A hybrid optimization algorithm for the optimal design …scientiairanica.sharif.edu/article_2135_1e9a6b6754376217304ee3f222...A hybrid optimization algorithm for the optimal design

Scientia Iranica A (2016) 23(2), 508{519

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

A hybrid optimization algorithm for the optimal designof laterally-supported castellated beams

A. Kaveh� and F. Shokohi

Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak,Tehran, P.O. Box 16846-13114, Iran.

Received 10 January 2015; received in revised form 19 May 2015; accepted 22 June 2015

KEYWORDSColliding BodiesOptimization (CBO);Particle SwarmOptimization (PSO);A hybrid CBO-PSOalgorithm;Optimum design ofcastellated beams;Hexagonal openingbeam;Cellular openingbeam.

Abstract. In this paper, a hybrid method is developed for optimum design ofcastellated beams by combining two meta-heuristic algorithms, namely the Colliding BodiesOptimization (CBO) and Particle Swarm Optimization (PSO). In this hybrid algorithm(CBO-PSO), positive features of PSO are added to the CBO. Two common types oflaterally supported castellated beams are considered to be the design problems: beamswith hexagonal openings and beams with circular openings. These beams have foundwidespread usage in buildings because of great savings in materials and construction costs.Here, the minimum cost is taken as the design objective function and the new hybridmethod is utilized for obtaining the solution of some benchmark problems. Comparisonof the results of the CBO-PSO with those of some other meta-heuristics demonstrates thecapability of the presented optimization algorithm. For most of the examples, the resultsobtained by CBO-PSO have less cost than those of the other considered methods. It is alsoconcluded that the beams with hexagonal openings require smaller amount of steel, hencebeing superior to the beams with cellular openings.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, due to huge extension in size and dimensionof the structures, there has been a great increase inweight and cost of construction materials used forstructures. Therefore, it is not surprising that a lot ofattention is being paid by engineers to optimal designof the structures which lead to a signi�cant decrease intheir cost.

Since the 1940s, the production of structuralbeams with higher strength and lower cost has beenan asset to engineers in their e�orts to design moree�cient steel structures. Due to the limitations onmaximum allowable de ections, the use of sectionswith heavy weight and high capacity in the design

*. Corresponding author. Tel.: +98 21 77240104;Fax: +98 21 77240398E-mail address: [email protected] (A. Kaveh)

problem cannot always be bene�cial. As a result,several new methods have been created for increasingthe sti�ness of steel beams without increase in weightof the required steel. Hence, castellated and cellularbeams have been utilized extensively in recent years [1].

In design of steel structures, beams with web-opening are widely used to pass the under oor servicesducts such as water pipes and air ducts. Castellatedbeam is created from a standard wide- ange beam bycutting it longitudinally in a zigzag or semi-circularpattern, separating and o�setting the two halves, andwelding them back together. The resulting holes in thewebs permit mechanical ducts, plumbing, and electricallines to pass through the beam rather than beneath thebeam. Web-openings have been used for many years instructural steel beams in a great variety of applicationsbecause of the necessity and economic advantage. Theimportant advantage of the steel beam castellationprocess is that the designer can increase the depth of

A. Kaveh and F. Shokohi/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 508{519 509

a beam to raise its strength without adding steel. Theresulting castellated beam is approximately 50% deeperand much stronger than the original unaltered beam [2-5].

In recent years, a great deal of progress has beenmade in the design of steel beams with web-openingsand cellular beam is one of them. Cellular beam is themodern form of the traditional castellated beam, butwith a far wider range of applications for oor beams,in particular. Cellular beams are steel sections withcircular openings that are made by cutting a rolledbeam web in a half-circular pattern along its centerlineand re-welding the two halves of hot rolled steel sectionsas shown in Figure 1. This opening increases the overall

Figure 1. (a) A castellated beam with hexagonalopening. (b) A castellated beam with circular opening.

beam depth, moment of inertia, and section moduluswithout increasing the overall weight of the beam [6].

The Colliding Bodies Optimization method(CBO) is one of the recently developed meta-heuristicalgorithms that utilizes simple formulation and it re-quires no parameter tuning. This algorithm is based onone-dimensional collisions between two bodies, whereeach agent solution is modeled as a body [7,8].

