Engineering Structures 32 (2010) 1760–1768

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

An algorithm for grouping members in a structureRichard Walls, Alex Elvin ∗School of Civil and Environmental Engineering, University of the Witwatersrand, Johannesburg, South Africa

a r t i c l e i n f o

Article history:Received 30 November 2009Received in revised form23 February 2010Accepted 25 February 2010Available online 8 April 2010

Keywords:Variable linkingCommonalityStandardizationEconomic steel designVirtual Work Optimisation

a b s t r a c t

An automated method for grouping discrete structural members is presented in this paper. The numberof groups is specified by the user, and members are grouped according to their mass per unit length. Themethod first optimizes a structure assuming that every member can have a different section. This is doneusing the Virtual Work Optimization (VWO) method, but any method can be used. The initial solutionis the lightest possible, but the number of sections required makes it impractical and uneconomical toconstruct. Next, an exhaustive search of all possible grouping permutations is carried out. The mass ofthe structure is predicted for each permutation. The permutation which results in the lightest structure isselected. The structure is passed through the optimization process one last time. The solution producedsatisfies all strength and deflection criteria. A method for reducing computational cost is proposed toaddress very large structures. Four case studies are presented to demonstrate the effectiveness of thegrouping algorithm. A stepped cantilever, a 15 storey frame, a truss and a warehouse are investigated. Asan example of results, the 195 members of the 15 storey frame are placed into 25 groups to produce astructure only 2.4% heavier than the ungrouped solution. The configuration computed is 5.9% lighter thangrouping the structure using the method found in the literature.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

This paper presents an automatedmethod for grouping discretestructural members. A group is defined as all members in a struc-ture which have the same section. Grouping is related to the prin-ciple of commonality [1]. The fewer section types a structure has,and the more similar the members are, the lower the constructioncosts become. The process of grouping elements is also known asvariable linking [2]. In a structure each time variables are linkedthe optimization problem changes, producing different solutions.It is unclear how a structure’s behaviour will change once it hasbeen grouped, making it difficult to develop generalized groupingmethods.For construction purposes engineers group members together

based on past experience, personal preferences and fabricationrequirements. This is ad hoc grouping. In complex structures itmaynot be apparent how sections should be linked to reduce materialcosts. Inexperienced designers can create poor groupings.A few grouping algorithms can be found in the literature. Krish-

namoorthy et al. [3] and Toğan and Doloğlu [4,5] have developedmethods which group members in trusses according to the mag-nitude of axial forces in members. A second method suggested by

∗ Corresponding author. Tel.: +27 11 717 7145; fax: +27 11 717 7045.E-mail address: Alex.Elvin@wits.ac.za (A. Elvin).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.02.027

Toğan and Doloğlu [5] is to group tension members together ac-cording to internal axial forces, and to group compression mem-bers according to slenderness ratios. Biedermann and Grierson [6]group beams based on member lengths; beams with spans within20% of each other are assigned a common section. Shea et al. [7]group truss members according to similar sectional areas. Barbosaand Lemonge [8] and Barbosa et al. [9] have developed methodsfor variable linking using an adaptive penalty scheme. In generalthe methods in the literature suffer from either being only suit-able for specific types of structures, such as trusses, or not tak-ing both deflection and strength requirements into account. Mostmethods cannot considermultiple load cases. Theseweaknesses ofthe grouping techniques are addressed in this paper.The algorithm presented in this paper determines how a user-

specified number of groups can be created to minimise the massof the structure. The number of groups is an independent variableand should be chosen to satisfy fabrication and constructionrequirements. Structures with fixed geometric topologies andloading conditions subject to multiple load cases are considered.The Virtual Work Optimization (VWO) method [10,11] has beenadopted in the grouping algorithm, but any optimization methodcan be used.This paper is arranged as follows: first, the theories and

limitations regarding various grouping techniques are discussed.The newmethod for groupingmembers is then presented. A simpleframe is grouped to illustrate the algorithm. Four case studies areshown to demonstrate the effectiveness of the method. A simple,stepped cantilever is considered first. A 15-storey 5-bay frame,

R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768 1761

and a truss, are considered to compare ad hoc grouping to theresults produced by the algorithm. A warehouse, as designed byprofessional engineers, is investigated and the results compared.

