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HomeMy WebLinkAboutWind-Induced Vibration of Tubular Poles, May 1974Alaska Energy Authority LIBRARY COPY PRINTED IN U.S.A. ° e a wu - a = mT HIGHSMITH #42-222! Wind-Induced Vibration of Tubular Poles by Leon Kempner Jr. A REPORT submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science May 1974 ENG O35 APPROVED: Professor of Civil Engineering in charge of major Head of Department of Civil Engineering Date report is presented ACKNOWLEDGMENT I wish to express my sincere gratitude to Professor T. J. McClellan for his encouragement and guidance during my graduate program. TABLE OF CONTENTS Section Page INTRODUCTION- - --- ---2---2--- <2 - 22 - nnn reenter enn n n-ne az HISTORY - - ------ 2-2 nn nnn nnn nnn nn nnn nnn rene nn e-- 3 BASIC THEORY-------- 9-92-2220 nnn enn nn nnn enna nn nn nee 6 SELECTION OF PARAMETERS -----------------------2--%------ 20 GENERAL CONSIDERATIONS - ------------------ n-ne ne ne nnn n= 27 CALCULATED RESULTS ----------------c2-- 20 e-em nnn nnn- 34 RECOMMENDATIONS - --------- 2-2-2 222-2 nnn nnn nnn nnn -- 45 HIGHER MODE VIBRATIONS ----------2---- 0-2-9 2-2-2 ------2- 50 APPENDIX I - REFERENCES II - REFERENCES (SUBJECT) WIND-INDUCED VIBRATION OF TUBULAR POLES INTRODUCTION This report was initiated during the summer of 1973. The Bonneville Power Administration at that time was in the process of designing two new lines utilizing tubular pole transmission structures. It was felt that an investigation to determine the effects of potential wind-induced vibration on these structures was needed. The reason for concern was due to the fact that another power company reported (1) having difficulty with tubular transmission structures similar in design to the proposed Bonneville Power Administration's transmission poles. This problem being wind-induced vibrations which resulted in actual structural failure of the crossarm components of the ' pole structure. The ieee of this report is to introduce the reader to one aspect of structural design which is becoming more important in modern structures. This is wind effects on structures. Selected from this broad subject is one minis- cule aspect, that being wind-induced vibrations sometimes referred to as aeolian vibrations. The development of this problem can be seen from the case history, part of which is included in this paper. Then to better understand this phenomena a basic introduction to the fundamental theory that applies to aeolian vibration is 2, presented. With this basic understanding the doors start to open to a field as yet unconquered. Presented are the parameters used in the structural analysis. There are two major points of interest in the structural analysis when considering wind-induced vibrations. These are the fatigue analysis and the determination of the critical wind velocity. Of these two, the investigation presented here deals with the determination of the critical wind velocity. With this value the designer can ascertain whether there exists the problem of wind-induced vibrations. Once this is established, then further investigation would lead to a fatigue analysis to determine the effects to the structure due to this type of oscillation. An example demonstrating the procedure used to obtain the critical wind velocity for one type of tubular trans- mission pole being used by Bonneville Power Administration is presented. This example makes use of the basic theory and parameters introduced in the previous sections. This is followed by a discussion on various methods used for the rectification of such problems. All of the information contained in this report was obtained through intensive research of current and past literature extending from 1931 to 1973. It is presented here with the hope that it will help to give a better per- spective into the problem of wind-induced vibration of tubular poles. HISTORY The problem of wind-induced vibrations has been plaguing the engineering profession for many years. The earlier prob- lems dealt with wind-excited vibrations in bridges and chimneys. These problems are new and are the result of modern dovetingante, The phenomena that cause the vibration is analogous to fluid-induced motion vhich is caused by eddy formation also known as vortex shedding. This phenomena can be traced as far back as Leonardo da Vinci, who describes eddy formations with drawings in his notebook.(2) In 1911, Von Karman published a paper on eddy formation explaining in great detail this phenomena. His contribution was so enlightening that this phenomena has been named after him, Von Karman vortices. The seriousness of these eddies applied to civil engi- neering structures had not been realized until the dramatic collapse of the Tacoma Narrows bridge, 1940. Following this failure extensive research was conducted with respect to dynamic wind forces. Prior to this, the design effects of wind were considered with the use of static wind loads which were based on an arbitrary force constant, ignoring the shape of the structure, the actual value of the wind velocity and direction of the site. From this time on there have been numerous cases of steel stacks oscillating due to vortex shedding. Some 4 examples are: Oleum steam plant, 1941, wind velocity 45 mph; Moss Landing steam plant, 1949, wind velocity 30-40 mph, vibrations occurred before the gunite lining was applied. It was found in most cases which dealt with tall steel stacks that the application of the gunite lining resulted in the elimination of the vibration problem. The gunite lining increased the inherent dampening thereby reducing the potential wind-induced vibrations. To reduce this prob- lem the Contra Costa stacks were guyed immediately upon their ‘erection, the guys being removed upon the completion of the lining. Den Hartog concluded in an article in 1958, (C25 that tall welded stacks were unsafe