Particle Swarm Optimization (PSO) is based onthe behavior of a colony or swarm of insects (such asants, termites, bees, and wasps), a ock of birds, or aschool of �sh. This algorithm was originally proposedby Kennedy and Eberhart in 1995. A basic variantof the PSO algorithm works by having a population(swarm) of candidate solutions (particles). Theseparticles move around in the search-space using a fewsimple formulas. Each particle iteratively moves acrossthe search-space and is attracted to the position ofthe best �tness (evaluation of the objective function)historically achieved by the particle itself (local best)and by the best among the neighbors of the particle(global best) [9,10].

The main objective of this paper is to present ahybrid CBO-based algorithm (combined with the PSO)to �nd an optimum design of castellated beams. Forthis purpose, the positive properties of the PSO will beadded to the CBO.

According to the above-mentioned content, thepresent study is organized as follows: In Section 2, thedesign of castellated beam is introduced. Statementof the optimization design problem is formulated inSection 3, based on The Steel Construction InstitutePublication Number 100 and Euro code 3. In Section 4,the CBO algorithm and the PSO approach are brie yintroduced. Also, the new hybrid method is presentedin this section. In Section 5, the cost of castellatedbeam as the design objective function is minimized,and �nally, Section 6 concludes the paper.

2. Design of castellated beams

Beams must be su�ciently strong to carry the appliedbending moments and shear forces. The performanceof any beam is dependent upon the physical dimensionsas well as the cross-section geometry and shape. Dueto the presence of holes in the web, the structuralbehavior of castellated steel beam is di�erent fromthat of the solid web beams. At present, there is nota prescribed design method due to the complexity ofthe behavior of castellated beams and their associatedmodes of failure [2]. The strength of a beam withvarious web openings is determined by considering theinteraction of the exure and shear at the openings.There are many failure modes to be considered inthe design of a beam with web opening consistingof lateral-torsional buckling, Vierendeel mechanism,

510 A. Kaveh and F. Shokohi/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 508{519

exural mechanism, rupture of welded joints, and webpost buckling. Lateral-torsional buckling may occurin an unrestrained beam. A beam is considered tobe unrestrained when its compression ange is freeto displace laterally and rotate. In this paper, it isassumed that the compression ange of the castellatedbeam is restrained by the oor system. Therefore,the overall buckling strength of the castellated beam isomitted from the design considerations. These modesare closely associated with beam geometry, shapeparameters, type of loading, and provision of lateralsupports. In the design of castellated beams, thesecriteria should be considered [11-17].

2.1. Overall beam exural capacityThis mode of failure can occur when a section issubjected to pure bending. In the span subjectedto pure bending moment, the tee-sections above andbelow the holes yield in a manner similar to thatof a plain webbed beam. Therefore, under appliedload combinations, the castellated beam should havesu�cient exural capacity to be able to resist theexternal loading [12,13]:

MU �MP = ALTPYHU ; (1)

where ALT is the area of lower tee, PY is the designstrength of steel, and HU is the distance betweencenters of gravities of upper and lower tees.

2.2. Beam shear capacityIn the design of castellated beams, it is necessary tocontrol two modes of shear failure. The �rst one is thevertical shear capacity and the upper and lower teesshould undergo it. The sum of the shear capacities ofthe upper and lower tees is checked using the followingequations [2,13]:

PV Y = 0:6PY (0:9AWUL) for circular opening,

PV Y =p

33PY (AWUL) for hexagonal openning, (2)

where AWUL is the total area of the webs of tees.The second one is the horizontal shear capacity. It

is developed in the web post due to the change in axialforces in the tee-section as shown in Figure 2. Webpost with too short mid-depth welded joints may failprematurely when horizontal shear exceeds the yieldstrength. The horizontal shear capacity is checkedusing the following equations [2,13]:

PV H = 0:6PY (0:9AWP ) for circular opening

PV H =p

33PY (AWP ) for hexagonal openning; (3)

where AWP is the minimum area of web post.

Figure 2. Horizontal shear in the web post of castellatedbeams: (a) Hexagonal opening; and (b) circular opening.

2.3. Flexural and buckling strength of web postAs mentioned above, it is assumed that the compres-sion ange of the castellated beam is restrained by the oor system. Thus, the overall buckling of the castel-lated beam is omitted from the design considerations.The web post buckling capacity in castellated beam isgiven by [2,13]:

MMAX

ME=�C1:�� C2:�2 � C3

�; (4)

where � = S2d for hexagonal openings, and � = S

D0for circular openings, also MMAX is the maximumallowable web post moment and ME is the web postcapacity at critical section A-A shown in Figure 2. C1,C2, and C3 are constants obtained by the followingexpressions:

C1 = 5:097 + 0:1464(�)� 0:00174(�)2; (5)

C2 = 1:441 + 0:0625(�)� 0:000683(�)2; (6)

C3 = 3:645 + 0:0853(�)� 0:00108(�)2; (7)

where � = 2dtw for hexagonal openings, and � = D0

tw forcircular openings, S is the spacing between the centersof holes, d is the cutting depth of hexagonal opening,D0 is the holes diameter, and tw is the web thickness.


