STEEL TOWER ECONOMICS*By P. J. RYLE, B.Sc.(Eng.), Member.t

(The paper was first received 25th May, and in revised form Uth September, 1945. // was read before the TRANSMISSION SECTION 9th January,1946, and before the NORTH-EASTERN CENTRE 22nd October, 1945.)

SUMMARYPartly from theoretical considerations, but mainly from practical

data, an empirical formula is derived giving the approximate weightof any tower in terms of its height and maximum working over-turning moment at the base. The tower weight above ground line isshown to be satisfactorily represented by kHy/M tons, where H isthe overall tower height above ground line, in feet; M is the over-turning moment at ground line, in thousand lb-ft; and A: is a "constant'which varies within an extreme range of about 0-0014 to 0 0029throughout the whole series of towers; most towers have k values wellwithin this range.

The towers investigated covered ranges of about 16 to 1 in height,3 000 to 1 in overturning moment, and 1 200 to 1 in tower weight.

Design factors which explain variations in the "constant" k are dis-cussed at some length on a qualitative basis.

It is shown that an economic value for the base width of any toweris in the neighbourhood of 0 • 5 y/Mft.

On the assumption that, for any given general type of tower, thetotal erected cost may be regarded as proportional to the above-ground weight, the tower-weight formula, kH^/M, can be used withconvenience and reasonable accuracy for a wide field of cost estima-tions, especially those involving comparisons of a number of alterna-tives of a generally similar nature.

Such investigations include the following:Ordinary transmission lines: (a) Economic span lengths. (6) Rela-

tive costs of lines for different voltages, conductor sizes and conductormaterials, (c) Number of circuits, (d) Number and height of earthconductors, (e) Abnormal wind and ice loading assumptions and/orfactors of safety. (/) Different clearances between conductors andground, (g) Suspended auxiliary cables, (h) Hypothetical "dog-leg" lines. 0 ) Hypothetical "super e.h.v." lines.

Navigable river crossings: (a) General layout, (b) Clearance tohigh water; Air Ministry height restrictions, (c) Span lengths andriver bank configuration, (d) Foundation conditions, (e) Numberof circuits. (/) Long spans; economic conductor, (g) Narrowwaterways; economic number of towers. (A) Terminal-type crossingtowers; economic conductor tension.

(1) TOWER-WEIGHT ESTIMATION

(1.1) Theoretical Approach to Tower-Weight Formula,

The fundamental function of an overhead transmission-linetower is simply that of a dual-purpose distance piece—to keepthe conductors apart and at not less than some prescribedminimum height above ground. The chief duty of a tower,however, is to resist the mechanical loads which may be appliedto it by the wind acting on the conductors (ice-covered) and onthe tower itself, by the resultants of the conductor tensions, andby the dead weights of the (ice-covered) conductors and of thetower itself. Usually the effects of dead weight are comparativelyunimportant, and a tower may be considered basically as avertical cantilever resisting one or more horizontal forces appliedat or near its top.

For a given tower height and loading, it might at a first glancebe expected that there must be some particular design of towerwhich would be the most economical in weight of steelwork.This would, however, be true only so long as definite minimumthicknesses of metal in the various members were specified.

* Transmission Section paper. t Merz and McLellan.

Theoretically, if the designer were free and satisfied to use metalindefinitely thin, and if such construction were practicable, therewould appear to be no lower limit to the tower weight.

In default of a rigid proof of these assertions, take a simplecalculable case, namely that of an ordinary cylindrical tubularmast of constant wall thickness, assumed small in comparisonwith the diameter. If the diameter is d, the thickness /, theheight h and the overturning moment at the base w, in anyconvenient units, the modulus of section of the tube will be pro-portional to d2t. For a given working stress, m will then beproportional to d2t, or dec y/imlt). The area of cross-section ofthe tube metal will be proportional to td, thus making the totalmast weight proportional to htd, hence to ht^irnft) or to hy/(jnt).

Hence, for given height and overturning moment, the mastweight can always be reduced by reducing t. The diameterwould, of course, have to be increased inversely as the squareroot of the wall thickness, but no lower limit to the mast weightis indicated.

On the other hand, if the thickness / is fixed at some prescribedminimum, the mast weight is determined and will be of the formkh^/m. It will also be seen that, for a fixed value of t, thediameter or base dimension is proportional to y/m.

Throughout the paper, H is the total height of a tower aboveground line, in feet; and M is the total overturning moment(usually transverse) at the ground line, under maximum workingload, in thousands of lb-ft. Tower weights above ground lineare given in tons; and, unless otherwise stated or implied, alltower designs referred to are based on a factor of safety of 2-5under maximum working loads. Also, except where otherwiseindicated, the effects of broken-conductor design assumptionsare ignored.

In Appendix 7.1, a simplified mathematical basis is given forarriving at the economic proportions and weight of an elementarysquare-based, parallel-sided lattice steel mast, without cross-arms,for a given height and given single horizontal load at the top.Such mathematical treatment depends on certain simplifying as-sumptions of which some, though of a practical order, are some-what arbitrary and debatable. However, the results are given ina very simple form and may even be of some practical use in thepreliminary design of small lattice masts.

The chief result of the investigation in Appendix 7.1 is that theweight of the economic elementary lattice mast is kH^/M tons,where k, on the particular assumptions made, is 0-00153.

(1.2) Empirical Confirmation of Tower-Weight Formula: Rangeof Application

For lattice steel overhead-line towers of normally tapering out-line necessitating bracing members of varying lengths and anglesof inclination, with leg-member sizes graded in steps up the tower,and with different loadings applied at different heights, no suchsimplified approach to a mathematical basis for economicaldesign or tower-weight estimation is feasible. However, fromthe foregoing considerations, it is not unreasonable to expectthat the weight of any tower of commercially economic designand constructed, as is usual, of members of defined minimumthicknesses, may be satisfactorily represented by an empiricalformula of the form kHy/M. This is found to be true.

[263]

264 RYLE: STEEL TOWER ECONOMICS

500

450

River Thamessing tower

Typicalriver-crossing tower

Typicall.i kV tower

Typical r55'kVtowiT .'

RBmast $*•

400

350

300

250

200

150

100

50

Fig. 1.— Comparative sizes of typical towers.

Fig. 1 shows, drawn to scale, the scope of tower sizes and dutiesinvestigated; it covers ranges of about 16 to 1 in height, 3 000 to1 in overturning moment, and 1 200 to 1 in tower weight. Withcomparatively small variations in k, which are discussed below,it is found that the formula for tower weight, kHy/M, is ap-plicable over the whole range.

(1.3) Weights of Ordinary Transmission-Line Towers

Fig. 2 plots tower weights down to ground line, against Hy/M,for some 60 different designs of ordinary transmission-linetowers. Values near the origin apply to small "PB" masts, andthose at the upper end of the curve to 132-kV double-circuittowers for large angles of deviation. Between these limits arepoints representative of single- and double-circuit, straight-lineand angle, towers for various conductor numbers and sizes andconductor loadings, and for voltages of 33, 66, 88, 110 and132 kV. The actual designs are due to the designers of half-a-dozen firms. A reliable average figure for tower weight may betaken as 0-0016 H-\/M tons, for nearly all the towers haveweights between 0 0014#VA/and 0 0018i/VMtons.

