Research ArticleOptimization of Rigidly Supported Guyed Masts

R BeleviIius1 D Jatulis2 D RusakeviIius3 and D MaIi0nas3

1Department of Information Technologies Vilnius Gediminas Technical University Sauletekio Al 11 LT-10223 Vilnius Lithuania2Department of Bridges and Special Structures Vilnius Gediminas Technical University Sauletekio Al 11 LT-10223 Vilnius Lithuania3Department of Engineering Mechanics Vilnius Gediminas Technical University Sauletekio Al 11 LT-10223 Vilnius Lithuania

Correspondence should be addressed to R Belevicius rimantasbeleviciusvgtult

Received 9 February 2017 Accepted 23 April 2017 Published 29 May 2017

Academic Editor Luigi Di Sarno

Copyright copy 2017 R Belevicius et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A technique for simultaneous topology shape and sizing optimization of tall guyed masts is presented in this paper The typicalscheme of telecommunication masts that are triangular in plan and supported by a certain number of guysrsquo levels is examinedTheobjective function is the total mass of the mast including the mass of the guys The mast structure is optimized for self-weightand wind loading that is evaluated according to Eurocodes The nonlinear behaviour of the guyed mast is simplified idealizingthe nonlinear guys as approximate boundary conditions for the mast A comparison of the simplified solution with the resultsof nonlinear analysis with Ansys shows small discrepancies that are on the safe side The constraints involve all strength localand global stability and slenderness requirements The optimization problem is solved using evolutionary algorithm with originalgenome repair procedure As an example a typical 96m guyed broadcasting antenna for mobile-phone networks was designedemploying the proposed optimization technique and taking up to 10 guysrsquo levels It is shown that the optimal mast is supported by3ndash5 guysrsquo levels The optimal ranges of all remaining design variables were also obtained

1 Introduction

In recent years the development of telecommunication net-works has stimulated the amount of design and constructionof tall steel-guyed masts The mast structures are usuallyproduced and constructed in certain quantities as typicalstructures depending on the terrain conditions and requiredarea of antennas Therefore research innovation and costoptimization of guyed masts are a relevant engineeringproblem

The guyed masts are supported by prestressed guy cablesattached to the shaft of the mast at different levels Thesecables exhibit nonlinear behaviour and are very sensitive totheir geometry and prestress levels therefore despite theelastic behaviour of the mast shaft these structures should beanalysed by a nonlinear analysis programThe shaft is usuallycomposed of a number of sections of identical structureand dimensions one section comprises shaft legs stiffenersbetween legs of the shaft and oblique bracing elements Oneexample of triangular mast and the structure of the shaft areshown in Figures 1 and 2

The problem of analysis of a guyed mast is even morecomplicated since the dominant loading on the mast isturbulent wind and the dynamic effects cannot be neglectedin a reliable analysis

Those nonlinear prestressed structures were analysed ina number of research papers for example Gantes et al[1] and Smith [2] Different analytical and computationalmethods for evaluation of the strain-stress behaviour weresuggested by Wahba et al [3] He et al [4] and Gioffre etal [5] Several researchers pursued improvements of maststructure seeking the least possible weight (synonymous withcost) of the structure (Jasim and Galeb [6]) In most casesrefinement of the structure was understood exclusively asselection of the geometrical scheme of shaft elements that isleg bracing and stiffener members and the dimensioning ofcross sections However the maximum effectiveness of thosestructures can be achieved by simultaneously tuning all thetopological geometric and physical parameters

The topology of the guyed mast is identified by thenumber of the guysrsquo clusters the number of guysrsquo anchors tothe ground the geometric scheme of a typical shaft section

HindawiAdvances in Civil EngineeringVolume 2017 Article ID 4561376 9 pageshttpsdoiorg10115520174561376

2 Advances in Civil Engineering

h

a

Mean wind loading

z

i = 1 2 3 4 5 6

Patch loads

Figure 1 Typical mast with two clusters of guys Horizontal plan of the mast

g g

b

s

a

L

Typicalsection

Figure 2 Structure of the shaft 119871 leg 119887 bracing 119904 stiffenermembers 119892 guy and 119886 auxiliary equipment

and the shaft-ground connection scheme Some of theseparameters were chosen before the optimization For exam-ple it is obvious (and it is explicitly shown in [7]) that themastof the triangular horizontal plan is lighter than the squareplan mastThus only the triangular mast is considered As tothe shaft-ground connection scheme two schemes prevail inthe engineering practice pinned at the foundation and fixedIn the pinned masts the stresses in the leg are distributedmore evenly In fixed masts the horizontal displacements ofthe shaft cause significant bending moments at the supportand the maximum stresses develop in the lowest sections ofthe mast Such uneven distribution of stresses is not rationaland results in a substantially heavier structure comparedwith the pinned masts However pinned masts have inherentdeficiencies Firstly the shaftmust be strutted at the assemblystage of mast construction Secondly the hinged connectionat the support does not provide the torsional stiffness of themast therefore either the antitwist tackle (ldquomounting starrdquo)or sophisticated support construction that does not transferthe bending moment but assures the required torsionalstiffness must be provided [8] Due to these drawbacks of thepinned scheme in this paper the mast scheme that is fixed atthe foundation is considered

The shape of the mast is defined by the vertical locationsof the guy attachments to the shaft the horizontal locationsof the guy anchors on the ground the width of the shaft andthe height of a typical section The sizing parameters includecross sections of all structural elements of the shaft and crosssections of guys All leg bracing and stiffener elements areround solid steel bars and are of constant diameter along themast height In addition the pretension forces have to bechosen in guys at all levels

The aim of this paper is twofold to suggest the techniqueenabling the use of a classical EA with special genome repairprocedure for optimization of guyed masts and to obtain theoptimal scheme of a typical 96m tall mast The suggestedtechnique should yield the optimal that is lightest schemefor a mast with ldquoone button clickrdquo using the following initialdata for the structure

(i) Height of the mast(ii) The geometrical scheme of the typical mast section(iii) The geometrical dimensions of auxiliary equipment

and antennas(iv) Material data of structural elements and guys (Youngrsquos

moduli specific weights material is treated asisotropic)

(v) Maximum allowable stresses in elements and guys(vi) Maximum allowable deflection at the top of the mast(vii) Loading data(viii) Lower and upper limits for radii of leg bracing

stiffener members and guys

In mathematical terms the mast optimization problemis a mixed integer global optimization problem since thelandscape of objective function is not known and cannotbe obtained in a closed form The results of numericalexperiments presented in this paper definitely show that theproblem is really multimodal Since the number of designparameters is large we use the stochastic optimization algo-rithms which do not require the sensitivity informationThestochastic algorithms do not guarantee the global solutionof a problem but in the engineering practice it is moreimportant to find a rational solution that is better than thesolution currently known

In [9 10] we optimized the triangular mast with twoclusters of guys using genetic algorithms and simulated

Advances in Civil Engineering 3

annealing and revealed the dominant parameters for optimaldesign In [11] the typical 96m broadcasting antenna formobile-phone networks with an arbitrary number of guysrsquoclusters employing an evolutionary algorithm (EA) fromMatlab [12] was optimized Comparison of optimized maststructure with the typical industrial design revealed possibil-ities for significant reduction of steel expenditure Howeverthe simplified algorithm for evaluation of computationallength between adjacent clusters of guys led to a so-calledldquocheckerboard effectrdquo finer structural elements conditionedbetter objective function values and some threshold valueswere needed to prevent excessive refinement of the structureIn this paper the exact computational lengths were usedfor checking the global stability constraints and a specialgenome repair procedure was employed in EA It is shownthat the mass of the mast is significantly reduced by installingat least three guysrsquo clusters the reduction of mass is upto 40 for a three-guy-level mast compared to a two-levelone but starting from the five-level mast the mass slightlyincreases Also the optimal ranges of the determinant designparameters are discussed

2 Problem Formulation

21 Idealizations The structural behaviour of guyed mastsis extremely complicated The guys exhibit geometricallynonlinear behaviour and more at low pretensioning levelsIncreasing the pretension forces decreases the nonlinearityand enhances lateral stiffness but simultaneously increasescompressive loads and herewith the buckling probability forthe mast The mast itself can be also geometrically nonlineardue its slender structure In addition to the substantial windloads the loads due to the self-weight of themast with all aux-iliary equipment and possible icing should be accounted for

The use of optimization algorithms with tens of designparameters supposes analysing the mast structure andobtaining the objective function for any number of timesTherefore a very fast and reliable analysis tool is a purenecessity and we restrict the analysis only to the linearstage approximately substituting the guys by the springs andcompression forces on the shaft of the mast The horizontalstiffness of springs is proportional to the displacementsat the guy attachment node and the vertical compressionforces depend first of all on the prestress in the guy Thegeometrically nonlinear verification solution of the samemast structure via finite element package Ansys proves theseapproximate assumptions are always on the safe side In theAnsys solution the guys are represented by cable elementseach guy is divided into 10 elements In Figure 3 bothsolutions for the horizontal displacements in the oppositedirection to the wind are compared for the obtained bestmast scheme the 5 guysrsquo cluster mast All geometrical dataare presented in Table 2 The discrepancies in horizontaldisplacements are higher at the middle levels of the mastbut the differences in displacements of nodes at the masttop do not exceed 2 (Figure 3) Since higher displacementsobtained by linearized simplified solution also cause higherstresses in the rigidly supported mast the constraints (see(6)ndash(8) Section 22) slightly narrow the search space of

0 10 20 30 40 50 60 70 80 90 100minus1

minus08

minus06

minus04

minus02

0

02

04

Height (m)

Hor

izon

tal d

ispla

cem

ents

(m)

