Optimum Design of Braced Steel Space Frames including …downloads.hindawi.com/journals/ace/2018/3854620.pdf · in accordance with BS5950. Degertekin and Hayalioglu [9] applied the

Research ArticleOptimum Design of Braced Steel Space Frames includingSoil-Structure Interaction via Teaching-Learning-BasedOptimization and Harmony Search Algorithms

Ayse T Daloglu 1 Musa Artar 2 Korhan Ozgan 1 and Ali I Karakas 1

1Department of Civil Engineering Karadeniz Technical University Trabzon Turkey2Department of Civil Engineering Bayburt University Bayburt Turkey

Correspondence should be addressed to Korhan Ozgan korhanozganyahoocom

Received 19 August 2017 Revised 13 November 2017 Accepted 6 December 2017 Published 3 April 2018

Academic Editor Moacir Kripka

Copyright copy 2018 Ayse T Daloglu et al -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Optimum design of braced steel space frames including soil-structure interaction is studied by using harmony search (HS) andteaching-learning-based optimization (TLBO) algorithms A three-parameter elastic foundation model is used to incorporate thesoil-structure interaction effect A 10-storey braced steel space frame example taken from literature is investigated according tofour different bracing types for the cases withwithout soil-structure interaction X V Z and eccentric V-shaped bracing types areconsidered in the study Optimum solutions of examples are carried out by a computer program coded in MATLAB interactingwith SAP2000-OAPI for two-way data exchange -e stress constraints according to AISC-ASD (American Institute of SteelConstruction-Allowable Stress Design) maximum lateral displacement constraints interstorey drift constraints and beam-to-column connection constraints are taken into consideration in the optimum design process -e parameters of the foundationmodel are calculated depending on soil surface displacements by using an iterative approach -e results obtained in the studyshow that bracing types and soil-structure interaction play very important roles in the optimum design of steel space framesFinally the techniques used in the optimum design seem to be quite suitable for practical applications

1 Introduction

Optimum design of steel structures prevents excessiveconsumption of the steel material Suitable cross sectionsmust be selected automatically from a predefined listMoreover selected profiles should satisfy some requiredconstraints such as stress displacement and geometric sizeMetaheuristic search techniques are highly preferred forproblems with discrete design variables -ere are manymetaheuristic techniques developed recently Some of themare genetic algorithm harmony search algorithm tabu searchalgorithm particle swarm optimization ant colony algo-rithm artificial bee colony algorithm teaching-learning-based optimization simulated annealing algorithm bat-inspired algorithm cuckoo search algorithm and evolu-tionary structural optimization In literature there are manystudies available for the optimum design of structures using

these algorithms For example Daloglu and Armutcu [1]used the genetic algorithm method for the optimum designof plane steel frames Kameshki and Saka [2] carried out theoptimum design of nonlinear steel frames with semirigidconnections using the genetic algorithm Lee and Geem [3]developed a new structural optimization method based onthe harmony search algorithm Hayalioglu and Degertekin[4] applied genetic optimization on minimum cost designof steel frames with semirigid connections and columnbases Kelesoglu and Ulker [5] searched for multiobjectivefuzzy optimization of space trusses by MS Excel Degertekin[6] compared simulated annealing and genetic algorithmsfor the optimum design of nonlinear steel space framesEsen and Ulker [7] optimized multistorey space steelframes considering the nonlinear material and geometricalproperties Saka [8] used the harmony search algorithmmethod to get the optimum design of steel sway frames

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 3854620 16 pageshttpsdoiorg10115520183854620

in accordance with BS5950 Degertekin and Hayalioglu[9] applied the harmony search algorithm for minimumcost design of steel frames with semirigid connections andcolumn bases Hasancebi et al [10] investigated non-deterministic search techniques in the optimum design ofreal-size steel frames Hasanccedilebi et al [11] used the simulatedannealing algorithm in structural optimization Hasancebiet al [12] investigated the optimum design of high-rise steelbuildings using an evolutionary strategy integrated withparallel algorithm Togan [13] used one of the latest sto-chastic methods teaching-learning-based optimization fordesign of planar steel frames Aydogdu and Saka [14] usedant colony optimization for irregular steel frames includingthe elemental warping effect Dede and Ayvaz [15] studiedstructural optimization problems using the teaching-learning-based optimization algorithm Dede [16] appliedteaching-learning-based optimization on the optimumdesign of grillage structures with respect to LRFD-AISCHasanccedilebi et al [17] used a bat-inspired algorithm forstructural optimization Saka and Geem [18] prepared anextensive review study on mathematical and metaheuristicapplications in design optimization of steel frame structuresHasanccedilebi and Ccedilarbas [19] studied the bat-inspired algo-rithm for discrete-size optimization of steel frames Dede[20] focused on the application of the teaching-learning-based optimization algorithm for the discrete optimizationof truss structures Azad and Hasancebi [21] focused ondiscrete-size optimization of steel trusses under multipledisplacement constraints and load case using the guidedstochastic search technique Artar and Daloglu [22] obtainedthe optimum design of composite steel frames with semi-rigid connections and column bases Artar [23] used theharmony search algorithm for the optimum design of steelspace frames under earthquake loading Artar [24] used theteaching-learning-based optimization algorithm for theoptimum design of braced steel frames Carbas [25] studieddesign optimization of steel frames using an enhanced fireflyalgorithm Daloglu et al [26] investigated the optimumdesign of steel space frames including soil-structure in-teraction Saka et al [27] researched metaheuristics instructural optimization and discussions on the harmonysearch algorithm Aydogdu [28] used a biogeography-basedoptimization algorithm with Levy flights for cost optimi-zation of reinforced concrete cantilever retaining wallsunder seismic loading

In literature there are several researches available foroptimum structural design as mentioned above On the otherhand there are a few researches on the optimum design ofbraced steel space frames including soil-structure interactionSo this study investigates a 10-storey braced steel space framestructure studied previously in literature which is investigatedfor four different bracing types and soil-structure interaction-ese bracing types are X V Z and eccentric V-shapedbracings Optimum design solutions are obtained usinga computer program developed in MATLAB [29] interactingwith SAP2000-OAPI (open application programming in-terface) [30] Suitable cross sections are automatically selectedfrom a list including 128W profiles taken from AISC(American Institute of Steel Construction) -e frame model

is subjected to wind loads according to ASCE7-05 [31] as wellas dead live and snow loads-e analysis results are found tobe quite consistent with the literature results In this study thevertical displacements on soil surfaces are also calculated It isobserved that minimum weights of space frames varydepending on the bracing type Also it can be concluded thatincorporation of soil-structure interaction results in heaviersteel weight

2 Optimum Design Formulation

-e optimum design problem of braced steel space frames iscalculated as follows

minW 1113944

ng

k1Ak 1113944

nk

i1ρiLi (1)

where W is the weight of the frame Ak is the cross-sectionalarea of group k ρi and Li are the density and length ofmember i ng is the total number of groups and nk is thetotal number of members in group k

