analysis and design of guyed transmission towers—case study in kuwait

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ANALYSIS AND DESIGN OF GUYED TRANSMISSION TOWERS-CASE STUDY IN KUWAIT

H. A. El-Ghazalyt and H. A. Al-Khaiatz

TDepartment of Civil Engineering, Cairo University, Fayoum Campus, Cairo, Egypt

IDepartment of Civil Engineering, Kuwait University, PO Box 5969, Safat 13060, Kuwait

(Received 21 December 1993)

Abstract-Guyed towers are special nonlinear structures requiring special techniques for analysis and design. The various aspects of geometric nonlinearity are discussed and the energy search method is explained, which is an algorithm believed to be suitable for the analysis of guyed towers. A two-dimensional example tower is investigated which demonstrates the effect of prebuckling displacements on the resulting critical wind load. A three-dimensional 600-m guyed tower case study is also analysed and designed using ANSYS and STAAD-III computer packages. Certain modelling assumption techniques are introduced in order to be able to use the two packages effectively for the analysis and design of guyed towers. The effects of accidental guy rupture and temperature variation have been studied as well.

NOTATION

cross-sectional area and moment of inertia, respectively jth displacement component modulus of elasticity number of elements number of degrees of freedom axial-bending coupling constants initial member length end moment subscripts indicating element terminals deformed chord length displacement components in the .X,JJ directions, respectively strain energy end force normal to chord line external work Eulerian coordinates local coordinates reference global coordinates angle between reference and local axes wind load multiplier strain prestrain potential energy angle of rotation

INTRODUCTION

Guyed towers are special structures that normally exhibit geometrical nonlinear behaviour. Such towers are frequently designed to heights in the range of 2000 feet to transmit and/or receive high frequency signals for various electronic communication systems. More recently, tall towers have been designed and utilized for supporting collectors in solar energy applications

and have been proposed for off-shore oil operations. The nonlinear behaviour of a guyed tower may significantly complicate the analysis of this structural system; it is this nonlinear aspect which generates the interest in the problem.

The mathematical model for a guyed tower is essentially a flexible beam-column with elastic supports. Guyed towers exhibit most, if not all, of the geometrically nonlinear aspects. The amplification of deflections and bending stresses due to the beam- column action is evident. At the same time, the tower may undergo large deformations under severe wind conditions, which may necessitate studying the equilibrium in the deformed configuration. The tower is usually prestressed in the unloaded state due to pretentioning the guys. Finally the change in the structural configuration, due to slackening of some guys on the leeward side, may have to be taken into consideration. In Ref. [I]. Rowe investigated the amplification of stresses and displacements in guyed towers when changes in the geometry are included in the analysis. Analytical charts were included in the same reference which show when refined methods of analysis are necessary in the design and what modifi- cations should be made so that the ordinary methods of structural analysis give adequate results. Dean [2] gave the necessary information to consider the sag of the hanging cable which takes the form of the catenary under its own weight. A stability analysis of guyed towers was presented in Ref. [3] by Hull, who tried to find the most critical moment of inertia that corresponds to specified wind forces. Hull suggested that increasing the stiffness of the guys is the most efficient means of increasing the buckling capacity of the tower up to the limit when the tower starts to buckle into a number of sine waves with nodes at the

413

414 H. A. El-Ghazaly and H. A. Al-Kharat

guyed levels. At that stage, increasing the guys’ stiffness will be ineffective, and the only way to increase the buckling capacity is to increase the moment of inertia of the mast itself.

Goldberg and Meyers [4] presented a method of analysis for guyed towers where the nonlinear behaviour was considered and the effect of the wind on the cable stiffness was also investigated. The technique employed was based on transforming the nonlinear algebraic equilibrium equations into a corresponding set of ordinary differential equations which were then integrated numerically. Reference [5] covered the complete analysis and construction aspects of the cylindrical television mast which was built for the Independent Television Authority at Winter Hill, Emley Moor and Belmont in the U.K. Unfortunately, the structure collapsed a few years later and was replaced by a high con- crete television tower of 1084 ft. whose description is given in Ref. [6]. The failure of the previously mentioned cylindrical mast was discussed by Williamson [7] in a stability study of guyed towers under ice loads.

Miklosfsky and Abegg [8] presented a simplified systematic procedure for the design of guyed towers using interaction diagrams which provide the designer with a graphical visualization of the design range without resorting to a trial and error procedure.

In an attempt to analyze guyed towers, Odley [9] presented a method of solution in which some secondary effects, such as the effect of ice loads and insulators located on the guys, shear deformations. initial imperfections in tower shaft etc., were included. Odley started the solution by assuming a set of displacements at each joint to calculate the spring constants of the guys, which were then used to obtain the tower deflections. The procedure was repeated until all assumed and computed values of deflections were in satisfactory agreement.

In a study of shear effects in the design of guyed towers [IO], Williamson and Margolin stressed the fact that, in the cases where the secondary moments and deflections, due to the bearncolumn action. significantly affect the final moments and deflections. the shear deformations should be considered to achieve a safe design. They also presented a means for modifying the conventional moment distribution factors when the axial thrust and web flexibility arc considered. Finally a formula was given to find the thickness of the fictious solid web, which has the same shear rigidity as a flexible trussed web.

Livesley [I I] attributed the nonlinearity in guyed towers to both the stiffening of the guy cables with increasing tension and to the destablilizing effects of the axial thrust on the mast iself. In the same paper, a procedure was described for calculating the guy tension in the cases where specified detlections arc not to be exceeded under a number of different loading conditions.