ANSYS solutionCurrent solutionGuy attachment

Figure 3 Comparison of the idealized solution with the Ansyssolution

optimization problem to the safer side More important hereis the coincidence of displacements at the mast top analysisof the intermediate results of optimization reveals that theconstraint most difficult to satisfy is the given maximumlateral displacement of the mast top

The next simplifying assumption concerns the turbulentwind loading on the structure Instead of dynamic loadingwe evaluate the corresponding static wind loads by the patch-load method (Eurocode 1 Part 1-4 [13] and Eurocode 3 Part3-1 [14]) which adequately mimics the dynamic behaviourof the structure and solve the static problem According tothe Eurocodes the designer has to consider several otherloadings in addition to the wind loading of different direc-tions including icing and several of their combinations Inthis paperwe choose to optimize themast for themost criticaland determinant case of wind loading when the direction ofthe wind is at a right angle to one side of the mast The loadsdue to the self-weight of the structure and equipment areaccounted for but not the ice loads Simultaneous icing andextreme winds are more topical for Nordic and mountainouscountries only However our meshing program has thestraightforward capability of including icing loads of therequired type and class according to the European standardon atmospheric icing [15]

Subsequently after rounding the obtained values of con-tinuous design parameters to suitable values (eg geometri-cal dimensions to in-stock values) the designer may verifythe scheme employing the more accurate nonlinear analysisand taking into account other loading cases Therefore theoptimization technique presented can be useful during thepreliminary mast design stage supplying the designer withhints for the topology and shape of the mast as well asgeometrical data on all structural elements

22 Optimization Problem and Technique The complete setof programs consists of four independent parts the opti-mization algorithm meshing program that renders the setof design parameters to the discretized model of the mast

4 Advances in Civil Engineering

Table 1 Set of design parameters

Parameter Type Quantity of parameters Lower bound Upper boundNumber of guysrsquo clusters 119899119892 Integer 1 1 10Number of typical shaft sections 119899119904 Integer 1 6 120Levels of guysrsquo clusters in sections from the ground 119897 Integer 119899119892 11989911990420 119899119904Distances of guy anchors from the shaft 119889 (m) Continuous 119899119892 5 100Width of the shaft 119908 (m) Continuous 1 020 240Radius of the leg elements 119903119897 (m) Continuous 1 0008 0040Radius of the bracing and stiffener elements 119903119887 (m) Continuous 1 0008 0030Radius of guys 119903119892 (m) Continuous 1 0001 0010Pretension forces in guys 119891 (N) Continuous 119899119892 5000 30000

structure finite element analysis program and program forevaluation of objective function value

The genetic algorithm with preprocessor for genomerepair is used as the primary optimization engine Afterthe best solution from the set of EA solutions is selectedthe pattern search algorithm [16] is employed to explorethe neighbourhood of the solution in depth and (probably)improve it

The meshing program is used as a preprocessor for thefinite element program It inputs the set of design parametersproduced by the optimization algorithm together with datawhich do not vary in the optimization process (eg mastheight wind velocity and material characteristics) and pre-pares the complete dataset for the finite element programAll loadings including self-weight and wind loads on thestructure are recalculated using new values of all geometricalparameters

A fast finite element program is required hence anoriginal problem-oriented Fortran program was developedfor the static analysis All the structural elements of theshaft are represented by beam elements Instead of guys atthe guy attachment nodes the horizontal stiffness boundaryconditions and corresponding vertical compression forces areapplied

After the finite element analysis the postprocessor pro-gram evaluates the value of the objective function verifies allconstraints and transmits the objective function value to theoptimization engine which prepares a new estimate of designparameters set on its basis

In mathematical terms the optimization problem isformulated as follows

119891lowast = min119909isin119863

119891 (119909) (1)

subject to

(i) structure equilibrium constraints(ii) strength constraints in all structural elements of the

shaft and guys(iii) local stability constraints in all leg and bracing ele-

ments(iv) global stability constraints between the clusters of

guys and between the lower cluster and mast foun-dation

(v) slenderness constraints in all leg and bracing ele-ments

(vi) lateral stability constraint of the mast top

119891(119909) in (1) is a nonlinear objective function of continuousand integer variables 119891R119899 rarr R 119899 is the number ofdesign parameters 119909 and 119863 isin R119899 is a feasible region ofdesign parametersThe globalminimum should be foundNoassumptions on unimodality are included into formulation ofthe problem that is many local minima may exist

The total mass of themast including themass of the guysis considered in the objective function Since the materialof the guys is more expensive the mass of the guys ispremultiplied by a factor given by the program user Thecost of guy anchors and the cost of connections can also beincluded into the objective function in a similar manner

The complete set of design parameters along with theircharacteristics and bounds is listed in Section 3 Table 1

All strength stability and slenderness constraints areassessed according to Eurocode 3 Part 3-1 [14] and Eurocode3 Part 1-1 [17] The overall structure equilibrium constraintsare checked solving the static problem with the finite elementprogram The typical 2-node beam elements with 12 degreesof freedom are used to represent the structural elements ofthe shaft The main statics equation is

[119870]119886 119906119886 = 119865119886 (2)

where [119870]119886 is the stiffness matrix 119906119886 are nodal displace-ments and 119865119886 are the active forces of the ensemble ofelements The element stiffness matrix can be found in manytextbooks for example [18] The stress resultants in all finiteelements according to [14] are

119878TM = 119878119872 plusmn 119878119901 (3)

where 119878119872 is stress due to the static wind load on the wholemast height plus the dead weight of the mast structure and119878119901 is cumulative stress due to the pseudo-dynamic loads onthe patches of the mast according to [13] the number of loadcases for 119899119892 guysrsquo clusters is 2(119899119892 + 1) + 1 Thus (2) has to besolved for 2(119899119892 + 1) + 2 times

The influence of guys on the behaviour of the shaft ismodelled by linear stiffness boundary conditions The totalstiffness estimation of the guys in a guysrsquo cluster when the

Advances in Civil Engineering 5

Table 2 Optimal mast parameters at 1ndash10 guysrsquo clusters

119899119892119898119864119860 119908 (m) 119899119904 119903119897 (m) 119903119887 (m) 119903119892 (m) 119873119891 119879 (h)119898119864119860+119875119878(kg)

1 21713 199 38 00430 00184 0010 613030 48721712

2 11018 118 57 00281 00123 00081 629430 111910983

3 6974 086 72 00226 00101 00055 692630 12686940

4 5913 069 82 00206 000855 000489 696230 13905862

5 5538 069 91 00191 000810 000459 683230 15405459

6 5860 068 83 00195 000811 000447 719430 17545475

7 5984 0625 83 00195 00081 000464 749630 21275682

8 6043 086 90 00178 00080 000437 772430 23725828

9 6396 079 88 00183 00080 000441 810430 26266005

10 6891 083 90 00182 00080 000444 784430 26806371

wind blows toward one guy is given by Gantes et al [1]However this case is not critical for themast fixed at the foun-dation The shaft of the mast receives larger displacements inthe opposite wind direction Now two stressed guys absorball wind action while the third guy is slack and provides onlycompression forcemainly due its pretension force In this casewe obtain different stiffnessrsquo expression of one windward guy

119896 = 12119897119909119860119892119864

1198882 (1 + (119898119892119897119909119879119901)2 (11986011989211986412119879119901)) (4)

where 119897119909 is the distance of the guy foundation from the axisof the shaft 119860119892 is the guy cross section area 119864 is Youngrsquosmodulus of guy material 119888 is the length of the guy 119898119892 isthe dead weight of the guy per unit length and 119879119901 is thepretension force in the guy

The compressive effects of one set of guys are idealized bythe vertical compressive forces applied to the guy attachmentnodes

119901 = 119899119892119888119879119901119867119888 (5)

where 119899119892119888 is the number of guys in one guysrsquo cluster andH is the height of the mast Our assumptions (4) and (5)are checked via nonlinear modelling of the mast with guysand are of sufficient precision (Figure 3) The spring stiffnessaccording to (4) is slightly less than the stiffness of the realguy but the error is to the safe side

The first of the inequality constraints the strength con-straints are checked in all structural elements of the mast

accounting for both the axial force N and bending moment119872 for the allowable stress 120590120590 = 1003816100381610038161003816100381610038161003816119873119860 plusmn 119872119882

1003816100381610038161003816100381610038161003816 le 120590 (6)

The stability requirements for the leg and bracing mem-bers are posed according to Eurocode 3 Part 3-1 [14]

|119873| le 1205941198601198911199101205741198721 (7)

where 119860 is the cross section area of the member and 119891119910is the steel yield point (dependent on the diameter of themember) According to [14] the partial factor of resistanceof the members to member buckling 1205741198721 is taken to be 10and reduction factor coefficient 120594 is evaluated via complexengineering formulas depending on the nondimensionalslenderness of structural elements

The global stability requirements between adjacent clus-ters of guys are expressed in the form of a mast segmentbuckling condition due to the equivalent compression force119873119864119889 and the equivalent bending moment 119872119864119889 in the crosssection of the mast

1198731198641198891205941199101198731198771198961205741198721 +119872119864119889120594119871119905 (1198721199101198771198961205741198721) le 1 (8)

where 120594119910 is the reduction factor for the relevant bucklingmode 119873119877119896 is the cross section axial resistance and 119872119910119877119896is the cross section moment resistance (based on either the

6 Advances in Civil Engineering

plastic elastic or effective section modulus depending onclassification) Details on the evaluation of these coefficientsmay be found in Eurocode 3 Part 1-1 [14]

The slenderness constraints for the mast segmentsbetween clusters of guys and slenderness constraints forbracing members correspondingly are

120582 le 120120582 le 180 (9)