2. Limitations of grouping methods found in the literature

One aim of a good design is to satisfy strength and deflectionconstraints whilst being as economical as possible. To standardizedesigns and reduce fabrication and erection costs members haveto be grouped together. It is necessary to determine whichparameters should be used as a basis for specifying groups. Eitherthe geometric properties of members, or stresses induced by loads,have been considered. Specific properties which have been usedinclude: axial forces in members [3–5,7], or member lengths [6].Other parameters which could be considered, but have not beenexplored in the literature, include the second moment of areas,locations of members within the structure, stresses, or memberenergies per unit volume.A major weakness of grouping members according to internal

stresses, forces or energies is that in general only a single load casecan be considered at a time. For grouping according to internalforces,membersmust have only one dominant type of force: eitheraxial, bending, or torsion. It is difficult to combine multiple forcesfor grouping members. Further, a strength dependent membermay have its section governed by a combination of internalforces, while a deflection dependent member’s size is not onlygovernedby the load it carries. Compressionmembers and laterallyunsupported beams require extra factors to take buckling intoaccount.When members are grouped together based on their length

then geometric properties, forces inmembers, stress requirementsand deflection criteria might not be accounted for. A member’slength does not adequately represent its geometric properties.If members are grouped together based on second moments

of area, then implicitly, only bending forces are considered. Thesame limitations as using cross-sectional areas are encountered,as discussed above. There is a large variation in second moment ofarea in section databasesmaking it difficult to group sections basedon this parameter alone.

3. Grouping members according to mass per unit length

It is proposed that members should be grouped according totheir mass per unit length, i.e. their cross-sectional area.1 For astructure in which all design constraints have been satisfied it isassumed that members with a similar mass per unit length havecomparable section properties. Grouping members, which havebeen selected to satisfy all design criteria, according to sectionproperties solves the problems associatedwithmultiple load casesand strength requirements. It is important to note that whenoptimizing structures for weight, the mass per unit length ofmembers serves as part of the objective function.

4. Single and multi step grouping

It is possible to group members in either a single or multiplesteps. For a multi-step process the number of sections used in

1 Please note that the grouping algorithm presented here is very different tothat of Shea et al. [7] who also used cross-sectional areas as the basis of grouping.This reference considered only trusses with members grouped according to pre-specified ranges. The proposed algorithm is more general and does not have theselimitations.

the structure is reduced by one in each iteration, until the user-defined number of groups has been produced. Groups that arecreated are linked either with other members or groups. Theproblem encountered is that in one iteration, it may be optimalto group certain members together, but in a later iteration sucha group may need to be split to create a different, but moreeffective, configuration. It was found that a single step procedureis less computationally expensive, more effective and easier toimplement. For these reasons the algorithm presented is based ona single step grouping method.

5. The single step grouping algorithm

The aim of the presented algorithm is to determine a groupingconfiguration which will result in the lightest structure. Anoverview of the grouping process is: first, an ungrouped structureis optimized to produce an initial solution. Second, all possiblegrouping configurations are investigated. The lightest, predictedconfiguration is chosen for the structure. The structure is thenoptimized again to satisfy all design criteria and produce thesolution.Step 0—Setting grouping parametersThe following information is required for the grouping algo-

rithm: the structure’s geometric topology, loading, load combina-tions, deflection requirements, design code and the properties ofthe materials to be used. The user must define howmany differentgroups, n, need to be created. The method will group the membersin the structures such that the maximum number of groups is lim-ited to n.Step 1—Obtaining the initial, ungrouped solutionIf a structure in which every member can have a different