unless precautions were taken to eliminate the potential of wind-induced vibrations. With respect to this remark, it has been felt that riveted stacks were safer due to the dampening action of the rivets. Steel stacks are not the only structures that are affected by vortex shedding. Transmission power lines, street lighting structures, cable systems and now tubular transmission towers have become victims of wind-induced oscillations. The two most current problems to date are tubular transmission towers and bridge cable systems. An example of a cable system would be the Fremont Bridge just completed in Portland during the summer of 1973. This is a double-deck tied arch bridge. After completion it was reported that the cable system was experiencing wind-induced vibrations. The solution was to apply spacer dampers such 5 as are used on transmission lines to prevent vortex-induced motion. Almost all of the structures just mentioned can be classified as slender structures. These types of structures usually exhibit low inherent dampening thereby being more susceptible to wind-induced vibrations. In all cases correc- tive upasuxes have to be taken to eliminate or reduce this potential problem. An example of stacks was given for the Contra Costa stacks where guys were used until the gunite lining was applied. This report will give some procedures that could be used on tubular transmission poles. BASIC THEORY Two dynamic forces, drag force and vortex shedding, are a source of concern with respect to wind response (Fig. 1). Vortex shedding produces a force perpendicular to the direc- tion of flow, while the drag force produces a force parallel to the flow. The drag force is represented by a mean force calculated from a randomly distributed gust fluctuation curve. The vortex shedding force can be described by a steady state force related to the mean wind velocity. This representation ignores the unsteady component due to the random variation of the wind velocity. This procedure is as accurate as current state-of-the-art methods allow. WIND VORTEX STREET ae VORTEX. FORCE / FIG.1.-DYNAMIC FORCES In this report we are concerned only with the vortex shedding forces. If there is a need for more information about drag forces, consult Reference 3. Vortex shedding forces are a result of aerodynamic instabilities caused by the geometric shape of the structure. As flow of air passes the structure, vortices are produced on alternate sides of the cross section. These vortices are of equal strength, but opposite rotation. This alternate shedding gives rise to an alternating lift force which tends to move the structure in a direction transverse to the wind stream. This phenomena was initially described by Theodor Von Karman and is now commonly termed aeolian vibration. Figure 2 and Table I will help explain the phenomena of vortex shedding. As the flow of air approaches the windward side of the cylinder, it is deflected at 90 degrees to the wind direction, point "A". At this point, the velocity is equal to zero. This position is called the stagnation point and Bernoulli's theorem gives the pressure at this point. 1/2 V,7p +P = WU2V-p +P=C 1. Vy and F - Velocity and Pressure of approaching free stream. V and P ~- Velocity and Pressure at any given point. C - Total pressure of fluid at infinity. P - Mass density Ps =F + 1/2p\y? @ Vs = 0, Point “A” a FLUID DIVIDES AT CONFLUENT AT "C" oa" FIG. 2.- IDEAL FLOW TABLE |.- KEY POINTS FOR IDEAL FLOW Since we are interested in only the difference in pressure, the above equation can be expressed as: - iz BR ~% = 129% ™ Therefore, the difference in pressure is a function of the velocity of the free air stream squared and the mass density. This expression is usually denoted as "¢ . ra 2 G = Wzyt, 4. As the flow proceeds around the cylinder, point "A" to "B" Figure 2, more fluid is deflected, thus causing a pro- gressive increase in velocity. Corresponding to this increase in velocity is a decrease in pressure. Now using Bernoul1is equation and experimental results for a circular cylinder, the velocity at a position 90 degrees from the stagnation point, point "B", is twice that of the free stream. Using Equation 1.: Pop = MER in U2 atagy? ) = 1/29 (vy, * - av, = -3 7 / 5. Therefore, for an ideal fluid a suction three times as great as the stagnation pressure exists. Figures showing various pressure patterns for different geometric shapes can be found in Reference 4. 10 Bernou1118 equation was developed for incompressible flow conditions. In assuming that air is incompressible, very little error is induced. It has been shown that at 175 mph the error is about 2 1/2 percent, and at average velocities it is negligible (4). Several stages of flow patterns around a circular cylinder are shown in Reference 5. Initially the potential flow which exists briefly after the beginning of motion of the fluid corresponds to the pressure diagram for ideal flow. The forces that are required to deflect the wind around the cylinder, and the orientation of these forces are directed toward the center of the curvature of the stream lines. The stagnation pressure, which developed on the windward side, is opposed by an equal force on the leeward surface. There- fore, the ideal flow case has no net force, tending to move the cylinder. If the ideal case is applied to Figure 2, a fluid particle flowing from "A" to "B" attains its increase in velocity due to the decrease in pressure between "A" and "B". The kinetic energy would be just enough to allow the fluid particle to reach point "C" against the increase in pressure from "B-C". But after the initial stage it is necessary to consider the effects of viscosity. Viscosity causes the air flow near the surface of the cylinder to slow down. Conse- quently, the fluid particles at the surface do not reach the point "C". More and more retarded fluid is being continually 11 added and