Goldberg and Gaunt [I21 presented a method for determining the response of guyed towers to increasing lateral wind loads until the conditions of instability are reached. The criterion for buckling was the occurrence of relatively large increase in deformations for small increase in the applied loads. A valuable study was also given to illustrate the influ- ence of certain system parameters on the critical load of the tower.

Williamson [7] examined the icing effects on special types of tall guyed communication tower called “top- loaded” towers, where the uppermost level of guys consists of an array of conducting cables which serve as a radiating element for an antenna system. The result of the study is expressed as a critical ice thickness which corresponds to the occurrence of the instability conditions in the tower.

More recently. Romstad and Chiesa [I 31 proposed an equivalent one-dimensional beam element to re- place the actual truss tower element, which dramati- cally reduced the number of degrees of freedom included in the analysis.

Recently Magued et al. [I41 concluded that the failure rate for guyed towers designed to earlier versions of the Canadian Standards is generally unac- ceptably high. Many of these collapses were due to excessive environmental loads, which exceeded those values believed to be maxima when the towers were designed. They also concluded that a new “ice-only” load case appears desirable, as some towers have been reported to collapse under excessive ice loading under conditions of light or no wind. Bruneau et al. [I51 further published guidlines for upgrading existing towers where structural reliability studies were conducted to develop rational guidelines applicable to the current edition of the standard. Three reliability classes were defined with increasing greater probabil- ities of failure corresponding to reductions in the load factors to be used in the analysis.

ANALYSIS CONSIDERATIONS

For a realistic static analysis of guyed towers. the following considerations should be accounted for:

(a) Equilibrium is to be considered in the deformed configuration to account for the additional bending moments resulting from horizontal drift due to wind loads.

(b) The reduction in the flexural stiffness as a result of the axial compression in the mast should be considered. The high tension in the supporting cables causes significant compression in the mast and con- tributes to the instability of the tower.

(c) The curved nature of the supporting cables should be accounted for. which necessitates the formulation of a special element stiffness matrix for the cable. This special element is not available in most commercially available finite element codes.

(d) Wind loads on the cables could have an effect on the overall behaviour of the shaft. This effect is

Analysis and design of guyed transmission towers 415

even augmented in the case of accumulation of ice on the cable, thus increasing the cable projected area subjected to wind. Needless to say that in the local environment of Kuwait snow loads are rather rare, but severe sand storms are not unusual.

(e) Pretension in the cables and precompression in the tower shaft should be accounted for in order to properly describe the stiffness of the structure under service loads. The energy search method [16] con- siders the member prestrain and evaluates the member strain energy accordingly.

(f) A successful analytical procedure should be capable of accounting, internally, for the changes in the structural stiffness, even in the linear range, due to members going out of service, depending on the state of deformations. In guyed towers, the cables are usually pretensioned and some or all of the cables on the leeward side may go out of service and will have no contribution to the structural stiffness under increasing wind loads. In conventional finite element codes this feature may require either assiging very small stiffness to slackened cables or renumbering the nodes to eliminate the “out of service” members, thus avoiding singularity of the resulting stiffness matrix. In this paper the energy search method, as detailed in Ref. [16], is used to solve an example of a planar guyed tower. This method needs no special provisions to account for members going out of service, since only active members contribute to the structural strain energy during the search for the proper displacement configuration. This will be explained later and has been throughly illustrated in Ref. [l6].

In the following the formulation of geometric nonlinear analysis planar frame element is presented for incorporation within an energy search approach.

Expressions for the potential energy of the element, as well as its analytic gradient, are obtained since both are required for implementation of the function minimization technique employed in the sequel. A Eulerian local coordinate system attached to the deformed element is used in conjunction with a nonlinear strain-diplacement relationship, which reflects the coupling action between bending and axial stiffnesses.

A typical frame element in the undeformed and deformed positions is shown in Fig. 1. The R and p axes represent a reference coordinate system with - - displacements in X, Y directions, denoted by U, 5, respectively. The undeformed length of the member is

L = [(Fq - Xp)’ + (F<, - Pp))?]’ ? , (1)

where (F,,, F,,) and (X,, yq) define the coordinates of the p and q ends of the member before deformation. The distance between the ends of the element in the deformed state defines the chord length s given by

+ [( y<, + z;,, - ( yp + t:, )I?} ’ ? , (2)

where C,,, U, and &,, L;, are the displacement components at the ends of the member measured in the R and P reference directions.

The local coordinates 2 and p define the direction of the element in the underformed position and displacements in the .C,_Q directions are denoted by zi,,, z?,, and zZ, ~7, at ends p and q, respectively; cor- resondingly, rotations of the p and q ends, measured anticlockwise with respect to the .? axis, are denoted

Fig. I. Beamxolumn discrete element.

416 H. A. El-Ghazaly and H. A. Al-Khaiat

by @,, and 6,. The angle or, measured from the ,? axis to the chord line of the deformed element, is given by

and the chord length may be expressed as

s = [(L + 27, l&y + (z;,

f (.X..,~) = fp + td (6)

where t (x, J) is the strain at any point in the element, tp is the initial centric prestrain due to prestressing, and cd is the strain of deformation, which is given as

I- +

3600 in-

70 kips

r

-3600 in -I

Fig. 2. Two-dimensional guyed tower under reference load (; = 1.0).

The introduction of the non-linear term iri reflects the coupling between the transverse and the axial stiffnesses.