-e stress constraints according to AISC-ASD [32] aredefined as follows

gi(x) fa

Fa+

Cmxfbx

1minus faFexprime( 1113857( 1113857Fbx

1113890 1113891i

minus 10le 0 i 1 nc

gi(x) fa

060Fy+

fbx

Fbx

1113890 1113891i

minus 10le 0 i 1 nc

(2)

If (faFa)le 015 instead of using (2) the stress con-straint is calculated as follows

gi(x) fa

Fa+

fbx

Fbx

1113890 1113891i

minus 10le 0 i 1 nc (3)

where nc is the total number of members subjected to bothaxial compression and bending stresses fa is the computedaxial stress Fa is the allowable axial stress under axialcompression force alone fbx is the computed bending stressdue to bending of the member about its major (x) Fbx is theallowable compressive bending stress about major Fexprime is theEuler stress Fy is the yield stress of the steel and Cmx isa factor It is calculated from Cmx 06minus 04(M1M2) forthe braced frame member without transverse loading be-tween the ends andCmx 1 + ψ(faFeprime) for the braced framemember with transverse loading

-e effective length factors of columns in braced framesare calculated as follows [33]

K 3GAGB + 14 GA + GB( 1113857 + 0643GAGB + 20 GA + GB( 1113857 + 128

(4)

where GA and GB are the relative stiffness factors at the Athand Bth ends of columns

-e maximum lateral displacement and interstorey driftconstraints are defined as follows

2 Advances in Civil Engineering

gjl(x) δjl

δju

minus 1le 0 j 1 m l 1 nl (5)

where δjl is the displacement of the jth degree of freedomunder load case l δju is the displacement at the upper boundm is the number of restricted displacements and nl is thetotal number of loading cases

gjil(x) Δjil

Δju

minus 1le 0 j 1 ns i 1 nsc

l 1 nl(6)

where Δjil is the interstorey drift of the ith column in the jthstorey under load case l Δju is the limit value ns is thenumber of storeys and nsc is the number of columns ina storey

-e beam-to-column connection geometric constraint isdetermined as follows

gbf i(x) bfbkiprime

dci minus 2tfliminus 1le 0 i 1 nbw

gbbi(x) bfbki

bfcki

minus 1le 0 i 1 nbf

(7)

where nbw is the number of joints where beams are con-nected to the web of the column bfbki

prime is the flange width ofthe beam dci is the depth of the column tfli is the flangethickness of the column nbf is the number of joints wherebeams are connected to the flange of the column and bfbki

and bfcki are flange widths of the beam and column re-spectively (Figure 1)

3 Three-Parameter Vlasov ElasticFoundation Model

-e soil reaction exerted on a structure resting on a two-parameter elastic soil is expressed in

qz kwminus 2tnabla2w (8)

-e reaction depends on the soil surface vertical dis-placement w soil reaction modulus k and soil shear pa-rameter 2t -ese two soil parameters k and 2t can bedefined by

k 1113946H

0

Es 1minus υs( 1113857

1 + υs( 1113857 1minus 2υs( 1113857middot

zφ(z)

zz1113888 1113889

2

dz

2t 1113946H

0Gsφ(z)

2dz

(9)

in which H υs and Gs are the depth Poissonrsquos ratio andshear modulus of the soil respectively In most of theclassical two-parameter soil foundation models such asPasternak Hetenyi and Vlasov models the soil parametersare constants obtained by experimental tests or arbitrarilydefined However it is highly difficult to determine these

parameters experimentally -erefore Vallabhan andDaloglu [34] developed an additional parameter c tocharacterize the vertical displacement profile within subsoil-ey called this model including the third parameter c asa three-parameter Vlasov model -is model eliminates thenecessity of experimental tests to determine soil parameterssince these values are determined iteratively in terms of thenew parameter c -e vertical deformation profile of thesubsoil is described via a mode shape function as given in

ϕ(z) sinh c(1minus(zH))

sinh c (10)

-e boundary values of ϕ(z) are assumed to be ϕ(0) 1and ϕ(H) 0 as shown in Figure 2 -e c parameter can becalculated using

c

H1113874 1113875

2

1minus 2]s( 1113857

2 1minus ]s( 1113857middot

1113938+infinminusinfin 1113938

+infinminusinfin (nablaw)2 dx dy


+infinminusinfin w2 dx dy

(11)

Equation (9) indicates that the soil parameters (k and 2t)are calculated based on the material properties and modeshape function (φ(z)) Also it is necessary to compute the c

parameter to calculate the mode shape function It is nec-essary to know the soil vertical surface displacements ob-tained from the structural analysis to calculate the c

parameter So it can be stated that k 2t ϕ c and w areinterdependent -at is why the analysis requires an iterativeprocedure For this purpose a computer program is coded inMATLAB interacting with SAP2000 structural analysisprogram via OAPI (open application programming in-terface) to perform this iterative procedure in the three-parameter foundation model

bfck

dcl

tfl

B2

B1

bfbk

bprimefbk

Figure 1 Beam-to-column connection geometric constraints

Advances in Civil Engineering 3

Using the coded program a soil model is generated suchthat the soil reaction modulus k is represented by elastic areasprings e interaction between springs is taken into ac-count using shell elements connecting the top of springsesoil shell element with one degree of freedom at each nodereects only shear behavior of the soil e c parameter iscomputed numerically in the coded program using thevertical displacements of soil shell elements To determinethe soil parameters iteratively c 1 is assumed initially andk and 2t values are calculated en the structural model isanalyzed using SAP2000 and the soil surface vertical dis-placements are retrieved to compute a new c value edierence between successive values of c is calculated andchecked whether it is within a prescribed tolerance or not Ifit is smaller than the tolerance the iteration is terminatedOtherwise the next iteration is performed and the pro-cedure is repeated until the convergence is fullled

4 Optimization Algorithms

41 Harmony Search Algorithm Harmony search (HS) al-gorithm method is developed by Lee and Geem [3] andmimics improvisation procedures of musical harmony Itconsists of three basic procedures Operations are conductedby the harmony memory (HM) matrix In the rst step HMis randomly and automatically lled by the program codedinMATLABe form of harmonymemorymatrix is shownas follows

H

x11 x12 middot middot middot x1nminus1 x1n


⋮ ⋮ ⋮ ⋮ ⋮

xHMSminus11 xHMSminus1

2 middot middot middot xHMSminus1nminus1 xHMSminus1

n

xHMS1 xHMS

2 middot middot middot xHMSnminus1 xHMS

n

rarr

rarr

rarr

rarr

rarr

φ x1( )φ x2( )⋮

φ xHMSminus1( )φ xHMS( )

(12)

where xji is the ith design variable of the jth solution vector nis the total number of design variables φ(xj) is the jthobjective function value and HMS (harmony memory size)indicates a specied number of solutions In the harmonymemory matrix each row presents design variables

In the second step the objective function values(φ(x1)φ(x2) middot middot middotφ(xHMSminus1)φ(xHMS)) of solution vectors inthe harmony memory matrix are determined In the thirdstep a new solution vector (xnh [xnh1 xnh2 xnhn ]) isprepared by selecting each design variable from either theharmony memory matrix or the entire section list XslHarmony memory consideration rate (HMCR) is appliedas follows

xnhi isin x1i x2i xHMSi with probability of HMCR

xnhi isin Xsl with probability of (1minusHMCR)(13)