section is optimized, the lightest solution is produced. The aim ofthe grouping method is to create a configuration that weighs asclose to the ungrouped solution as possible.The Virtual Work Optimization (VWO) method [10,11] is used

to obtain the initial, ungrouped solution. All the load cases actingon the structure have to be input, including loads that canchange direction. The VWO method is based on the principle ofvirtual work, and selects members to satisfy both strength anddeflection criteria to produce the lightest structure. Sections arechosen from standard databases by determining which sectionsprovide the highest deformation and strength resistance per unitmass. The VWO method is chosen because it requires feweriterations than other methods, and is influenced linearly by thenumber of optimization variables. It must be emphasised thatany optimization method can be incorporated into the groupingalgorithm.Once sections have been selected for the ungrouped structure

they are ordered from largest to smallest according to their massper unit length. Members with identical sections are groupedtogether. This configuration is still referred to as the ungroupedstructure, or the initial grouping configuration. The total numberof different sections, i, selected for the ungrouped solution is lessthan or equal to the number of members in the structure.Step 2—Investigating grouping configurationsAn exhaustive search is performed on the ungrouped structure

by computing all possible member groupings. For each permuta-tion the mass of the structure is calculated. The assumption in thisstep is that the section of a heaviermemberwill satisfy the strengthanddeflection constraints of a lightermember. Thus, in any permu-tation a member cannot have its section size reduced from the oneinitially selected in Step 1. Also, the largest section of all the mem-bers in a group will be selected for each member in that group.A structure with i initial sections will be reduced to the user-

defined number of groups, n, where 1 ≤ n ≤ i. Thus, the number

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Table 1Possible grouping configurations for creating n groups from i sections.

of sectionsmust be reduced by (i−n). The total number of groupingpermutations, N , is defined by the binomial coefficient:

N =(i− 1n− 1

)=

(i− 1)!(n− 1)!(i− n)!

. (1)

The permutation process is illustrated in Table 1. The memberswith the sections listed on the left are placed into groups numberedon the right of the table (the un-shaded region). In the tablem1 >m2 > · · · > mn > · · · > mi, where m denotes the mass per unitlength of a section. Members are distributed progressively intoeach group until all permutations have been investigated.In the first permutation, k = 1, all members retain their initial,

ungrouped section size, except for the last i− n sections which areincorporated into the nth group. For the second permutation, k =2, section number n is incorporated into group n − 1 rather thangroup n. This process continues until the size of group n reaches1, at permutation k = i − n + 1. Then the size of group n − 2increases by 1, and groups n and n − 1 move one lower than theywere in permutation k = 1. This process of regrouping progressesuntil permutation k = N , where the additional (i− n) sections areincorporated into the 1st group. As a numerical example considerhow 7 sections can be placed into 3 groups, creating 15 groupingpermutations (i = 7, n = 3, and N = 15), as shown in Table 2.The total predicted mass of the structure, Mk, for each

permutation, k, is given by:

Mk =J∑j=1

Ljmj,new (2)

where J is the number of members in the structure and j is themember index. Lj is the length of each member. The newmass perunit length ismj,new. The newmass of each member is taken as themass of the largest section in its group.Step 3—Selecting a new grouping configurationThemember grouping selected out of all the permutations is the

one that produces the minimummass,Mmin,, where:

Mmin = min {M1,M2, . . . ,MN} (3)

Mmin is the estimatedmass of the grouped structure.Step 4—Ensuring design constraints are satisfiedThe lightest grouped structure obtained in Step 3 may violate

strength and/or design deflection criteria. This might occurbecause of the redistribution of forces resulting from changingmembers in indeterminate structures. Alternatively, a groupedstructure may be over-designed because of the increase in sectionsize of many members. The latter situation occurs more often.Thus, it is necessary to optimize the grouped structure once again.Although any method can be used, the VWO method is employedto give the final grouped solution.The difference in the estimated and final masses will depend

on numerous factors. In statically determinate structures, withstrength dependent members, the estimation will be accurate.In statically indeterminate structures, which are predominantlydeflection dependent, the estimated mass is usually inaccurate,and probably an over-estimate. However, the estimation providesan effective method for determining groups, and not the finalmembers sizes.