Figure 3. Olander's curved beam approach.

2.4. Vierendeel bending of upper and lowertees

Vierendeel mechanism is the most common failure forperforated steel beams that results in the formation offour plastic hinges above and below the web opening.It is always critical in steel beams with web openings,where global shear force is transferred across theopening length, and the Vierendeel moment is resistedby the local moment resistances of the tee-sectionsabove and below the web openings.

The overall Vierendeel bending resistance de-pends on the local bending resistance of the web- ange sections. This mode of failure is associated withhigh shear forces acting on the beam. The Vierendeelbending stresses in the circular opening is obtainedby using the Olander's approach in Figure 3. Theinteraction between Vierendeel bending moment andaxial force for the critical section in the tee should bechecked as follows [13]:

P0

PU+

MMP

� 1:0; (8)

where P0 and M are the force and the bending momenton the section, respectively; PU is equal to the area ofcritical section �PY ; MP is calculated as the plasticmodulus of critical section in plastic section or elasticsection modulus of critical section for other sections.

The plastic moment capacity of the tee-sectionsin castellated beams with hexagonal opening are calcu-lated independently. The total of the plastic momentsis equal to the sum of the Vierendeel resistances ofthe above and below tee-sections [2]. The interactionbetween Vierendeel moment and shear forces should bechecked by the following expression:

VOMAX:e� 4MTP � 0; (9)

where VOMAX and MTP are the maximum shear forceand the moment capacity of tee-section, respectively.

2.5. De ection of castellated beamServiceability checks are of high importance in design,especially for beams with web opening where the de- ection due to shear forces is signi�cant. The de ection

of a castellated beam under applied load combinationsshould not exceed span/360. In castellated beamswith circular opening, the de ection at each point iscalculated by the following expression:

YTOT = YMT + YWP + YAT + YST + YSWP ; (10)

where YMT , YWP , YAT , YST , and YSWP are de ectiondue to bending moment in tee, de ection due tobending moment in web post of beam, de ection dueto axial force in tee, de ection due to shear in tee, andde ection due to shear in web post, respectively. Theseequations are provided in [13].

For a castellated beam with hexagonal openingand length L subjected to transverse loading, the totalde ection is composed by two terms: the �rst termcorresponds to pure moment action, fb, and the secondone corresponds to shear action, fs. Thus, the totalde ection can be calculated by the following expression:

f = fb + fs = c1L3 + c2L; (11)

c1 and c2 are determined by means of a curve �ttingtechnique [15].

3. Formulation of the optimization problem

The main initiative for producing and using castellatedbeam is to suppress the cost of material by apply-ing more e�cient cross sectional shapes made fromstandard pro�les in combination with aesthetic andarchitectural design considerations.

In a castellated beam, there are many factors thatrequire special considerations when estimating the costof beam, such as man-hours of fabrication, weight, priceof web cutting, and welding process. At this study, itis assumed that the costs associated with man-hours offabrication for hexagonal and circular opening are iden-tical. Thus, the objective function includes three parts:the beam weight, price of the cutting, and price of thewelding. The objective function can be expressed as:

Fcost = �Ainitial(L0):P1 + Lcut:P2 + Lweld:P3; (12)

where P1, P2, and P3 are the price of the weightof the beam per unit weight, length of cutting andwelding for per unit length, respectively, L0 is theinitial length of the beam before castellation process,� is the density of steel, Ainitial is the area of theselected universal beam section, Lcut and Lweld arethe cutting length and welding length, respectively.The length of cutting is di�erent for hexagonal andcircular web-openings. The dimension of the cuttinglength is described by the following equations:

For circular opening:

Lcut = �D0:NH + 2e(NH + 1) +�D0

2+ e: (13)


For hexagonal opening:

Lcut = 2NH�e+

dsin(�)

�+ 2e+

dsin(�)

; (14)

where NH is the total number of holes, e is the lengthof horizontal cutting of web, D0 is the diameter of holes,d is the cutting depth, and � is the cutting angle.