Also the new value of the design variable selected fromthe harmony memory matrix is checked whether this valueshould be pitch adjusted or not depending on the pitchadjustment ratio (PAR) is decision is determined asfollows

Yes with probability of PAR

No with probability of 1minusPAR(14)

Detailed information about the HS algorithm can befound in the literature [3 8 9 23]

42Teaching-Learning-BasedOptimization Teaching-learning-based optimization (TLBO) was developed by Rao et al [35]is method mimics teaching and learning processes be-tween a teacher and students in classroom e personhaving the highest information in the class is selected asa teacher e teacher gives hisher information to the other

Frame

Raft

Soil

ϕ(0) = 1

ϕ(z)

ϕ(H) = 0

(a)

Shear layer

Spring

(b)

Figure 2 A space frame on three-parameter elastic foundation (a) 3D frame on elastic subsoil (b) Mathematical model


people (students) in the class ese procedures providesuitable solutions in structural optimizations e teaching-learning-based optimization method consists of two basicsteps such as teaching and learning In the rst step theteaching step the rst population (class) is randomly lled inthe matrix form presented as follows

class(population)



⋮ ⋮ ⋮ ⋮ ⋮

xSminus11 xSminus12 middot middot middot xSminus1nminus1 xSminus1n

xS1 xS2 middot middot middot xSnminus1 xSn

rarr

rarr

rarr

rarr

rarr

f x1( )f x2( )⋮

f xSminus1( )f xS( )

(15)

where each row represents a student and gives a designsolution S is the population size (the number of students) nis the number of design variables and f(x12S) is theunconstrained objective function value of each student inthe classe student in a class having the best information isselected as a teacher of the class His or her objectivefunction value is the minimum in the class e informationupdate of students in the class is carried out with the help ofthe teacher as follows

xnewi xi + r xteacher minusTFxmean( ) (16)

where xnewi is the new student xi is the current student r isa random number in the range [01] and TF a teachingfactor is either 1 or 2 xmean is the mean of the class denedas xmean (mean(x1)mean(xS)) If the new student hasbetter information (f(xnewi)) the new student is replacedwith the current student In the second step the learning stepinformation is shared between students is step is similar tothe rst step If the new student presents a better informationheshe is replaced with the current student e informationupdate of students in the class is carried out as follows

if f xi( )ltf xj( )rArrxnewi xi + r xi minus xj( )if f xi( )gtf xj( )rArrxnewi xi + r xj minus xi( )

(17)

e detailed information about the TLBO algorithm canbe obtained from [13 15 16 20 24 35] e owchart ofprocesses in MATLAB-SAP2000 OAPI developed to getoptimum solutions is presented in Figure 3

5 Design Example

A 10-storey braced steel space frame example taken fromliterature [36] is studied considering four dierent types of

Determine new soilparameters γ k and 2t

Is the convergencecriteria satisfied

Calculate the objectivevalue of

each correspondingstructural model

Check convergencecriteria

Is it satisfied

Present results

Prepare the nextgroup of solution

vectors

Data feed(W profile sections

and soil parameters)Determine the valuesof k and 2t for gamma

(γ) = 1

Result retrieval(deflections)

Data feed(new soil parameters)

No

Yes

Analyze the structuralmodel for new soil

parameters

SAP2000 OAPIYes

Get analysis results(element number and

sections) Result retrieval(element number and

sections)

No

MATLAB OAPI SAP2000

Create and analyzethe corresponding

structural model anddetermine deflections

Start with initial dataand create the first group

of solution vectors

Figure 3 Flowchart for the optimum design algorithm by HS and TLBO for space frames on elastic foundation


bracing such as X V Z and eccentric V e behavior of theframe is investigated with and without considering the eectof soil-structure interaction e frame example is exposedto wind loads according to ASCE7-05 [31] in additionto dead live and snow loads Optimum cross sections arepractically selected from a predened list of 128W prolestaken from AISC e stress constraints according to AISC-ASD [32] maximum lateral displacement constraint(H400) interstorey drift constraint (h400) and beam-to-column geometric constraints are subjected to the optimumdesign of the braced steel space frames In the analyses thesteel modulus of elasticity E and yield stress Fy are taken as

CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)

Figure 4 Typical plane view of a 10-storey steel frame

Figure 5 ree dimensional view of a V-braced frame

Table 1 Gravity loading on beams of roof and oors [36]

Beam type Outer spanbeams (kNm)

Inner spanbeams (kNm)

Long span oor beams 979 1959Short span oor beams 804 1607Long span roof beams 675 1350Short span oor beams 554 1107

Table 2 Wind loads calculated for the 10-storey braced frame [36]

Floor

X-direction Y-directionDistributedwindward

force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)

Figure 6 2D view of the X-braced steel space frame without andwith soil-structure interaction (a) e case without soil-structureinteraction (b) e case with soil-structure interaction


29000 ksi (2038936MPa) and 36 ksi (2531MPa) re-spectively Figure 4 shows the typical plane view of a 10-storey steel frame Also Figure 5 represents the three di-mensional view of a V-braced frame Each storey hasa height of 366m (12 ft) Modulus of elasticity for theconcrete is taken as 32000000 kNm2 Poissonrsquos ratio is 02and weight per unit volume is 25 kNm3

All floors excluding the roof are exposed to a dead load of288 kNm2 and a live load of 239 kNm2 -e roof floor isexposed to a dead load of 288 kNm2 and a snow load of075 kNm2 -e total gravity loading on the beams of roofand floors is tabulated in Table 1 [36] Wind loads areapplied to the frame according to ASCE7-05 [31] Windloads calculated for the 10-storey braced frame are presentedin Table 2 [36] Modulus of elasticity of the soil Es is takento be equal to 80000 kNm2 -e depth of the soil stratum to

the rigid base is taken asHs 20m and Poissonrsquos ratio of thesoil is equal to 025

51 X-Braced Steel Space Frame Figure 6 shows the typical2D view of an X-braced steel space frame without and withsoil-structure interaction Optimum results of the X-bracedspace frame are given in Table 3 Soil parameters for the

Table 4 Soil parameters for 10-storey X-braced steel space frameon elastic foundation

Algorithm c k(kNm3) 2t(kNm3)

Teaching-learning-basedoptimization 452365 10882438 70605337

Harmony search algorithm 451999 10873807 70661621

Table 3 Optimum results of the X-braced space frame

Storeys Membergroups

Literatureresearch

Hasancebi [36]

-is studyTLBO withoutsoil-structureinteraction

TLBO withsoil-structureinteraction

HS withoutsoil-structureinteraction

HS withsoil-structureinteraction

1-2

CC W16times 36 W8times 24 W8times 28 W8times 35 W8times 31OCl W12times 65 W21times83 W14times 99 W14times 43 W12times 50OCs W24times 94 W44times 224 W44times 224 W12times 50 W18times 50IC W12times120 W14times 74 W16times 89 W18times 97 W18times119OB W8times18 W8times 24 W10times 26 W8times 24 W8times 24IB W18times 35 W10times 33 W14times 48 W10times 33 W16times 40BR W5times19 W8times15 W12times16 W8times 21 W8times15