The strain energy density of a general prestrained frame element, induced due to the application of external loads. is given as

10

)o Ain

” “. ‘, . ” 0 200 400 600 600 1000 1200 1400 1600 1800 2000

Fig. 3. Load-deflection curves for the guyed tower


0.8

-.+Gurj2.4&6 slackmed{b.=0.!3}

-Guys 2 13 4 slackened {a -~0.3}

0 Ain 0 20 40 60 80 loo 120 140 160

Fig. 4 Load+teRection curves for the guyed tower within the reference range.

Under the assumption of linear elastic material behaviour and upon integrating eqn (8) to obtain US,, which is then integrated over the cross-sectional area A, the following strain energy expression results:

dx. (9)

The work done by the element end forces is express- ible as

U’, = Pquq + M/)0, + M<,O,. (10)

Under the assumption of small rotation with respect to the x axis, 0, and 0, may be replaced by u,,, and L’,~, respectively; therefore eqn (10) reads

W, = Pp, + M,c, + Mq~~,y. (11)

Note that P, is the component of force along the chord at end y (Fig. I), while M,,, M, are end moments. The forces P,,, V, and V, do not contribute to eqns (10) and (1 I), since the corresponding displacements (u,,, P,, and cy) are zero with respect to the Eulerian system.

The total potential energy of the frame element in Fig. 1 is thus given as

7rp= lJ,- w,. (12)

Substituting the expressions in eqns (9) and (I 1) into eqn (12) gives

- {P, u<, + M, L,,,’ + M,, t,,<, 1. ( 13)

Application of the variational principle of stationary potential energy (Srrp = 0) results in the following

governing differential equations for the element:

& (ep + Ll, + ;L+, = 0 (t4a)

I --c A .~,,,-~[(Lp.tU,+iC:)~,l=O. (14b)

Equation (14a) states that the axial strain along the chord (x-direction) of the element is constant;

I-- 7

3600 in. __

0 kips

+-. 50 kips L

Fig. 5. Buckling mode ignoring prebuckling displacements.


therefore, the force component F directed along the Note that the solution to eqn (17) depends on chord is whether the axial-bending coupling action produces

F=EA(~,+u,+foz)=k*EI, (15)

tension (k* > 0), compression (k’ < 0) or zero (k’ = 0) axial strain.

where k is the “coupling constant” given by Using eqn (I 7) the strain energy of the element (9) reduces to

k’ =; (c, + u, + fut). (16)

Substitution of eqn (16) into eqn (14b) gives +[;k’+/;a:,d,+ (19)

I’,,,, - k’r,, = 0. (I 7) Substitution of the solution to eqn (17) into eqn (19)

The transverse displacement, L’, can be expressed in allows the strain energy to be expressed as

terms of k, s and the rotations of the element ends, 0, and 0,. by solving eqn (17) while applying the following boundary conditions (Fig. 1):

End p(x = Oj: c = 0; L:, = eP (tga)

Endq(.w =s): n =0: c,=O,. (lgbj

ELEV (11)

1968.5 -

1810 --

1645 -.-

1480 -_

1315 --

1150 --

985 --

620 --

655 --

490 --

325 --

160 __

00 --

Solid I e

Tower Elevation

Cross Section

Fig. 6. General dimensions for the 600 m guyed tower.


where the expressions for k,, (i. j = 1,2,3) in eqn (20) are functions of k and s and depend upon the value of k2 being positive, negative or zero.

According to the principle of stationary potential energy, the structural analysis may be viewed as an unconstrained minimization problem where the displacement degrees of freedom {D} assume values that minimize the potential energy function 7cp as follows:

given ncp({D 1)

find ID},

such that nP{D) is the minimum where

n,(P)}) = i %({DJ) - i P,D,, (21) ,=I ,=I

where I is the number of elements contributing to the total strain energy of the structure. If any element goes out of service during the search process, such as a slackened cable, the contribution of that element to the strain energy is simply neglected. J in eqn (21) is the total number of independent displacements. P, and D, are thejth load and the corresponding displacement, respectively.

The necessary condition for the occurrence of a local minimum of 7~~ at {D} = {D}* is

%(iD)*) dD:

=O; j=l,2 ,...., J. (22)

Table I. Two-dimensional mast properties (E = 30,000 ksi)

Moment of Area inertia Initial prestrain

Span no. (in’) (in4) (in/in)

I 60 300,000 - I.16 x lO-4 2 60 300,000 -0.924 x IO-“ 3 60 300,000 -0.527 x IO-’

Since 7rP is a highly nonlinear function of the displacement {D}, eqn (22) is a set of J nonlinear equations representing the first derivative of the potential energy function with respect to each of the general- ized coordinates {D). The most obvious approach to finding the displacements is to solve eqn (22). Unfortunately, the task of solving a large set of nonlinear equations may be very difficult. The function xP may be so complex, such as that encountered in the present study, that it is virtually impossible to write the equations in closed form. The use of mathematical programming techniques in direct minimization of the potential energy function allows powerful numerical methods of unconstrained minimization to be used. The method of conjugate gradi- ents known as the Fletcher and Reeves method was used in this study because of it modest storage requirement. In addition, the incorporation of a scaling transformation technique effectively improved convergence.

Vertical Loads

1

+ Uniform

Elements

i + Own weight Appurtenances

Horizontal Loads

1

1 Uniform Concentrated

Elements

t Wind on projecied

t Drag force on

area

Wind on projected of appurtenances

cables

Wind on projected area of members area of appurte-

nances

Fig. 7. Loads on tower.