6. Using multiple section types—a further constraint

In most structures a further constraint can be imposed byselecting the type of section to be used for eachmember (I-section,angle, channel). These sections must be grouped separately, andare treated as subgroups. The user must specify the number ofgroups to be created for each type of section: n1, n2, . . . , nα whereα is the number of different types of sections in the structure. Thenumber of sections of each type in the initial ungrouped structureis i1, i2, . . . , iα . The total number of sections in the ungroupedstructure, i, is the summation of i1, i2, . . . , iα . The number ofpermutations to be investigated for each section type is calculatedusing Eq. (1), with the values of il and nl of each type, where l

R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768 1763

Table 2The possible permutations for creating 3 groups from 7 members.

Table 3Mass and lengths of members for the ungrouped, optimized structure shown inFig. 1.

Section number Length (m) Ungrouped member Ungroupedmass(kg/m)

1 3 W16× 26 40.72 5 W14× 22 33.33 5 W8× 18 274 3 W8× 13 19.85 3 W6× 12 18.26 3 W10× 12 18.2

Mass (kg) 592.2

is the section type index of each subgroup. The total number ofpermutations to be investigated is:

N =α∑l=1

Nl. (4)

The predicted mass of the structure is the summation of theminimummass permutation of each section type:

Mmin =α∑l=1

Mmin,l. (5)

By considering different section types as subgroups the numberof permutations to be investigated is limited. Please note thatmembers having different section types, but the same mass perunit length, cannot be grouped together because of the possiblelarge variation in geometric properties.

7. Illustrative example

To illustrate the grouping method, the two-storey, 6 memberframe (i = 6), shown in Fig. 1, will have 3 groups (n = 3) created.The loading is as shown and is not symmetrical. The structuremustsatisfy the SouthAfrican steel code requirements, SANS 10162 [12],using grade 350 W steel and AISC sections [13]. Inter-storey driftis limited to L/300 (10 mm). The VWO method calculates theungrouped structure to have the mass per unit length shown inTable 3, and depicted in Fig. 2. In Fig. 2 the thickness of the lineis proportional to the mass per unit length of the member. Fig. 2provides a graphical representation of themass distribution whichis used to assign member groups. The total mass of the ungroupedstructure is 592.2 kg.There are 10 possible grouping permutations for the structure

(1). Table 4 shows how the members are placed into different

5m

3m3m

Fig. 1. Two-storey frame to be grouped.

Fig. 2. Mass distribution in the two-storey ungrouped frame.

groups for eachpermutation. For each configuration the three extrasections (i− n = 3) are included progressively in different groups.The lightest grouped structure obtained is estimated to be

633 kg (bold entries in Table 4).Members 2 and 3, aswell as 4, 5 and6 are grouped together. When the grouped structure is optimizedusing the VWO method the new mass obtained is 618.9 kg. This is4.5% heavier than the ungrouped structure, but contains 50% fewersections. The predicted mass is 2.3% greater than the optimizedmass.

1764 R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768

Table 4Possible grouping configurations for the 2 storey frame and their mass estimates.