3-4

CC W10times 33 W8times 28 W8times 28 W14times 48 W10times 60OCl W10times 54 W12times 87 W10times 49 W10times 33 W12times 65OCs W12times 72 W10times 49 W12times 53 W24times 68 W24times 76IC W30times 90 W24times 62 W12times 72 W14times 68 W24times 94OB W8times18 W12times 30 W8times 24 W18times 50 W18times 50IB W12times 26 W14times 34 W8times 35 W10times 33 W12times 35BR W6times15 W12times14 W12times14 W10times15 W6times15

5-6

CC W16times 31 W16times 36 W8times 31 W14times 34 W14times 43OCl W10times 49 W8times 35 W14times 34 W8times 28 W12times 45OCs W10times 54 W8times 40 W10times 60 W14times 38 W16times 50IC W12times 58 W24times 55 W14times 53 W16times 57 W18times 65OB W8times 21 W16times 36 W10times 30 W10times 30 W10times 30IB W12times 26 W16times 36 W14times 38 W16times 45 W18times 50BR W6times15 W10times15 W12times14 W12times 22 W6times15

7-8

CC W16times 26 W8times 24 W12times 53 W10times 45 W10times 45OCl W10times 49 W8times 31 W10times 49 W12times 45 W12times 45OCs W10times 49 W10times 54 W12times 53 W10times 45 W14times 48IC W14times 38 W16times 36 W14times 43 W14times 48 W10times 68OB W8times 21 W8times 24 W10times 30 W8times 35 W12times 35IB W12times 26 W14times 34 W18times 40 W10times 33 W10times 33BR W6times 9 W12times14 W12times14 W12times14 W12times14

9-10

CC W12times 26 W8times 28 W8times 24 W8times 31 W12times 40OCl W8times 31 W10times 49 W14times 30 W10times 33 W14times 53OCs W8times 40 W8times 31 W12times 40 W14times 30 W14times 30IC W8times 28 W12times 30 W12times 40 W16times 36 W12times 45OB W8times18 W8times 24 W8times 24 W21times44 W8times 24IB W12times 30 W21times44 W10times 33 W16times 36 W10times 39BR W6times 9 W10times15 W10times15 W12times14 W12times14

Weight (kN) 109291 117003 121791 120113 126550Maximum lateral displacement (cm) mdash 636 759 792 829Interstorey drift (cm) mdash 081 091 0915 0915Maximum settlement (cm) mdash mdash minus0657 mdash minus0670


ndash700 ndash646 ndash592 ndash538 ndash485 ndash431 ndash377 ndash323 ndash269 ndash215 ndash162 ndash108 ndash54 0 E ndash 3

(a)


(b)

Figure 7 Settlements of soil surface for the X-braced steel space frame with two different algorithm methods (cm) (a) TLBO (b) HS

0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV

Figure 8 Design histories of the 10-storey X-braced steel space frame I HS with soil-structure interaction II TLBO with soil-structureinteraction III HS without soil-structure interaction IV TLBO without soil-structure interaction

(a) (b)

Figure 9 2D view of the V-braced steel space frame without and with soil-structure interaction (a) -e case without soil-structureinteraction (b) -e case with soil-structure interaction


space frame on elastic foundation are presented in Table 4Moreover Figures 7 and 8 display settlements of the soilsurface for two different algorithm methods and designhistories of optimum solutions respectively

52 V-Braced Steel Space Frame Figure 9 shows the typical2D view of a V-braced steel space frame without and withsoil-structure interaction Optimum results of the V-braced

space frame are shown in Table 5 Soil parameters for thespace frame on elastic foundation are given in Table 6 AlsoFigures 10 and 11 show the soil surface settlements obtainedusing two different algorithms and design histories of op-timum solutions respectively

53 Z-Braced Steel Space Frame Figure 12 represents the 2Dside view of a Z-braced steel space frame without and with

Table 5 Optimum results of the V-braced space frame


Literatureresearch

Hasancebi [36]





1-2

CC W8times 28 W8times 31 W8times 28 W8times 28OCl W14times 43 W16times 67 W16times 50 W12times 50OCs W44times 224 W27times161 W16times 67 W18times106IC W10times 88 W18times106 W18times106 W14times132OB W8times 28 W14times 34 W8times 24 W8times 24IB W10times 33 W10times 33 W8times 35 W10times 33BR W12times14 W6times15 W8times 21 W12times14

3-4

CC W8times 24 W8times 24 W8times 31 W8times 24OCl W12times 45 W10times 49 W12times 35 W21times44OCs W10times 45 W18times 86 W10times 45 W16times 40IC W21times62 W21times68 W14times 68 W18times 65OB W8times 24 W10times 30 W8times 24 W8times 24IB W12times 35 W8times 35 W12times 35 W16times 40BR W5times16 W8times 21 W14times 22 W10times 26

5-6

CC W12times 30 W8times 24 W10times 39 W10times 33OCl W14times 34 W8times 31 W16times 36 W16times 36OCs W18times 46 W14times 48 W18times 55 W18times 50IC W12times 50 W18times 55 W24times 55 W21times62OB W10times 30 W8times 24 W14times 38 W14times 38IB W8times 35 W16times 36 W14times 38 W18times 50BR W10times15 W12times19 W12times 22 W6times15

7-8

CC W8times 28 W8times 24 W14times 53 W16times 77OCl W8times 28 W8times 35 W12times 45 W8times 35OCs W8times 24 W12times 30 W10times 45 W8times 35IC W16times 40 W16times 45 W10times 39 W12times 40OB W10times 30 W8times 24 W8times 31 W8times 35IB W16times 36 W14times 34 W10times 33 W18times 40BR W12times14 W12times16 W8times 21 W12times 22

9-10

CC W12times 30 W8times 28 W14times 48 W12times 40OCl W10times 33 W8times 24 W8times 35 W8times 40OCs W8times 31 W8times 24 W8times 35 W8times 35IC W24times 55 W12times 26 W16times 36 W16times 36OB W8times 24 W8times 24 W8times 31 W8times 31IB W16times 36 W18times 40 W14times 34 W16times 36BR W10times15 W10times15 W12times14 W12times14


Table 6 Soil parameters for the 10-storey V-braced steel space frame on elastic foundation


Teaching-learning-based optimization 470318 11306947 67945018Harmony search algorithm 470957 11322086 67853886


soil-structure interaction Optimum results of the Z-bracedspace frame are presented in Table 7 Soil parameters for thespace frame on elastic foundation are given in Table 8Moreover Figures 13 and 14 represent settlements of the soilsurface carried out with two different algorithm methodsand design histories of optimum solutions respectively

54 Eccentric V-Braced Steel Space Frame Figure 15 showsthe 2D side view of an eccentric V-braced steel space framewithout and with soil-structure interaction -e braces areconnected to the beam from one-third of the beam lengthOptimum results of the eccentric V-braced space frame arepresented in Table 9 Soil parameters for the space frame onelastic foundation are presented in Table 10 Moreover