Also, the welding length for both circular andhexagonal openings is determined by Eq. (15):

Lweld = e(NH + 1): (15)

As an example, in Figure 1(a), the number of holes isequal to 3. Therefore, the total length of cutting canbe expressed by the following equation:

Lcut = 8e+ 7�

dsin(�)

�: (16)

Similarly, for cellular beams, the same equations can beobtained. Lcut is shown for both circular and hexagonalopenings in Figure 1.

3.1. Design of castellated beam with circularopening

Design process of a cellular beam consists of threephases: selection of a rolled beam, selection of adiameter, and spacing between the centers of holes ortotal number of holes in the beam as shown in Fig-ure 1, [13,14]. Hence, the sequence number of the rolledbeam section in the standard steel sections tables, thecircular holes diameter, and the total number of holesare taken as design variables in the optimum designproblem. The optimum design problem formulated byconsidering the constraints explained in the previoussections can be expressed as follows.

Find an integer design vector fXg = fx1; x2; x3gTwhere x1 is the sequence number of the rolled steelpro�le in the standard sections list, x2 is the sequencenumber for the hole diameter which contains variousdiameter values, and x3 is the total number of holesfor the cellular beam [13]. Hence, the design problemcan be expressed as:

Minimize Eq. (12):

Subjected to:

g1 = 1:08�D0 � S � 0; (17)

g2 = S � 1:60�D0 � 0; (18)

g3 = 1:25�D0 �Hs � 0; (19)

g4 = Hs � 1:75�D0 � 0; (20)

g5 = MU �MP � 0; (21)

g6 = VMAXSUP � PV � 0; (22)

g7 = VOMAX � PV Y � 0; (23)

g8 = VHMAX � PV H � 0; (24)

g9 = VA-AMAX �MWMAX � 0; (25)

g10 = VTEE � 0:50� PV Y � 0; (26)

g11 =P0

PU+

MMP� 1:0 � 0; (27)

g12 = YMAX � L=360 � 0; (28)

where tW is the web thickness; Hs and L are theoverall depth and the span of the cellular beam; Sis the distance between centers of holes; MU is themaximum moment under the applied loading; MPis the plastic moment capacity of the cellular beam;VMAXSUP is the maximum shear at support; VOMAXis the maximum shear at the opening; VHMAX is themaximum horizontal shear; MA-AMAX is the maximummoment at A-A section shown in Figure 2. MWMAXis the maximum allowable web post moment; VTEErepresents the vertical shear on the tee at � = 0 ofweb opening; P0 and M are the internal forces on theweb section as shown in Figure 3; and YMAX denotesthe maximum de ection of the cellular beam [13,17].

3.2. Design of castellated beam with hexagonalopening

In design of castellated beams with hexagonal open-ings, the design vector includes four design variables:selection of a rolled beam, selection of a cutting depth,spacing between the centers of holes or total number ofholes in the beam, and cutting the angle as shown inFigure 2. Hence, the optimum design problem can beexpressed as follows.

Find an integer design vector fXg = fx1; x2; x3;x4gT , where x1 is the sequence number of the rolledsteel pro�le in the standard sections list, x2 is thesequence number for the cutting depth which containsvarious values, x3 is the total number of holes for thecastellated beam, and x4 is the cutting angle. So, thedesign problem turns out to be as follows:

Minimize Eq. (12).Subjected to:

g1 = d� 38: (Hs � 2tf ) � 0; (29)

g2 = (Hs � 2tf )� 10� (dT � tf ) � 0; (30)

g3 =23:d: cot�� e � 0; (31)

g4 = e� 2d: cot� � 0; (32)

g5 = 2d: cot�+ e� 2d � 0; (33)


g6 = 45� � � � 0; (34)

g7 = �� 64� � 0; (35)

g8 = MU �MP � 0; (36)

g9 = VMAXSUP � PV � 0; (37)

g10 = VOMAX � PV Y � 0; (38)

g11 = VHMAX � PV H � 0; (39)

g12 = MA-AMAX �MWMAX � 0; (40)

g13 = VTEE � 0:50� PV Y � 0; (41)

g14 = VOMAX:e� 4MTP � 0; (42)

g15 = YMAX � L=360 � 0; (43)

where tf is the ange thickness, dT is the depth ofthe tee-section, MP is the plastic moment capacityof the castellated beam, MA-AMAX is the maximummoment at A-A section, shown in Figure 2, MWMAXis the maximum allowable web post moment, VTEErepresents the vertical shear on the tee, MTP is themoment capacity of tee-section, and YMAX denotesthe maximum de ection of the castellated beam withhexagonal opening [2].