suffers the same fate, a steadily increasing accumulation is formed between "B" and "C". This accumula- tion sets itself into motion towards the region of minimum pressure and pushes away the outer stream lines. The release of this accumulation of air causes the periodic discharge of vortices. The movement of the accumulation of air towards the point of minimum pressure is sometimes called "reverse flow". The flow approaching from the front leaves the boundary at a point somewhat behind the point of minimum pressure. This point is referred to as the point of separation. This point will be shown on a diagram after the discussion on Reynolds number. While a vortex is forming, the fluid on the side opposite the eddy has a higher velocity as it passes the cylinder. In accordance with Bernoullis theorem, this increased velocity must be accompanied by a decrease in pressure. There is, consequently, a lift force exerted upon the cylinder acting in a direction away from the side on which the eddy is formed. The alternating vortices result in an alternating lift force which causes wind-induced vibra- tions, the direction of vibration being perpendicular to that of the wind flow. The shedding of vortices can be expressed in terms of the Strouhal number: F' - Shedding frequency 5 aan D - Projected diameter V- Wind velocity Another parameter describing the es a structure to fluid (air) flow is the Reynolds number: = WD R=" 7. V - Wind velocity D - Projected diameter “V - Kinematic viscosity Both the Strouhal and Reynolds numbers are pure numbers; that is to say, they are nondimensional, Another way to represent the Reynolds number is: Re Inertial force 8 Viscous force e With the substitution of the kinematic viscosity at approxi- mately 60 degrees Fahrenheit into Equation 7., the Reynolds number becomes: R = 362.5 VD 9. V - in units of mph D - in units of inches The relationship of the Strouhal number to the Reynolds number is shown in Figure 3. This figure is for a typical a 13 circular cylinder. Note that there is a distinct range in which the curve has certain characteristics. For the sub- critical range, 300¢R£3(10)>, the structure experiences a predominate frequency, but the amplitude of the alternating forces become random. For the critical range, R>3(10)°, the frequency and amplitude of the alternating force become random. Therefore, within the subcritical range there exists a constant Strouhal number for a continuous cross section. When the air flow approaches the critical range, the trailing air stream becomes turbulent. Figure 4, a to e, illustrates the relationship between the Reynolds numbers and the repre- sentative flow patterns. Figure 4a represents the initial flow conditions. Figure 4b indicates that there exists two stationary vortices with the separation points Sj} just forward of the center section. These separation points Sy are classified as laminar separation. Figure 4c shows that as the velocity is increased the vortices detach themselves and form a vortex street. Von Karman experimentally determined the given characteristics. The velocity of the vortex street is 86 percent of the free stream velocity and these character- istics have been measured six to seven diameters downstream from the cylinder. Figure 4d illustrates that the vortex street is no longer constant and that the separation point oscillates between Sj and Sos The width of the vortex street and pitch of the vortices oscillate in a similar manner. DRAG COEFFICIENT SUBCRITICAL CRITICAL 10° 10° 10* 10 10° REYNOLDS NUMBER (R) FIG.3.-REYNOLDS NUMBER, STROUHAL NUMBER,DRAG COEFFICIENT STROUHAL NUMBER 15 ¢ | d— diameter (A ———>{ |, | ———-~ SS) = FIG4a.-R<5 REGIME OF UNSEPARATED FLOW wae ee 6to 7 diameter down stream F1G.4c.- 40<R<(10)? (L.S.) ¢ TRANSITION TO TURBULENCE FIG. 4D -(10)5< R < 3(10)§ ¢ SP acs ae ig ae >, », on ee FIG. 4E- R>3(10)6 16 17 For R~=3(10)6 the flow changes from laminar to turbulent with the point of separation at Sj. At this time there is a sharp drop in the drag coefficient and an abrupt rise in the Strouhal number (Fig. 3). The wake narrows with regular vortex formation no longer visible. This region is known as the critical point and the Reynolds number is called the "Critical Reynold's number". Figure 4e shows that the sepa- ration point remains at S, and that there is a turbulent boundary layer with a thin wake. Similar characteristics can be shown for a flat plate, except that sharp edges fix the point of separation of the vortices. The regular process of vortex shedding is inde- pendent of Reynolds number above R = (10)3, Figure 5 shows the vortex street characteristics. H=0.2806D D=5.5D FIG. 5.- VORTEX STREET FLAT PLATE 18 When the frequency of vortex shedding equals the natural frequency of the structure, resonant vibration occurs. There- fore, by knowing the natural frequency and section properties, the critical wind velocity can be determined for a particular section. F V critical = wi 2 ee Fy - Natural frequency of the structural system At this point, the structure will be experiencing maximum displacement due to vortex shedding. If we restrict the subject to the study of vortex shed- ding within the subcritical range we can make some simplifying assumptions. 1) The dynamic response will be its grestest within this range resulting from the constant nature of the forcing frequencies and their correspondence to one of the lower natural frequencies of the structure. . 