Some variation in the value of k is to be expected, because of(a) type and number of cross-arms, which are not part of thebasic-cantilever structure, but which may contribute appreciably tothe total weight of the tower; (b) the adoption of different speci-fications for assumed broken-conductor conditions (if any), whichwill affect design in respect of longitudinal loadings and torsion;and (c) differences in technique of different designers.

On the whole, the general range of k values is unexpectedlysmall. Assuming, for the sake of argument, that there is somephysically acceptable mean value, there is of course more scopefor upward than downward variation. For specified minimummember thicknesses, etc., there must be for any given duty some

9

8

7

*~>.O

$3 5

p °

I4

3

2

1

•7

a

• /

ft •* • • •

* >

•

/

/•

V*

/ •/

•

1000 2000 3000 4000 5000 6000

Fig. 2.—Weights of ordinary transmission-line towers.

one hypothetical design of minimum weight; on the other hand,for the same duty, there is no limit to the amount by which atower could, without increasing factors of safety, be made heavierthan the most economical design by adopting structurallyinefficient proportions.

From his investigations the author would regard with somesuspicion the adequacy of design of any ordinary transmission-line tower giving a weight of less than, say, 0-0013//\AM; onthe other hand, he would hazard that any ordinary tower weigh-ing more than, say, 0-QOlHy/M must be, for some reason, ofuneconomic design and therefore capable of improvement.These assertions only refer to towers of conventional types,particularly as regards number and disposition of conductors.For instance, a single tower of normal height, but to carry four132-kV circuits, would need a very abnormal type of "top-hamper," and would undoubtedly have a weight in excess ofany figure suggested above, although probably not more thanone of the order of 0 0024/f\/M.

(1.4) Weights of River-Crossing Towers

Fig. 3, which is similar to Fig. 2 but to different scales, showsthe plot of tower weight down to ground line against Hy/M fora number of special river-crossing towers of heights from, say,120 ft upward. The upper limit is the C.E.B. River Thamescrossing tower, which is 487 ft high. To demonstrate the widerange over which the empirical formula, kH\JM, is found to begenerally applicable, all the points in Fig. 2 would occupy thesmall black triangular area near the origin in Fig. 3.

It will be seen from Fig. 3 that for the tall towers the valuefor k is somewhat higher than for towers of more normal height.

(1.5) General Design Features Affecting Value of fc

A qualitative investigation will now be made of generaldesign features which may be expected to affect the relativeweights of towers; variations from empirical mean-weight for-mulae will also be considered.

RYLE: STEEL TOWER ECONOMICS 265

River Rodin6 tower J(4-circuit^* , /

/

/

/

Rive •Thanest )wer-/

t—

-

10 W 30 40 50 60 70 80 90 iOO 110 120Thousands

280

260

240

220

200

£ 180

I: 160

f 140s3 120

^ 100

80

60

40

20

Fig. 3.—Weights of river-crossing towers.

(1.5.1) Cross-Arms and " Top-Hamper."As pointed out above, a tower is fundamentally a dual-

purpose distance piece, of which one function is to keep theconductors at prescribed distances apart. For ordinary 3-phaselines there may be 2, 3 or 6 separate arms projecting from themain cantilever structure. For reasons of electrical andmechanical clearances, the general dimensions of the cross-armsdepend on the line voltage, the conductor material and size, andthe span length. The proportion of the total tower weightrepresented by cross-arms may therefore show appreciablevariations.

When the voltage range of, say, 220-330 kV is reached, thepower-transmitting capacity per circuit approaches such highvalues that the tendency is to avoid double-circuit constructionin order to eliminate the risk of simultaneous 2-circuit faults. Atthe same time the insulation level is approaching a value where,with low tower-footing resistance and with adequate overheadearth-conductor protection, the line may be regarded as nearly"lightning-proof." These considerations, of which a detaileddiscussion would be outside the scope of the paper, favour theadoption of the single-circuit arrangement, in which the three lineconductors are in one horizontal plane with two widely-spacedearth conductors well above them. The "top-hamper" of thetower is then forced, by mechanical and electrical spacing re-quirements, to take the general form shown as ABCD in Fig. 4.It is clear that the purely "top-hamper" steelwork must representa comparatively large proportion of the total.

Table 1 gives for typical towers the approximate percentages

Table 1

Type of tower(ordinary line)

33-kV single circuit132-kV double circuit . .220/330-kV single circuit

I

Approximate cross-armor "top-hamper" weightas percentage of total

tower weight

8%18%

Probably 25-30%

H

(a) (b)Fig. 4»—220/330-kV single-circuit tower arrangements.

of total tower weight represented by cross-arms or equivalent"top-hamper."

(1.5.2) Bracing Members and Tower Outline.The bracing members in the main body of a tower should be

considered under the two heads of stressed and unstressedbracings. Stressed braces are those which perform a mechanicalduty, generally in resisting the horizontal shear forces due to theapplied loads. Unstressed bracings are those which have noreciprocal counterpart in the stress diagram and which are onlyrequired for stiffening (by reducing their unsupported lengths)other members which are called upon to act as struts.

In Appendix 7.2 it is shown that the weight of a stressed com-pression brace may be considered as roughly proportional toPi + c/2, where P is the design load, c a constant, and / theunsupported length. The weight of an unstressed brace is thenroughly proportional to /2. The general arrangement of atower, provided that the leg-member weight is not appreciablyincreased thereby, should therefore be such that the loads instressed braces, and the individual and total lengths of bothstressed and unstressed braces, are made as small as possible.

The simplest lattice-cantilever frame, that to resist a single loadat the top^ is naturally of triangular outline with the lines of thelegs meeting at the load point. There is then no load in any ofthe braces. Extending this, it can be shown that, at any sectionof a tower body, if the lines of the leg members meet at the centreof gravity of all the applied horizontal loads above that section,the braces in that section will be unstressed. Owing to the needfor an appreciable width of tower body at the upper levels forfirm attachment of cross-arms, and to meet the requirements oftorsional strength under broken-conductor conditions, thisprinciple cannot often be fully applied.

A good example of partial application, however, is that of theoutline design of the main body part of the typical 220/330-kVsingle-circuit tower. The requirements of line and earth-conductor spacing, as mentioned above, fix the "top-hamper" onthe lines depicted as ABCD in Fig. 4. At first sight, the mainbody below the line-conductor cross-arm level would obviouslybe set out as a vertical-sided structure EFGH as in Fig. 4(a).This arrangement, however, would involve an inordinate totallength of bracing members, and, since the leg-member lines wouldnot approach the desirable condition of meeting at the centre ofgravity of the loads, the stressed braces would be heavily loaded.The total weight of braces would therefore be very high.

The general design adopted in practice is shown in Fig. 4(b).As far as possible the leg lines are directed towards the loadcentre of gravity, although a "waist line" JK is reached, abovewhich the leg members must branch out again. However, fromground line up to the waist level the ideal condition is approached;

266 RYLE: STEEL TOWER ECONOMICS

the leg members will in general be little, if at all, increased in size,stressed braces will be lightly loaded, and the lengths of bothstressed and unstressed braces will obviously be much less thanin Fig. 4(a). In spite of this, the "top-hamper" of such towersrenders them relatively heavy in comparison with towers of morecommon outline having ordinary cross-arms. Tower weightsmay be expected to be of the order of 0 0022#VAf tons.