4. A hybrid colliding bodies optimization andparticle swarm optimization

In this section, the hybrid CBO and PSO methods ispresented. In order to create the hybrid approach,the CBO is used as the main algorithm and thepositive features of the PSO are added to it. Thehybrid algorithm utilizes the location of the global andlocal best points to improve the searching process. Asummary of these methods is described in the followingsubsections.

4.1. Particle swarm optimization algorithmParticle swarm optimization, �rst developed byKennedy and Eberhart [9], is a population-based meta-heuristic method. The development of this algorithmfollows from observations of social behaviors of animals,such as bird ocking and �sh schooling. The theory ofPSO describes a solution process in which each particle ies through the multidimensional search space whilethe velocity and position of the particle are constantlyupdated according to the best previous performanceof the particle or of the particle's neighbors, as wellas the best performance of the particles in the entirepopulation [18].

The velocity vector is used to update the currentposition of each particle in the swarm. Likewise, the

velocity vector is updated utilizing a memory in whichthe best position of each particle and the best positionamong all particles are stored.

The position of the ith particle at iteration k + 1is calculated using the following equation:

xik+1 = xik + vik+1; (44)

where, xik+1 is the new position, xik is the positionat the kth iteration, and vik+1 is the updated velocityvector of the ith particle. The velocity vector of eachparticle is determined by:

vik+1 = !:vik + c1:r1:�P ik �Xi

k�

+ c2:r2:�P gk �Xi

k�;

(45)

where, vik is the velocity vector at iteration k; r1 andr2 are two random numbers between 0 and 1; P ikrepresents the best ever position of particle i, localbest; P gk is the best position among all particles in theswarm up to iteration k; c1 and c2 are two accelerationconstants; and ! is the inertia weight.

4.2. Colliding bodies optimization algorithmThe Colliding bodies optimization algorithm is oneof the recently developed meta-heuristic search algo-rithms [7]. It is a population-based search approach,where each agent is considered as a Colliding Body(CB) with mass m. The idea of the CBO algorithm isbased on observation of a collision between two objectsin one dimension, in which one object collides withanother object and they move toward minimum energylevel [8].

In the CBO algorithm, each solution candidate,Xi, is considered to be a Colliding Body (CB). Themassed objects are composed of two main equal groups,i.e. stationary and moving objects, where the movingobjects move to follow stationary objects and a collisionoccurs between pairs of objects. This is done fortwo purposes: (i) to improve the positions of movingobjects; (ii) to push stationary objects towards betterpositions. After the collision, the new positions ofthe colliding bodies are updated based on their newvelocities.

The pseudo-code for the CBO algorithm can besummarized as follows:

Step 1. Initialization. The initial positions of CBsare determined randomly in the search space:

x0i = xmin + rand: (xmax � xmin) i = 1; 2; :::; n;

(46)

where x0i determines the initial value vector of the

ith CB; xmin and xmax are the minimum and themaximum allowable value vectors of variables, respec-tively; rand is a random number in the interval [0,1],and n is the number of CBs.


Figure 4. (a) The sorted CBs in an increasing order. (b)The pairs of objects for the collision.

Step 2. Determination of the body mass for eachCB. The magnitude of the body mass for each CB isde�ned as:

mk =1

fit(k)Pni=1

1fit(i)

; k = 1; 2; :::; n; (47)

where fit(i) represents the objective function value ofthe agent i, and n is the population size. Obviously,a CB with good values exerts a larger mass thanthe bad ones. Also, for maximizing the objectivefunction, the term 1

fit(i) is replaced by fit(i).

Step 3. Arrangement of the CBs. The arrangementof the CBs objective function values is performed inascending order (Figure 4(a)). The sorted CBs areequally divided into two groups:� The lower half of CBs (stationary CBs): These

CBs are good agents which are stationary and thevelocity of these bodies before collision is zero.Thus:

vi = 0 i = 1; 2; :::;n2: (48)

� The upper half of CBs (moving CBs): These CBsmove toward the lower half. Then, accordingto Figure 4(b), the better and worse CBs, i.e.agents with upper �tness value in each group, willcollide with each other. The change of the bodyposition represents the velocity of these bodiesbefore collision as:

vi = xi � xi�n2 i =n2

+ 1; :::; n; (49)

where vi and xi are the velocity and positionvectors of the ith CB in this group, respectively;and xi�n2 is the ith CB pair position of xi in theprevious group.