2) The excitation forces can be approximated by harmonic functions. With these assumptions in mind, the lateral forces produced by the shedding vortices can. be expressed as follows: F's 1/2 p Vac, Sin(2-w £t) ll. £ - Mass density V - Wind velocity Projected area of the cylinder Lift coefficient Frequency of vortex shedding Time LS 20 SELECTION OF PARAMETERS The majority of the research discussed above was completed considering a uniform cross section and a station- ary member. Therefore, when analyzing a tapered cantilever section, special precautions must be taken into account in the selection of parameters. The Strouhal number, Reynolds number and the lateral force coefficient will all vary with the changing diameter of the tapered section. Also the magnitude of the alternating forces will be greater for the vibrating cylinder than for the stationary cylinder (6, 7). Wind tunnel testing is needed for the determination of the lateral lift force coefficient (c,) and the Strouhal number. These parameters have already been determined for basic shapes with constant cross sections such as circular, rectangular, square, Tee and I-shaped sections (1, 5, 8). After vibration starts, the frequency of vortex shedding becomes locked to the frequency of the structure. At this point, the structure possesses the ability of self-amplifi- cation and may be considered self-excited for a limited range of wind velocities at and around the resonant wind speed. As a result of this phenomena, a constant value of C, is not possible (6). L Because of the randomly varying value of Crs studies have been conducted to determine the appropriate value to be used. 21 Weaver (6) suggests the use of a RMS value of Cc, which can be related to the Reynolds number. Due to the difficulty in determining the Cc. value, it was found that most studies used Cc. = 1, and this was felt to be conservative (10, 11, 12). The use of C, = 1 is to say that the maximum Von Karman force is about the same as the stagnation force. Other investigations of circular cross sections have revealed the range of C; to be between 0.65 and 1.71 depending on the value of the Reynolds number (11). In another report it varied from 0.15 to 1.3 (13). All of these values were determined on stationary, constant cross-sectional circular shapes. Reference 1 established two values of Cc for a constant octagonal-shaped member. These two values were calculated for the positions of the crossarms as shown below in Figure 6. These positions will be referred to as the flat-up and point- up orientations. c.= 0.21 c= 0.36 FLAT- UP ORIENTATION POINT-UP ORIENTATION FIG.6.-C, FOR OCTAGONAL CROSS SECTION 22 The preceding values correspond to the maximum value of Cy (Cc, = 0.4) which was established by Vickery and Clark for a tapered circular section (14). This value occurred within the upper one third height of the stack. It can be concluded from the lift coefficients that the response to wind excita- tion for the octagonal cross section is 70 percent greater in the point-up orientation than in the flat-up position. The research study by Vickery and Clark (14) considered the effects of a tapered section on the Strouhal number, Reynolds number and the lateral force coefficient. Figure 7 shows the relationship of the shedding frequency in respect to a 4 percent tapering height. There are two curves for each flow condition, one being for a constant Strouhal number and the other for actual measured values. In the case of turbulent flow, the two curves are in relative agreement for the lower three fourths of the structure and there is some discontinuity for the upper one fourth. The conclusion would be that for turbulent flow, the structure can be characterized by a constant Strouhal number and varying frequency; whereas for the smooth flow, the measured values follow the general shape of the constant Strouhal number but with distinct continuous steps. This would indicate that a constant frequency does exist over certain intervals. A major finding in the Vickery and Clark report was that the maximum response in the fundamental mode occurred when the shedding frequency at approximately one third of the stack HEIGHT —~ (INCHES) TIP 36 Pi af oe 32 28 TURB. FLOW 24 20 16 SHEDDING FREQUENC SMOOTH 12 FLOW 8 AAA oO 10 20 30 40 50 60 70 SHEDDING FREQUENCY - F, (cps) FIG.7.-STROUHAL VS. HEIGHT 28 HEIGHT — (INCHES) TIP 2u0)*< R < 710) ee " ” SMOOTH FLOW 0.2 0.4 0.6 0-8 Cie (RMS) FIG.6.-C, VS. HEIGHT 1.0 24 25 height was equal to the natural frequency. This conclusion was in contrast to earlier works (14) where the tip of the structure was critical. Figure 8 shows the variation of Cy for the tapered cross section. Again the maximum C, (Cc, =0.4) occurred in the upper one third of the stack height. The Strouhal number was established in Reference 1 for the octagonal shapes; S = 0.19, point-up, and S = 0.17, flat-up. This indicates that the point-up orientation would respond to a slightly lower wind speed than the flat-up. The above Strouhal numbers were established on constant octagonal- shaped cross sections. Table 2 shows some of the suggested values for Strouhal numbers and lift coefficients for various shapes. These values have been taken from various sources and it is suggested here that they not be used without further investigation of a particular problem. The values given are limited to within the subcritical range, (10)2<R<(10)>. CIRCULAR O.15- 171 TABLE 2,- STROUHAL & LIFT COEFF- ICIENT VALUES 26 The flow diagram shown below indicates the steps used in determining the Strouhal number. Wind Tunnel tests Vortex shedding frequencies are measured : (Hot-wire anemometer) Power Spectrum Density of signal coming from the anemometer is measured and the prevalent frequency is determined. Knowing the wind velocity and characteristic dimension D of the section, the Strouhal number can be determined. In the analysis that follows, only the Strouhal number will be used. The lift coefficient is necessary when con- ducting fatigue analysis and can be determined from energy relationships after wind tunnel tests have been performed to determine necessary parameters. OU GENERAL CONSIDERATIONS The plane exhibiting the most severe vibrations is perpendicular to the direction of wind flow (10). In the case of transmission poles and similar lighting structures, this