Obtaining the buckling length of a mast segment neededfor assessment of the stability constraint takes significantnumerical effort In [11] we used an approximate value ofbuckling length that was equal to the segment length this isadequately accurate for amast supported by 1ndash4 guysrsquo clustersFor a mast supported by a larger number of guysrsquo clusters itled to unrealistic designs when the mass of the mast steadilydecreases with the increasing number of guysrsquo clusters In thisresearch the ith segment buckling length is obtained with anempirical computational length coefficient

120583119894 = radic089005775 + 74379952119898119894 (10)

where 119898119894 is the coefficient of the segment support stiffnesswith regard to the shaft stiffness 119864119868 dependent on the totalstiffness provided by the guys ]119894

119898119894 = ]1198941198971198943119864119868 (11)

where 119897119894 is the length of the ith segment between adjacentguysrsquo clusters The coefficient 1205831 for the segment betweenfoundation and the first guysrsquo level is obtained with a moreawkward procedure

1205831 = 120583avg119896 (12)

where

120583avg = radic089005775 + 74379952119898avg

119898avg = sum119894119898119894119899119904119892

119896 = (1 + 11990611199062 + 11990611199063 + sdot sdot sdot + 1199061119906119899)119899119904119892

119906119894 = 1198971198942radic119873119864119889119864119868

(13)

with 119899119904119892 being the number of segmentsFinally the lateral displacement 119889 of the mast top is

constrained to one percent of the height of the mastThe classical real coded genetic algorithm for solving

mixed integer optimization problems MI-LXPM [19] wasemployed as a primary optimization engine The meshing

FEM analysis and postprocessor programs are connectedto the optimization algorithm as ldquoblack boxesrdquo The maingenetic parameters population size number of generationsprobabilities of mutation and crossover and so on are tunedto the problem solvedThe complete list of genetic parametersis presented in Section 3

The list of design parameters along with their lowerand upper bounds is presented in Table 1 Some of theparameters are interdependent (eg the guys cannot intersector be attached to a nonexisting shaft section) thereforein the initial stage of optimizationmdashrandom generationof individualsmdashonly the viable or repaired individuals areincluded into the initial population For example at therandom generation of one particular individual of the initialpopulation the following data was obtained the number ofguysrsquo clusters 119899119892 = 3 number of sections 119899119904 = 50 (let the upperbound for 119899119904 be 100) and levels of guysrsquo clusters 1198971 = 10 1198972 =60 and 1198973 = 80 In order to repair this infeasible individualthe levels are scaled down proportionally 1198971 = 5 1198972 = 30 and1198973 = 40

Different conditions hold for the invalid individuals of thenext generations obtained due to the mutation or crossoverSince the main idea of optimization technique to keep allconstituent programs independent the obtained infeasibleguysrsquo levels are repaired but the individuals with intersectingguys are retained However with an additional sufficientpenalty given by the preprocessor of the finite elementprogram the finite element analysis for these individuals isnot executed The penalty is proportional to the number ofsuch violations and acts like a ldquodeadrdquo penalty fairly reducingthe chances of infeasible individuals surviving the selection

After the best EA solution is selected the neighbourhoodof the solution is explored with the pattern search (PS)algorithm For optimization problems with a larger numberof design parameters it slightly improves the solution by3ndash6

3 Numerical Examples and Discussion

As an example we optimize the 96m tall guyed telecom-munication mast that is produced by the steel industriesThe industrial mast of triangular horizontal plan has fourguysrsquo clusters anchored at two points 32 and 595m from theshaft The width of shaft is 120m All the leg bracing andstiffener elements are round solid steel bars At the radii ofthe structural elements 002m (leg) 00125m (bracing) and0007m (guy) the total mass of the shaft is 10410 kg while themass of all guys is 1230 kg

The optimal mast scheme is sought retaining the schemeof an industrial mast but varying the topology shape andsizing parameters of themastThe range of guysrsquo clusters from1 to 10 is explored Constant data specific to this mast whichdo not vary in the optimization process are as follows

(i) Height of the mast 96m(ii) Area of antennas distributed in the top 15 of mast

height 1034m2

(iii) Diameter of auxiliary equipment along the wholemast height 005m

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

Advances in Civil Engineering 3

annealing and revealed the dominant parameters for optimaldesign In [11] the typical 96m broadcasting antenna formobile-phone networks with an arbitrary number of guysrsquoclusters employing an evolutionary algorithm (EA) fromMatlab [12] was optimized Comparison of optimized maststructure with the typical industrial design revealed possibil-ities for significant reduction of steel expenditure Howeverthe simplified algorithm for evaluation of computationallength between adjacent clusters of guys led to a so-calledldquocheckerboard effectrdquo finer structural elements conditionedbetter objective function values and some threshold valueswere needed to prevent excessive refinement of the structureIn this paper the exact computational lengths were usedfor checking the global stability constraints and a specialgenome repair procedure was employed in EA It is shownthat the mass of the mast is significantly reduced by installingat least three guysrsquo clusters the reduction of mass is upto 40 for a three-guy-level mast compared to a two-levelone but starting from the five-level mast the mass slightlyincreases Also the optimal ranges of the determinant designparameters are discussed

2 Problem Formulation

21 Idealizations The structural behaviour of guyed mastsis extremely complicated The guys exhibit geometricallynonlinear behaviour and more at low pretensioning levelsIncreasing the pretension forces decreases the nonlinearityand enhances lateral stiffness but simultaneously increasescompressive loads and herewith the buckling probability forthe mast The mast itself can be also geometrically nonlineardue its slender structure In addition to the substantial windloads the loads due to the self-weight of themast with all aux-iliary equipment and possible icing should be accounted for

The use of optimization algorithms with tens of designparameters supposes analysing the mast structure andobtaining the objective function for any number of timesTherefore a very fast and reliable analysis tool is a purenecessity and we restrict the analysis only to the linearstage approximately substituting the guys by the springs andcompression forces on the shaft of the mast The horizontalstiffness of springs is proportional to the displacementsat the guy attachment node and the vertical compressionforces depend first of all on the prestress in the guy Thegeometrically nonlinear verification solution of the samemast structure via finite element package Ansys proves theseapproximate assumptions are always on the safe side In theAnsys solution the guys are represented by cable elementseach guy is divided into 10 elements In Figure 3 bothsolutions for the horizontal displacements in the oppositedirection to the wind are compared for the obtained bestmast scheme the 5 guysrsquo cluster mast All geometrical dataare presented in Table 2 The discrepancies in horizontaldisplacements are higher at the middle levels of the mastbut the differences in displacements of nodes at the masttop do not exceed 2 (Figure 3) Since higher displacementsobtained by linearized simplified solution also cause higherstresses in the rigidly supported mast the constraints (see(6)ndash(8) Section 22) slightly narrow the search space of

0 10 20 30 40 50 60 70 80 90 100minus1

minus08

minus06

minus04

minus02

0

02

04

Height (m)

Hor

izon

tal d

ispla

cem

ents

(m)

ANSYS solutionCurrent solutionGuy attachment

Figure 3 Comparison of the idealized solution with the Ansyssolution

optimization problem to the safer side More important hereis the coincidence of displacements at the mast top analysisof the intermediate results of optimization reveals that theconstraint most difficult to satisfy is the given maximumlateral displacement of the mast top

The next simplifying assumption concerns the turbulentwind loading on the structure Instead of dynamic loadingwe evaluate the corresponding static wind loads by the patch-load method (Eurocode 1 Part 1-4 [13] and Eurocode 3 Part3-1 [14]) which adequately mimics the dynamic behaviourof the structure and solve the static problem According tothe Eurocodes the designer has to consider several otherloadings in addition to the wind loading of different direc-tions including icing and several of their combinations Inthis paperwe choose to optimize themast for themost criticaland determinant case of wind loading when the direction ofthe wind is at a right angle to one side of the mast The loadsdue to the self-weight of the structure and equipment areaccounted for but not the ice loads Simultaneous icing andextreme winds are more topical for Nordic and mountainouscountries only However our meshing program has thestraightforward capability of including icing loads of therequired type and class according to the European standardon atmospheric icing [15]

Subsequently after rounding the obtained values of con-tinuous design parameters to suitable values (eg geometri-cal dimensions to in-stock values) the designer may verifythe scheme employing the more accurate nonlinear analysisand taking into account other loading cases Therefore theoptimization technique presented can be useful during thepreliminary mast design stage supplying the designer withhints for the topology and shape of the mast as well asgeometrical data on all structural elements

22 Optimization Problem and Technique The complete setof programs consists of four independent parts the opti-mization algorithm meshing program that renders the setof design parameters to the discretized model of the mast

4 Advances in Civil Engineering

Table 1 Set of design parameters

Parameter Type Quantity of parameters Lower bound Upper boundNumber of guysrsquo clusters 119899119892 Integer 1 1 10Number of typical shaft sections 119899119904 Integer 1 6 120Levels of guysrsquo clusters in sections from the ground 119897 Integer 119899119892 11989911990420 119899119904Distances of guy anchors from the shaft 119889 (m) Continuous 119899119892 5 100Width of the shaft 119908 (m) Continuous 1 020 240Radius of the leg elements 119903119897 (m) Continuous 1 0008 0040Radius of the bracing and stiffener elements 119903119887 (m) Continuous 1 0008 0030Radius of guys 119903119892 (m) Continuous 1 0001 0010Pretension forces in guys 119891 (N) Continuous 119899119892 5000 30000

structure finite element analysis program and program forevaluation of objective function value

The genetic algorithm with preprocessor for genomerepair is used as the primary optimization engine Afterthe best solution from the set of EA solutions is selectedthe pattern search algorithm [16] is employed to explorethe neighbourhood of the solution in depth and (probably)improve it