Table 2. Guy properties (E = 20,000 ksi)

Initial Area prestrain

Level no. (in’) (in/in)

I 1.0 0.15 x lo- 2

2 I.5 0.13 x 10m?

3 2.0 0.125 x IO-’

TWO-DIMENSIONAL GUYED TOWER

The nonlinear analysis of a planar three-level guyed tower is presented to demonstrate the response of the tower under increasing wind loads (Fig. 2). The tower is modelled using three beam-column elements, pinned at the base and supported at the three levels by prestressed elastic guys. The mathematical model for a guyed tower is essentially a flexible beam- column with elastic supports. The tower shaft is assumed to have infinite shear rigidity to justify

neglecting shear strains. The guys are considered straight elastic cable elements.

The dimensions and material propeties of the tower shaft are given in Table 1 for each of the three spans. The structural properties of the guys, as well as the initial pretension in each guy, are given in Table 2. The complete structure under the reference load is shown in Fig. 2, where the constant vertical loads represent the weight of the tower shaft as well as any equipment that may be mounted on the tower. The horizontal loads represent increasing wind loads. A load parameter y is used as a multiplier to relate the wind intensity to the reference wind intensity. The three curves in Fig. 3 represent the load-deflection curves for the three guyed levels. No solution was obtained at 7 = IO and the instability limit is esti-

mated to lie between ;I = 9.75 and IO. The load-deflection curves were also plotted in

the range :,’ = O-1, which represents normal wind

Wind Direction

(b Fig. 8. Geometry and wind loads on a cable.


conditions. Figure 4 shows the load&deflection curves method [17] was used to trace the value of the in this range. The sudden change in the direction of smallest eigenvalue to detect the critical wind load the loaddeflection curve of level 1 is attributed to the intensity corresponding to vanishing of the smallest slackening of the leeward guys at the first and second eigenvalue. When this technique was applied to the levels, which takes place at y = 0.37; the planar guyed tower example, it was found that load-deflection curves of the second and the third levels abruptly change their directions at y = 0.37, due to the slackening of the leeward guys at the first

yCr r 13.2,

and the second levels and again change their directions at y = 0.5 1 due to the slackening of the leeward which is about 32% higher than the critical wind

guys at the third level. force when nonlinear stability analysis was con-

In order to show the effect of certain design ducted, and shows that yCr 1 10. This example illus-

assumptions, the same example was solved using trates that, in such slender structures exhibiting large

conventional linear stability analysis, where the horizontal drift, nonlinear stability analysis account-

effects of prestressing and prebuckling displacements ing for prebuckling displacements and prestressing

have been ignored. All guys on the leeward side were forces is a must. The buckling mode according to the

assumed inactive. Vertical loads were first applied linear buckling analysis is shown in Fig. 5.

and the stability of the tower was checked and the tower was found stable. The wind loads were then 600 m GUYED TOWER-CASE STUDY

increased incrementally until instability has been detected mathematically according to the known

A job was assigned to the authors which involved

criterion: checking the analysis and design of a 600m guyed tower to be constructed at a site in Kuwait. The

det. [KE + KG] = 0, (23) original design has been prepared by a specialized steel design company. It was also required to ensure

which designates the wind loads corresponding to the the conformity of the design with the EIA critical intensity. The scaled inverse power standards [18].

1

Tower project

ANSYS 4.4 UNIV VERSION OCT 23 1989 15:03:05 PLOT NO. 1 POST1 LINE STRESS STEP = 1 ITER = 40 FXl FX MAX = 0.8398 + 07 ELEM = 1

YV=l DIST = 14112 XF = 10793

y “,,I669

m 0.186E + 07

0.466E + 07

m 0.559E + 07

m 0.6528 + 07

0.7458 + 07

0.8398 + 07

Fig. 9. Axial force distribution. as plotted by ANSYS, linear analysis.


The designing company provided analysis and design data for a 12-ft square latticed steel tower with round legs. The tower is anchored at 12 levels and pinned at the base. Radial cables were suggested meeting along the tower axis. Figure 6 shows the overall dimensions of the proposed guyed tower.

Two computer packages have been exclusively used in the checking of the analysis and design aspects. ANSYS was primarly used to check the soundness of the linear and nonlinear analysis and to investigate the stability of the tower. STAAD-III (Version 8) on the other hand, was employed to check the design of the tower members.

LOUdS

Loads consist mainly of dead and wind loads. Dead loads represent the weight of members, guy wires and appurtenances. Wind load effects on the projected member areas and guys should be considered. The design wind velocity was taken as 180 km h-‘. For this tower two wind directions were considered, one perpendicular to the face of the tower (wind direction I) and another along the diagonal of the cross-section of the tower (wind direction 2). Notice that the design of this tower includes the installation of an elevator within the core of the tower

and a ladder for equipment installation and mainten- ance purposes, The tower is divided into I2 spans between successive guyed levels. Figure 7 summarizes the classification of the vertical and horizontal loads on the elements and nodes. EIA [18] gives the necessary formulation for calculation of wind loads on all tower components and appurtenances.

Linear analysis

ANSYS (Version 4.4) was used for the analysis of the tower. The tower shaft was modelled as 73 (three-dimensional) prismatic beam elements. The element cross-sectional area and moment of inertia were calculated, conservatively, considering the area of the solid legs only. A compressive prestrain in each beam element was considered due to cable pretension. ANSYS (4.4) does not have a special cable element but has a (three-dimensional) bracing (tension-only) element which has full stiffness when the net strain is tensile and zero stiffness when the net strain is either zero or compressive. This element is, therefore, a bilinear stiffness element and is classified as a nonlinear element even under small strains and deformations. Under gravity loads and wind blowing perpendicular to the tower face (wind direction I), a linear analysis was carried out.