A similar "waisted" effect below the bottom cross-arm hasbeen adopted on certain rather unusual river-crossing towers suchas the C.E.B. River Witham (3-circuit, 33-kV) and River Roding(4-circuit, 66-kV), of which the upper parts are shown in outlinein Fig. 5.

River Withamcrossing

.5-circuit. 55-kV)

River Rodingcrossing

(4-circuit. 66k V)

Fig. 5.—River-crossing towers with special "top-hamper."

For river-crossing towers of considerable height, the principlealso leads to the "Eiffel Tower" type of outline, in which the leglines appear curved, but are actually made in three or morestraight-legged sections of slightly different leg slopes. On suchtowers the wind load on the tower itself may be of the sameorder as the total conductor wind load; on the River Thamestower the overturning moment at the base due to tower wind isnearly double that due to the conductor wind. Table 2 givesthe approximate proportion of the tower-wind overturningmoment for typical suspension, or straight-line, towers.

Table 2

Tower type

33-kV double circuit . .132-kV double circuit . ."Average" river crossing

Thames crossing

Tower-wind overturning moment asapproximate percentage of total overturning

moment

10%20%

Between 25% and 50% depending onnumber of conductors and relationbetween tower heights and spanlengths

62%

On a tall tower, the position of the centre of gravity of allhorizontal loads above each tower section then varies consider-ably and is lower for lower sections. This obviously favours the"Eiffel" outline, as indicated in Fig. 6, in which, for clarity, thetower base and body widths have been exaggerated in size.

Even if the conductor loads were the only ones acting, somedegree of "Eiffel" outline would probably prove economical forthe taller towers, because of the quite considerable reductions

C.G.of all loadsabove CC

C.G. of all loadsabove BB

C.G. of all loadsabove AA

c —

I

Fig. 6.—Typical "Eiffel" outline (exaggerated).

which could be made in the individual and total lengths ofbracing members, notably of the unstressed braces. Fig. 7 showstypical sections of bracings, legs and stressed braces being shownas full lines and unstressed braces as broken lines. In the lowerparts of a tall tower, the unstressed braces represent a very con-siderable proportion of the steelwork weight; besides bracings inthe plane of the tower face, further unstressed bracings may berequired in inclined planes "across the corners," in order tostiffen the face bracings against movement normal to the face(Fig. 7). In the bottom panel section [Fig. 8(a)] of the RiverThames tower, i.e. from ground level up to the 87-ft level, thetotal weight of unstressed braces is nearly four times that of thestressed braces, and actually represents nearly 10% of the wholetower weight. It is not impossible, in the author's opinion, thatfor a tower of this order of size a bottom panel arrangementgenerally as shown in Fig. 8(6) might prove economical. Thetotal weight of the stressed braces EF, FG, GH and FH wouldprobably be higher than that of AB and BC in the existing ar-rangement, but the saving in total weight of unstressed bracesshould more than counterbalance this.

Another type of (normally) unstressed bracing which must con-tribute considerably to the weight of a tall tower is that com-monly described as "plan" bracing. At several levels up thetower, bracings are put in to ensure that horizontal sections ofthe tower as a whole remain square under the effects of un-balanced torsion due to broken conductors, and to distribute theeffects of such torsion. A typical frame of such "plan" bracingis shown in Fig. 9. The total weight of such a frame may beconsiderable, but can obviously be much reduced wherever thetransverse dimensions of the tower body can be decreased by"Eiffel" outline. How many frames of "plan" bracing arenecessary in a tower is debatable.

RYLE: STEEL TOWER ECONOMICS 267

Table 3 shows, for typical towerbodies (omitting cross-arms), how theproportional weights of legs andstressed and unstressed braces varywith different types of tower. It willbe seen that legs and total bracesroughly share the weight in anytower, but that unstressed braces re-present a percentage increasing rapidlywith tower size. On the Thamestower the unstressed braces amountto considerably more than the stressedbraces and represent one-third of thetotal tower-body weight.

In a very tall tower, the total designload in the bottom section of the legmember, and therefore the cross-section and weight of the member,are markedly affected by the deadweight of the tower itself; this effectis usually small in an ordinarytransmission-line tower. Table 4gives for some typical towers theapproximate proportion of the totalload in the bottom-leg section, due todead weight of the tower itself.

Even "on tallj towers the weight ofunusual types of "top-hamper" (e.g.Fig. 5) may account for a relativelylarge part of the total weight. Ter-minal-type river-crossing towers forlong spans may also be relativelyheavy; the main overturning momentis longitudinal owing to the con-ductor tensions, but there is, in

Table 3TOWER BODIES, APPROXIMATE PERCENTAGE WEIGHTS

87 ft

Plan on A B

8' '.—Typical stressed andunstressed bracings.

f* 120 ft *\(a) (b)

Fig. 8.—River Thames crossing tower, bottom panel.

addition, a considerable transverseoverturning moment due to the windon the conductors and tower. Theauthor has not, so far, establishedany entirely satisfactory basis for asimple kH\/M weight formula forsuch towers.

In Fig. 3, points marked with across represent certain 3- or 4-circuittowers with heavy "top-hamper" andcertain long-span terminal towers.It will be seen that their weight factor k

Fig. 9.—Typical frame of' 'plan" bracing.

tends to be high.Tall towers also include a not inappreciable weight represented

by climbing ladders, railings and platforms.Taking all things into consideration, it is probably fair to say

that the weight of a river-crossing suspension tower with normalcross-arms lies between 0Q0l7H\/M for towers of about only120 ft in height, and 0-0Q25HVM for towers similar to the

LegsStressed braces ..

Unstressed braces

Typical33-kV

.(narrow-basetype tower)

57431

r4 3

Typical132-kVtower

50441

^506J

"Average"river-cross-ing tower

49301

2lJ

Thamescrossingtower

47201

r5 3

33j

Table 4

Tower type

33-kV large-angle tower33-kV straight-line tower132-kV large-angle tower132-kV straight-line tower"Average" river-crossing towerThames crossing tower

Load in bottom leg, due totower weight, as percentage

of total load in leg

375

1015-30

40

487-ft Thames tower. Towers with special "top-hamper" orlong-span terminal-type towers may be from 10-20% heavier.

Though not strictly within the intended scope of the paper,the author would anticipate that tall self-supporting radio towersof the type not designed for any transverse loading, except thewind on the tower itself, would reveal quite a low weight-factor kin comparison with similarly tall transmission-line towers. Atthe top, a tall transmission-line tower is subject to heavy trans-verse loads, and a little lower the tower body has to resist a con-siderable moment, so that even the upper members are quiteheavy. On the other hand, a tower designed to resist only thewind on itself could in the upper sections consist theoreticallyof nothing more than a thin tapering rod or spike. For equalheight and overturning moment at the base and equal factor ofsafety, a tall radio tower of the type referred to is therefore likelyto be considerably lighter than a transmission tower. Theauthor has no experience of radio towers, but from admittedlyvery rough calculations, he estimates that the weights of suchtowers, for a factor of safety of 2-5, might not much exceedabout 00012HVM tons.