The meshing program is used as a preprocessor for thefinite element program It inputs the set of design parametersproduced by the optimization algorithm together with datawhich do not vary in the optimization process (eg mastheight wind velocity and material characteristics) and pre-pares the complete dataset for the finite element programAll loadings including self-weight and wind loads on thestructure are recalculated using new values of all geometricalparameters

A fast finite element program is required hence anoriginal problem-oriented Fortran program was developedfor the static analysis All the structural elements of theshaft are represented by beam elements Instead of guys atthe guy attachment nodes the horizontal stiffness boundaryconditions and corresponding vertical compression forces areapplied

After the finite element analysis the postprocessor pro-gram evaluates the value of the objective function verifies allconstraints and transmits the objective function value to theoptimization engine which prepares a new estimate of designparameters set on its basis

In mathematical terms the optimization problem isformulated as follows

119891lowast = min119909isin119863

119891 (119909) (1)

subject to

(i) structure equilibrium constraints(ii) strength constraints in all structural elements of the

shaft and guys(iii) local stability constraints in all leg and bracing ele-

ments(iv) global stability constraints between the clusters of

guys and between the lower cluster and mast foun-dation

(v) slenderness constraints in all leg and bracing ele-ments

(vi) lateral stability constraint of the mast top

119891(119909) in (1) is a nonlinear objective function of continuousand integer variables 119891R119899 rarr R 119899 is the number ofdesign parameters 119909 and 119863 isin R119899 is a feasible region ofdesign parametersThe globalminimum should be foundNoassumptions on unimodality are included into formulation ofthe problem that is many local minima may exist

The total mass of themast including themass of the guysis considered in the objective function Since the materialof the guys is more expensive the mass of the guys ispremultiplied by a factor given by the program user Thecost of guy anchors and the cost of connections can also beincluded into the objective function in a similar manner

The complete set of design parameters along with theircharacteristics and bounds is listed in Section 3 Table 1

All strength stability and slenderness constraints areassessed according to Eurocode 3 Part 3-1 [14] and Eurocode3 Part 1-1 [17] The overall structure equilibrium constraintsare checked solving the static problem with the finite elementprogram The typical 2-node beam elements with 12 degreesof freedom are used to represent the structural elements ofthe shaft The main statics equation is

[119870]119886 119906119886 = 119865119886 (2)

where [119870]119886 is the stiffness matrix 119906119886 are nodal displace-ments and 119865119886 are the active forces of the ensemble ofelements The element stiffness matrix can be found in manytextbooks for example [18] The stress resultants in all finiteelements according to [14] are

119878TM = 119878119872 plusmn 119878119901 (3)

where 119878119872 is stress due to the static wind load on the wholemast height plus the dead weight of the mast structure and119878119901 is cumulative stress due to the pseudo-dynamic loads onthe patches of the mast according to [13] the number of loadcases for 119899119892 guysrsquo clusters is 2(119899119892 + 1) + 1 Thus (2) has to besolved for 2(119899119892 + 1) + 2 times

The influence of guys on the behaviour of the shaft ismodelled by linear stiffness boundary conditions The totalstiffness estimation of the guys in a guysrsquo cluster when the

Advances in Civil Engineering 5

Table 2 Optimal mast parameters at 1ndash10 guysrsquo clusters

119899119892119898119864119860 119908 (m) 119899119904 119903119897 (m) 119903119887 (m) 119903119892 (m) 119873119891 119879 (h)119898119864119860+119875119878(kg)

1 21713 199 38 00430 00184 0010 613030 48721712

2 11018 118 57 00281 00123 00081 629430 111910983

3 6974 086 72 00226 00101 00055 692630 12686940

4 5913 069 82 00206 000855 000489 696230 13905862

5 5538 069 91 00191 000810 000459 683230 15405459

6 5860 068 83 00195 000811 000447 719430 17545475

7 5984 0625 83 00195 00081 000464 749630 21275682

8 6043 086 90 00178 00080 000437 772430 23725828

9 6396 079 88 00183 00080 000441 810430 26266005

10 6891 083 90 00182 00080 000444 784430 26806371

wind blows toward one guy is given by Gantes et al [1]However this case is not critical for themast fixed at the foun-dation The shaft of the mast receives larger displacements inthe opposite wind direction Now two stressed guys absorball wind action while the third guy is slack and provides onlycompression forcemainly due its pretension force In this casewe obtain different stiffnessrsquo expression of one windward guy

119896 = 12119897119909119860119892119864

1198882 (1 + (119898119892119897119909119879119901)2 (11986011989211986412119879119901)) (4)

where 119897119909 is the distance of the guy foundation from the axisof the shaft 119860119892 is the guy cross section area 119864 is Youngrsquosmodulus of guy material 119888 is the length of the guy 119898119892 isthe dead weight of the guy per unit length and 119879119901 is thepretension force in the guy

The compressive effects of one set of guys are idealized bythe vertical compressive forces applied to the guy attachmentnodes

119901 = 119899119892119888119879119901119867119888 (5)

where 119899119892119888 is the number of guys in one guysrsquo cluster andH is the height of the mast Our assumptions (4) and (5)are checked via nonlinear modelling of the mast with guysand are of sufficient precision (Figure 3) The spring stiffnessaccording to (4) is slightly less than the stiffness of the realguy but the error is to the safe side

The first of the inequality constraints the strength con-straints are checked in all structural elements of the mast

accounting for both the axial force N and bending moment119872 for the allowable stress 120590120590 = 1003816100381610038161003816100381610038161003816119873119860 plusmn 119872119882

1003816100381610038161003816100381610038161003816 le 120590 (6)

The stability requirements for the leg and bracing mem-bers are posed according to Eurocode 3 Part 3-1 [14]

|119873| le 1205941198601198911199101205741198721 (7)

where 119860 is the cross section area of the member and 119891119910is the steel yield point (dependent on the diameter of themember) According to [14] the partial factor of resistanceof the members to member buckling 1205741198721 is taken to be 10and reduction factor coefficient 120594 is evaluated via complexengineering formulas depending on the nondimensionalslenderness of structural elements

The global stability requirements between adjacent clus-ters of guys are expressed in the form of a mast segmentbuckling condition due to the equivalent compression force119873119864119889 and the equivalent bending moment 119872119864119889 in the crosssection of the mast

1198731198641198891205941199101198731198771198961205741198721 +119872119864119889120594119871119905 (1198721199101198771198961205741198721) le 1 (8)

where 120594119910 is the reduction factor for the relevant bucklingmode 119873119877119896 is the cross section axial resistance and 119872119910119877119896is the cross section moment resistance (based on either the

6 Advances in Civil Engineering

plastic elastic or effective section modulus depending onclassification) Details on the evaluation of these coefficientsmay be found in Eurocode 3 Part 1-1 [14]

The slenderness constraints for the mast segmentsbetween clusters of guys and slenderness constraints forbracing members correspondingly are

120582 le 120120582 le 180 (9)

Obtaining the buckling length of a mast segment neededfor assessment of the stability constraint takes significantnumerical effort In [11] we used an approximate value ofbuckling length that was equal to the segment length this isadequately accurate for amast supported by 1ndash4 guysrsquo clustersFor a mast supported by a larger number of guysrsquo clusters itled to unrealistic designs when the mass of the mast steadilydecreases with the increasing number of guysrsquo clusters In thisresearch the ith segment buckling length is obtained with anempirical computational length coefficient

120583119894 = radic089005775 + 74379952119898119894 (10)

where 119898119894 is the coefficient of the segment support stiffnesswith regard to the shaft stiffness 119864119868 dependent on the totalstiffness provided by the guys ]119894

119898119894 = ]1198941198971198943119864119868 (11)

where 119897119894 is the length of the ith segment between adjacentguysrsquo clusters The coefficient 1205831 for the segment betweenfoundation and the first guysrsquo level is obtained with a moreawkward procedure

1205831 = 120583avg119896 (12)

where

120583avg = radic089005775 + 74379952119898avg

119898avg = sum119894119898119894119899119904119892

119896 = (1 + 11990611199062 + 11990611199063 + sdot sdot sdot + 1199061119906119899)119899119904119892

119906119894 = 1198971198942radic119873119864119889119864119868

(13)

with 119899119904119892 being the number of segmentsFinally the lateral displacement 119889 of the mast top is

constrained to one percent of the height of the mastThe classical real coded genetic algorithm for solving

mixed integer optimization problems MI-LXPM [19] wasemployed as a primary optimization engine The meshing

FEM analysis and postprocessor programs are connectedto the optimization algorithm as ldquoblack boxesrdquo The maingenetic parameters population size number of generationsprobabilities of mutation and crossover and so on are tunedto the problem solvedThe complete list of genetic parametersis presented in Section 3

The list of design parameters along with their lowerand upper bounds is presented in Table 1 Some of theparameters are interdependent (eg the guys cannot intersector be attached to a nonexisting shaft section) thereforein the initial stage of optimizationmdashrandom generationof individualsmdashonly the viable or repaired individuals areincluded into the initial population For example at therandom generation of one particular individual of the initialpopulation the following data was obtained the number ofguysrsquo clusters 119899119892 = 3 number of sections 119899119904 = 50 (let the upperbound for 119899119904 be 100) and levels of guysrsquo clusters 1198971 = 10 1198972 =60 and 1198973 = 80 In order to repair this infeasible individualthe levels are scaled down proportionally 1198971 = 5 1198972 = 30 and1198973 = 40

Different conditions hold for the invalid individuals of thenext generations obtained due to the mutation or crossoverSince the main idea of optimization technique to keep allconstituent programs independent the obtained infeasibleguysrsquo levels are repaired but the individuals with intersectingguys are retained However with an additional sufficientpenalty given by the preprocessor of the finite elementprogram the finite element analysis for these individuals isnot executed The penalty is proportional to the number ofsuch violations and acts like a ldquodeadrdquo penalty fairly reducingthe chances of infeasible individuals surviving the selection