ANSYS 4.4 UNIV VERSION OCT 23 1989 15:03:16 PLOT NO. 2 POST1 LINE STRESS STEP = 1 ITER = 40 FZll FZ MAX = 161966 ELEM = I

YV = 1 DIST = 14112 XF = 10793

Tower project

Fig. IO. Shear force distribution. as plotted by ANSYS. linear analysis.


Cables were initially modelled as tension-only members directed along the chord line of the actual cable. Linear analysis using the previous model for cables yielded a stiffer structure where the deflection at the tower top was found to be 6.9 ft. A geometric nonlinear analysis was also tried using ANSYS, increasing the deflection to 7.02 ft. These values are still much smaller than the value obtained by the design company, which was 14.92 ft. The prime reason for the discrepancy is in directing the cable tension along the chord line. The following approximate analysis indicates that the direction of the cable reaction at the point of attachment with the tower differs significantly from the direction of the chord line. Referring to Fig. 8a,

let h = r, cos yav, conservatively representing the horizontal component of the force in the cable, and let

GJ=;, (24) 0

where W is the cable weight and L, is the unstressed length. It can be shown that the catenary equation of the cable takes the form

(25)

where

a, = {sinh-‘[ 2h(zLh r,]> - y (26)

and

qA y=- 2h

-h

(27)

az = - cash a,. 4

(28)

It can be easily shown that

y, = tan-‘[sinh(2r + a,)]. (29)

When the previous analysis is applied to any of the cables at the top level, it was found that

yav = 51.48”,

while y, = 58.18”.

Tower project

ANSYS 4.4 JNIV VERSION )CT 23 1989 5:03:31

‘LOT NO. 3 POST1 LINE STRESS STEP = 1 ITER = 40 MYI MY MAX = 0.852E + 08 3LEM = 6 1

YV = 1 3IST = 14112 <F = 10793

-0.393E + 08

-0.254E + 08

-0.116E + 08

0.221E + 07

0.160E + 07

0.299E + 08

0.4378 + 08

0.57% + 08

0.713E + 08

0.8528 + 08

Fig. 11. Bending moment distribution, as plotted by ANSYS, linear analysis.


When modelling the structure with ANSYS, the tension only elements, representing the supporting cables, were directed along the new direction defined by the angle y, , rather than along the original chord line. This modification necessitated relocating the ground anchor point for each cable such that the member radiating from each anchor point on the tower is directed along the line defined by ;I,, while maintaining the original chord length L unchanged. This, eventually, serves the purpose of keeping the axial stiffness of the simulated cable (tension only member) 154/L unchanged, but directing the reaction at the tower anchor point along the same direction of the actual cable. This, of course, is a suggested approximate means of representing catenary cables as tension-only truss members. The small price for this approximation was more ground anchor points (48 nodes) instead of the original points (I 2 nodes). This is why in Figs 9-l 7, generated by ANSYS, cables which in reality meet at the same ground anchor point, Fig. 6, intersect at various points.

Direct loading on the cables due to own weight and wind is evaluated and divided between top and bottom anchor points, assuming a straight cable. Secondary moments due to cable eccentricity were

accounted for in an approximately iterative manner as follows:

(a) Firstly, these moments were ignored and AN- SYS was allowed to run in order to yield the solution including final cable forces.

(b) Secondary moments due to cable eccentricity were calculated and added to the external load vector and a final solution by ANSYS was then obtained. If a more accurate solution is sought the procedure should be repeated for a few cycles. Figures 9-12 represent the axial, shear, bending moment and deflection diagrams, respectively. As a result of the foregoing assumptions the deflection at the tower top has increased to about I1 ft.

It must be realized that wind loads on the cable were calculated and lumped at the top-bottom anchor points. Under wind loads the actual active cable will sag more, thus increasing the angle y,(Fig. 8a), which in turn increases the tower deflections. It may be concluded that the analysis results are rather sensitive to the magnitude of the angle y,

Wind loads on a cable cause, typically, drag and lift forces Fn and FL, respectively, in the directions shown in Fig. 8b. Notice that, because of the orientation of the supporting cables, the horizontal component of the lift force on all cables

1

Tower project

Fig. 12. Deflection distribution, as plotted by ANSYS, linear analysis

ANSYS 4.4 UNIV VERSION OCT 23 1989 15:03:39 PLOT NO. 4 POST1 DISPL. STEP = 1 ITER = 40 DMX = 131.992

DSCA = 10.691 YV = 1 DIST = 14112 XF = 10793

Analysis and design of gu lyed transmission towers

will always vanish. Only the axial component of FL exists.

EfSect of accidental guy rupture

The effect of accidental rupture of active guys was considered in the present study. Figure 13 shows the bending moments if an active guy at the upper end of the sixth span is ruptured. It is noticed that the positive moment has increased substantially in the sixth and seventh span. The maximum moment increased by about 11% from the comparison between Figs 11 and 13. The deflection at the top, however, remained practically unchanged.