plane would be normal to the crossarm orientation. It was found in Reference 1 that the maximum amplitude was in the vertical direction and that the horizontal amplitude was generally less than 10 percent of the vertical amplitude. Also the maximum response of any pairs of opposite arms occurred when both arms were moving up and down at the same time. This motion can be described as an in-phase condition. Research to date shows that the maximum response does not continue to increase with wind velocity and that maximum amplitudes occur at certain critical velocities. These velocities correspond to those associated with the Von Karman vortex street (15). Tests conducted in Reference 1 showed that the amplitude increased to the first critical wind velocity and then began to drop with increasing velocity. The velocity was increased to three times the critical wind velocity with no further increase in amplitude above that, at the critical wind velocity. This does not always seem to be the case. There have been reports of structures experiencing severe vibrations at higher wind speeds, but the cause of this phenomena is not presently understood. 28 The possibility of having large amplitude displacements under resonant vibration conditions depends on the value of the energy introduced by the wind in comparison to that dissipated by the system. The energy per cycle induced by the wind can be expressed by the following equation: Wy = 7 Px 12. P - Equivalent lift force produced by the wind X - Amplitude of vibration The energy dissipation per cycle due to structural damping can be expressed as: W. = Thowx? 4. W - Frequency of the system C - Damping coefficient X - Amplitude of vibration At resonance this equation becomes: wo= 2$E 14. & - Logarithmic decrement E - Total energy of the system (either kinetic or elastic) The energy input due to the wind acting on a cantilever section can be expressed as: L 1S5V"D We 2g y dx 15. 29 Cc. - Lift coefficient vt "ec Dynamic head of wind where: V - Wind velocity Y - Specific weight of air g - Acceleration of gravity and _ D - Project diameter L - Length of specimen subjected to the wind y - Displacement at a dx distance along the axis of the structure Reference 11 gives an approximate equation for the energy per cycle applied to an elastic cantilever by a simple harmonic force acting uniformly along its length at resonance which is: w= (2/5) Pa 16. Yo - Elastic deflection of end of cantilever W - Uniform static load per unit length Z a Cy v°D 28 L - Length of cantilever The above equations can be used to determine the amplitude in respect to the energy input and dissipation. Oscillations occur when the energy input due to the wind exceeds that dissipated by damping and the amplitude will increase indefinitely if both excitation and damping are 30 independent of amplitude. Since both quantities are amplitude-dependent the oscillations are limited. This phenomena can be seen in Figure 9. The energy input is ENERGY INPUT uw a © > o - AMPLITUDE AMPLITUDE ae > aa INCREASE DECREASE © « i DAMPING ENERGY DISSIPATION wW oO AMPLITUDE FIG.9- AMPLITUDE VS. ENERGY greater than the energy dissipation from point "0" to "A" and the amplitude will increase to this point. Once passing through point "A" the dissipating energy becomes larger and the amplitude starts to decrease (16). C. Scruton developed a stability diagram which relates the critical parameter in determining the potential of wind- induced vibration (17). Referring to Figure 10,while the structure is at rest, the vortex frequency is controlled by 10 INSTABILITY REGION \—1ine "a" ght 92 Wd oe Se Oo 0.02 0.04 0.06 0-08 LOGARITHMIC DECREMENT (§) FIG.10.- STABILITY DIAGRAM Si 32. ORDER OF INCREASE IN LARGE RANGE PREFERENCE SECTION STRUCT. DAMPING | OF OSCILLATION NEEDED PERIOD CIRCULAR C.S. |_| eancuan x8 a DODECAGONAL TABLE 3-PREFERENCE OF CROSS SECTION DoD. UNSTABLE UNSTABLE UNSTABLE STRUCTURAL DAMPING FIG.11- STABILITY DIAGRAM VARIES CROSS - SECTIONS 33 the wind velocity. At certain discrete wind velocities, designated as critical velocities, the vortex frequency will lock in on the natural frequency of the structure. This corresponds to line Ae Once oscillation starts it will cont inue to a high wind velocity at which time it will start to decrease. It is shown that the range of instability is a function of the structural damping or logarithmic decrement « The higher the inherent damping the smaller the insta- bility range to a point where the structure is stable until the next critical wind velocity. Another study conducted by Scruton and Flint (13) shows the stability diagram for various shapes (Fig. 11). It was found that circular constant sections (C.C.) needed less structural damping to suppress oscillations. The order of preference for three structural shapes are summed up in Table 3. The damping characteristics can be determined on a percentage of critical damping basis. Typical values for metal poles range from about 0.05 percent to 0.5 percent (18). Damping ratios can be determined by starting the structure oscillating and counting the number of cycles required for the oscillation to decay to half the initial amplitude. 34 CALCULATED RESULTS The problem that will be dealt with here applies to Bonneville Power Administration's 500 KV Custer-Ingledow transmission line. This line consists of three tubular H-frame structures. The geometry of these structures is octagonal-shaped crossarms and dodecagon-shaped vertical poles. Dimensions are shown in Figure 12. All three towers have the same basic shape except that the height varies by 25 ft. The classification of these towers is as follows: 18 Hf-L Type 1 tower, 115"; and 18 Hf-L Type 2 tower, 90°. Type 1 tower will be dealt with first. 