Step 4. Calculation of the new position of theCBs. After the collision, the velocity of bodiesin each group is evaluated using collision laws andthe velocities before collision. The velocity of eachmoving CB after the collision is:

v0i =(mi � "mi�n2 )vimi +mi�n2

i =n2

+ 1; :::; n; (50)

where vi and v0i are the velocity of the ith moving CBbefore and after the collision, respectively; mi is themass of the ith CB; and mi�n2 is mass of the ith CBpair. Also, the velocity of each stationary CB afterthe collision is:

v0i =�mi+n

2+ "mi+n

2

�vi+n

2

mi +mi+n2

; i = 1; :::;n2; (51)

where vi+n2

and vi are the velocity of the ith movingCB pair before and the ith stationary CB after thecollision, respectively; mi is mass of the ith CB;mi+n

2is mass of the ith moving CB pair. As mentionedpreviously, " is the Coe�cient Of Restitution (COR)and for most of the real objects, its value is between0 and 1. It is de�ned as the ratio of the separationvelocity of two agents after collision to the approachvelocity of two agents before collision. In the CBOalgorithm, this index is used to control the explo-ration and exploitation rates. For this goal, the CORdecreases linearly from unit to zero. Thus, " is de�nedas:

" = 1� iteritermax

; (52)

where iter is the actual iteration number, and itermaxis the maximum number of iterations, with CORbeing equal to unit and zero representing the globaland local search, respectively.

New positions of CBs are obtained using thegenerated velocities after the collision in position ofstationary CBs.

The new position of each moving CB is:

xnewi = xi�n2 + rand � v0i i =

n2

+ 1; :::n; (53)

where xnewi and v0i are the new position and the

velocity after the collision of the ith moving CB,respectively; xi�n2 is the old position of the ithstationary CB pair. Also, the new positions ofstationary CBs are obtained by:

xnewi = xi + rand � v0i i = 1; :::;

n2; (54)

where xnewi , xi, and v0i are the new position, old

position, and velocity after the collision of the ithstationary CB, respectively. rand is a random vectoruniformly distributed in the range (-1,1) and the sign\�" denotes an element-by-element multiplication.


Step 5. Termination criterion control. Steps 2-4 are repeated until a termination criterion as themaximum number of iterations is satis�ed.

4.3. Hybrid CBO and PSOBoth of the previously introduced methods (CBOand PSO) are population-based algorithms and �ndoptimum solutions by changing the position of theagents. However, the movement strategies are di�erentfor the CBO and PSO. The PSO algorithm utilizesthe local best and the global best to determine thedirection of the movement, while the CBO approachuses the collision laws and the velocities before collisionto determine the new positions. Using the local bestand the global best are the main reasons for the successof PSO.

However, in spite of having the above-mentionedadvantages, the standard PSO is infamous for prema-ture convergence. This algorithm has some problemsin controlling the balance between the exploration andexploitation due to ignoring the e�ect of other agents.

Similar to the PSO method, the CBO algorithmuses the previous velocities, when the upper half ofCBs (moving CBS) moves toward the stationary CBs.As it is mentioned in the previous section, after thecollision, the velocity of all CBs is evaluated using thevelocity of moving CBs before the collision and themass of the paired CBs. This will lead to loss of thebest position of particles which is found in the previousiteration. Therefore, in the present hybrid algorithm,the advantages of the PSO consist of the local best andthe global best is added to the CBO algorithm. For thispurpose, the best position of the stationary particles issaved in a memory called Stationary Bodies Memory(SBM). Also, another memory is considered to save thebetter position of each particle that has been found sofar. This memory, so-called Particles Memory (PM), istreated as the local best in the PSO, and it is updatedby the following expression:

PM ik+1 =

8<:PM ik ! F (Xi

k+1) � F (Xik)

Xik+1 ! F (Xi

k+1) � F (Xik);

(55)

in which the �rst term identi�es that when the newposition is not better than the previous one, updatingwill not be performed, while when the new position isbetter than the so far stored good position, the newsolution vector is replaced. With the above de�nitions,and considering the above-mentioned new memories,the velocity of CBs after collision is modi�ed by thefollowing equations:

For moving particles:

v0i =�mi � "mi�n2

�vi

mi +mi�n2+ r1:c1: (PMi � xi)

Figure 5. Flowchart of the CBO.

+r2:c2: (SPM� xi) i =n2

+ 1; :::; n: (56)

For stationary particles:

v0i =�mi+n

2+ "mi+n

2

�vi+n

2

mi +mi+n2

+ r1:c1: (PMi � xi)

+ r2:c2: (SPM� xi) i = 1; :::;n2: (57)

Figure 5 shows the schematic procedure of the CBO-PSO algorithm.