(a)


(b)

Figure 10 Settlements of soil surface for the V-braced steel spaceframe with two different algorithmmethods (cm) (a) TLBO (b) HS

0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV

Figure 11 Design histories of the 10-storey V-braced steel spaceframe I HS with soil-structure interaction II HS without soil-structure interaction III TLBO with soil-structure interaction IVTLBO without soil-structure interaction

(a)

(b)

Figure 12 2D view of the Z-braced steel space frame without andwith soil-structure interaction (a) -e case without soil-structureinteraction (b) -e case with soil-structure interaction


Figures 16 and 17 show settlements of the soil surface carriedout with two different algorithm methods and design his-tories of optimum solutions respectively

It is observed from Tables 3 5 and 7 that the minimumweights of the braced frames for the case without soil-structure interaction are very similar to the ones availablein literature research [36] In this study the V-braced

type provides the lowest steel weight of 108380 kN byusing teaching-learning-based optimization Z-bracedand X-braced types provide the second and third lowweights 109527 kN and 117003 respectively Moreoverthe minimum weight of the eccentric V-braced frame127501 kN is nearly 15 141 and 82 heavier thanthe minimum steel weights of the V- Z- and X-bracedframes respectively On the other hand harmony searchalgorithm presents 26ndash47 heavier minimum steel weightsthan the ones obtained from teaching-learning-based opti-mization for the case without soil-structure interaction -etables including optimum results also show that the value ofinterstorey drift is very close to the limit value (h400)-erefore the displacement constraints play very crucialroles in the optimum design of the braced frames Five

Table 7 Optimum results of the Z-braced space frame


Literatureresearch

Hasancebi [36]





1-2

CC W8times 31 W8times 31 W8times 35 W8times 31OCl W21times83 W16times 67 W12times 65 W10times 68OCs W24times103 W30times108 W14times132 W30times108IC W12times106 W14times 99 W18times106 W14times109OB W8times 24 W8times 24 W8times 24 W8times 28IB W10times 33 W12times 35 W12times 35 W14times 43BR W6times15 W8times 24 W12times 30 W8times15

3-4

CC W10times 54 W8times 35 W16times 36 W14times 30OCl W10times 49 W14times 90 W12times 35 W12times 65OCs W14times 74 W30times108 W12times 72 W18times 55IC W14times 74 W21times68 W16times 77 W21times73OB W16times 36 W8times 24 W8times 24 W8times 24IB W16times 40 W12times 35 W12times 35 W14times 38BR W5times16 W12times16 W12times19 W8times 24

5-6

CC W8times 24 W8times 31 W8times 35 W8times 31OCl W8times 31 W10times 49 W10times 54 W8times 35OCs W24times 55 W16times 45 W16times 50 W21times68IC W18times 65 W24times 55 W18times 60 W24times 62OB W8times 24 W16times 36 W12times 30 W16times 36IB W14times 38 W14times 34 W12times 35 W14times 34BR W8times15 W12times 22 W6times15 W8times 28

7-8


9-10



Table 8 Soil parameters for the 10-storey Z-braced steel spaceframe on elastic foundation

Algorithm methods c k(kNm3) 2t(kNm3)




independent runs are performed for each braced type for thecase without soil-structure interaction

In the case with soil-structure interaction the minimumweights of all braced frames increased depending on set-tlements on the soil surfaces It is observed from Tables 4 68 and 10 that the soil parameters of 10-storey braced steelframes on elastic foundation are similar for all cases -eminimum steel weights are mostly obtained by teaching-learning-based optimization For the X-braced frame theminimum weight obtained by TLBO for the case with soil-structure interaction is 401 heavier than the weight of theframe excluding soil-structure interaction -is ratio is532 for the harmony search algorithm Moreover set-tlement values on the soil surfaces are nearly minus066 cm asseen in Figure 7 For the V-braced frame including soil-structure interaction TLBO and HS present 166 and 59heavier weights respectively -e settlements in this braced

ndash700 ndash646 ndash592 ndash538 ndash485 ndash431 ndash377 ndash323 ndash269 ndash215 ndash162 ndash108 ndash54 E ndash 30

(a)


(b)

Figure 13 Settlements of soil surface for the Z-braced steel spaceframe with two different algorithmmethods (cm) (a) TLBO (b) HS

0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV

Figure 14 Design histories of the 10-storey Z-braced steel spaceframe I HS with soil-structure interaction II TLBO with soil-structure interaction III HS without soil-structure interaction IVTLBO without soil-structure interaction

(a)

(b)

Figure 15 2D view of the eccentric V-braced steel space framewithout and with soil-structure interaction (a) -e case withoutsoil-structure interaction (b) -e case with soil-structureinteraction


frame are nearly minus065 cm as given in Figure 10 For theZ-braced frame with soil-structure interaction the mini-mum weights obtained are 511 and 373 heavier byusing TLBO and HS respectively -e settlements in thisbraced frame are similar to the ones of the other bracedframes For the eccentric V-braced frame with soil-structureinteraction the minimum steel weights obtained are 627 and794 heavier by using TLBO andHS respectivelyMoreover

the convergences of optimum solutions with iteration steps areseen in Figures 8 11 13 and 17 in detail

6 Summary and Conclusions

In this study the optimum design of a 10-storey steel spaceframe braced with X V Z and eccentric V-shaped bracingsincluding soil-structure interaction is investigated Opti-mum solutions are obtained using two different meta-heuristic algorithm methods teaching-learning-basedoptimization (TLBO) and harmony search (HS) For thispurpose a code is developed in MATLAB computer pro-gram incorporated with SAP2000-OAPI (open applica-tion programming interface) Required cross sections areautomatically selected from a list of 128W profiles takenfrom AISC (American Institute of Steel Construction) -e

Table 9 Optimum results of the eccentric V-braced space frame






1-2

CC W24times 68 W14times14 W14times14 W10times10OCl W24times 24 W12times12 W14times14 W12times12OCs W36times 36 W14times14 W10times10 W40times 40IC W14times14 W10times10 W18times18 W12times12OB W10times10 W18times18 W18times18 W16times16IB W18times18 W16times16 W8times 8 W16times16BR W10times10 W16times16 W10times10 W8times 8

3-4

CC W10times10 W12times12 W18times18 W14times14OCl W12times12 W12times12 W14times14 W12times12OCs W10times10 W12times12 W12times12 W21times21IC W27times 27 W14times14 W14times14 W14times14OB W18times18 W16times16 W16times16 W16times16IB W12times12 W24times 24 W10times10 W16times16BR W12times12 W10times10 W10times10 W12times12

5-6

CC W12times12 W12times12 W16times16 W10times10OCl W12times12 W10times10 W12times12 W8times 8OCs W24times 24 W12times12 W12times12 W12times12IC W14times14 W16times16 W24times 24 W24times 24OB W18times18 W12times12 W14times14 W16times16IB W14times14 W24times 24 W16times16 W14times14BR W10times10 W6times 6 W12times12 W6times 6