18 Hf-L Type 1 Tower, 115' The natural frequencies and characteristic shapes were obtained through the use of the computer program, SNAP- Dynamics, developed by W. D. Wetstone of Lockheed Corporation. .This program uses the Rayleigh-Ritz analysis to obtain initial approximations of the first modes and frequencies of the system. After the initial approximation of the modes has been computed, the program executes an iterative procedure to compute initially the first lowest frequency mode, then the second mode, and so forth. In computing higher modes a process based on orthogonal relations is used to ‘sweep out’ lower frequency components of each mode. 35 [ 31'-1-3/8" 30' 3I'-1-378" ie" ie" u 0 0 a \ : ; as OCTAGON OCTAGO DODECAGON us’ 39-1/2 39-1/2" FIG.I2.- TOWER DIMENSIONS 36 The effect of the pole to the total vibrating system can be neglected if the critical wind velocity is relatively small. This can be proven by the relationship F' (perpen- dicular force)av (critical wind velocity). Therefore, the system can be partitioned into individual systems consisting of crossarms and the vertical pole. These sec- tions were analyzed on a lump mass basis. Because these are variable sections, the average values over a chosen interval were used for calculating the section properties (Fig. 13). The number of intervals selected to obtain the appropriate accuracy can be determined from experience or by a number of trial runs.. It can be seen from Table 4 that with a large difference between the number of intervals the change in accuracy is small. INCREMENT FREQUENCY 9.67093859E - Ol 3 9.70228766E- Ol Uf 73 9.7086 OI87E- Ol TABLE 4. INCREMENT Vs. FREQUENCY The vibration in the vertical plane was assumed to be critical. The cantilever arms were considered fixed at the pole connections. Connections to the midsection crossarms ae r- wiesttack vas LUMP MASSES A, = Davg. uj pam FIG.13.- LUMP MASS & AVERAGE DIMENSIONS 37 < 38 are considered pinned. Because there was no available data for the Strouhal number for a dodecagon shape, S was assumed to be 0.20, which corresponds to a circular cross section. The error induced by this value is small because the larger the number of sides, the closer the response resembles that of a circular cross section. The parameters used to calculate the critical wind velocities are shown below: Vertical Cantilever Midsection Pole Crossarm Crossarm Shape dodecagon octagon octagon Strouhal number 0.20 0.17 0.17 Diameter Miedo Oe 50 caus 143° For this tower design the crossarms are oriented with a flat side up. The upper one third of the height of the vertical pole was used to obtain an average diameter. The smallest diameter of each crossarm was used because of the upswept shape. The base of the vertical pole was assumed to be fixed; but for a more accurate analysis, the interaction between the pole and the foundation should be considered. This relation- ship would have some effect on the natural frequency of the system, which would change the critical wind velocity. Natural frequencies obtained fromSNAP -Dynamics are shown below with their corresponding critical wind velocities. 39) Vertical Cantilever Midsection Pole Arm Arm Natural frequencies 0.962 cps 11.044 cps 33.080 cps Critical Wind Velocities 4.41 mph 22.93 mph 180.65 mph Comparison of these values with those obtained from Refer- ence 1 would indicate that a potential problem does exist. Vertical Cantilever Pole Arm Natural frequencies 0.6 cps 7.0 - 9.0 cps Critical Wind Velocities 4.0 mph 15.0 - 30.0 mph At low wind velocities, 4.41 mph for the pole, very little energy is available in the wind to maintain resonant vibrations of the pole structure. Since the energy input from the wind is porportional to the applied force, which is proportional to the velocity squared, the smaller the veloc- ity the less energy produced. Test data from Reference 1 indicate that for the poles the maximum amplitude measured was less than 0.5 inches peak to peak. At this point the vibration of the vertical pole can be neglected (1). The midsection crossarm has a critical wind velocity of 180.65 mph. Since this velocity is significantly larger than the average expected velocity, there is no poten- tial problem for this section. A potential problem does exist for the cantilever arm. For this arm the calculated critical wind velocity is 22.93 mph. Because of the assumptions made in selecting the 40 parameters, it would probably be safer to assume a critical wind velocity between 20-25 mph. Wind Rose charts can be used to establish the possibility of experiencing wind velocities within this range (19). The 18 Hf-L line will run between Custer, Washington and Ingledow, Canada, in a northerly direction. Shown below are the average wind veloc- ity, direction and percent duration for January, July and on an annual basis. Month Velocity (mph) Direction % Time January 16 - 31 SE 5 16 - 31 S 3 16 - 31 SW 2 4 - 15 -- 72 0-3 -- 18 July 16 - 31 s el 16 = 31 SW 2 4-15 : -- 74 0-3 -- 23 Annual 16 - 31 SE 2 16 - 31 Ss 3 16 - 31 SW x 4-15 -- 13 0-3 -- 21 41 It can be concluded from this data that 6 percent of the time the 18 Hf-L cantilever arm will be subjected to wind-induced resonant vibrations. The accuracy of this con- clusion depends on the reliability of the above wind data in regard to the location of the 18 Hf-L line. The data used was collected at the Bellingham Station, which is approximately 15 miles from Custer. Therefore, to apply these values to the Custer-Ingledow line, the influence of mountains, hills, valleys, gorges and even timbered areas must be considered. These geographical characteristics can change both wind speed and direction. To correct the vibration problem one of two things can be done: 1) Install the insulator assemblies immediately after erection of the structure. 