TEACHING-LEARNING-BASED OPTIMIZATION ALGORITHM FOR SHAPE AND SIZE OPTIMIZATION OF TRUSS STRUCTURES WITH DYNAMIC FREQUENCY CONSTRAINTS *

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1 IJST, Transactons of Cvl Engneerng, Vol. 37, No. C +, pp Prnted n The Islamc Republc of Iran, 2013 Shraz Unversty TEACHING-LEARNING-BASED OPTIMIZATION ALGORITHM FOR SHAPE AND SIZE OPTIMIZATION OF TRUSS STRUCTURES WITH DYNAMIC FREQUENCY CONSTRAINTS * A. BAGHLANI 1** AND M. H. MAKIABADI 2 1 Faculty of Cvl and Envronmental Engneerng, Shraz Unversty of Technology, Shraz, I. R. of Iran Emal: baghlan@sutech.ac.r 2 Faculty of Cvl Engneerng, Shraz Unversty of Technology, I. R. of Shraz, Iran Abstract The complcated problem of truss shape and sze optmzaton wth multple frequency constrants s nvestgated n ths paper. A recently developed metaheurstcs called teachnglearnng-based optmzaton (TLBO) algorthm s used for the frst tme to solve ths knd of problem. Contrary to other metaheurstcs, the procedure of TLBO s smple to mplement snce no tunng parameters need to be adjusted. Analyses of structures are performed by a fnte element code n MATLAB whch s used n conjuncton wth an optmzaton code based on TLBO. Varous benchmark problems are solved wth ths technque and the results are compared wth those found by other methods ncludng metaheurstcs such as PSO, and. In all test cases, the results show that TLBO leads to very satsfactory results.e. lghter structures whch satsfy all frequency constrants. The results of ths ndcate excellent nherent capacty of the approach n dealng wth complcated dynamc non-lnear optmzaton problems. Keywords Truss structures, non-lnear dynamc optmzaton, frequency constrants, teachng-learnng-based optmzaton (TLBO) 1. INTRODUCTION Truss optmzaton wth frequency constrants s a hghly non-lnear dynamc optmzaton problem. In spte of dffcultes n addressng ths type of problem, consderable progress has been acheved n soluton methods where the geometry of the structure s prescrbed and cross-sectonal areas have to be optmzed,.e. sze optmzaton of trusses [1-4]. However, t s well known that the structural shape has a great nfluence on the sze of elements. Hence, takng both shape and szng optmzaton of trusses wth frequency constrants nto account ncreases the complexty of the problem extensvely. Most of the dffcultes n shape and szng optmzaton of trusses smultaneously wth multple frequency constrants, are attrbutable to hgh non-lnearty of the problem wth respect to desgn varables and entrely dfferent physcal representaton of shape and szng varables, snce ther values are dfferent orders of magntude. Thus, couplng these varables sometmes leads to ll-condtonng problems and dvergence [5-7]. In spte of ts dffculty, from an engneerng applcaton pont of vew, ths type of problem s very useful. As has been mentoned by Grandh [8], the problem s advantageous for structural desgners n manpulatng the selected frequency to mprove the performance of the structure under dynamc exctaton. In other words, the desgner can control the selected frequences n a desred manner n order to mprove the dynamc characterstcs of the structure. Because of ths mportant applcaton n correct desgn of structures under dynamc loads, varous efforts have been made n the lterature to deal wth ths knd of problem [1, 3, 5, 7, 9, 10]. Receved by the edtors November 11, 2012; Accepted February 5, Correspondng author