After the best EA solution is selected the neighbourhoodof the solution is explored with the pattern search (PS)algorithm For optimization problems with a larger numberof design parameters it slightly improves the solution by3ndash6

3 Numerical Examples and Discussion

As an example we optimize the 96m tall guyed telecom-munication mast that is produced by the steel industriesThe industrial mast of triangular horizontal plan has fourguysrsquo clusters anchored at two points 32 and 595m from theshaft The width of shaft is 120m All the leg bracing andstiffener elements are round solid steel bars At the radii ofthe structural elements 002m (leg) 00125m (bracing) and0007m (guy) the total mass of the shaft is 10410 kg while themass of all guys is 1230 kg

The optimal mast scheme is sought retaining the schemeof an industrial mast but varying the topology shape andsizing parameters of themastThe range of guysrsquo clusters from1 to 10 is explored Constant data specific to this mast whichdo not vary in the optimization process are as follows

(i) Height of the mast 96m(ii) Area of antennas distributed in the top 15 of mast

height 1034m2

(iii) Diameter of auxiliary equipment along the wholemast height 005m

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

4 Advances in Civil Engineering

Table 1 Set of design parameters

Parameter Type Quantity of parameters Lower bound Upper boundNumber of guysrsquo clusters 119899119892 Integer 1 1 10Number of typical shaft sections 119899119904 Integer 1 6 120Levels of guysrsquo clusters in sections from the ground 119897 Integer 119899119892 11989911990420 119899119904Distances of guy anchors from the shaft 119889 (m) Continuous 119899119892 5 100Width of the shaft 119908 (m) Continuous 1 020 240Radius of the leg elements 119903119897 (m) Continuous 1 0008 0040Radius of the bracing and stiffener elements 119903119887 (m) Continuous 1 0008 0030Radius of guys 119903119892 (m) Continuous 1 0001 0010Pretension forces in guys 119891 (N) Continuous 119899119892 5000 30000

structure finite element analysis program and program forevaluation of objective function value

The genetic algorithm with preprocessor for genomerepair is used as the primary optimization engine Afterthe best solution from the set of EA solutions is selectedthe pattern search algorithm [16] is employed to explorethe neighbourhood of the solution in depth and (probably)improve it

The meshing program is used as a preprocessor for thefinite element program It inputs the set of design parametersproduced by the optimization algorithm together with datawhich do not vary in the optimization process (eg mastheight wind velocity and material characteristics) and pre-pares the complete dataset for the finite element programAll loadings including self-weight and wind loads on thestructure are recalculated using new values of all geometricalparameters

A fast finite element program is required hence anoriginal problem-oriented Fortran program was developedfor the static analysis All the structural elements of theshaft are represented by beam elements Instead of guys atthe guy attachment nodes the horizontal stiffness boundaryconditions and corresponding vertical compression forces areapplied

After the finite element analysis the postprocessor pro-gram evaluates the value of the objective function verifies allconstraints and transmits the objective function value to theoptimization engine which prepares a new estimate of designparameters set on its basis

In mathematical terms the optimization problem isformulated as follows

119891lowast = min119909isin119863

119891 (119909) (1)

subject to

(i) structure equilibrium constraints(ii) strength constraints in all structural elements of the

shaft and guys(iii) local stability constraints in all leg and bracing ele-

ments(iv) global stability constraints between the clusters of

guys and between the lower cluster and mast foun-dation

(v) slenderness constraints in all leg and bracing ele-ments

(vi) lateral stability constraint of the mast top

119891(119909) in (1) is a nonlinear objective function of continuousand integer variables 119891R119899 rarr R 119899 is the number ofdesign parameters 119909 and 119863 isin R119899 is a feasible region ofdesign parametersThe globalminimum should be foundNoassumptions on unimodality are included into formulation ofthe problem that is many local minima may exist

The total mass of themast including themass of the guysis considered in the objective function Since the materialof the guys is more expensive the mass of the guys ispremultiplied by a factor given by the program user Thecost of guy anchors and the cost of connections can also beincluded into the objective function in a similar manner

The complete set of design parameters along with theircharacteristics and bounds is listed in Section 3 Table 1

All strength stability and slenderness constraints areassessed according to Eurocode 3 Part 3-1 [14] and Eurocode3 Part 1-1 [17] The overall structure equilibrium constraintsare checked solving the static problem with the finite elementprogram The typical 2-node beam elements with 12 degreesof freedom are used to represent the structural elements ofthe shaft The main statics equation is

[119870]119886 119906119886 = 119865119886 (2)

where [119870]119886 is the stiffness matrix 119906119886 are nodal displace-ments and 119865119886 are the active forces of the ensemble ofelements The element stiffness matrix can be found in manytextbooks for example [18] The stress resultants in all finiteelements according to [14] are

119878TM = 119878119872 plusmn 119878119901 (3)

where 119878119872 is stress due to the static wind load on the wholemast height plus the dead weight of the mast structure and119878119901 is cumulative stress due to the pseudo-dynamic loads onthe patches of the mast according to [13] the number of loadcases for 119899119892 guysrsquo clusters is 2(119899119892 + 1) + 1 Thus (2) has to besolved for 2(119899119892 + 1) + 2 times

The influence of guys on the behaviour of the shaft ismodelled by linear stiffness boundary conditions The totalstiffness estimation of the guys in a guysrsquo cluster when the

Advances in Civil Engineering 5

Table 2 Optimal mast parameters at 1ndash10 guysrsquo clusters

119899119892119898119864119860 119908 (m) 119899119904 119903119897 (m) 119903119887 (m) 119903119892 (m) 119873119891 119879 (h)119898119864119860+119875119878(kg)

1 21713 199 38 00430 00184 0010 613030 48721712

2 11018 118 57 00281 00123 00081 629430 111910983

3 6974 086 72 00226 00101 00055 692630 12686940

4 5913 069 82 00206 000855 000489 696230 13905862

5 5538 069 91 00191 000810 000459 683230 15405459

6 5860 068 83 00195 000811 000447 719430 17545475

7 5984 0625 83 00195 00081 000464 749630 21275682

8 6043 086 90 00178 00080 000437 772430 23725828

9 6396 079 88 00183 00080 000441 810430 26266005

10 6891 083 90 00182 00080 000444 784430 26806371

wind blows toward one guy is given by Gantes et al [1]However this case is not critical for themast fixed at the foun-dation The shaft of the mast receives larger displacements inthe opposite wind direction Now two stressed guys absorball wind action while the third guy is slack and provides onlycompression forcemainly due its pretension force In this casewe obtain different stiffnessrsquo expression of one windward guy

119896 = 12119897119909119860119892119864

1198882 (1 + (119898119892119897119909119879119901)2 (11986011989211986412119879119901)) (4)

where 119897119909 is the distance of the guy foundation from the axisof the shaft 119860119892 is the guy cross section area 119864 is Youngrsquosmodulus of guy material 119888 is the length of the guy 119898119892 isthe dead weight of the guy per unit length and 119879119901 is thepretension force in the guy

The compressive effects of one set of guys are idealized bythe vertical compressive forces applied to the guy attachmentnodes

119901 = 119899119892119888119879119901119867119888 (5)

where 119899119892119888 is the number of guys in one guysrsquo cluster andH is the height of the mast Our assumptions (4) and (5)are checked via nonlinear modelling of the mast with guysand are of sufficient precision (Figure 3) The spring stiffnessaccording to (4) is slightly less than the stiffness of the realguy but the error is to the safe side

The first of the inequality constraints the strength con-straints are checked in all structural elements of the mast

accounting for both the axial force N and bending moment119872 for the allowable stress 120590120590 = 1003816100381610038161003816100381610038161003816119873119860 plusmn 119872119882

1003816100381610038161003816100381610038161003816 le 120590 (6)

The stability requirements for the leg and bracing mem-bers are posed according to Eurocode 3 Part 3-1 [14]

|119873| le 1205941198601198911199101205741198721 (7)

where 119860 is the cross section area of the member and 119891119910is the steel yield point (dependent on the diameter of themember) According to [14] the partial factor of resistanceof the members to member buckling 1205741198721 is taken to be 10and reduction factor coefficient 120594 is evaluated via complexengineering formulas depending on the nondimensionalslenderness of structural elements

The global stability requirements between adjacent clus-ters of guys are expressed in the form of a mast segmentbuckling condition due to the equivalent compression force119873119864119889 and the equivalent bending moment 119872119864119889 in the crosssection of the mast

1198731198641198891205941199101198731198771198961205741198721 +119872119864119889120594119871119905 (1198721199101198771198961205741198721) le 1 (8)

where 120594119910 is the reduction factor for the relevant bucklingmode 119873119877119896 is the cross section axial resistance and 119872119910119877119896is the cross section moment resistance (based on either the

6 Advances in Civil Engineering

plastic elastic or effective section modulus depending onclassification) Details on the evaluation of these coefficientsmay be found in Eurocode 3 Part 1-1 [14]

The slenderness constraints for the mast segmentsbetween clusters of guys and slenderness constraints forbracing members correspondingly are

120582 le 120120582 le 180 (9)

Obtaining the buckling length of a mast segment neededfor assessment of the stability constraint takes significantnumerical effort In [11] we used an approximate value ofbuckling length that was equal to the segment length this isadequately accurate for amast supported by 1ndash4 guysrsquo clustersFor a mast supported by a larger number of guysrsquo clusters itled to unrealistic designs when the mass of the mast steadilydecreases with the increasing number of guysrsquo clusters In thisresearch the ith segment buckling length is obtained with anempirical computational length coefficient

