981), Prickett et al. (1983), Mueller et al. (1985, 1986) and Bathon et al. (1990). The purpose of these missions were to perform tests on single members commonly found in transmission towers with the goal of determining a member performance data base, specially, for the quantification of the post buckling performance of single compression members. The parameters studied included: intermediate supports end restraints slenderness ratio (L/r) Load eccentricity These are the parameters which vary for members in a transmission tower. All of these parameters contribute to determining the effective L/r ratio of the member. A computer model was developed to predict member performance in the post buckling region. It was based purely on the results obtained by the above investigators. To accomplish this a regression analysis was done on five member performance curves. The five different members were chosen to give the maximum difference in values of effective L/r and a relatively uniform spacing between these extremes. Figure 5.7 is a typical three dimensional plot of these five member performance curves using L/r as the third axis. They are presented as load Vs. deflection curves. However, since the area is not the same for all the members, the

18 75 NORMALIZED AXIAL COMPRESSION AXIAL C O M M IS S IO N IN KN AXIAL DEFORMATION IN cm FIGURE 5*7 3-0 PLOT OF MEMBER PERFORMANCE CURVE BY REGRESSION M00ELUN6 FI6URE 5-8 NORMALIZED MEMBER PERFORMANCE CURVES

19 76 regression analysis was done using a normalized load term i.e. changing the load term to stress. The normalized, member performance curves are shown in Figure 5.8. The regression analysis was done to quantify the surface defined by these five curves. To simplify the process the surface was defined by a series of equations relating L/r and axial compression while holding the value of axial shortening constant at each of nineteen values. The values of axial shortening started at 1.27 mm and continued to mm increments of 1.27 mm. Thus, the surface was represented by nineteen independent equations. The equation with the best fit was P = Area * (A * eb * V*) (5.12) Where P = Axial load Area = Area A = Regression constant B = Regression constant L/r = Effective slenderness ratio The values of A and B are shown in Table 5.2 as reported by Prickett et al. (1983). A member performance curve is a line described by the intersection of the surface and a plane described by the axial compression and axial shortening with L/r a constant.

20 77 Table 5.2: Regression Coefficients for Member Performance Model (as In reference Prickett et at 1983) Axial Shortening A B*10" , To determine the validity of the above procedure a special test configuration was devised. In the test configuration the angles had effective L/r ratios equal to 193,112 and 60 with a section of 76x76x6mm. The comparison is given in Figures 5.9 through A study of these show excellent agreement for the members with an L/r value of 193 and 112. Figure 5.11 has a difference between the predicted and the test results in the post buckling region. This discrepancy is because the regression data base uses test results for members with high L/r values, i.e. where Euler buckling dominates. An L/r value of 60 puts it in a range where crushing plays an important role in the results. The member performance projections from the regression analysis should be used with caution when the L/r value is less than 120. It should also be noted that the regression equations are from actual test data and thus include a normal framing eccentricity in the calculation of member performance.

21 78 AXIAL COMPRESSION IN K N ^ AXIAL COMPRESSION IN KN FIGURE 5$ COMPARISON OF CALCULATED VALUES VERSUS TEST RESULTS (L/r=193) FIGURE 5-10 COMPARISON OF CALCULATED VALUES VERSUS TEST RESULTS (L/r = 112)

22 79 AXIAL COMPRESSION IN KN FIGURE 5.11 COMPARISON OF CALCULATED VALUES Vs TEST RESULTS {Ur = 60)

23 APPROXIMATE STRUCTURAL ANALYSIS METHOD INCORPORATING NONLINEAR BEHAVIOUR OF TOWER MEMBERS Structural nonlinearity at member-level is introduced by modeling elements with piece-wise linear force-deformation characteristics in the truss representation. The structural analysis methods are also simplified to reduce computational effort. As we know from the previous discussions, the member performance curve i.e. force deformation curves plays a great role on the structural analysis. The member performance curves for members in tension and compression are different. Typical idealized force-deformation curves for tension and compression members are shown in Figure The simplest way to model nonlinear compression member is by a bi-level or 2-state semi-brittle force-deformation relationship as shown in Figure 5.12 {Guenard, 1984 and Karamchandani; 1987, 1990). This bi-level model is an approximate idealized force-deformation curve made mainly for the simplified analysis procedure. Another approach to represent or to model nonlinear compression member is by a multi-state, specially, 3-state force-deformation model. This 3-stage model is far more versatile than semi-brittle model and can be used for a more accurate representation of tower members under compressive load. A typical three-state representation of a tower component is shown in Figure Member Replacement Method The response of the tower structures using semi-brittle member performance curve in compression has been obtained by this simplified method by Alam et al. (1993a). In this method, after a member fails the member is removed from the structure and its effects are accounted for by applying a set of forces (equal in magnitude to the post buckling force in the member) at the nodes at the ends of the member. The forces are maintained at the same level for the subsequent steps of the analysis.

24 FIGURE 5.12 TYPICAL IDEALIZED BI-LEVEL FORCE-DEFORMATION CURVES FOR MEMBERS UNDER TENSION AND COMPRESSION. FIGURE 5.13 TYPICAL IDEALIZED 3-STAGE MEMBER PERFORMANCE CURVE FOR MEMBERS UNDER COMPRESSION

25 82 This model and analysis method is attractive from a conventional systems reliability view point because each member can be represented by a component which is safe when the member is in the linear elastic state and failed when it is in die post-buckling constant-force state. Once a component fails, it remains in the failed state. We can identify sequences of component failures leading to system failure. For a 3-state model, when the force in a member reaches the end of a segment, the member is assumed to transit into the adjacent segment and its stiffness contribution to the structure stiffness is defined by the slope of the adjacent segment. The deformation is not monitored and therefore a member cannot unload. For example, in Figure 5.13, after the member fails in compression, it moves into state II. After this it moves into State III. The member deformation is not monitored and therefore it does not move into state IV even if the deformation starts to decrease in State III. Similarly, if the member fails in tension, the force in the member assumed to stay constant at the tensile-yield. Even if the deformation changes direction, the member is not allowed to unload.