Figures 14 and 15 show the tower bending moments and deflections if one of the active top guys is ruptured. The deflection at the top experienced a substantial increase and reached about 22 ft. The maximum moment, on the other hand, increased by about 180%. Remarkably, the tension in the remain- ing active cable at the top level increased by only 4%. It may, therefore, be concluded that if guy rupture happens to one of the top guys, it will have a detrimental effect on the static performance of the tower. This conclusion may warrant adopting a higher safety factor for top guys than for lower guys. Table 3 gives information about the supporting

425

direction 2, along the tower diagonal, gives more force in the cables than direction 1, perpendicular to tower force. The safety factor for the supporting cables ranges between 2.47 and 3.74, with a safety factor of 2.51 for cables in the top level.

Geometric no&near analysis

As indicated earlier, guyed towers exhibit almost all sources of geometric nonlinearity, such as:

(1) Large deflection under wind loads, which requires equilibrium based on the deformed configuration.

(2) Axial-bending stiffness coupling which eventually leads to bifurcation by buckling.

(3) Configuration-dependent behaviour for cables which are active only in tension.

(4) Nonlinearity of the supporting cables. ANSYS nonlinear options were activated in order

to obtain the internal forces for the tower under gravity and wind loads. Figure 16 shows the bending moment due to nonlinear analysis, to be compared with Fig. 11 for the results of the linear analysis. Geometric nonlinear analysis caused a 6.6% increase in the maximum bending moment in the tower. The top deflection according to the nonlinear analysis was

cables as given by the designing company. Wind found to be about 11.3 ft.

1

Tower project

Fig. 13. Bending moment distribution when one active guy at level 6 is ruptured

ANSYS 4.4 UNIV VERSION OCT 24 1989 145258 PLOT NO. 3 POST1 LINE STRESS STEP = 1 ITER = 50 MY1 MY MAX = 0.948E + 08 ELEM=61

YV=l DIST = 14112 XF = 10793

m -0.9’9E + OS E3 -0.712E + O8 E ::“,::“, : “,: 1 -;.“““E 0 118E+08 + 07

m 0.3258 + 09

m 0.533E + 09

0.740E + 09

0.9488 + 09

426 H. A. El-Chazaly and H. A. Al-Khaiat

Table 3. 600 m guyed tower--cable properties

Cable Initial Breakage Cable maximum Cable maximum Level diameter tension force Breakage force force for wind force for wind FS = breakage force.

IlO. (in.) (bs) (bs) initial tension direction I direction I cable maximum force

I 2; 2,

170 988 5.8 317.35 387.94 2.55 2 93 988 10.6 295.5 I 389.78 2.53 3 3 86 1076 12.5 312.70 414.35 2.59 4 3 88 IO76 12.2 3 16.65 412.57 2.6 5 2!

2, J 104 904 8.7 283.04 346.45 2.6

6 102 988 9.7 306.2 I 380.12 2.6 7 3 105 1076 10.2 340.10 422.85 2.54 8 3, ’ II5 1168 10.15 382.4 470.08 2.48 9 3 130 1076 8.3 374.00 435.26 2.47

IO 3; 145 1250 X.6 435.66 505.32 2.47 I1 3.5 165 I448 8.77 51 1.26 586.62 3.74 12 2; 104 988 9.5 360.40 393.07 2.51

The design company used its own in-house special- All of the aforementioned approximations are ac- ized programs for the analysis of the guyed tower. counted for by the in-house program of the designing Using ANSYS should, however, involve certain ap- company. The most pronounced approximation is proximations as a result of the following: believed to be involved in the means of modelling of

(1) Neglecting the effect of direct wind load on the the supporting cables. sag of the supporting cables.

(2) Approximation in calculating the additional e&t of temperuture variation

moment at guyed levels due to cable eccentricity. Two cases were considered where the gravity loads (3) Replacing the nonlinear cable support by a and design wind loads perpendicular to the tower face

linear tension-only member. were assumed to occur concurrently with a tempera- (4) Neglecting shear deformations. ture variation of + IOO’F. It was noted that, for

ANSYS 4.4 UNIV VERSION OCT 24 1989 14:29:37 PLOT NO. 3 POST1 LINE STRESS STEP = 1 ITER = 50 MYI MY MAX = 0.244E + 09 ELEM = 67

YV = 1 DIST = 14112 XF = 10793

Tower project

Fig. 14. Bending moment distribution when one active guy at top level is ruptured.

Analysis and design of gu lyed transmission towers

AT = + lOO”F, the lateral deflection at the tower top increased by about 16%, while the maximum moment decreased by about 4%. On the other hand, if the tower experiences a temperature drop of AT = - lOO”F, the maximum deflection decreased by about 14%, and the maximum moment increased by about 4%.

Stability analysis

For the design of the tower shaft, a buckling parameter K is needed in order to calculate the effective length of each span (KI). To evaluate K a linear buckling analysis (usually referred to as eigenvalue stability analysis) was carried out. Loading on the tower generally consists of

(1) prestressing loads due to cable pretension; (2) gravity vertical loads due to members and

appurtenances’ own weight; (3) horizontal wind load on the structure. The first two types of loading cause direct compres-

sive stresses in the tower shaft. These loads are predetermined and are usually well below the buckling load of the tower shaft. Wind load, on the other hand, will cause the tower to deflect laterally, thus exerting more tension on the windward side cables while causing the leeward side cables to slacken. For

421

equilibrium, the added tension in the cables must be balanced by exerting more compressive stresses in the tower shaft. As wind loads are increased, compressive stresses in the tower shaft increase too, leading to either:

(a) rupture of the over stressed cables; (b) buckling of the tower shaft due to increased

compressive stresses. The model for tower buckling analysis is a

beam-column over yielding supports (active cables). The stiffer the supports, the larger will be the tower buckling capacity. Therefore, even if some of the cables at certain levels ruptured under high wind loads, the support stiffness drops drastically at those levels leading to buckling of the tower. It can be concluded from the previous discussion that bifurcation by buckling is a potential failure mode for such towers and deserves special attention both in analysis and design.