2 410 A. Baghlan and M. H. Makabad Due to the aforementoned obstacles n optmzaton of shape and szng of structures wth multple frequency constrants, the choce of the soluton method s of great mportance and local search algorthms are not approprate. As ndcated by, tradtonal optmzaton methods based on gradents also have dffcultes n problems wth repeated egenvalues and may trap nto local optma. As powerful alternatve methods, modern metaheurstc algorthms, whch are not based on gradents, can be successfully employed to solve the problem. They are also capable of solvng hghly non-lnear problems wth complex objectve functons. Due to ther effectveness n dealng wth real-lfe complcated problems, metaheurstcs are frequently proposed. For example, among the most recently developed metaheurstcs, Water Cycle Algorthm (WCA) [12], Mne Blast Algorthm (MBA) [13], Cuckoo Optmzaton Algorthm (COA) [14] can be mentoned. Some hybrd optmzaton algorthms have also been presented whch make use of two or several metaheurstcs and local search methods n order to mprove ther performance n dealng wth trusses and other structures [15, 16]. However, use of metaheurstc optmzaton methods n optmzaton of shape and sze of trusses wth multple frequency constrants have receved lttle attenton n the lterature. Recently, mplemented Partcle Swarm Optmzaton (PSO) algorthm to optmze the shape and sze of truss structures wth multple frequency constrants. Later, utlzed two other metaheurstcs,.e. Harmony Search () method and Frefly Algorthm () to solve ths knd of problem. A combnaton of the Charged System Search (CSS) and the Bg Bang-Bg Crunch (BBC) algorthms has been recently employed by Kaveh and Zolghadr [18] for truss optmzaton wth natural frequency constrants. In ths paper, a recently developed metaheurstc, called teachng-learnng-based optmzaton (TLBO) algorthm s used for the frst tme to solve the problem of truss shape and sze optmzaton wth multple frequency constrants. TLBO has some nherent capabltes and advantages compared to other metaheurstc approaches. It s reported that t outperforms most metaheurstcs regardng constraned benchmark functons, constraned mechancal desgn, and contnuous non-lnear numercal optmzaton problems [19]. However, the effectveness of TLBO n shape and sze optmzaton of truss structures wth multple frequency constrants has not been nvestgated up to now. In ths paper, the effectveness of TLBO n solvng the complex problem of shape and sze optmzaton of truss structures wth multple frequency constrants s nvestgated through optmzng shape and sze of some benchmark trusses. The results are compared wth those found by other methods n the lterature, ncludng other metaheurstcs such as harmony search (), partcle swarm optmzaton (PSO) and frefly algorthm (). The results show that TLBO results n lghter structures whch satsfy all frequency constrants, ndcatng outstandng capabltes of ths modern approach. The results of applyng TLBO n shape and sze optmzaton of truss structures wth frequency constrants show the excellent capacty of the method n dealng wth complcated structural engneerng problems. 2. PROBLEM FORMULATION In dealng wth truss shape and sze optmzaton problems, the truss topology s prescrbed and t s assumed to be unchanged durng the optmzaton procedure. However, the cross-sectonal areas of elements and nodal coordnates are consdered as desgn varables whch should be optmzed. The natural frequences are consdered as desgn constrants to avod resonance wth the external exctatons. The weght of the structure should be mnmzed subject to some prescrbed constrants. The problem can be mathematcally represented as follows: Mnmze: n W L A (1) e1 e e e IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

3 Teachng-learnng-based optmzaton algorthm for 411 Subject to, 1,...,q (2) * 1, q 1,...,q (3) * 1, j 1,..., k (4) low j up where W s the total weght of structure, L e, e and A e are length, materal densty, and cross sectonal area of the eth element, respectvely. Total number of elements s denoted by n and the number of ndependent desgn varables s denoted by k. Frequency constrant (2) represents that some natural frequences, numberng q 1, should exceed the prescrbed lower lmts. Frequency constrant (3) represents that other natural frequences, numberng q q1, should be less than the prescrbed upper lmts. Inequalty (4) ndcates that the desgn varables j, ncludng ether a shape or szng varable must take a value between ts lower bound low and upper bound up, respectvely. It s worthy to note that the natural frequency s hghly non-lnear and mplct wth respect to desgn varables. 3. TEACHING-LEARNING-BASED- OPTIMIZATION (TLBO) ALGORITHM One of the most recently developed metaheurstcs s teachng-learnng-based- optmzaton (TLBO) algorthm [20]. TLBO has many smlartes to evolutonary algorthms (EAs): an ntal populaton s randomly selected, movng on the way to the teacher and classmates s comparable to mutaton operator n EA, and selecton s based on comparng two solutons n whch the better one always survves [19]. Smlar to most other evolutonary optmzaton methods, TLBO s a populaton-based algorthm nspred by learnng process n a classroom. The searchng process conssts of two phases,.e. Teacher Phase and Learner Phase. In teacher phase, learners frst get knowledge from a teacher and then from classmates n learner phase. In the entre populaton, the best soluton s consdered as the teacher (X teacher ). On the other hand, learners learn from the teacher n the teacher phase. In ths phase, the teacher tres to enhance the results of other ndvduals (X ) by ncreasng the mean result of the classroom (X mean ) towards hs/her poston X teacher. In order to mantan stochastc features of the search, two randomly-generated parameters r and T F are appled n update formula for the soluton X as: X X r.( X teacher T F.X mean ) (5) where r s a randomly selected number n the range of 0 and 1 and T F s a teachng factor whch can be ether 1 or 2: TF round 1 rand(0,1) 21 (6) Moreover, X and X are the and exstng soluton of, [20-21]. In the second phase,.e. the learner phase, the learners attempt to ncrease ther nformaton by nteractng wth others. Therefore, an ndvdual learns knowledge f the other ndvduals have more knowledge than hm/her. Throughout ths phase, the student X nteracts randomly wth another student X j ( j ) n order to mprove hs/her knowledge. In the case that X j s better than X (.e. f (X ) f(x ) for mnmzaton problems), X s moved toward X j. Otherwse t s moved away from X j : j X X r.( X j X ) f f ( X ) f ( X j ) (7) December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +

4 412 A. Baghlan and M. H. Makabad X X r.( X X j ) f f ( X ) f ( X j ) (8) If the soluton X s better, t s accepted n the populaton. The algorthm wll contnue untl the termnaton condton s met. The pseudo code shown n Table 1 demonstrates the TLBO algorthm stepby-step. Table 1. The pseudo code for TLBO Set k 1 ; Objectve functon f ( X), X ( x1, x2,..., x ) T d d=no. of desgn varables Generate ntal students of the classroom randomly X, 1,2,..., n n=no. of students Calculate objectve functon f ( X ) for whole students of the classroom WHILE (the termnaton condtons are not met) Teacher Phase X Mean Calculate the mean of each desgn varable Identfy the best soluton (teacher) FOR 1 n Calculate teachng factor TF round 1 rand(0,1) 21 Modfy soluton based on best soluton(teacher) X (0,1) ( X rand X teacher TF X mean) Calculate objectve functon for mapped student f ( X ) IF X sbetterthan X, e. f ( X ) f ( X ) X X END IF End of Teacher Phase Student Phase j Randomly select another learner X, such that j IF X sbetterthan X j, e. f ( X ) f ( X j ) j X X rand(0,1) X X Else j X X rand(0,1) X X END IF IF X sbetterthan X, e. f ( X ) f ( X ) X X END IF End of Student Phase END FOR Set k k 1 END WHILE Postprocess results and vsualzaton 4. BENCHMARK DESIGN EXAMPLES To nvestgate the effectveness of TLBO algorthm n shape and sze optmzaton of truss structures wth frequency constrants, four benchmark problems have been solved. A fnte element code n MATLAB s developed for analyss of structures whch s used wth an optmzaton code based on TLBO. The results are compared wth the results found by other researchers and those found by other metaheurstc approaches. Standard penalty functon method has been used to handle the frequency constrants. a) 10 bar plane truss For the frst example, the 10 bar planar truss shown n Fg. 1 s consdered. Ths truss has been already nvestgated by Wang et al. [7], Grandh [8] usng evolutonary node shft methods, et al. [5] usng Nche Hybrd Genetc Algorthm (NHGA), et al. [22] based on parallel genetc algorthm, usng PSO, Zuo et al. [23] usng adaptve egenvalue re-analyss methods, and IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