(1.5.3) Broken Conductor Conditions.It will hardly be disputed that, for lines of major importance,

steel towers should be designed with some margin of safetyagainst the effects of unbalanced longitudinal loads due to brokenconductors. In Great Britain, the majority of towers are designedfor a factor of safety of 1^ under such conditions, the numberof conductors assumed to be broken ranging from only one onsuspension towers up to four on 132-kV double-circuit large-angle towers. Service experience has shown the value of thispractice, since the number of tower failures under excessive windand ice loadings accompanied by conductor breakages has beenvery small.

No attempt is made here to estimate the degree to whichbroken-conductor design requirements affect tower weight, andin Figs. 2 and 3, and in corresponding recommended average-weight formulae, no differentiation is made between towersdesigned for light or onerous broken-conductor assumptions.

In general terms it may be said, however, that the provision,in design, for one broken conductor may be more decisive inthe weight of a suspension tower than the provision for even

268 RYLE: STEEL TOWER ECONOMICS

three or four broken conductors in the weight of a large-angletower.

(1.6) Weights of Reinforced-Concrete PolesAs a digression from the subject of the paper, it appears that

the empirical weight formula kH^JM may also apply satis-factorily to reinforced-concrete poles, with a suitable modi-fication in the value of k. In the paper1 by Neate and Bowling,design loadings, heights and weights are given for various re-inforced-concrete poles of a single general basis of design.Fig. 10, in which the total concrete-pole weight (no cross-

fy^

100 200 500 400

10.—Reinforced-concrete-pole weights.

arms) is plotted against H^/M, indicates that the weight can bevery fairly represented by 0-QOSHy/M tons. This value ofk = 0-005 is to be compared with 0-0016 taken by the authoras representative for ordinary steel towers. For the same duty,therefore, a reinforced-concrete pole may be taken to weighabout three times as much as a lattice steel tower. Now theweight of reinforcing steel in a concrete pole is usually aboutone quarter of the total pole weight, so that for the same dutya reinforced-concrete pole uses about three quarters as muchsteel as a lattice steel tower; considered purely as a means ofsaving steel, it cannot therefore be regarded as a very impressivealternative.

(2) TOWER-BASE DIMENSIONSIn Section 1.1 it was pointed out that, for defined unit stress

and fixed wall thickness, the diameter or base dimension of acylindrical tubular mast is proportional to ^m.

In Appendix 7.1 it is shown that, on the simplifying assump-tions made, the economical base-width of the elementary latticetower is also simply proportional to the square root of the over-turning moment and appears as 0 • 333 M ft. This is, of course,for minimum weight of the elementary tower itself and takes noaccount of foundation cost, which will, in general, decrease withincrease of base width. For foundations of the single massconcrete-block type the base-width formula arrived at wouldprobably be quite sound as a basis of design, and would giveresults comparable with actual narrow-base tower designs, whichare much used, probably from wayleave considerations, on theContinent. British and American practice generally favours theso-called wide-base type of design, for which the total cost oftower plus foundations is presumably a minimum. In theUnited States a continuous, wide strip of land called the "rightof way" has usually to be acquired along the line route; in GreatBritain the payments made for individual-tower wayleaves areusually reasonably small and not greatly affected by tower-basedimensions; for both, therefore, a truly economical base-width isfree to be adopted.

Taking account of the expression for economical base-width in

Appendix 7.1, it is not unreasonable to expect that actual tower-base dimensions might be approximately proportional to y/M.In Fig. 11 the base width is plotted against ^/M for the majority

2/2s

// y

///

/

//

/

//

// !

Thames crossing/

/

^0

|4x

r'1

/•

/

20 40 60 80 100 J20 140 160 180 200 220 240

120

110

100

90

I 8°4 70

? 60

! * >40

30

20

10

Fig. 11.—Tower base widths.

of the towers whose weights are represented in Figs. 2 and 3.Base widths are seen to lie mostly between 0-35^M and0-65^M; a good overall average figure would be Q-5y/M.Marked divergencies from a mean value are only to be expected.If, for any given tower duty, the true total cost of tower plusfoundations could be plotted against base width, the curve whoseminimum represented the economical base would probably bevery flat over a considerable range. Again, the actual fixing ofthe base width for a given tower duty must often be largely amatter of engineering convenience; for example, a variation inwidth might just permit the economical use of some convenientor easily obtainable size of B.S. angle for the leg member.

In spite of the obvious "spread" of the points in Fig. 11, it isclear that the empirical proportionality of base width to y/Mcan be justified, and it is suggested that, for the quick pre-liminary design of any new tower, or for, say, the pre-estimationof the ground space likely to be taken up by a projected river-crossing tower, a base width of 0-5\/Mft would be a veryreasonable starting point.

(3) ECONOMIC APPLICATIONS(3.1) General Application of Weight Formulae to Cost

EstimationThe formulae recommended for use in average tower-weight

estimation from Sections 1.3 and 1.5.2 are:

Ordinary transmission-line0 • 0016H\/M tons to ground linetower

220/330-kVtower

River-crossing tower

single-circuit0 • QOllH^M tons to ground line0-0017 to 0-0025#V^ tons to

ground line(depending on height and loading,

and with upward adjustmentfor special "top-hamper" orfor long-span terminal-typetowers).

The author does not suggest that tower weights given by

RYLE: STEEL TOWER ECONOMICS 269

these formulae would be sufficiently accurate for tendering for aparticular contract, without checking by approximate design; butthey should at least be accurate enough for preliminary estimatesof, say, the total tonnage of steel likely to be required. Thechief usefulness of the formulae is, however, for estimating pur-poses when time, or number and nature of estimates required, oruncertainty of data, makes it impossible to invite tenders or evento prepare rough designs.

The formulae merely give the estimated tower weights aboveground line. The total cost of an erected tower involves supply,transport and erection of the tower itself and of the foundations,and various overhead charges, profit, etc. Many of these itemswill not be proportional to the tower weight, but the authorfinds that for estimating purposes it is reasonable to assume thatthe complete erected cost of a tower of given general type isproportional to its above-ground weight. The assumed totalerected cost per ton will, of course, vary with the general type oftower and contract concerned, but from previous contracts asuitable unit cost can be chosen. It is obvious that the cost ofan erected ton of steel 300 ft above ground, on a contract in-volving two towers only a few hundred feet apart, will not bethe same as that of a ton of steel when only 60 ft above groundon a contract involving 600 towers spread over 100 miles ofcountry.

The following are examples of the types of investigation inwhich the weight formulae prove extremely useful and in which,without them, consistent and sufficiently accurate answers wouldbe almost unobtainable, short of actually designing a largenumber of different towers.

(3.2) Ordinary Transmission Lines(3.2.1) Economic Span Length and General Line Estimates.

The economic span length of a steel tower line dependsprincipally on:—

(a) The size, material and number of conductors.(b) The assumed wind and ice loadings.(<•) The specified factors of safety.

depend upon (d), (e) and (/). The tower loadings due to theconductors, per foot of span, depend upon (a) and (b). Thecosts per tower, independent of span length, depend upon (/)and (/*).

Consider, as an example, a 132-kV double-circuit line with0-25-in2 (copper equivalent) steel-cored aluminium conductorsand one earth conductor, erected generally in accordance withexisting Grid (0- 175-in2) line practice.