5. Design examples

In this section, in order to compare fabrication costsof the castellated beams with circular and hexagonalholes, three benchmark examples from the literatureare selected. Also, these beams are used in thissection to show the e�ciency of the new optimizationalgorithm. Among the steel section list of BritishStandards, 64 Universal Beam (UB) sections startingfrom 254 � 102 � 28 UB to 914 � 419 � 388 UB arechosen to constitute the discrete set for steel sectionsfrom which the design algorithm selects the sectionaldesignations for the castellated beams. In the designpool of holes diameters, 421 values are arranged whichvary between 180 and 600 mm with increment of 1 mm.Also, for cutting depth of hexagonal opening, 351values are considered which vary between 50 and 400mm with increment of 1 mm and cutting angle changesfrom 45 to 64. Another discrete set is arranged for the


Figure 6. Simply supported beam with 4 m span.

number of holes. Likewise, in all the design problems,the modulus of elasticity is equal to 205 kN/mm2 andGrade 50 is selected for the steel of the beam which hasthe design strength of 355 MPa [13,14]. The coe�cientsP1, P2, and P3 in the objective function are considered0.85, 0.30, and 1.00, respectively.

5.1. Castellated beam with 4 m spanA simply supported beam with 4 m span, shown inFigure 6, is selected as the �rst design example. Thebeam is subjected to 5 kN/m dead load includingits own weight. A concentrated live load of 50 kNalso acts at mid-span of the beam and the allowabledisplacement of the beam is limited to 12 mm. Thenumber of CBs is taken as 50 and the maximumnumber of iterations is considered 200.

Castellated beams with hexagonal and circularopenings are separately designed by using the newalgorithm. The optimum results obtained by CBO-PSO are given in Table 1. It is apparent from the sametable that the optimum cost for castellated beam withhexagonal hole is equal to 89.78$, which is obtainedby three methods, but the CBO-PSO algorithm givesbetter results than those of other approaches for cellu-lar beams [19-20]. Also, these results indicate that thecastellated beam with hexagonal opening has low costas compared to the cellular beam. In this problem, thelength of the span is short; hence, shear capacity is veryimportant in optimum design of this beam and it is themost e�ective factor in the design of this example.

Figure 7 shows the convergence of CBO-PSOalgorithm for design of castellated beams with di�erentopenings.

5.2. Castellated beam with 8 m spanIn the second problem, the CBO-PSO algorithm isused to design a simply supported castellated beam

Figure 7. Variation of minimum cost versus the numberof iterations for 4 m span castellated beam: (a)Castellated beam with hexagonal opening; and (b)castellated beam with circular opening.

with 8 m span. The beam carries a uniform dead load0.40 kN/m, which includes its own weight. In addition,it is subjected to two concentrated loads; dead load of70 kN and live load of 70 kN, as shown in Figure 8.The allowable displacement of the beam is limited to23 mm. The number of CBs is taken as 50. Themaximum number of iterations is considered 200.

This beam is also designed by three optimizationmethods and the optimum results are given in Ta-ble 2. In design of the beam with hexagonal hole,the corresponding costs obtained by the ECSS and

Table 1. Optimum designs of the castellated beams with 4 m span.

Method OptimumUB section

Hole diameteror cutting

depth (mm)

Total numberof holes

Cuttingangle

Minimumcost ($)

Type ofthe hole

CBO-PSO algorithm UB 305�102�25 125 14 57 89.78HexagonalCBO algorithm [20] UB 305�102�25 125 14 57 89.78

ECSS algorithm [19] UB 305�102�25 125 14 57 89.78

CBO-PSO algorithm UB 305�102�25 243 14 { 91.08CircularCBO algorithm [20] UB 305�102�25 244 14 { 91.14

ECSS algorithm [19] UB 305�102�25 248 14 { 96.32





depth (mm)


Cuttingangle

Minimumcost ($)

Type ofthe hole



CBO-PSO algorithm UB 610�229�101 487 14 { 721.55CircularCBO algorithm [20] UB 610�229�101 487 14 { 721.55





depth (mm)


Cuttingangle

Minimumcost ($)

Type ofthe hole



CBO-PSO algorithm UB 684�254�125 538 14 { 998.58CircularCBO algorithm[20] UB 684�254�125 538 14 { 997.57


Figure 8. A simply supported beam with 8 m span.