8. Optimization considerations

In symmetric structures with symmetric loading, two optionsare possible to obtain the optimized member selection. Either (a)all load cases must be applied and considered separately, or (b)symmetric members can be constrained to have the same sections,and separate symmetric load cases need not be considered. Ithas been found that linking symmetric members produces moreconsistent results with lower computational costs. Symmetricmembers in case studies 2–4 have been constrained to be the same.In un-symmetric structures (e.g. case study 1), or symmetric

structureswith un-symmetric loading (example in Fig. 1), the VWOmethod produces un-symmetric solutions. The grouped structurewill thus also be un-symmetric (see Figs. 1 and 2).The number of sections in a database will influence the initial

solution’s number of sections, i, that have to be grouped together.The larger the database, the closer i will be to the numberof members in the structure. If databases are small, numerousmembers may have the same section after the initial optimizationprocess, and will be pre-grouped together (see Step 1). To preventthis, it is recommended that a large database is used in theinitial selection process to minimize any initial grouping. Forconstructability only the available section database can then beused in Step 4.Members with the same section type, but different require-

ments, can be isolated and grouped separately. This creates addi-tional subgroups,which are addressed in the samemanner as usingdifferent section types. An example of such a requirement is spec-ifying that the chords of a truss must not be grouped with bracingor diagonal members (see case study 3).Linking existing groups is possible, and performed in an

identical manner to linking individual members. In Step 1 thenumber of sections, i, is set to the number of existing groups.

9. Reducing computational costs

Large search spaces are rare because of the size and natureof existing section databases. It is unusual to find more than20 different sections of each type in an optimized, ungroupedstructure; this produces less than 100,000 permutations for eachsection type. To give an indication as to cost, the computationof 106 permutations takes approximately 1.1 s on a PC runningat 2.1 GHz clock speed. In the case studies it was not necessaryto reduce the search spaces. However, if large or continuous,synthetic section databases had been used to obtain the ungroupedsolutions, it would have been essential to decrease computertime. If search spaces become too large, two ways to reduce

computational costs are proposed: (a) creating subgroups, and (b)investigating permutations only within a viable ‘radius’.Creating subgroups introduces extra constraints but reduces

computational cost. For example if 80 sections have to be placedinto 10 groups there would be 2.06 × 1011 permutations (1).However, creating 2 subgroups of 40 members and placing theminto 5 groups each would only result in 1.64× 105 permutations.Reducing computational cost by performing a radius search

is based on the following observation: the lightest and heaviestmembers in a group are separated by only a few section sizes foundin the initial solution. Permutations can thus be performed only auser defined radius, X , away from any one entry.The two limits of the radius X , are: (a) i − n, and (b) the larger

of 1 and i/n.2 If X is equal to i− n (or larger) then all permutationsare performed (1). If X is less than i − n then the number ofpermutations to be performed reduces. Fig. 3 plots the number ofpermutations versus the number of initial sections for 10 groupswith various radii X . Fig. 3 shows that the number of permutationsdecreases rapidly as the radius, X , decreases. However, if X is settoo low it is possible that the optimal solution may be missed.When grouping a large structure it may be necessary to test forthe convergence of solutions by investigating several values of X .More research is required to understand how to choose X .Consider the example of 80 sections placed into 10 groups. If

sections are not allowed to increase by more than 10 section sizes,the search space reduces from 2.06× 1011 to 1.29× 106.

10. Advantages of the algorithm

The grouping algorithm proposed is straightforward to imple-ment and canbeused for any structure. Themethod can group indi-vidual members or existing groups. Multiple internal forces arisingfrom different load cases are considered. Strength and deflectioncriteria are satisfied by the optimization method.The method is computationally inexpensive, even though large

numbers of configurations are investigated. Structures are notanalysed for each permutation, rather the algorithm predicts thestructure’s masses. Almost the entire computational cost is spentoptimizing the structure in Steps 1 and 4. However, if necessary,the grouping computational cost of Step 2 can be decreased asexplained above.

2 The uninteresting case of X = 0 produces no permutations and the structureremains ungrouped.

R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768 1765

Fig. 3. Comparison of the number of initial sections to the number of configurations to be investigated for fixed values of i and n.

Fig. 4. Stepped cantilever beam [14].

11. Limitations of the method

The assumption that all section properties can be representedby the mass per unit length is an oversimplification. Large, non-linear variations in sectional properties relative to cross-sectionalareas may cause members to be grouped incorrectly. These points,and how they interact, require further research.In the method presented, each member is assumed to have

a constant cross section along its length. The effect of tailoringmembers to have continuous varying sections is not considered.