7-8

CC W10times10 W12times12 W8times 8 W14times14OCl W16times16 W8times 8 W12times12 W12times12OCs W10times10 W8times 8 W16times16 W14times14IC W10times10 W18times18 W10times10 W10times10OB W12times12 W8times 8 W16times16 W8times 8IB W14times14 W14times14 W10times10 W10times10BR W12times12 W12times12 W12times12 W12times12

9-10

CC W12times12 W8times 8 W14times14 W14times14OCl W8times 8 W10times10 W10times10 W8times 8OCs W8times 8 W12times12 W10times10 W10times10IC W14times14 W10times10 W14times14 W12times12OB W8times 8 W8times 8 W10times10 W14times14IB W12times12 W10times10 W14times14 W10times10BR W10times10 W12times12 W12times12 W6times 6

Weight (kN) 127501 135505 133523 144158Maximum lateral displacement (cm) 776 809 751 766Interstorey drift (cm) 091 0915 091 0915Maximum settlement (cm) mdash minus0609 mdash minus0609

Table 10 Soil parameters for the 10-storey eccentric V-braced steelspace frame on elastic foundation





frame model is exposed to wind loads according to ASCE7-05 in addition to dead live and snow loads -e stressconstraints in accordance with AISC-ASD (American In-stitute of Steel Construction-Allowable Stress Design)maximum lateral displacement constraints interstoreydrift constraints and beam-to-column connection con-straints are applied in analyses A three-parameterVlasov elastic foundation model is used to consider thesoil-structure interaction effect -e summary of the resultsobtained in this study are briefly listed below

(i) It is observed from analyses that the minimumweight of the space frame varies by the types ofbracing -e lowest steel weight 108380 kN isobtained for the V-braced steel frame by usingTLBO Z-braced and X-braced types provide the

second and third low weights 109527 kN and117003 respectively -ese results are similar to theones available in literature [36] -e heaviest amongthem is the minimum weight of the eccentricV-braced frame 127501 kN

(ii) Harmony search algorithm presents 26ndash47heavier steel weights than the ones obtained fromteaching-learning-based optimization for theframes without soil-structure interaction Althoughthe lighter analysis results are obtained in TLBOa representative structure model in TLBO is ana-lyzed twice in an iteration step by SAP2000 pro-gramming On the other hand it is enough toanalyze the system once in HS -is situation re-quires longer time for the analysis in TLBO

(iii) Interstorey drift values are very close to its limitvalue of 0915 cm (h400) -erefore the constraintsare important determinants of the optimum designof the braced frames

(iv) Consideration of soil-structure interaction results inheavier steel weight For the X-braced frame in-cluding soil-structure interaction the minimumweights are obtained to be 401 and 532 heavier byusing TLBO and HS respectively For the V-bracedframe these values are calculated to be 166 and 59heavier respectively For the Z-braced frame thesevalues are obtained to be 511 and 373 heavierrespectively Moreover for the eccentric V-bracedframe the minimum weights are obtained to be 627and 794 heavier by these algorithm methods

(v) Settlement values on the soil surfaces are nearly061ndash067 cm for all braced frames

(vi) Finally the techniques used in optimizations seemto be quite suitable for practical applications Anadaptive setting for the parameters will be veryuseful and user-friendly especially for the structureswith a large number of members as in the case here-is will be considered in future studies


(a)

0 E ndash 3ndash700 ndash646 ndash592 ndash538 ndash485 ndash431 ndash377 ndash323 ndash269 ndash215 ndash162 ndash108 ndash54

(b)

Figure 16 Settlements of soil surface for the eccentric V-bracedsteel space frame with two different algorithm methods (cm) (a)TLBO (b) HS

0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV

Figure 17 Design histories of the 10-storey eccentric V-bracedsteel space frame I HS with soil-structure interaction II TLBOwith soil-structure interaction III HS without soil-structure in-teraction IV TLBO without soil-structure interaction


Conflicts of Interest

-e authors declare that they have no conflicts of interest

References

[1] A Daloglu and M Armutcu ldquoOptimum design of plane steelframes using genetic algorithmrdquo Teknik Dergi vol 9pp 483ndash487 1998

[2] E S Kameshki andM P Saka ldquoOptimum design of nonlinearsteel frames with semi-rigid connections using a genetic al-gorithmrdquo Computers amp Structures vol 79 no 17 pp 1593ndash1604 2001

[3] K S Lee and Z W Geem ldquoA new structural optimizationmethod based on the harmony search algorithmrdquo Computersamp Structures vol 82 no 9-10 pp 781ndash798 2004

[4] M S Hayalioglu and S O Degertekin ldquoMinimum cost designof steel frames with semi-rigid connections and column basesvia genetic optimizationrdquo Computers amp Structures vol 83no 21-22 pp 1849ndash1863 2005

[5] O Kelesoglu and M Ulker ldquoMulti-objective fuzzy optimi-zation of space trusses by Ms-Excelrdquo Advances in EngineeringSoftware vol 36 no 8 pp 549ndash553 2005

[6] S O Degertekin ldquoA comparison of simulated annealing andgenetic algorithm for optimum design of nonlinear steel spaceframesrdquo Structural and Multidisciplinary Optimizationvol 34 no 4 pp 347ndash359 2007

[7] Y Esen and M Ulker ldquoOptimization of multi storey spacesteel frames materially and geometrically properties non-linearrdquo Journal of the Faculty of Engineering and Architectureof Gazi University vol 23 pp 485ndash494 2008

[8] M P Saka ldquoOptimum design of steel sway frames to BS5950using harmony search algorithmrdquo Journal of ConstructionalSteel Research vol 65 no 1 pp 36ndash43 2009

[9] S O Degertekin and M S Hayalioglu ldquoHarmony searchalgorithm for minimum cost design of steel frames withsemi-rigid connections and column basesrdquo Structural andMultidisciplinary Optimization vol 42 no 5 pp 755ndash7682010

[10] O Hasancebi S Carbas E Dogan F Erdal and M P SakaldquoComparison of non-deterministic search techniques in theoptimum design of real size steel framesrdquo Computers ampStructures vol 88 no 17-18 pp 1033ndash1048 2010

[11] O Hasanccedilebi S Ccedilarbas and M P Saka ldquoImproving theperformance of simulated annealing in structural optimiza-tionrdquo Structural and Multidisciplinary Optimization vol 41no 2 pp 189ndash203 2010

[12] O Hasancebi T Bahcecioglu O Kurc and M P SakaldquoOptimum design of high-rise steel buildings using an evo-lution strategy integrated parallel algorithmrdquo Computers ampStructures vol 89 no 21-22 pp 2037ndash2051 2011

[13] V Togan ldquoDesign of planar steel frames using teachingndashlearning based optimizationrdquo Engineering Structures vol 34pp 225ndash232 2012

[14] I Aydogdu and M P Saka ldquoAnt colony optimization ofirregular steel frames including elemental warping effectrdquoAdvances in Engineering Software vol 44 no 1 pp 150ndash1692012