Assume on a rough pre-war basis a cost of £45 per ton forerected towers, and £50 per tower for erected insulators andfittings and capitalized wayleaves. With principal tower-height dimensions as indicated in Fig. 12, the total tower

Fig. 12.—132-kV tower, showing principal height dimensions.

height H, in feet, will then be 58 + sag, and the height, in feet,of the centre of gravity of conductor loads 42* 5 + sag.

The calculations are best set out as in Table 5.

Table 5

Span length, ft

(1) Sag (approximately) at 122° F ft(2) H - 58 -f- sag ft(3) Height of centre of gravity of conductors = 42-5 + sag . . . . ft(4) Total transverse conductor load lb, , . _ , , ^ , . . item (4) x item (3) , ^ ^ ,, r.(5) Conductor overturning moment = • ^ • • * °°° lb"ft

(6) Tower-wind overturning moment (estimated) 1 000 Ib-ft(7) M = item (5) + item (6) 1 000 lb-ft(8) VM(9) Tower weight - QW>\6H\/M tons

(10) Erected cost per tower at £45 per ton(11) Cost, per tower, of insulators, fittings and capitalized wayleaves(12) Total cost per tower position = item (10) + item (11)(13) Towers per mile . .(14) Total cost per mile, variable with span length = item (12) x item (13) . .

600

967

51-55 750

296

126422

20-62-21

£99-5£50£149-5

8-8£1 315

800

1674

58-57 680

449

153602

24-62-91

£131£50

£1816-6

£1 195

1000

2583

67-59600

648

193841

293-84

£173£50

£2235-28

£1 180

1200

3694

78-511500

904

2471 15133-95 1

£230£50

£2804-4

£1 230

1400

4910791-5

13 400

1 225

3201 545

39-46-75

£304£50

£3543-77

£1 330

(d) The minimum conductor clearance to ground at highesttemperature.

(e) The general conductor disposition.If) The line voltage.(g) The cost of erected towers per ton.(h) The costs of erected insulators and fittings and of capi-

talized wayleaves.The conductor sag/span relation depends upon (a), (b) and (c).

The components of total tower height, independent of sag,

The tower-wind overturning moment, item (6) in Table 5, canbe estimated from a rough analysis of designs of existing,generally similar towers. From these previous designs an ap-proximate estimate can be made of the likely projected area ofsteel members expressed as so many square feet per foot of towerheight. The actual figure will vary somewhat from base totower top, and will depend, of course, on the general type andsize of tower.

The total cost per mile variable with span length is plotted,

270 RYLE: STEEL TOWER ECONOMICS

vari

i

S

C"

'c0

Sa.

1400

1300

1200

1100

\.—

y

1000

200 400 600 800 1000 1200 1400Spanlpngth.fcet

Fig. 13.—Economic span length (132-kV, double-circuit, 0-25 in2s.c.u.).

with a displaced zero, in Fig. 13. It will be seen that over thespan range considered the curve is comparatively flat, and thatthe variable cost changes only about 6% between span lengthsof 700 and 1 300 ft. Between, say, 800 and 1 100 ft the changedoes not exceed about 2%. The corresponding percentages ofthe true total line cost would, of course, be very much less.The economic span length could be put at 900 or 1 000 ft; theauthor favours the latter. The majority of line faults occur atinsulators, so that the fewer insulators there are the better; asan insulator, air is cheap, reliable and self-repairing. Ingeneral, too, wayleave considerations and amenities favourmaximum spans or, in other words, minimum number of towersvisible from any one spot.

It may be pointed out that the economic span length as obtainedabove would be little affected by quite considerable changes inthe relative assumed unit costs. For instance, if the insulator-plus-wayleave costs per tower were halved, the economic spanlength would still be about 800 feet; if doubled, about 1100 ft.

When preparing preliminary estimates for projected large-scaletransmission schemes, it is frequently necessary to examine twoor three different line voltages, and for each voltage possibly twoor three conductor sizes.

The chief effects of voltage on line cost, apart from the actualcost of insulators, are on account of conductor spacing; insulatorstring lengths, clearances from live metal to steelwork, andcorona considerations decide the cross-arm dimensions andspacing and the height of the bottom cross-arm above ground.Below say 66 kV, except for the "step" at the change from pin-type to suspension insulators, the effects of voltage on towerdimensions and cost are relatively small, but at the highervoltages the proportion of total tower height determined byvoltage requirements becomes increasingly important. On a132-kV double-circuit tower, approximately one-third of thetotal tower height is above the bottom cross-arm; a 330-kVdouble-circuit tower, if such an unlikely structure were used,would have well over half its total height above the bottomcross-arm.

The tower-weight formulae enable estimates of total line costto be made fairly quickly, and the various alternatives are allknown to be on a common, sound basis for comparison. With-out some such common basis it is usually necessary, in estimatingfor a number of alternatives, to make somewhat arbitraryadjustments from costs of previous contracts which may havebeen carried out by different contractors, at different dates,under different labour and transport conditions, and possiblyeven in different parts of the world, and it is very easy to arrive

at a list of line-cost estimates which appear reasonable at firstsight, but which reveal awkward inconsistencies among them-selves.

(3.2.2) Effects of Conductor Material.The voltage and conductor size of a projected line may be

fixed, and comparative line costs for different conductor materialsmay then be required. For important lines, especially on newschemes, possibly abroad, it may be justifiable for reasons offinance, policy or even of prejudice to examine the pros andcons of any or all of the four present-day alternatives, i.e. copper,cadmium-copper, steel-cored copper or steel-cored aluminium.Each, apart from the conductor cost itself, will mean differentvalues of sag and tower height, tower loadings, economic spanlength, and therefore different total costs.

(3.2.3) Number of Circuits.A comparison of costs of single- and double-circuit lines is,

of course, very frequently required. Sometimes congestion ofroute space may justify consideration of sections of line carryingthree or more circuits on one set of towers. The leading-out ofa large number of circuits from a major power station in a moreor less built-up area is one condition which may warrant suchan investigation; the impracticability of more than one line oftowers along the bottom of a steep-sided gorge through amountainous area is another.

Generally speaking, a tower designed to carry a number ofcircuits should cost less per circuit than one designed to carryfewer circuits. At first sight multi-circuit lines are thereforeattractive, but the basket rapidly becomes too full of eggs.The more important a line is, the less can one tolerate even thetemporary loss of more than one circuit at a time. Directlightning strokes to tower or earth conductor are likely to affectsimultaneously more than one circuit on a tower. This alsoapplies to crashed or misdirected aircraft (a hazard likely to in-crease, even in peace time), to the occasional blown tree branchor similar object, or to the admittedly very rare failure of a towerdue, for example, to blizzard overloading, flood "wash-out" offoundations, subsidence, or malicious or accidental damage.

(3.2.4) Number and Height of Earth Conductors.The voltage and line-conductor size and material of a pro-

jected line may have been settled, but, especially in certainregions abroad, it may be desirable to give special attention tolightning considerations. Comparative estimates may then berequired for otherwise identical lines with one, two or even threeoverhead earth conductors. Such earth conductors, being atthe tower top, seriously affect the tower overturning moment,and if there is more than one they call for additional weightowing to special "top-hamper." Theoretically, the higher theearth conductors can be put above the line conductors thebetter from the point of view of lightning protection, but thereare obviously practical limits. Estimates may thus be requiredfor earth conductors at various heights.