120583119894 = radic089005775 + 74379952119898119894 (10)

where 119898119894 is the coefficient of the segment support stiffnesswith regard to the shaft stiffness 119864119868 dependent on the totalstiffness provided by the guys ]119894

119898119894 = ]1198941198971198943119864119868 (11)

where 119897119894 is the length of the ith segment between adjacentguysrsquo clusters The coefficient 1205831 for the segment betweenfoundation and the first guysrsquo level is obtained with a moreawkward procedure

1205831 = 120583avg119896 (12)

where

120583avg = radic089005775 + 74379952119898avg

119898avg = sum119894119898119894119899119904119892

119896 = (1 + 11990611199062 + 11990611199063 + sdot sdot sdot + 1199061119906119899)119899119904119892

119906119894 = 1198971198942radic119873119864119889119864119868

(13)

with 119899119904119892 being the number of segmentsFinally the lateral displacement 119889 of the mast top is

constrained to one percent of the height of the mastThe classical real coded genetic algorithm for solving

mixed integer optimization problems MI-LXPM [19] wasemployed as a primary optimization engine The meshing

FEM analysis and postprocessor programs are connectedto the optimization algorithm as ldquoblack boxesrdquo The maingenetic parameters population size number of generationsprobabilities of mutation and crossover and so on are tunedto the problem solvedThe complete list of genetic parametersis presented in Section 3

The list of design parameters along with their lowerand upper bounds is presented in Table 1 Some of theparameters are interdependent (eg the guys cannot intersector be attached to a nonexisting shaft section) thereforein the initial stage of optimizationmdashrandom generationof individualsmdashonly the viable or repaired individuals areincluded into the initial population For example at therandom generation of one particular individual of the initialpopulation the following data was obtained the number ofguysrsquo clusters 119899119892 = 3 number of sections 119899119904 = 50 (let the upperbound for 119899119904 be 100) and levels of guysrsquo clusters 1198971 = 10 1198972 =60 and 1198973 = 80 In order to repair this infeasible individualthe levels are scaled down proportionally 1198971 = 5 1198972 = 30 and1198973 = 40

Different conditions hold for the invalid individuals of thenext generations obtained due to the mutation or crossoverSince the main idea of optimization technique to keep allconstituent programs independent the obtained infeasibleguysrsquo levels are repaired but the individuals with intersectingguys are retained However with an additional sufficientpenalty given by the preprocessor of the finite elementprogram the finite element analysis for these individuals isnot executed The penalty is proportional to the number ofsuch violations and acts like a ldquodeadrdquo penalty fairly reducingthe chances of infeasible individuals surviving the selection

After the best EA solution is selected the neighbourhoodof the solution is explored with the pattern search (PS)algorithm For optimization problems with a larger numberof design parameters it slightly improves the solution by3ndash6

3 Numerical Examples and Discussion

As an example we optimize the 96m tall guyed telecom-munication mast that is produced by the steel industriesThe industrial mast of triangular horizontal plan has fourguysrsquo clusters anchored at two points 32 and 595m from theshaft The width of shaft is 120m All the leg bracing andstiffener elements are round solid steel bars At the radii ofthe structural elements 002m (leg) 00125m (bracing) and0007m (guy) the total mass of the shaft is 10410 kg while themass of all guys is 1230 kg

The optimal mast scheme is sought retaining the schemeof an industrial mast but varying the topology shape andsizing parameters of themastThe range of guysrsquo clusters from1 to 10 is explored Constant data specific to this mast whichdo not vary in the optimization process are as follows

(i) Height of the mast 96m(ii) Area of antennas distributed in the top 15 of mast

height 1034m2

(iii) Diameter of auxiliary equipment along the wholemast height 005m

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

Advances in Civil Engineering 5

Table 2 Optimal mast parameters at 1ndash10 guysrsquo clusters

119899119892119898119864119860 119908 (m) 119899119904 119903119897 (m) 119903119887 (m) 119903119892 (m) 119873119891 119879 (h)119898119864119860+119875119878(kg)

1 21713 199 38 00430 00184 0010 613030 48721712

2 11018 118 57 00281 00123 00081 629430 111910983

3 6974 086 72 00226 00101 00055 692630 12686940

4 5913 069 82 00206 000855 000489 696230 13905862

5 5538 069 91 00191 000810 000459 683230 15405459

6 5860 068 83 00195 000811 000447 719430 17545475

7 5984 0625 83 00195 00081 000464 749630 21275682

8 6043 086 90 00178 00080 000437 772430 23725828

9 6396 079 88 00183 00080 000441 810430 26266005

10 6891 083 90 00182 00080 000444 784430 26806371

wind blows toward one guy is given by Gantes et al [1]However this case is not critical for themast fixed at the foun-dation The shaft of the mast receives larger displacements inthe opposite wind direction Now two stressed guys absorball wind action while the third guy is slack and provides onlycompression forcemainly due its pretension force In this casewe obtain different stiffnessrsquo expression of one windward guy

119896 = 12119897119909119860119892119864

1198882 (1 + (119898119892119897119909119879119901)2 (11986011989211986412119879119901)) (4)

where 119897119909 is the distance of the guy foundation from the axisof the shaft 119860119892 is the guy cross section area 119864 is Youngrsquosmodulus of guy material 119888 is the length of the guy 119898119892 isthe dead weight of the guy per unit length and 119879119901 is thepretension force in the guy

The compressive effects of one set of guys are idealized bythe vertical compressive forces applied to the guy attachmentnodes

119901 = 119899119892119888119879119901119867119888 (5)

where 119899119892119888 is the number of guys in one guysrsquo cluster andH is the height of the mast Our assumptions (4) and (5)are checked via nonlinear modelling of the mast with guysand are of sufficient precision (Figure 3) The spring stiffnessaccording to (4) is slightly less than the stiffness of the realguy but the error is to the safe side

The first of the inequality constraints the strength con-straints are checked in all structural elements of the mast

accounting for both the axial force N and bending moment119872 for the allowable stress 120590120590 = 1003816100381610038161003816100381610038161003816119873119860 plusmn 119872119882

1003816100381610038161003816100381610038161003816 le 120590 (6)

The stability requirements for the leg and bracing mem-bers are posed according to Eurocode 3 Part 3-1 [14]

|119873| le 1205941198601198911199101205741198721 (7)

where 119860 is the cross section area of the member and 119891119910is the steel yield point (dependent on the diameter of themember) According to [14] the partial factor of resistanceof the members to member buckling 1205741198721 is taken to be 10and reduction factor coefficient 120594 is evaluated via complexengineering formulas depending on the nondimensionalslenderness of structural elements

The global stability requirements between adjacent clus-ters of guys are expressed in the form of a mast segmentbuckling condition due to the equivalent compression force119873119864119889 and the equivalent bending moment 119872119864119889 in the crosssection of the mast

1198731198641198891205941199101198731198771198961205741198721 +119872119864119889120594119871119905 (1198721199101198771198961205741198721) le 1 (8)

where 120594119910 is the reduction factor for the relevant bucklingmode 119873119877119896 is the cross section axial resistance and 119872119910119877119896is the cross section moment resistance (based on either the

6 Advances in Civil Engineering

plastic elastic or effective section modulus depending onclassification) Details on the evaluation of these coefficientsmay be found in Eurocode 3 Part 1-1 [14]

The slenderness constraints for the mast segmentsbetween clusters of guys and slenderness constraints forbracing members correspondingly are

120582 le 120120582 le 180 (9)

Obtaining the buckling length of a mast segment neededfor assessment of the stability constraint takes significantnumerical effort In [11] we used an approximate value ofbuckling length that was equal to the segment length this isadequately accurate for amast supported by 1ndash4 guysrsquo clustersFor a mast supported by a larger number of guysrsquo clusters itled to unrealistic designs when the mass of the mast steadilydecreases with the increasing number of guysrsquo clusters In thisresearch the ith segment buckling length is obtained with anempirical computational length coefficient

120583119894 = radic089005775 + 74379952119898119894 (10)

where 119898119894 is the coefficient of the segment support stiffnesswith regard to the shaft stiffness 119864119868 dependent on the totalstiffness provided by the guys ]119894

119898119894 = ]1198941198971198943119864119868 (11)

where 119897119894 is the length of the ith segment between adjacentguysrsquo clusters The coefficient 1205831 for the segment betweenfoundation and the first guysrsquo level is obtained with a moreawkward procedure

1205831 = 120583avg119896 (12)

where

120583avg = radic089005775 + 74379952119898avg

119898avg = sum119894119898119894119899119904119892

119896 = (1 + 11990611199062 + 11990611199063 + sdot sdot sdot + 1199061119906119899)119899119904119892

119906119894 = 1198971198942radic119873119864119889119864119868

(13)

with 119899119904119892 being the number of segmentsFinally the lateral displacement 119889 of the mast top is

constrained to one percent of the height of the mastThe classical real coded genetic algorithm for solving

mixed integer optimization problems MI-LXPM [19] wasemployed as a primary optimization engine The meshing

FEM analysis and postprocessor programs are connectedto the optimization algorithm as ldquoblack boxesrdquo The maingenetic parameters population size number of generationsprobabilities of mutation and crossover and so on are tunedto the problem solvedThe complete list of genetic parametersis presented in Section 3

The list of design parameters along with their lowerand upper bounds is presented in Table 1 Some of theparameters are interdependent (eg the guys cannot intersector be attached to a nonexisting shaft section) thereforein the initial stage of optimizationmdashrandom generationof individualsmdashonly the viable or repaired individuals areincluded into the initial population For example at therandom generation of one particular individual of the initialpopulation the following data was obtained the number ofguysrsquo clusters 119899119892 = 3 number of sections 119899119904 = 50 (let the upperbound for 119899119904 be 100) and levels of guysrsquo clusters 1198971 = 10 1198972 =60 and 1198973 = 80 In order to repair this infeasible individualthe levels are scaled down proportionally 1198971 = 5 1198972 = 30 and1198973 = 40