ANSYS (Version 4.4) was used for the linear stability analysis. Gravity and prestressing loads were added together and the precompression in each element due to those loads was calculated and used as precompressive strain in the mast elements. Only active cables on the windward side were included in the stiffness of the structure, since the structure is

r

I’

Tower project

1 ANSYS 4.4 UNIV VERSION OCT 24 1989 14:29:39 PLOT NO. 4 POST1 DISPL. STEP = 1 ITER = 50 DMX = 295.241

DSCA = 4.78 YV=l DIST = 141 I2 XF = 10793

Fig. 15. Deflection distribution when one active guy at top level is ruptured.


perturbed about an equilibrium position when usually all cables on the leeward side have slackened near the buckling load. The stability criterion is thus

where [Ka] is the elastic structural stiffness matrix, [PC,;] is the geometric stiffness matrix for the known axial and prestressing part of forces, [KF] is the geometric stiffness matrix for a reference wind loads intensity, {6W} is the vector of perturbed displacements, and y is the unknown wind load multiplier. Equation (30) can be solved by eigenvalue analysis to yield the unknown wind load multiplier 7, corresponding to the vanishing of the determinant of the matrix on the left-hand side of eqn (30).

According to ANSYS the load factor was found to equal 5.6, which means that under wind load equal to 5.6 times the design wind loads, buckling is expected and the buckling mode will be as shown in Fig. 17. It must be realized that pre-buckling deformations (P-delta effects) are ignored in linear buckling analysis and, therefore, buckling is expected to occur practically at a load multiplier y less than 5.6. It may be reiterated that, in the two-dimensional guyed tower example discussed at the end of the energy

search section, the linear stability analysis overesti- mated the buckling load by about 32% when compared with the prediction of the geometric nonlinear analysis, as obtained by the energy search method.

In order to quantify the reduction in the critical load due to consideration of the pre-buckling displacements, a multi-stage geometric nonlinear analysis was carried out. Up to wind loads of 2.5 times the design load (y = 1 .O) the solution was stable and was found by ANSYS after 17 iterations. When the wind loads were doubled to become 5 times the design wind load (y = 5.0), no solution was found due to numerical instability and convergence problems. The true buckling load is therefore bounded by 2.5 and 5.0 times the design wind load. For an accurate determi- nation of the buckling load, the structural analyst must increment and decrement the load until a good estimate is obtained for the buckling load.

In order to obtain the buckling coefficient (K) for each span, we must first calculate the total compressive force (PC,) in each span corresponding to the buckling load. The well known Euler equation below may, then, be used to determine the (K) factor for each span:

ll2EI PC, = ~

(KI)’ ’ (31)

Tower project

Fig. 16. Bending moment distribution-geometric nonlinear analysis.

ANSYS 4.4 UNIV VERSION OCT 23 1989 14:33:59 PLOT NO. 3 POST1 LINE STRESS STEP = 1 ITER = 50 MY1 MY MAX = 0.918E + 08 ELEMz61

YV = 1 DIST = 14112 XF = 10793

m -0.460E + 08

E IO:3O7E + 08

0 154E + 08

0.918E + 08


where E, I and i refer to properties of the designated span.

Design by STAAD-III

STAAD-III was used by the design company and by the authors in order to design the members of the tower shaft, although different versions of the program were used in each case. A substructuring technique was used where the tower was approxi- mated by a series of beam-column spans (12 in this case), connected at the guyed levels. Each span was then re-analysed using STAAD-III under loads directly acting on it and also the end forces and moments, as obtained from the geometric nonlinear analysis explained earlier. The actual truss member properties, and not the equivalent properties, were used when analysing and designing each span by STAAD-III. Following the analysis stage, design parameters were activated to allow member selection by STAAD-III. Figure 18 summarizes the modelling stages of the tower for design using STAAD-III. The figure also shows the displacement boundary condition used when analysing and designing each span. The weight of steel involved in the tower construction was found to be slightly more

than 2100 kips. This weight includes the legs, diag- onals and struts. It does not include the gusset plate or any special hardware fittings. The weight of the supporting cables was found to be slightly larger than 1400 kips.

CONCLUSIONS AND RECOMMENDATIONS

(I) A comprehensive review is presented to the sources of nonlinearity in the static analysis of guyed towers.

(2) The energy search method renders itself as a suitable algorithm for the geometric nonlinear analysis of guyed towers, mainly because of its ability to exclude elements that no longer contribute to the structural stiffness under certain displacement patterns.

(3) The two-dimensional example presented re- veals clearly that the stability limit of a guyed tower and similar slender structures subject to high axial forces should be predicted using geometric nonlinear analysis rather than the conventional eigenvalue analysis, because of the pronounced effects of the prebuckling displacements on the buckling capacity.

Tower proiect !

ANSYS 4.4 UNIV VERSION OCT 24 1989 15:08:04 PLOT NO. I POST1 DISPL. STEP = 1 ITER = 1 FACT = 5.601 DMX = I

DSCA = 1411 YV = 1 DIST = 14112 XF = 10793 ZF = -59.43

Fig. 17. Buckling mode at critical wind velocity.