12. Case studies

Various aspects of the automated grouping algorithm aredemonstrated by considering four case studies. First, the steppedcantilever illustrates how masses increase when decreasing thenumber of groups. The 15-storey 5-bay frame and truss demon-strates how the grouping algorithmproduces lighter solutions thanad hoc grouping. Finally, the results of the grouping algorithm arecompared with a warehouse designed by professional engineers.In all the case studies the structures are steel with a density of

7850 kg/m3.

12.1. Stepped cantilever

The cantilever shown in Fig. 4 was optimized by Thanedar andVanderplaats [14] and Walls and Elvin [11] with the followingconstraints. The tip of the cantilever is restricted to deflect amaximum of 2.7 cm. The section of each member is rectangularand the maximum height, H , to breadth, B, ratio is limited to20. Section dimensions must be integer centimeter values. Themaximum allowable stress is 140 MPa.First, the ungrouped structure was optimized with the VWO

method to produce a solution of 531.3 kg. The developed algorithmwas applied to the stepped cantilever; 4 to 1 groupswere specified.

Table 5Final masses for various grouping configurations of the cantilever.

No. ofgroups

Final mass(kg)

% Mass increase from 5groups

Members grouped

5 534.6 – –4 534.6 0.0 1–23 555.8 3.9 1–2, 3–42 602.9 12.8 1–2–3, 4–51 706.5 32.2 1–2–3–4–5

Table 6Final masses and section lengths for the cantilever. The structure was split into 25members and regrouped.

No. ofgroups

Finalmass (kg)

% Mass savingcompared to thesame no. of groups inTable 5

Lengths grouped

100 511.8 – Each member 0.05 m long5 521.0 2.5 0–2 m, 2–2.95 m,

2.95–3.55 m, 3.55–4.45 m,4.45–5 m

4 533.9 0.1 0–2 m, 2–2.95 m,2.95–3.55 m, 3.55–5 m

3 549.3 1.2 0–2.4 m, 2.4–3.55 m,3.55–5 m

2 592.7 1.7 0–3.55 m, 3.55–5 m

The results are summarized in Table 5. As expected, as the numberof sections decrease so the structure’s mass increases.The member lengths specified in Fig. 4 by Thanedar and Van-

derplaats [14] introduce extra constraints. Lighter solutions can befound if the cantilever has more steps. To demonstrate this, thecantilever is discretized into 100 equal lengths, and then linkedto form from 2 to 5 new groups. The results are summarized inTable 6. Comparing the solutions for the two levels of discretiza-tion, shows that the finer discretization produces lighter can-tilevers for all levels of grouping.

12.2. 15 storey 5 bay frame

The 15 storey 5 bay frame, shown in Fig. 5, has been includedto compare the ad hoc grouping method found in the literatureto configurations computed by the algorithm. The structure issubject to both strength and deflection constraints. Membersmustsatisfy the South African steel code, SANS 10162 [12], using grade350W steel. Interstorey drift is limited to 9mm. Standard AISC I, Hand angle sections are chosen for the beams, columns and bracesrespectively [13].

1766 R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768

5 Bays @ 6.0m = 30.0m

15 s

tore

ys @

3.5

0m =

52.

50m

Fig. 5. 20 storey 5 bay frame case study.