[15] T Dede and Y Ayvaz ldquoStructural optimization withteaching-learning-based optimization algorithmrdquo Struc-tural Engineering and Mechanics vol 47 no 4 pp 495ndash5112013

[16] T Dede ldquoOptimum design of grillage structures to LRFD-AISC with teaching-learning based optimizationrdquo

Structural and Multidisciplinary Optimization vol 48no 5 pp 955ndash964 2013

[17] O Hasanccedilebi T Teke and O Pekcan ldquoA bat-inspired al-gorithm for structural optimizationrdquo Computers amp Structuresvol 128 pp 77ndash90 2013

[18] M P Saka and Z W Geem ldquoMathematical and metaheuristicapplications in design optimization of steel frame structuresan extensive reviewrdquo Mathematical Problems in Engineeringvol 2013 Article ID 271031 33 pages 2013

[19] O Hasanccedilebi and S Ccedilarbas ldquoBat inspired algorithm fordiscrete size optimization of steel framesrdquo Advances in En-gineering Software vol 67 pp 173ndash185 2014

[20] T Dede ldquoApplication of teaching-learning-based-optimizationalgorithm for the discrete optimization of truss structuresrdquoKSCE Journal of Civil Engineering vol 18 no 6 pp 1759ndash1767 2014

[21] S K Azad and O Hasancebi ldquoDiscrete sizing optimization ofsteel trusses under multiple displacement constraints and loadcase using guided stochastic search techniquerdquo Structural andMultidisciplinary Optimization vol 52 no 2 pp 383ndash4042015

[22] M Artar and A T Daloglu ldquoOptimum design of compositesteel frames with semi-rigid connections and column bases viagenetic algorithmrdquo Steel and Composite Structures vol 19no 4 pp 1035ndash1053 2015

[23] M Artar ldquoOptimum design of steel space framesunder earthquake effect using harmony searchrdquo StructuralEngineering and Mechanics vol 58 no 3 pp 597ndash6122016

[24] M Artar ldquoOptimum design of braced steel frames viateaching learning based optimizationrdquo Steel and CompositeStructures vol 22 no 4 pp 733ndash744 2016

[25] S Carbas ldquoDesign optimization of steel frames using anenhanced firefly algorithmrdquo Engineering Optimizationvol 48 no 12 pp 2007ndash2025 2016

[26] A T Daloglu M Artar K Ozgan and A I Karakas ldquoOp-timum design of steel space frames including soilndashstructureinteractionrdquo Structural and Multidisciplinary Optimizationvol 54 no 1 pp 117ndash131 2016

[27] M P Saka O Hasanccedilebi and ZW Geem ldquoMetaheuristics instructural optimization and discussions on harmony searchalgorithmrdquo Swarm and Evolutionary Computation vol 28pp 88ndash97 2016

[28] I Aydogdu ldquoCost optimization of reinforced concretecantilever retaining walls under seismic loading usinga biogeography-based optimization algorithm with Levyflightsrdquo Engineering Optimization vol 49 no 3 pp 381ndash400 2017

[29] MATLAB gte Language of Technical Computing -eMathworks Inc Natick MA USA 2009

[30] SAP2000 Integrated Finite Elements Analysis and Design ofStructures Computers and Structures Inc Berkeley CAUSA 2008

[31] ASCE7-05 Minimum Design Loads for Building and OtherStructures American Society of Civil Engineering RestonVA USA 2005

[32] AISCndashASD Manual of Steel Construction Allowable StressDesign American Institute of Steel Construction Chicago ILUSA 1989

[33] P Dumonteil ldquoSimple equations for effective length factorsrdquoEngineering Journal AISC vol 29 pp 111ndash115 1992

[34] C V G Vallabhan and A T Daloglu ldquoConsistent FEM-Vlasov model for plates on layered soilrdquo Journal of StructuralEngineering vol 125 no 1 pp 108ndash113 1999


[35] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novel method for constrainedmechanical design optimization problemsrdquo Computer-AidedDesign vol 43 no 3 pp 303ndash315 2011

[36] O Hasancebi ldquoCost efficiency analyses of steel frameworksfor economical design of multi-storey buildingsrdquo Journalof Constructional Steel Research vol 128 pp 380ndash3962017


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

in accordance with BS5950 Degertekin and Hayalioglu[9] applied the harmony search algorithm for minimumcost design of steel frames with semirigid connections andcolumn bases Hasancebi et al [10] investigated non-deterministic search techniques in the optimum design ofreal-size steel frames Hasanccedilebi et al [11] used the simulatedannealing algorithm in structural optimization Hasancebiet al [12] investigated the optimum design of high-rise steelbuildings using an evolutionary strategy integrated withparallel algorithm Togan [13] used one of the latest sto-chastic methods teaching-learning-based optimization fordesign of planar steel frames Aydogdu and Saka [14] usedant colony optimization for irregular steel frames includingthe elemental warping effect Dede and Ayvaz [15] studiedstructural optimization problems using the teaching-learning-based optimization algorithm Dede [16] appliedteaching-learning-based optimization on the optimumdesign of grillage structures with respect to LRFD-AISCHasanccedilebi et al [17] used a bat-inspired algorithm forstructural optimization Saka and Geem [18] prepared anextensive review study on mathematical and metaheuristicapplications in design optimization of steel frame structuresHasanccedilebi and Ccedilarbas [19] studied the bat-inspired algo-rithm for discrete-size optimization of steel frames Dede[20] focused on the application of the teaching-learning-based optimization algorithm for the discrete optimizationof truss structures Azad and Hasancebi [21] focused ondiscrete-size optimization of steel trusses under multipledisplacement constraints and load case using the guidedstochastic search technique Artar and Daloglu [22] obtainedthe optimum design of composite steel frames with semi-rigid connections and column bases Artar [23] used theharmony search algorithm for the optimum design of steelspace frames under earthquake loading Artar [24] used theteaching-learning-based optimization algorithm for theoptimum design of braced steel frames Carbas [25] studieddesign optimization of steel frames using an enhanced fireflyalgorithm Daloglu et al [26] investigated the optimumdesign of steel space frames including soil-structure in-teraction Saka et al [27] researched metaheuristics instructural optimization and discussions on the harmonysearch algorithm Aydogdu [28] used a biogeography-basedoptimization algorithm with Levy flights for cost optimi-zation of reinforced concrete cantilever retaining wallsunder seismic loading

In literature there are several researches available foroptimum structural design as mentioned above On the otherhand there are a few researches on the optimum design ofbraced steel space frames including soil-structure interactionSo this study investigates a 10-storey braced steel space framestructure studied previously in literature which is investigatedfor four different bracing types and soil-structure interaction-ese bracing types are X V Z and eccentric V-shapedbracings Optimum design solutions are obtained usinga computer program developed in MATLAB [29] interactingwith SAP2000-OAPI (open application programming in-terface) [30] Suitable cross sections are automatically selectedfrom a list including 128W profiles taken from AISC(American Institute of Steel Construction) -e frame model

is subjected to wind loads according to ASCE7-05 [31] as wellas dead live and snow loads-e analysis results are found tobe quite consistent with the literature results In this study thevertical displacements on soil surfaces are also calculated It isobserved that minimum weights of space frames varydepending on the bracing type Also it can be concluded thatincorporation of soil-structure interaction results in heaviersteel weight