2) A force could be applied to the critical arms equal to the insulator assembly weight. This load should also Wd. applied immediately after the structure is erected. Both procedures would lower the critical wind velocity to the point where the structural damping is significantly higher than the wind energy input which results in smaller vibration amplitudes. The insulator assembly has two attachment points, one at each end of the cantilever arm. The approximate weight of this assembly is 930 pounds. The vertically applied force 42 at the attachments would be 465 pounds. This applied force reduced the natural frequency to 1.557 cps, and the critical wind velocity to 3.12 mph. This would reduce the maximum amplitude and duration. The natural frequencies and critical wind velocities for the individual sections, considering the second modes, are shown below. Vertical Cantilever Cantilever Arm Pole Arm with Insulators Natural frequency 2nd Mode 4.063 cps 35.456 cps 14.626 cps Critical Wind Velocity 23.8 mph 70.0 mph 29.3 mph The critical wind velocities have substantially increased. There is no question that the cantilever arm without insula- tors is out of the critical range. Both the vertical pole and cantilever arm with insulators are in the range of an average wind velocity. Calculation of the Reynolds numbers gives R = 1.75(10)° and 6(10)*, respectively. For the ver- tical pole, the Reynolds number can be assumed in the turbu- lent range, at which point the vortex shedding changes from periodic to aperiodic. It can be assumed in this case that there is no potential problem. The Reynolds number for the cantilever arm with the insulator assembly is still in the subcritical range. This would indicate that there is still a problem. A crude approximation was used to determine the effects of the 43 insulator assembly. A more accurate model would include the interaction of the arm and the assembly. This would have some added damping characteristics and would affect the vibra- tion amplitude. Also at no time in the literature was there mention of a case which exhibited second mode vibrations. It is believed that this condition will not be critical and that no further modification will be required. The use of an applied force was not modeled, but the effect would be the same as for the insulator assembly. One other consideration when considering the applied force method is the change in fixity of the end of the arm. With this method the beam no longer acts as a cantilever. Partial fixity exists at the end due to the elastic properties of the member used to apply the force. This should help to lower the natural frequency. 18 Hf£-L Type 2 Tower, 90' The results for this tower are the same as for the Type 1 tower except for the vertical pole, which is shorter than the Type 1 tower. Shown below are the parameters used for this section, natural frequencies and critical wind velocities. Vertical Pole Shape dodecagon Strouhal number 0.200 Average 1/3 Height Diameter 1.646 ft 44 Natural frequencies: lst 1.358 cps 2nd 5.958 cps Critical wind velocities: lst 7.62 mph 2nd 33.50 mph Reynolds number: 2nd 2.40(10)°? These results compare with the Type 1 tower--low energy input at the first mode and turbulent flow conditions for the second mode. In these problems the wind velocity was assumed constant over the cross section. It can be shown that for poles 100 ft or less in height, the variation in wind velocity is not significant. For poles 200 ft high the velocity at the top may be 30 percent greatér than the velocity at the ground elevation. And for poles 500 ft high the velocity at the top may be 50 percent greater than at the ground level (7). Therefore, for the two poles used in the above example, the constant wind velocity approximation would be reasonable. 45 RECOMMENDATIONS Slender tubular members subjected to moderate winds will exhibit fatigue-type failures in weld-heated areas at points of maximum flexural stress. One example of this is indicated in Reference 1 where the vibrations, which were principally in the vertical direction, produced cracks at the arm shaft-to-flange plate welds. These cracks occurred in the heat-affected zones. Metallurgical examinations of the fractured surfaces disclosed that these failures were being caused by fatigue, which in turn was attributed to wind-induced resonant vibrations of the arms. Because the vertical pole experiences small amplitudes over a large dis- tance, the flexural stresses at the base can be neglected. This would account for the reason why the base-welded areas have not experienced fatigue-type failures. Listed below are several methods by which wind-induced vibrations can be suppressed. 1) Increase the flexural stiffness of the member so that the critical velocity is above the range of moderate wind. 2) Reduce the effective length of the member through the introduction of intermediate supports. 3) Use damping devices to restrict the amplitude of vibration. 46 4) Attach a device that would disrupt the flow near the surface and to interfere with the regular formation of vortices, thereby destroying the cause of vibrations. Increasing the flexural stiffmess would be a considera- tion during the design procedure. An example would be to increase the wall thickness or change the cross section which in either. case would mean an increase in material and cost. If the structure is already in service, this method would be difficult to apply and one of the other methods should be used. Changing the effective length of a tubular pole would be difficult since this structure in general is a single member system. Also, the aesthetic value would be impaired by changing the simple tubular configuration of this type of structure. One indirect method to chante the effective length would be the use of guy cables, but the true value of this type of system would be from the impact damping of the cable more than changing the effective length. This type of system would also change the aesthetic appearance. Damping devices can be used to eliminate wind-induced vibration. Two such devices are the Stockbridge and Fanner dampers. Attaching these dampers to the ends of the members effectively reduces the problem of vibrations. These dampers are used on transmission lines to eliminate oscilla- tions. Again this type of damper can affect the aesthetic appearance of the structure. 