The following members in the frame are grouped together (a)all the beams in three consecutive stories, and (b) symmetriccolumns over 3 stories. This grouping was used by Camp [15],while a similar grouping was performed by Chan [16]. This ad hocmethod produces a structure with an initial, assumed grouping of5 I-sections, 15 H-sections and 5 angles.Optimizing this ad hoc grouping using the VWO method

produces a structure with a mass of 32,954 kg. If groupingconstraints are removed, except for symmetry, a structure of30,371 kg is obtained (Step 1). This structure is then grouped tohave the same number of I-sections (15), H-sections (5) and angles(5) as the ad hoc grouping. The grouping algorithm solution is31,104 kg. Results are summarized in Table 7.Fig. 6 shows the final, optimized section selection for the

structure with ad hoc grouping. Beams with the same thicknessand shade of grey have been grouped together. The thickness ofthe line is proportional to the mass per unit length of the member.Fig. 7 shows the grouping calculated by the algorithm.The grouping algorithm produces a 5.9% lighter solution than

the structure with the ad hoc grouping. Comparing Figs. 6 and7 shows that the ad hoc and algorithm’s groupings and massdistributions are different. The distribution of mass is sufficientlyuniform to allow the algorithm grouped structure to be fabricated.

12.3. Truss

The truss shown in Fig. 8 has to be designed to satisfy service-ability and ultimate limit state criteria. Groups have been defined(a) in two ad hoc ways, and (b) using the automated groupingalgorithm. The maximum serviceability deflection is span/400 atthe mid-span. Angles (from BS4:Part 1 [17]) must be used for allmembers. Strength requirements must satisfy SANS 10162 [12]using grade 350 W steel.The number of groups in the structure is limited to 4:2 groups

for the chords, and 2 groups for the vertical and diagonalmembers.Two ad hoc groupings are defined. Ad hoc grouping 1 consistsof: setting the same member for the top and bottom chords inthe middle 8 bays, a separate section for the outer 4 bays, the 4verticals at each support are linked together, and the remainingmembers are grouped. Ad hoc grouping 2 consists of: the topchord, bottom chord, vertical members, and diagonal memberseach have a separate group. The optimized mass distributions of

Fig. 6. Optimized 15 storey structurewith groups across 3 floors (ad hoc grouping).

Fig. 7. Optimized 15 storey frame with groups computed by the developedalgorithm.

these grouping configurations are shown in Figs. 9 and 10. Thestructures’ final masses are 809.5 kg and 788.2 kg for ad hocgrouping 1 and 2 respectively.The 4 group requirement specified above forms the input to the

grouping algorithm. The ungrouped truss is optimized to createa structure of 660.2 kg (Step 1). The grouping calculated by thealgorithm is shown in Fig. 11. The optimized mass is 765.2 kg(Step 4). Table 8 summarizes the results obtained for the variousgrouping configurations.The grouping algorithm produces a structure 5.8% lighter than

ad hoc grouping 1 and 3% lighter than ad hoc grouping 2. Pleasenote that the algorithm has stiffened the mid-span to limitdeflections. Further, the algorithm has grouped the largest verticalsections in the end bays to resist the higher compressive forcesfound there.

R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768 1767

Table 7Results for the 15 storey frame.

Configuration Final mass (kg) % Greater than ungrouped Max no. of allowable groups:Beams Columns Braces

Ungrouped—symmetrical members the same 30371 – ∞ ∞ ∞

Single step grouping 31104 2.4 5 15 5Ad hoc grouping across 3 floors 32954 8.5 5 15 5

Fig. 8. Truss—geometry and loading.

Fig. 9. Ad hoc # 1—mass distribution. Optimized mass: 809.5 kg.

Fig. 10. Ad hoc # 2—mass distribution. Optimized mass: 788.2 kg.

Fig. 11. Algorithm grouping. Optimized mass: 765.2 kg.

Fig. 12. Warehouse with dead, live, crane and wind loads.

Table 8Results for the optimized the truss.

Configuration Mass (kg) % Greater than ungrouped

Ungrouped 660.2 –Ad hoc grouping # 1 809.5 22.6Ad hoc grouping # 2 788.2 19.4Algorithm grouping 765.2 15.9

12.4. Warehouse

The warehouse shown in Fig. 12 was designed by a SouthAfrican company of professional engineers. The simplified loadingis shown. Seven load combinations accounting for dead, live, craneand wind loads are considered. Fourteen deflection criteria are

imposed. The structure is to consist of I, H, channel and anglesection types (from BS4: Part 1 [17]). Sections are required tosatisfy SANS 10162 [12] strength requirements using grade 300Wsteel. Lateral buckling of latticed columns is taken into account.Initially the engineers designed this structure to have 24 groups.