Different conditions hold for the invalid individuals of thenext generations obtained due to the mutation or crossoverSince the main idea of optimization technique to keep allconstituent programs independent the obtained infeasibleguysrsquo levels are repaired but the individuals with intersectingguys are retained However with an additional sufficientpenalty given by the preprocessor of the finite elementprogram the finite element analysis for these individuals isnot executed The penalty is proportional to the number ofsuch violations and acts like a ldquodeadrdquo penalty fairly reducingthe chances of infeasible individuals surviving the selection

After the best EA solution is selected the neighbourhoodof the solution is explored with the pattern search (PS)algorithm For optimization problems with a larger numberof design parameters it slightly improves the solution by3ndash6

3 Numerical Examples and Discussion

As an example we optimize the 96m tall guyed telecom-munication mast that is produced by the steel industriesThe industrial mast of triangular horizontal plan has fourguysrsquo clusters anchored at two points 32 and 595m from theshaft The width of shaft is 120m All the leg bracing andstiffener elements are round solid steel bars At the radii ofthe structural elements 002m (leg) 00125m (bracing) and0007m (guy) the total mass of the shaft is 10410 kg while themass of all guys is 1230 kg

The optimal mast scheme is sought retaining the schemeof an industrial mast but varying the topology shape andsizing parameters of themastThe range of guysrsquo clusters from1 to 10 is explored Constant data specific to this mast whichdo not vary in the optimization process are as follows

(i) Height of the mast 96m(ii) Area of antennas distributed in the top 15 of mast

height 1034m2

(iii) Diameter of auxiliary equipment along the wholemast height 005m

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

6 Advances in Civil Engineering

plastic elastic or effective section modulus depending onclassification) Details on the evaluation of these coefficientsmay be found in Eurocode 3 Part 1-1 [14]

The slenderness constraints for the mast segmentsbetween clusters of guys and slenderness constraints forbracing members correspondingly are

120582 le 120120582 le 180 (9)

Obtaining the buckling length of a mast segment neededfor assessment of the stability constraint takes significantnumerical effort In [11] we used an approximate value ofbuckling length that was equal to the segment length this isadequately accurate for amast supported by 1ndash4 guysrsquo clustersFor a mast supported by a larger number of guysrsquo clusters itled to unrealistic designs when the mass of the mast steadilydecreases with the increasing number of guysrsquo clusters In thisresearch the ith segment buckling length is obtained with anempirical computational length coefficient

120583119894 = radic089005775 + 74379952119898119894 (10)

where 119898119894 is the coefficient of the segment support stiffnesswith regard to the shaft stiffness 119864119868 dependent on the totalstiffness provided by the guys ]119894

119898119894 = ]1198941198971198943119864119868 (11)

where 119897119894 is the length of the ith segment between adjacentguysrsquo clusters The coefficient 1205831 for the segment betweenfoundation and the first guysrsquo level is obtained with a moreawkward procedure

1205831 = 120583avg119896 (12)

where

120583avg = radic089005775 + 74379952119898avg

119898avg = sum119894119898119894119899119904119892

119896 = (1 + 11990611199062 + 11990611199063 + sdot sdot sdot + 1199061119906119899)119899119904119892

119906119894 = 1198971198942radic119873119864119889119864119868

(13)

with 119899119904119892 being the number of segmentsFinally the lateral displacement 119889 of the mast top is

constrained to one percent of the height of the mastThe classical real coded genetic algorithm for solving

mixed integer optimization problems MI-LXPM [19] wasemployed as a primary optimization engine The meshing

FEM analysis and postprocessor programs are connectedto the optimization algorithm as ldquoblack boxesrdquo The maingenetic parameters population size number of generationsprobabilities of mutation and crossover and so on are tunedto the problem solvedThe complete list of genetic parametersis presented in Section 3

The list of design parameters along with their lowerand upper bounds is presented in Table 1 Some of theparameters are interdependent (eg the guys cannot intersector be attached to a nonexisting shaft section) thereforein the initial stage of optimizationmdashrandom generationof individualsmdashonly the viable or repaired individuals areincluded into the initial population For example at therandom generation of one particular individual of the initialpopulation the following data was obtained the number ofguysrsquo clusters 119899119892 = 3 number of sections 119899119904 = 50 (let the upperbound for 119899119904 be 100) and levels of guysrsquo clusters 1198971 = 10 1198972 =60 and 1198973 = 80 In order to repair this infeasible individualthe levels are scaled down proportionally 1198971 = 5 1198972 = 30 and1198973 = 40

Different conditions hold for the invalid individuals of thenext generations obtained due to the mutation or crossoverSince the main idea of optimization technique to keep allconstituent programs independent the obtained infeasibleguysrsquo levels are repaired but the individuals with intersectingguys are retained However with an additional sufficientpenalty given by the preprocessor of the finite elementprogram the finite element analysis for these individuals isnot executed The penalty is proportional to the number ofsuch violations and acts like a ldquodeadrdquo penalty fairly reducingthe chances of infeasible individuals surviving the selection

After the best EA solution is selected the neighbourhoodof the solution is explored with the pattern search (PS)algorithm For optimization problems with a larger numberof design parameters it slightly improves the solution by3ndash6

3 Numerical Examples and Discussion

As an example we optimize the 96m tall guyed telecom-munication mast that is produced by the steel industriesThe industrial mast of triangular horizontal plan has fourguysrsquo clusters anchored at two points 32 and 595m from theshaft The width of shaft is 120m All the leg bracing andstiffener elements are round solid steel bars At the radii ofthe structural elements 002m (leg) 00125m (bracing) and0007m (guy) the total mass of the shaft is 10410 kg while themass of all guys is 1230 kg

The optimal mast scheme is sought retaining the schemeof an industrial mast but varying the topology shape andsizing parameters of themastThe range of guysrsquo clusters from1 to 10 is explored Constant data specific to this mast whichdo not vary in the optimization process are as follows

(i) Height of the mast 96m(ii) Area of antennas distributed in the top 15 of mast

height 1034m2

(iii) Diameter of auxiliary equipment along the wholemast height 005m

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

Advances in Civil Engineering 7

(iv) Wind velocity 25msec(v) Allowable steel stresses 235 times 108Nm2

(vi) Specific weight of steel 770 times 104Nm3

(vii) Youngrsquos modulus of steel elements 210 times 1011Nm2

(viii) Youngrsquos modulus of guys 150 times 1011Nm2

(ix) Allowable guy stresses 100 times 109Nm2

(x) Weight factor for the mass of guys 3

The design parameters along with their characteristics(dimensions in m forces in N) are listed in Table 1

We treat all the cross section characteristics as continuousvariables After the solution is obtained the designer shouldround the radii of structural elements to the safe side and tothe available assortment or in-stock values Consequently themodified mast structure will be slightly heavier

The following genetic parameters of EAwere chosen aftera few numerical experiments

(i) Population size 200 individuals(ii) Number of generations 200(iii) Stopping criterion the first met the maximum num-

ber of generations 200 stall generations (ie if theweighted average change in the objective functionvalue over stall generations is less than functiontolerance 1119890 minus 6) 100 objective function tolerance(ie cumulative change in the objective functionvalue over stall generations is less than given functiontolerance)

(iv) Crossover probability 08(v) Crossover type Laplace crossover [19](vi) Mutation type power mutation [19](vii) Penalty type static proportional to the extent of

constraint violation

Since the EA is a stochastic algorithmwhose solution gen-erally cannot be repeated we take 30 independent numericalexperiments In this case the median solution stabilizes andadditional numerical experiments do not change the best andmedian results significantly After the EA solution the bestobtained result is refined with pattern search

Solution time depends on the number of finite elementmesh nodes that in turn is determined by the number oftypical sections 119899119904 119899 = 119899119904 sdot 6 + 3 Thirty runs of the programpackage usually take from 300 to 1600min

The final results of mast optimization successively with1 2 10 guysrsquo clusters that is the total mass of structure(119898119864119860+119875119878 in Table 2) width of shaft and radii of all structuralelements after solution with the EA and the subsequent PSare provided in Table 2 In order to reveal the influence of PSon the quality of the result the value of total mass obtainedonly using EA (119898119864119860 in Table 2) is also givenThemass is givenin kg while the geometrical parameters are in m Also somerelevant data on the optimization processmdashthe total numberof objective function evaluations (119873119891) in 30 independentnumerical experiments and the total time of computations

0 20 40 60 80 1000

102030405060708090

100

minus100 minus80 minus60 minus40 minus20

Figure 4 The best scheme obtained for the mast

1 2 3 4 5 6 7 8 9 1004

06

08

1

12

14

16

18

2

22

Number of guy clusters

Mas

s (kg

)

times104

Figure 5 Dependence of the mast mass on the number of guysrsquoclusters

(119879 in h)mdashare listed The pretension forces in the guysof different clusters are not provided in the table due tolack of space generally the pretension forces are highest atlowest guys and gradually diminish in higher clusters Thepretension forces (also in the form of prestress) for the mastsof 2ndash5 guysrsquo clusters are provided in Table 3 Some additionalinformation on the optimal pretensioning of guys may befound also in [10] The best scheme obtained is depicted inFigure 4

From the engineering point of view the most relevantones are the schemes with 2ndash5 guysrsquo clusters Figure 5definitely shows that the total mass of the mast reducessignificantly starting from 3 guysrsquo clusters the mass of a3-level mast is 37 less than the mass of a 2-level mastIntroducing more than 4 guysrsquo clusters does not produce alarge effectmdashthe mass of a 5-level mast is only 7 less Thusthe rational scheme for the analysed masts should possess 3to 5 guysrsquo clusters