47 For transmission poles, the installation of the insulator assembly or the hanging of the stringing blocks would be effective damping devices (1). This type of a system would increase the mass of the structure and this would produce a reduction in the natural frequencies. The end result would be a lower critical wind speed, and the possibility of bringing the second mode into the critical range. At this point it would have to be determined whether this would present more of a problem than it solved. There are many devices that can be used to disrupt the flow near the surface. The following discussion will cover two of the most common methods and one that is not often used for slender structures. The one that is not as common deals with elimination of the cause of vortex shedding. Under basic theory it was shown that the phenomena of reverse flow caused the shedding of vortices. Therefore, to elimi- nate vortices, eliminate reverse flow. Prandtl has proved experimentally that with the removal of retarded fluid there is a restabilization of fluid flow (20). This could be done by slotting the rear surface and creating a suction to draw the retarded fluid (air) from that surface. A system like this is not practical and therefore not commonly used. The enclosure of the structure with a concentric, perforated shroud provides effective suppression of wind- induced vibrations (21). It has been found that only the upper third of the structure need be enclosed. This type of system has found more use in stacks than in tubular poles. 48 The second most common system is the use of helical strakes. These are small tubular sections which spiral around the outer surface of the structure. This seems to be the most effective system. Strakes themselves do not prevent periodic vortices from being dischared. They are effective as stabilizers because they destory the correla- tion of vortex shedding along the height of the member. The variable parameters for the optimization of strakes are: 1) Number of windings 2) Size of windings 3) Pitch of windings 4) Portion of height over which windings are applied. Helical patterns are used because they are not sensitive to the direction of flow. It has also been found that sharp- edge rectangular strakes are more effective than strakes of circular or other rounded sections (22). Tests have been conducted on strakes for the range of Reynolds numbers between (10)4e R<(10)>. It is not presently known how they will function at lower values. It is recommended for cantilevers with the ratio of length to diameter greater than 20 (6) not to use strakes as a means for correcting aeolian vibration problems. Another possibility for reducing the amplitude would be the use of a foam solution which could be injected into the member after erection. The foam would provide damping to reduce the amplitude of vibration. 49 Reference 1 recommended the following methods for reducing the problem of aeolian vibrations when considering the design of tubular poles: 1) When possible, position the octagonal-shaped cross section in the flat-up position. This reduces the lift force by 70 percent in respect to a point-up orientation. 2) Eliminate drain holes and hermetically seal the cantilevered crossarms. 3) Use nongalvanized poles. Experiments showed that during fatigue tests of these poles, they experienced one million cycles as compared to 37,000 cycles for galvanized poles. 4) Stress relieving without galvanizing was recommended. This was the most effective method of extending the life of the arm. 5) Increase the stiffness in the design stage, which would raise the critical wind velocity above the average occurring value. 50 HIGHER MODE VIBRATIONS The conclusion made with respect to second mode vibrations of the 18 Hf-L Type 1 tower was based on experi- mental results conducted in Reference 1. Actual tests were conducted on tubular crossarms with insulator assemblies. The result was no noticeable vibration up to 45 mph. The indication from the report was that the insulator assemblies were considered as damping devices and there was no mention of what effect these devices would have on the natural frequency. In the case of the 18 Hf-L Type 1 tower there is a noticeable decrease in the natural frequency of the second mode which has lowered the critical velocity for that mode into a reasonable range of common wind speed. The statement made that this condition does not seem to be critical based on conclusions made in Reference 1 should be investigated further. Conclusions made in References 15, 18, and 22 would indicate there does exist a serious condition at the higher modes. These reports also did not take into account the damping characteristics of a damping device such as the insulator. Therefore two further possibilities should be studied: 1) What is the effect of the insulator assembly damping characteristics on the structure. 51 2) What is the magnitude of the inherent damping of the structure in the second mode as compared to that for the first mode. Statement one would mean determining a damping coefficient for the insulator assembly and comparing it to the damping characteristics of the structure to obtain the effect of the damping device on the overall system. Statement two would prove or disprove the assumption that the inherent damping would be greater for the second mode than for the first with respect to the energy input from the wind. The answers to the above statements would further prove the conclusion in Reference 1, that the installation of the insulator assemblies would be an effective means of damping wind-induced vibrations.