(3.2.5) Abnormal Conductor Loadings.Especially after any unusually heavy occurrence of wind

and/or ice loading, which has led to serious line failures (whetherof conductors or towers or both), the question is revived whetherlines should not be designed with greater margins against suchcontingencies. Such inquiries may take several different forms;for instance the effects on total line cost if, say, (a) the assumedwind pressures are doubled, or (b) the assumed radial thicknessesof ice-coating are doubled, or (c) the assumed loadings remainunaltered, but the factors of safety in conductors and towers aredoubled.

Each involves a different approach. For (a) and (b), the con-

RYLE: STEEL TOWER ECONOMICS 271

ductor wind loads and the sag and tower height would be in-creased, but in different ways and to different degrees. For (c),the conductor wind loads would be unaffected, but the sag andtower heights would be increased and the tower weights wouldbe increased owing to both the greater height and the greaterfactor of safety.

(3.2.6) Clearance to Ground.Estimates are sometimes required for the variations to be

expected in total cost of an otherwise standard design of line fora change in the specified minimum clearance from bottom con-ductor to ground. Typical examples are estimates of (a) thepercentage reduction in cost if ground clearance were reducedto 17 ft, or even to 15 ft in remote regions such as open moor-land, and (b) the percentage increase in cost if ground clearancewere increased to 40 or 50 ft for traversing suburban built-upareas.

(3.2.7) Catenary-Suspended Auxiliary Cables.It is sometimes necessary to make comparative estimates for a

line carrying a suspended auxiliary cable, and for the alternativeof a straightforward line with the auxiliary cable underground.The former is sometimes a convenient, but rarely an economical,arrangement. The suspended cable must, for a given span,have a sag usually very much greater than that of the conductorson the straightforward line. This means a much shortenedspan. The wind load on the catenary plus cable and suspendersis large, adding appreciably to the tower overturning moment.The costs of the suspending fittings and of actually running outand erecting the cable are very high, and the overall comparisonoften favours putting the cables underground and leaving theline free to assume its economical long-span construction.

(3.2.8) "Dog-leg" Lines in Congested Areas: Economic ConductorTension.

Considerable mileages of line in congested areas are oftenforced to take very indirect routes, usually involving shortenedspans and a very high proportion of angle towers. Althoughthere would rarely be justification for a distinctive range of towerdesigns for such routes, the author has, purely as a matter ofinterest, investigated the economics of a 132-kV, 0- 175-in2 steel-cored aluminium line on the quite hypothetical assumptions thatthe mean span is only 700 ft and that every tower will carry, onthe average, a 30-deg deviation. On such an angle tower, ofcourse, the major part of the conductor transverse loading is dueto the resultant of the conductor tensions, the wind loads beingrelatively unimportant. It appears that designing such a hypo-thetical line for the ordinary maximum working conductor tensionof 8 0001b might not be desirable, and that, in fact, it mightpay to adopt a lower tension. Reduced tension would reducethe tower loads, but would increase the sag and therefore thetower height, so that some economic value for the tension is tobe expected. On the quite arbitrary example chosen, thiseconomic tension works out at about 5 000 lb instead of thenormal 8 0001b.

(3.2.9) Hypothetical "Super E.H.V." Lines.Interest is occasionally shown in possible, but so far entirely

hypothetical, future lines at voltages above, say, 220 kV. Noprevious contracts or even designs are available as a basis forestimating, and the vague generalities of the type of problemrule out the invitation of tenders or the preparation of designs.However, the tower-weight formulae can, after sketching outprobable leading tower dimensions for each alternative, be usedto arrive at reasonably justifiable and at least comparable esti-mates. Typical questions may be: (1) What would be the ap-proximate relative costs of single-circuit lines at, say, 264, 330,

440 or 550 kV? (2) At, say, 440 kV, what would be the com-parative costs, on some particular basis of equal power-transmitting capacity, for single-circuit lines as follows: (a) a3-phase a.c. line, (b) a 2-wire d.c. line, and (c) a 1-wire, earrh-return, d.c. line?

(3.3) Navigable-River CrossingsThe main topographical features affecting the general layout,

economics and design of a navigable-river crossing are thefeasibility of suitable approach spans, the prescribed minimumclearance to high-water level, Air Ministry height restrictions,the length of the waterway crossing, the general contour andheight of the banks above high-water level, and foundationconditions.

(3.3.1) Approach Spans.The desirable arrangement is, of course, such that the tall

crossing towers do not have to carry any horizontal loadings dueto the conductor tensions, but only those due to the transversewind on the conductors. This means that the tall towers shouldpreferably carry no angle-of-route deviation and no terminaltension, the conductors being anchored off at anchor towers ofsubstantially normal height situated in line with the crossingspan. The Thames crossing (main span 3 060 ft, approach spanseach 1 530 ft, clearance at high water 250 ft, sag 172 ft, towerheight 487 ft) is a good example of this arrangement.2 Some-times, in congested districts, it is impossible to obtain suitableapproach spans, and the tall towers may have to carry an angle ofdeviation or, worse still, act as terminal or self-anchoring towers.These towers are, of course, heavy and costly.

(3.3.2) Clearance to High Water: Air Ministry Height Restrictions.It sometimes takes a long time to obtain a final decision from

the river authorities as to the prescribed minimum clearance to-high water; in the meantime, comparative estimates may have to bemade for crossings giving different clearances. It is frequently notfully realized that at an important crossing the river authoritieshave the power arbitrarily to increase very considerably the totalcrossing cost by specifying an excessive clearance. On a talltower, a large proportion of the total overturning moment is dueto wind on the tower itself, which, other things being equal,makes the tower weight approach direct proportionality to thesquare of the tower height.

Whilst the requirements of river authorities tend to make thetowers as tall as possible, the Air Ministry may require thatsome maximum overall height is not exceeded. To satisfy bothit may be necessary to adopt special measures to reduce the con-ductor sag, e.g. by using an uneconomically large conductor, orby shortening the crossing span by placing the towers in water-edge positions involving heavy special foundation costs.

(3.3.3) Span Length and River-Bank Configuration.For a given conductor, the sag may be regarded as propor-

tional to the square of the span, so that the siting of a crossingover a river of large but varying width may have a considerablebearing on the cost. It may sometimes pay to make a widediversion in the general run of a transmission line in order tosite the crossing at a narrow reach. On the other hand, acrossing over a narrow reach, where the towers have to besituated on flat ground only a few feet above the water, mayprove more expensive than a somewhat longer span crossingwhere the ground rises abruptly to a good height above thewater level and the towers themselves can be of comparativelylow height.

(3.3.4) Foundation Conditions.The foundations for the tall towers of a river crossing have,

272 RYLE: STEEL TOWER ECONOMICS

more often than not, to be piled. The cost of such foundationsis fairly high, but in the early estimating stages of an investiga-tion, which may involve considering the possibilities of a numberof different crossing sites, it is usually impracticable to carry outtrial borings. In any case a trial bore is of little value unlessmade at the precise location of the proposed foundations; 200 ftaway the underground conditions may be quite different. Pre-estimation of foundation costs can therefore sometimes be littlebetter than a well-informed guess based upon actual costs ofroughly similar previous jobs. However, with good approxima-tions for the tower overturning moment, tower base-width, andtower weight, and assuming likely lengths of piles required, it ispossible to arrive at a reasonable estimate of the foundationcost, and one which will give figures comparable with those foralternative crossings.