Actual Tower Equivalent Beam-Column

Nonlinear Analysis

Approximatmn ______)

Substructuring (12spans)

\ / n

Span stress Resultants

\ r ~~~~ I

All other Spans Including Span 1 span 12

STAAD 111 Analysis 8, Design of Members

Fig. 18. Modelling approximation for analysis and design.

(4) The case study presented is a full-size 600 m two packages. The approximate procedure seems guyed tower that is to be constructed on a site in adequate for the general static analysis of such towers Kuwait. An approximate analysis and design was in the absence of a more specialized package for the conducted using a combination of ANSYS and analysis and design of guyed towers. It must, however, STAAD-III computer packages after invoking certain be reiterated that the analysis result is rather sensitive simplifying assumptions to suit the limitations of the to the direction of the cable reaction at the tower.

Cables l-8 are flexure torsion cables

Fig. 19. Proposed guy arrangement for torsion resistance


(5) Although the EIA [18] standards do not, explicitly, require consideration of thermal loading on the structure, it is rather imperative for guyed towers constructed in Kuwait to include the extra loading on the tower and cables due to temperature variations simultaneously with maximum wind and gravity loadings.

(6) The gust response factor in the EIA specifica- tions may be taken as 1.25 rather than 1 .OO as a safety measure against severe sand storms which occur occasionally in Kuwait.

(7) The EIA standards clearly specify that anten- nas are also subjected to twisting moments due to wind; therefore any valid analysis should consider torsional loads on the structure. This would necessitate the inclusion of certain torsional springs at the guyed levels to counter-balance the torsional loads. Obviously, when the guys are arranged to meet along the tower axis, there will be no torsional resistance to these cables as far as linear analysis is concerned. It is advisable to revert occasionally to the cable arrangement shown in Fig. I9 to induce linear torsional springs at some levels. This new arrangement is as effective in resisting flexural loads as the radial arrangement (Fig. 6), but has significantly more torsional stiffness. Obviously, at the levels with torsional cables, each radial cable (Fig. 6) is to be replaced by two cables, each with half the area, arranged as shown in Fig. 19, thus causing no additional loading or expense.

(8) Whether the possibility of cable rupture should be considered in the design or not is a debatable subject. The EIA standard does not explicitly men- tion this possibility. Analysis, on the other hand, clearly shows that, if cable rupture occurs while wind speed is at its maximum, the consequences could be catastropic, expecially if rupture occurs to one of the cables at the top levels. If, however, rupture occurs during construction the effect will be less detrimental, since construction will, normally, stop during strong windy periods. It is also believed that a higher safety factor should be assigned to cables at top levels as compared to cables at lower levels.

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18.

REFERENCES

R. S. Rowe, Amplification of stress and displacement in guyed towers. J. Struct. Dir. ASCE ST6, 1821-l--1821- 20 (1958). D. L. Dean. Static and dynamic analysis of guy cables. J. Struct. Div. ASCE STi, I-21 (1961). - - F. H. Hull. Stability analysis of multi-level guved towers. J. Struct. Die: ASCk ST2, 61-80 (1962).- . J. E. Goldberg and V. J. Meyers, A study of guyed towers. J. Struct. Dir. ASCE ST4, 57-76 (1965). T. F. Mears and W. R. Charman, The design and construction of cylindrical television masts in Great Britain. Struct. Engr 44, 5-15 (1966). A. J. J. Bartak and M. Shears, The new tower for the Independent Television Authority at Emley Moor, Yorkshire. Struct. Engr 50, 67-80 (1972). R. A. Williamson, Stability study of guyed towers under ice loads. J. Struct. Dir. ASCE ST12,2391-2408 (1973). H. A. Miklofsky and M. G. Abegg, Design of guyed towers by interaction diagrams. J. Struct. Dir. ASCE STl, 245-266 (1966). E. G. Odley, Analysis of high guyed towers. J. Struct. Dit>. ASCE STI, 169-190 (1966). R. A. Williamson and M. N. Margolin, Shear effects in design of guyed towers. J. Struct. Die. ASCE STS, 213-235 (1966). R. K. Livesley, Automatic design of guyed masts subject to deflection constraints. ht. J. Numer. Meth. Engng 2, 33-43 (1970). J. E. Goldberg and T. J. Gaunt, Stability of guyed towers. J. Struct. Dir. ASCE ST4, 741L756 (1973). K. M. Romstad and M. Chiesa, Approximate analysis of tall guyed towers. ASCE Fall Convention and E.uhihit, San Francisco, CA (1977). M. H. Magued, M. Bruneau and R. B. Dryburgh. Evolution of design standards and recorded failures of guyed towers in Canada. Can. J. Civil Engng 16, 725-732 (I 989). M. Bruneau, M. H. Magued and R. B. Dryburgh Recommended guidelines for upgrading existing towers. Can. J. Civil Engnng 16, 733-742 (1989). H. A. El-Ghazaly and G. R. Monforton. Analysis of flexible plane frames by energy search. Comput. Strut/. 32, 75-86 (1989). H. A. El-Ghazaly and A. N. Sherbourne. Deformation theory for elastic--plastic buckling analysis of plates under non-proportional planar loading. Conrpur. Struct. 22, 131-149 (1986). ELA Standard ELA-222-D. Structural standards for steel antenna towers and antenna supporting structures. Electric Industries Association, Engineering Department (1986).

analysis and design of guyed transmission towers—case study in kuwait

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