A closer view of the best three results obtained for themasts with 2 to 5 guysrsquo clusters (Table 3) reveals that theoptimal parameters particularly geometrical are distributedin rather narrow bounds In many cases the differencesbetween corresponding parameters do not exceed 10 Oneof these determinant geometrical parameters is the width ofthe shaft Formasts clamped at the foundation this parameterconditions two important factors the bending stiffness and

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

8 Advances in Civil Engineering

Table 3Themain optimal parameters of the best 3 solutions for 2 to 5 guysrsquo clusters masts (m total mass of the mast kg119908 side of the mastm 119899119904 number of sections 120572 tilt angle of bracing elements deg 119897119894 attachment level of the ith guysrsquo cluster 119889119894 anchoring distance of the ithguysrsquo cluster m 120573119894 tilt angle of the guy at the ith level deg 119879119901119894 prestress force at the ith guy level kN 1205901199012 prestress at the ith guy level MPa119903119897119887119892 radii of the leg bracing and guy elements mm)

2 guysrsquo clusters 3 guysrsquo clusters 4 guysrsquo clusters 5 guysrsquo clusters1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

119898 10983 11007 11009 6941 6951 6956 5862 5904 5912 5459 5523 5523119908 118 118 114 086 087 090 069 075 075 069 067 063119899119904 57 57 59 72 74 71 82 81 77 91 85 85120572 355 355 355 377 368 370 402 382 399 333 401 4211198971 18 18 20 16 17 16 15 12 13 13 11 111198972 49 49 51 35 37 34 33 31 28 27 24 261198973 65 67 64 54 50 50 45 41 431198974 77 75 72 67 61 621198975 86 80 801198891 234 251 257 185 182 183 155 118 120 77 99 911198891 393 393 365 318 336 323 266 225 245 156 159 1901198893 574 523 572 441 401 517 355 366 3611198894 539 587 585 555 585 4981198895 608 704 6541205731 375 356 379 409 430 411 440 455 473 524 479 5041205732 513 513 544 477 478 465 512 541 488 581 564 5391205733 486 520 482 507 513 440 513 482 5001205734 550 520 509 512 462 5121205735 529 486 5071198791199011 277 255 289 106 125 121 910 930 200 801 230 2641198791199012 201 218 177 107 940 103 780 122 880 539 128 1401198791199013 128 115 127 960 880 111 899 137 9801198791199014 910 105 960 114 118 1011198791199015 890 100 1021205901199011 361 331 375 138 163 157 119 120 260 104 299 3431205901199012 262 283 230 139 122 134 101 159 114 698 166 1821205901199013 167 149 165 124 114 144 116 179 1271205901199014 118 136 125 147 154 1311205901199015 114 130 132119903119897 280 280 280 225 223 225 206 203 208 191 197 194119903119887 120 120 120 100 100 102 86 90 89 81 82 81119903119892 81 82 83 55 56 55 49 495 48 46 44 46

the magnitude of axial forces in the legs of the shaft Theaxial forces in the legs diminish with the increasing distancebetween the legs at the expense of increasing bendingstiffness Large bending stiffness is not acceptable since evensmall shaft displacements cause significant bendingmomentsat the foundation Due to this greater guy stiffness is requiredin the lower guys

The results also indicate that a small number of guysrsquoclusters determine very high prestress levels in the lower guysfor example for 2 guysrsquo clusters the values of prestress exceed300MPa The same holds for the 5-level mast the globalstability condition does not allow for diminishing bendingstiffness and this in turn requires larger prestress forces inthe lower guys for alignment of stresses A large prestress level

has an adverse influence onmetal fatigue due to the vibrationsof cables

As to the tilt angle of bracing elements in all of the bestsolutions for a small number of guysrsquo clusters it equals 355degreesmdashthis corresponds well to the optimal values of tiltangle reported in [8] This can be explained by the fact thatthe determinant factor in such structures is the bending ofthe shaft In larger numbers of guysrsquo clusters the shear forcewhich determines the stresses in bracing elements is relativelyhigher

Another interesting point is the fact that one of the moststable parameters is the tilt angle of the uppermost guysindependently of the number of guysrsquo clusters its value isclose to 50 degrees

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

Advances in Civil Engineering 9

4 Conclusions

A technique for simultaneous topologyshapesizing opti-mization of guyed telecommunication mast is suggested Itcomprises four independent parts the optimization algo-rithm finite element linear static analysis program meshingpreprocessor and postprocessor for evaluation of constraintsaccording to the Eurocodes and the objective functionmdashthetotal mass of the shaft and guys With ldquoone button clickrdquo theprograms yield an optimal scheme for the mast including alldimensions of structural elements and pretension forces inthe guys The whole optimization loop takes up to 1600minon a typical PC and provides the designer with an initialsuggestion for all elements of the mast structure thus accel-erating the design process

Using the proposed technique the typical 96m steel-guyed mast carrying a standard antenna cluster was opti-mized The optimization of the mast with different sets ofdesign parameters definitely shows that the most relevantschemes of the mast have three to five guysrsquo clusters withthe optimal scheme being the mast with five guysrsquo clustersIntroducing more guysrsquo clusters leads to a slight increase inthe mass of the mast

Analysis of the three best solutions for each 3- to 5-guy-cluster mast scheme reveals some stable ranges of designparametersThe optimal number of typical sections along theheight of the mast is around 57 for a 2-level mast 73 for a 3-level mast 81 for a 4-level mast and 85ndash90 for a 5-level mastThe optimal width of the mast side is around 118 087 072and 067m correspondingly The attachment levels of eachguysrsquo cluster and guysrsquo anchoring distances from the shaft alsoare very close for all of the best solutions

As stated in prior research [11] the comparison of theoptimized 96m tall mast with the typical industrial designshows possibilities for significant reduction in expenditureson steel

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] C Gantes R Khoury J J Connor and C Pouangare ldquoMod-eling loading and preliminary design considerations for tallguyed towersrdquo Computers and Structures vol 49 no 5 pp 797ndash805 1993

[2] B W Smith Communication Structures Thomas Telford Lon-don United Kingdom 2007

[3] Y M F Wahba M K S Madugula and G R MonfortonldquoEvaluation of non-linear analysis of guyed antenna towersrdquoComputers and Structures vol 68 no 1ndash3 pp 207ndash212 1998

[4] Y-L He X Ma and Z-M Wang ldquoNonlinear discrete analysismethod for random vibration of guyed masts under wind loadrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 4 pp 513ndash525 2003

[5] M Gioffre V Gusella A L Materazzi and I VenanzildquoRemovable guyed mast for mobile phone networks wind load

modeling and structural responserdquo Journal ofWind Engineeringand Industrial Aerodynamics vol 92 no 6 pp 463ndash475 2004

[6] N A Jasim and A C Galeb ldquoOptimum design of square free-standing communication towersrdquo Journal of ConstructionalSteel Research vol 58 no 3 pp 413ndash425 2002

[7] C Preeti and K Jagan Mohan ldquoAnalysis of transmissiontowers with different configurationsrdquo Jordan Journal of CivilEngineering vol 7 no 4 pp 450ndash460 2013

[8] D Jatulis A Juozapaitis and P Vainiunas ldquoOptimal design oflattice towers made up of solid round steel barsrdquo in Proceedingsof the 10th International Conference Modern Building MaterialsStructures and Techniques selected papers P Vainiunas and EK Zavadskas Eds pp 641ndash645 Technika Lithuania 2010

[9] R Belevicius D Jatulis and D Sesok ldquoOptimization of tallguyed masts using genetic algorithmsrdquo Engineering Structuresvol 56 pp 239ndash245 2013

[10] R Belevicius D Jatulis and D Sesok ldquoSome insights onthe optimal schemes of tall guyed mastsrdquo Journal of CivilEngineering and Management vol 19 no 5 pp 749ndash758 2013

[11] R Belevicius D Jatulis D Rusakevicius and D Sesok ldquoOpti-mal Schemes for Tall Guyed Telecommunication Mastsrdquo inProceedings of The Fifteenth International Conference on CivilStructural and Environmental Engineering Computing J KruisY Tsompanakis and B H V Topping Eds Civil-Comp PressPrague Czech Republic September 2015

[12] httpsemathworkscomproductsglobal-optimization[13] Eurocode 1 Actions on structuresmdashGeneral actionmdashPart 1-4

wind action European Committee for Standardization (CENBrussels 2005

[14] Eurocode 3 Design of steel structuresmdashTowers and mastsmdashPart3-1 towers masts and chimneys European Committee forStandardization (CEN) Brussels 2006

[15] ISO 124942001 ldquoAtmospheric Icing on Structuresrdquo 2001[16] T G Kolda R M Lewis and V J Torczon ldquoA generating set

direct search augmented Lagrangian algorithm for optimizationwith a combination of general and linear constraintsrdquo Tech RepSAND2006-5315 Sandia National Laboratories United States2006

[17] Eurocode 3 Design of steel structuresmdashPart 1-1 general struc-tural rules European Committee for Standardization (CEN)Brussels 2005

[18] O C Zienkiewicz R L Taylor and D D FoxThe finite elementmethod for solid and structural mechanics ElsevierButterworthHeinemann Amsterdam Seventh edition 2014

[19] K Deep K P Singh M L Kansal and C Mohan ldquoA realcoded genetic algorithm for solving integer and mixed integeroptimization problemsrdquo Applied Mathematics and Computa-tion vol 212 no 2 pp 505ndash518 2009

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

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VLSI Design

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

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