(3.3.5) Number of Circuits.Power stations are frequently situated on the banks of navigable

rivers, and it is therefore sometimes necessary, in the'neighbour-hood of a station, to run a number of circuits across the river.A decision has then to be made, usually as a compromise betweencost, reliability and site facilities, as to the number of circuits tobe carried on one set of towers. To take a fairly extreme case,suppose that six circuits have to be taken over a river. Possiblesymmetrical arrangements would be (a) six single-circuit crossings,(b) three double-circuit crossings, (c) two 3-circuit crossings, and(d) one 6-circuit crossing.

Purely from the aspect of operational reliability, (a) would befavoured since lightning, aircraft or other hazards would beunlikely to cause the loss of more than one out of the six circuitsat any one time; on the other hand, it would be by far the mostcostly and would, in practice, often be impracticable owing to siterestrictions. At the other extreme, (d) would almost certainlybe the cheapest and the easiest to site; but the risk of simultaneousfaults on several or even all of the six circuits would be a veryreal one. The choice might then lie between (b) and (c), ofwhich the latter would be the cheaper and the easier to accom-modate. The seriousness of possible multi-circuit faults would,of course, be minimized by ensuring that the two circuits of anyduplicate supply were split between the two separate crossings.

Power-station sites are now also studied from the aestheticangle, and the preservation of amenities may have a bearingon the number of separate crossings. Six, or even three, separatecrossings, probably of slightly different span lengths and thereforedifferent sags, would, if fairly close together, certainly be un-sightly.

(3.3.6) Long Spans: Economic Conductor.

If the crossing span is very long, say above 1 500 ft, the sagof a normal line conductor becomes very great and representsa large proportion of the total tower height. A conductor ofgreater tensile strength will permit smaller sag and reduced towerheight; for equal conductivity, however, this will usually meangreater conductor diameter and therefore greater conductorwind loads on the towers. The cost of the conductor itself andprobably of the supporting insulators and fittings will also behigher. Increased conductor tension means more expensiveanchor towers terminating the approach spans. The economicbalance is therefore complicated, but it may be financially im-portant for it to be fully explored.

The Thames crossing is an outstanding example. It forms,electrically, part of a standard 132-kV line with 0- 175-in2 steel-cored aluminium conductors, 0-77-in diameter, with a maximumworking tension of 8 000 lb. If this standard conductor hadbeen used at the crossing, the sag in the 3 060-ft crossing spanwould, under the increased wind pressure assumed for con-

ductors hundreds of feet above ground level, have been of theorder of 380 ft, which would have called for towers nearly 700 fthigh instead of the actual 487 ft. The total cost of the crossingwould have been very much increased.

The actual conductor used, 84 strands of phosphor-bronzeover 7 strands of cadmium-copper, with an overall diameter of0-942 in, a weight per foot of 2-07 lb, and a maximum workingtension of 25 650 lb, was chosen after an exhaustive investigationof the economics of this and other conductors. The otherconductors considered included: steel-cored aluminium of con-ventional make-up (2 layers of aluminium wires) but of sizeslarger than normal, steel-cored aluminium with extra large steelcore and only one layer of aluminium wires, cadmium-copper,and high-tensile steel alone.

Apart from economics, features such as freedom from corrosionand fatigue troubles, conductivity and current-carrying capacitybased on heating, and manufacture, transport and erection had,of course, also to be considered.

(3.3.7) Narrow Waterways: Number of Crossing Towers.The first mental picture of the general layout of a river crossing

is naturally of a tall tower each side of the river. However, fornarrow waterways it will often be found economical to use onlyone tall tower. Fig. 14A shows diagrammatically a two-tower

Fig. 14A.—Narrow waterway.Note: In Figs. 14A and 14B, the tall towers are represented by vertical lines carried

up to the bottom conductor only: hatched areas indicate requisite clearances to water.Heights and conductor sags are exaggerated for clearness.

crossing ABCD, and the alternative single-tower crossingAED. Depending chiefly on the waterway width, clearance towater, and sag/span relationship of the conductor used, the costof the single tower may be either less or more than that of thetwo shorter towers.

The River Roding crossing, with a span of 1 829 ft, a width ofwaterway of only 370 ft, and 150 ft clearance to water, is anexample of a single-tower crossing. The tower, Fig. 5, is 362 fthigh.

A fairly narrow river may have an island in mid-stream andsuitable for siting a tower, as in Fig. 14B. Depending again on

Fig. 14B.—Waterway with possible mid-stream tower site.

waterway widths, clearance to water, and sag/span relationship,the economic arrangement may be any one of the following:crossing ABCDE, three not very tall towers B, C and D; crossingAFGE, two rather tall towers F and G; or crossing AHE, onevery tall tower H.

RYLE: STEEL TOWER ECONOMICS 273

(3.3.8) Terminal-Type Crossing Towers: Short Spans: EconomicConductor Tension.

In congested districts it is sometimes impossible to find roomfor approach spans, and the tall crossing towers are forced tobe of the terminal type. The main overturning moment to beresisted is then the longitudinal one due to the conductor tensionsand, especially for double-circuit or multi-circuit towers, willbe large. If the crossing span is short, it may prove economicalto use considerably reduced conductor tensions at the expenseof slightly taller towers. For a particular conductor, theeconomic tension will depend chiefly on the relative proportionsof the span length and the clearance to water.

Take, as an example, a hypothetical 132-kV double-circuit,0-175 in2 steel-cored aluminium, 900-ft crossing span, clear-ance to water 150 ft, and tower bases substantially at high-water level. If the maximum working conductor tension werethe normal 8 000 lb, the 122° still-air sag would be about 25 ft,and the total tower height about 210 ft. If the conductortension were reduced to 6 000 lb, the longitudinal loads on thetowers would be reduced by 25 %, but the sag would only increaseto about 33 ft, adding 8 ft or only about 4% to the tower height.For this particular example, it can be shown that the economicconductor tension would be only about 4 000 lb. With a stillshorter span, the economic tension would be still slacker; at a500-ft span, with other particulars the same, the economictension would be only about 2 000 lb, or one-quarter of thenormal.

(4) CONCLUSIONGeneral problems calling for discussion are the following:(1) Experience of any actual tower designs showing marked

departure from the general limits of tower weight in terms ofH-\/M indicated in Figs. 2 and 3. Further data for amplifyingFig. 3 would also be valuable.

(2) The effect of broken-conductor design assumptions ontower weights.

(3) Whether a tower base width of 0 -5 \ / ^ f t approaches, ingeneral, the practical and economical dimension for any tower.

(4) How far the total cost of an erected tower of any singlegeneral type can be regarded as proportional to its above-ground weight, and to what extent actual cost depends on suchfactors as: («) number and simplicity of member joints; use oravoidance of gusset plates; bolt sizes, and number of bolts andholes per ton of steel; (b) number of members of different sizesand lengths, for example to what extent a parallel-sided towerwith all braces of the same size, length and inclination would becheaper, per ton, than one of normal design in which such a con-dition cannot usually be approached; (c) commercial availabilityor otherwise of the exact sizes of rolled-steel sections theoreticallydesirable; templates; fabrication; storage; galvanizing; transport,etc.; {(I) special erection costs dependent chiefly on tower height;some aspects of this have been discussed by Boyse and Hart,2

and Eve and Brown.3

It would be interesting to know how far the formulae fortower weight and base width given here have parallels in bridgeand girder design. For such structures, height would be re-placed by unsupported span, and base width by beam depth.In bridge and girder design, deflection is, of course, often amajor criterion, whereas a transmission-line tower is almostinvariably designed for strength alone, and stiffness or de-flection at working load is of no significance.