Using their group configuration the VWO method produced a3709.2 kg structure. To decrease the number of sections further,the 24 groups were then placed into 17 groups by the engineers.The new optimized structure has a mass of 3777.5 kg. Thesegroupings were defined by the engineers based on experience, andare thus ad hoc.The algorithm was applied to the structure with the 24 pre-

selected groups in order to reduce the number to 17 groups. Themass calculated is 3759.5 kg, or 0.5% lighter than the engineers’solution. When the ungrouped structure is optimized it has a

1768 R. Walls, A. Elvin / Engineering Structures 32 (2010) 1760–1768

Fig. 13. Warehouse with final grouping specified by the professional structural engineers.

Fig. 14. Warehouse with final grouping computed by the algorithm.

Table 9Results for the warehouse.

Configuration Optimized mass(kg)

%Saving

Max. no.of sections

Engineers—final 17 groups 3777.5 – 17Engineers—initial 24 groups 3709.2 1.8 24Ungrouped 3088.2 18.2 ∞

Algorithm—17 new groupsfrom the engineers’ 24sections

3759.5 0.5 17

Algorithm—17 new groupsfrom the ungroupedconfiguration

3605.1 4.6 17

mass of 3088.2 kg (Step 1). If 17 groups are now produced,the algorithm calculates a 3605.1 kg structure, which is 4.6%lighter than the engineers’ design. This shows that the algorithm’ssolution is dependent on the starting configuration, i.e. startingwith an ungrouped versus a pre-grouped structure. The results aresummarized in Table 9.Fig. 13 shows the mass distribution in the warehouse with the

17 groups defined by the engineers. Fig. 14 shows the warehousewith 17 groups computed by the algorithm, starting from the un-grouped configuration. It is interesting to note that the algorithmhas optimized the lattice columns by stiffening their lower por-tions. It has also grouped the chords of the roof trusses atmid-span.

13. Conclusion

Structural grouping is a complex task where solutions changewith every perturbation in the system. This paper presentedan automated algorithm for optimizing the grouping of discretestructural members. The algorithm groups members based ontheir mass per unit length. An exhaustive search of groupingpermutations is carried out and the grouping which producesthe lightest structure is selected. Any two-dimensional structurein which members carry axial and/or bending forces can beanalyzed. Multiple load cases, including loads that can reversedirection, are considered. The algorithm’s solution is a groupedstructure optimized for weight, which satisfies multiple strengthand deflection requirements.The algorithm is computationally inexpensive. Although the

number of permutations can be large, for each trial groupingonly the structure’s mass is estimated, the structure’s behaviouris not solved. If the search space is required to be reduced,as might happen for extremely large structures with membersselected from a large database, two methods were proposed: (a)creating subgroups, and (b) only investigating permutationswithina radius.Four cases studies were investigated to compare the algorithm

to ad hoc grouping configurations. In all cases lighter structures

were computed by the algorithm. As expected, the algorithmsolution is affected by the starting amount of pre-grouping, andwhether the problem is symmetrical.The following topics require further research. The uniqueness

of the solution obtained must be investigated. A multi-stepalgorithm should be developed further, and compared to thesingle stepmethod presented. The effects of using different sectiondatabases for the initial and final optimization steps must becharacterized. The algorithm’s framework is suitable for three-dimensional structures, but this needs to be implemented and theperformance investigated.

Acknowledgements

This research was supported in part by the Southern AfricanInstitute of Steel Construction (SAISC). The authors would like tothank Mr. Spencer Erling of SAISC for his helpful comments andsuggestions.

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