CBO are equal to 719.47$ and 718.93$, respectively,while this value is equal to 718.33$ for the CBO-PSO algorithm. As a result, the performance of theCBO-PSO method is better than other ways in thisdesign example. According to the obtained results, thedesigned beam with hexagonal opening in comparisonwith the cellular beam has low cost, and it is a moreappropriate option in this case. Also, the maximumvalue of the strength ratio is equal to 0.99 for bothhexagonal and circular beams, and it is shown thatthese constraints are dominant in the design.

Figure 9 shows the convergence history for opti-mum design of hexagonal beam, which is obtained bythree methods. As can be observed, the convergencerate of the CBO-PSO is nearly the same as that of theCBO and higher than ECSS.

5.3. Castellated beam with 9 m spanThe beam with 9 m span is considered to be thelast example of this study in order to compare theminimum cost of the castellated beams. The beam

Figure 9. Convergence history of the hexagonal beamwith 8 m span.

Figure 10. Simply supported beam with 9 m span.

caries a uniform load of 40 kN/m including its ownweight and two concentrated loads of 50 kN, as shownin Figure 10. The allowable displacement of the beamis limited to 25 mm. Similar to the two previousexamples, the number of CBs is taken as 50 and themaximum number of iterations is considered 200.

Table 3 compares the results obtained by theCBO-PSO with those of the other algorithms. In theoptimum design of castellated beam with hexagonal


Figure 11. Optimum pro�les of the castellated beamswith cellular and hexagonal openings.

hole, CBO-PSO algorithm selects 684� 254� 125 UBpro�le, 16 holes, and 230 mm for the cutting depth and56 degree for the cutting angle. The minimum cost ofdesign is equal to 990.33$. Also, in the optimum designof cellular beam, the CBO-PSO algorithm selects 684�254 � 125 UB pro�le, 14 holes, and 538 mm for theholes diameter. It is observed from Table 3 that theoptimal design has the minimum cost of 990.33$ forthe beam with hexagonal holes, which is obtained bythe CBO-PSO algorithm; however, the CBO methodgives better results for cellular beam. In cellularbeam, the maximum value of de ection of the beamis smaller than that of its upper bound. This showsthat the strength criteria are dominant in the designof this beam and they are related to the Vierendeelmechanism. Similar to the cellular beam, in castellatedbeam with hexagonal opening, the strength constraintsare dominant in the design process. The maximumratio of these criteria is equal to 0.99 for the Vierendeelmechanism.

The optimum shapes of the hexagonal and cir-cular openings are illustrated separately as shown inFigure 11.

6. Concluding remarks

In this paper, a new hybrid algorithm, called CBO-PSO, has been proposed for optimum design of castel-lated beams. This algorithm consists of hybridizationof the CBO and PSO methods and it synthesizestheir merits. For this purpose, the positive featuresof the PSO algorithm are added to the CBO. Threecastellated beams are selected from literature to designby the presented algorithm. Beams with hexagonaland circular openings are considered as web-openingsof castellated beams. Also, the cost of the beamis considered as the objective function. Comparingthe results obtained by CBO-PSO with those of otheroptimization methods demonstrates that the proposedapproach is superior to the other methods in the abilityto �nd the optimum solution. It is observed thatthe optimization results obtained by the CBO-PSOalgorithm for more design examples have low cost

in comparison to the results of the CBO and ECSSalgorithms. Also, from the results obtained in thispaper, it can be concluded that the use of the beamwith hexagonal opening can lead to the use of less steelmaterial and it is better than cellular beam from thepoint of view of the cost.

Acknowledgment

The �rst author is grateful to Iran National ScienceFoundation for the support.

References

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Biographies

Ali Kaveh was born in 1948 in Tabriz, Iran. Aftergraduation from the Department of Civil Engineeringat the University of Tabriz in 1969, he continued hisstudies on Structures at Imperial College of Science andTechnology at London University, and received his MS,DIC, and PhD degrees in 1970 and 1974, respectively.He then joined the Iran University of Science andTechnology in Tehran where he is presently Professor ofStructural Engineering. Professor Kaveh is the authorof 461 papers published in international journals and145 papers presented at international conferences. Hehas authored 23 books in Farsi and 7 books in Englishpublished by Wiley, the American Mechanical Society,Research Studies Press and Springer.

Farnoud Shokohi was born in 1988 in Naghadeh,Iran. He obtained his BS Degree in Civil Engineeringfrom Tabriz University, Iran, in 2010, and his MSDegree in Structural Engineering from Iran Universityof Science and Technology in 2012. At present, he isworking on the analysis, design, and optimization ofstructures.

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