(5) ACKNOWLEDGMENTSAcknowledgments and thanks are due to the Central Electricity

Board, Mr. C. O. Boyse (British Insulated Callender Cables,Ltd.), Mr. J. L. Eve (J. L. Eve Construction Co., Ltd.), Mr.

J. R. Harding (Pirelli-General Cable Works, Ltd.), for assistanceand the loan of slides, etc.; and to Merz and McLellan for per-mission to publish the paper.

(6) REFERENCES(1) NEATE, E. C , and BOWLING, W. F.: "Reinforced Concrete

Transmission Line Supports," Journal I.E.E., 1944, 91,Part II, p. 341.

(2) BOYSE, C. O., and HART, F. DE B.: "The 132-kV OverheadCrossing of the River Thames," The Paris H.T. Con-ference, 1933, Paper No. 53.

(3) EVE, J. L., and BROWN, R. C : "The Erection of TallTowers," Institution of Civil Engineers, Structural andBuilding Engineering Division, 1945, Pamphlet No. 11.

(7) APPENDICES(7.1) Approximate Economical Design of Elementary Lattice

MastFig. 15 represents the elevation of one face of an elementary

parallel-sided, lattice steel mast, square in plan, without cross-arms, and of modest dimensions. The single horizontal load P(in 1 000 lb) acting on the whole mast at the top, and the height h(in feet) are given. It is required to find the economical base-width b, the economical brace disposition and the weight of theeconomical mast.

It is assumed that:(1) Leg members are throughout of one size of B.S. equal

angle, i in thick.(2) Brace members are throughout of another size of B.S.

equal angle, -£g in thick.(3) All braces are at one inclination, 9, to the horizontal.(4) Slenderness ratio is 100 in leg members, and 150 in braces.(5) At IIr = 100, safe working stress in leg members is

10 000 lb/in2.(6) Brace-member size is fixed by slenderness only, and stress

in braces can be ignored.(7) The mast is a plain cantilever, and the effect of its own

weight on stress in leg members is negligible.(8) Weights of member joints, bolts, etc., are ignored.(9) Weight of steel, 0-284 lb/in*.Some of these assumptions are debatable, but they must be

made for the simplified treatment given, and in any case theyare not unrepresentative.

Load in leg member is 1 OOOPh/lb lb.At a stress of 10 000 lb/in2, the leg area required

1 OOOP/i Ph . .= ^cr in2 . . . . (1)10 000 x 2b 20b

PhVolume of steel in four legs = 4 x h x - - • x 12 in3

PhWeight of four legs = 0-284 X 4 X h x ^ -r x 12

lb (2)

From equation (1), the leg area is Ph/20b in2; for any B.S.equal angle, i in thick, it can be taken that the ratio (area inin2)/(radius of gyration in inches) = 2-41. The leg radius ofgyration is therefore Phl(20b x 2-41) in.

At ///• = 100, the unsupported leg length

a =lOOPh

20b x 2-41 x 120-173 Ph

(3)

274 RYLE: STEEL TOWER ECONOMICS

Substituting this value of 6 in (2) and (4) respectively, we obtain

04h\/(Pli) lb . (7)

1-82 x 126 0-1466. „

Length of one brace = 126/cos0in. At !/r = 150, braceradius of gyration is 126/150 cos 6 in; for any B.S. equal angle,-&- in thick, it can be taken that the ratio (area in in2)/(radius ofgyration in inches) = 1-82.

The brace area is therefore150 cos 6 cos 6

The total length of braces per mast face = A/sin 6 ft.The weight of all braces is therefore

1-986A4 x 0-284 x ~ a x ' _sin d cos 0 lb.

As sin 6 cos 6 —

sin 6 cos 9tan 6 a

f T t a n 2 0 ' a n d t a n ^ 2 6 ( F i g - 15) ' t h e

weight of all braces is therefore

1

a2b

lb.

P, in 1OOOI b

h ft

Fig. 15.—Elementary lattice mast.

From equation (3), a = 0- 173PA/6, therefore the weight of allbraces is

0-99A™*+?*». (4)

From equation (2), the weight of all legs is 0-68Ph2/b lb, sothat the total weight of the mast is

0-852Ph2 , 22-9634 In (5)

To find the economical base-width, differentiate with respectto 6 and equate to zero; this gives

0-852P//2

From which 6* = 0 0124PW

and 6 = 0- 333V(Ph) ft

Weight of legs = A-

Weight of braces =

0-333 V(Ph)

\12Ph*

(8)

Total economical mast weight, (7) + (8) = 3-4\h\/(Ph) lb or0• 00153hV(Ph) tons (9)

From (7), (8) and (9),2*04

Weight of legs = ^-^r, or 60% of total weight

1-37and weight of braces = ~-jr, or 40% of total weight

(10)

For the economical mast, these percentages are, on the as-sumptions made, constant and independent of P and h.

The slope of the braces is given by tan 6 = o/26

From (3) and (6), tan 6 = 0- 666 V(Ph)

Therefore = 38C (ID

(6)

On the assumptions made, this slope is constant for theeconomical mast and independent of P and ft.

The following are conclusions for the economical elementarymast on the basis of the preliminary assumptions made: Re-placing, in the expressions obtained above, Ph by M, where Mis the overturning moment at the ground line in 1 000 lb-ft,

Economical base width = O-333^/Mit.Economical mast weight = 0-00153 hy/Mtons.Leg weight = 60 % of total mast weight.Brace weight = 40% of total mast weight.Economical brace inclination to the horizontal = 38°.Variations in the preliminary assumptions within the limits

representative of normal practice will alter these results onlyslightly; it is therefore suggested that, as a basis for actual pre-liminary design of small narrow-base towers, they may be ofsome practical value. It should be remembered that the mast-weight formula represents a plain mast body above ground lineand without cross-arms.

(7.2) General Approximate Weight of a B.S. Equal-Angle Strutof Given Thickness

Let P represent the design compression load in the strut, inlb; a the area of cross-section, in in2; r the radius of gyration, ininches, and / the length in inches. Assume that the safe workingstress follows the straight-line strut formula, 18 000 - 80//rlb/in2.

Then P = a(18 000 - 80//r) = 18 000a - SOIa/r.Now for B.S. equal angles of any one thickness, afr may be

regarded as a constant n. Therefore, P = 18 000a — SOnl, orfl = (p+8O/7/)/18OOO.

The volume, and therefore the weight, of the strut is propor-tional to la, so that the weight is proportional to (PI+ 80A//2)/18 000, or to PI + cl\ where c is a constant.

The constant c will be different for other strut formulae, andwill, of course, vary for different thicknesses and again for shapesof cross-section different from B.S. equal angles. However, thenature of the general relationship between load, length andweight of a strut of given thickness should be approximately asgiven above.