2 Optimum Design Formulation

-e optimum design problem of braced steel space frames iscalculated as follows

minW 1113944

ng

k1Ak 1113944

nk

i1ρiLi (1)

where W is the weight of the frame Ak is the cross-sectionalarea of group k ρi and Li are the density and length ofmember i ng is the total number of groups and nk is thetotal number of members in group k

-e stress constraints according to AISC-ASD [32] aredefined as follows

gi(x) fa

Fa+

Cmxfbx

1minus faFexprime( 1113857( 1113857Fbx

1113890 1113891i

minus 10le 0 i 1 nc

gi(x) fa

060Fy+

fbx

Fbx

1113890 1113891i

minus 10le 0 i 1 nc

(2)

If (faFa)le 015 instead of using (2) the stress con-straint is calculated as follows

gi(x) fa

Fa+

fbx

Fbx

1113890 1113891i

minus 10le 0 i 1 nc (3)

where nc is the total number of members subjected to bothaxial compression and bending stresses fa is the computedaxial stress Fa is the allowable axial stress under axialcompression force alone fbx is the computed bending stressdue to bending of the member about its major (x) Fbx is theallowable compressive bending stress about major Fexprime is theEuler stress Fy is the yield stress of the steel and Cmx isa factor It is calculated from Cmx 06minus 04(M1M2) forthe braced frame member without transverse loading be-tween the ends andCmx 1 + ψ(faFeprime) for the braced framemember with transverse loading

-e effective length factors of columns in braced framesare calculated as follows [33]

K 3GAGB + 14 GA + GB( 1113857 + 0643GAGB + 20 GA + GB( 1113857 + 128

(4)

where GA and GB are the relative stiffness factors at the Athand Bth ends of columns

-e maximum lateral displacement and interstorey driftconstraints are defined as follows


gjl(x) δjl

δju



gjil(x) Δjil

Δju


l 1 nl(6)



gbf i(x) bfbkiprime


gbbi(x) bfbki

bfcki

minus 1le 0 i 1 nbf

(7)








k 1113946H

0



zφ(z)

zz1113888 1113889

2

dz

2t 1113946H

0Gsφ(z)

2dz

(9)




sinh c (10)


c

H1113874 1113875

2

1minus 2]s( 1113857






(11)




bfck

dcl

tfl

B2

B1

bfbk

bprimefbk






H



⋮ ⋮ ⋮ ⋮ ⋮



n

xHMS1 xHMS


n

rarr

rarr

rarr

rarr

rarr

φ x1( )φ x2( )⋮


(12)










Frame

Raft

Soil

ϕ(0) = 1

ϕ(z)

ϕ(H) = 0

(a)

Shear layer

Spring

(b)




class(population)



⋮ ⋮ ⋮ ⋮ ⋮



rarr

rarr

rarr

rarr

rarr

f x1( )f x2( )⋮


(15)





(17)


5 Design Example







Is it satisfied

Present results


vectors



(γ) = 1



No

Yes


parameters

SAP2000 OAPIYes



sections)

No

MATLAB OAPI SAP2000




of solution vectors




CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)








Floor


force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)













Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


gjl(x) δjl

δju



gjil(x) Δjil

Δju


l 1 nl(6)



gbf i(x) bfbkiprime


gbbi(x) bfbki

bfcki

minus 1le 0 i 1 nbf

(7)








k 1113946H

0



zφ(z)

zz1113888 1113889

2

dz

2t 1113946H

0Gsφ(z)

2dz

(9)




sinh c (10)


c

H1113874 1113875

2

1minus 2]s( 1113857






(11)




bfck

dcl

tfl

B2

B1

bfbk

bprimefbk






H



⋮ ⋮ ⋮ ⋮ ⋮



n

xHMS1 xHMS


n

rarr

rarr

rarr

rarr

rarr

φ x1( )φ x2( )⋮


(12)










Frame

Raft

Soil

ϕ(0) = 1

ϕ(z)

ϕ(H) = 0

(a)

Shear layer

Spring

(b)




class(population)



⋮ ⋮ ⋮ ⋮ ⋮



rarr

rarr

rarr

rarr

rarr

f x1( )f x2( )⋮


(15)





(17)


5 Design Example







Is it satisfied

Present results


vectors



(γ) = 1



No

Yes


parameters

SAP2000 OAPIYes



sections)

No

MATLAB OAPI SAP2000




of solution vectors




CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)








Floor


force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)













Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





H



⋮ ⋮ ⋮ ⋮ ⋮



n

xHMS1 xHMS


n

rarr

rarr

rarr

rarr

rarr

φ x1( )φ x2( )⋮


(12)










Frame

Raft

Soil

ϕ(0) = 1

ϕ(z)

ϕ(H) = 0

(a)

Shear layer

Spring

(b)




class(population)



⋮ ⋮ ⋮ ⋮ ⋮



rarr

rarr

rarr

rarr

rarr

f x1( )f x2( )⋮


(15)





(17)


5 Design Example







Is it satisfied

Present results


vectors



(γ) = 1



No

Yes


parameters

SAP2000 OAPIYes



sections)

No

MATLAB OAPI SAP2000




of solution vectors




CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)








Floor


force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)













Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



class(population)



⋮ ⋮ ⋮ ⋮ ⋮



rarr

rarr

rarr

rarr

rarr

f x1( )f x2( )⋮


(15)





(17)


5 Design Example







Is it satisfied

Present results


vectors



(γ) = 1



No

Yes


parameters

SAP2000 OAPIYes



sections)

No

MATLAB OAPI SAP2000




of solution vectors




CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)








Floor


force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)













Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



CC OCI OCI CC

OB OB OB

IB IB OB

IC IB IC IB OCs

YX IB OB

IC IB OCs

OB

3times20 ft (610 m)=60 ft (1829 m)

3times15 ft (457 m)=45 ft (1372 m)








Floor


force (kNm)

Distributedleeward

force (kNm)

Distributedwindward

force (kNm)

Distributedleeward

force (kNm)1 233 232 233 2632 266 232 266 2633 299 232 299 2634 324 232 324 2635 346 232 346 2636 364 232 364 2637 381 232 381 2638 395 232 395 2639 409 232 409 26310 211 116 211 132

(a)

(b)













Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a) (b)









Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10










(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

1000150020002500300035004000

45005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)








Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Literatureresearch

Hasancebi [36]





1-2


3-4


5-6


7-8


9-10











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV


(a)

(b)













1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












1-2


3-4


5-6


7-8


9-10

















(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











(a)


(b)


0 100 200 300 400 500 600 700 800Iterations

100015002000250030003500400045005000

Wei

ght (

kN)

III

IIIIV





References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




References










































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Optimum Design of Braced Steel Space Frames including …downloads.hindawi.com/journals/ace/2018/3854620.pdf · in accordance with BS5950. Degertekin and Hayalioglu [9] applied the