5 Teachng-learnng-based optmzaton algorthm for 413 usng both harmony search () and frefly algorthm () methods. Therefore, the results of ths can be compared wth varous approaches. Fg bar truss structure wth added masses 7 The structure s made of alumnum wth modulus of elastcty E Gpa (10 ps) and 3 3 materal densty kg m (0.1 lb n ). The lower bound of cross sectonal area s m (0.1 n ) for all elements. A nonstructural mass of kg (1000 lb ) s attached at each of the four free nodes. The frequency constrants are as follows: 1 7Hz, 2 15 Hz and 3 20 Hz. The optmal solutons of the cross-sectonal areas, structural weght and frequences obtaned from varous methods are tabulated n Tables 2 and 3, respectvely. As s clear from Table 2, TLBO gves excellent results compared to all other methods. The optmal weght of kg whch s found by TLBO s the best result among the aforementoned methods n Table 2. Table 3 shows that wth desgn varables obtaned by TLBO, all frequency constrants are met. It s noteworthy that the frst and the thrd frequences obtaned by TLBO are exactly the lower bounds of frequency constrants, smlar to the results found by PSO. However, the optmal weght of kg has been found by PSO whch s much heaver than the optmal soluton found by TLBO. Compared to two other metaheurstc methods,.e. and, TLBO shows a better performance as well. Table 4 gves statstcal results for fve ndependent runs of TLBO for ths example. A lttle standard devaton from the mean value of the ndependent runs (2.23 kg) shows that TLBO s very effectve n shape and sze optmzaton of truss structures wth frequency constrants. A value of standard devaton of 2.49 kg for and 3.64 kg for for fve ndependent runs has been reported by whch are greater than the value found by TLBO, ndcatng better performance of TLBO, compared to other metaheurstc methods as well. b) 37 bar planar truss Addtonal masses The smply supported 37 bar planar truss shown n Fg. 2 s nvestgated n ths example. The results of employng TLBO n mass mnmzaton of ths truss are compared wth other methods [5, 7, 17, 22]. The truss s made of steel wth modulus of elastcty of E 210Gpaand materal densty of kg m. The truss s optmzed on shape and sze for ts mass mnmzaton wth multple frequency constrants. Nodal coordnates n the upper chord and cross-sectonal areas of members are consdered as desgn varables. All members on the lower chord have fxed cross sectonal areas of m and the others have ntal cross sectonal areas of 1 10 m. A nonstructural mass of m=10 kg s attached at each of the nodes on the lower chord. In the optmzaton process, nodes on the upper December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +

6 414 A. Baghlan and M. H. Makabad chord can be shfted vertcally. In addton, nodal coordnates and member areas are lnked to mantan the structural symmetry about y-z plane. Therefore, only fve shape varables and fourteen szng varables 4 2 wll be redesgned for optmzaton. Moreover, the lower bounds on cross sectonal areas are 1 10 m for all bar elements. The natural frequency constrants are 1 20 Hz, 2 40 Hz and 3 60 Hz. Member Areas(cm 2 ) Grandh [8] Table 2. Optmum desgn of cross sectons (cm 2 ) for the 10 bar truss from varous methods Sedaghat et al. [3] Wang et al. [7] et al. [5] et al. [22] Zuo et al. [23] Mass (kg) Frequency No. Table 3. Optmum desgn of natural frequences (HZ) for the 10 bar truss from varous methods Natural frequences (HZ) Grandh Sedaghat [8] et al. [3] Wang et al. [7] et al. [5] et al. [22] Zuo et al. [23] Table 4. Statstcal results for fve ndependent runs of TLBO for the 10 bar truss structure Mean mass usng (kg) Standard devaton (kg) Coeffcent of varaton (%) Mean no. of searches Addtonal masses Fg. 2. Intal confguraton for the 37-bar truss structure wth added masses TLBO s used to optmze ths truss for ts shape and sze wth multple frequency constrants. The results are shown n Table 5 and the results obtaned by other researches wth other methods have been ncluded for the sake of comparson as well. Smlar to the prevous example, Table 5 shows that excellent results are obtaned by usng TLBO for ths example as well. The optmal weght of kg found by TLBO s the best result among others. Agan, the results show that TLBO outperforms other metaheurstcs such as PSO, and. Table 6 gves the natural frequences obtaned by the present IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

7 Teachng-learnng-based optmzaton algorthm for 415 work and other methods. As the table ndcates, all frequency constrants are satsfed wth desgn varables shown n Table 5. In Table 7, statstcal results for fve ndependent runs are reported. A slght value of standard devaton from the mean value of the ndependent runs shows that TLBO s very effcent n shape and sze optmzaton of truss structures wth frequency constrants. Fnal confguraton of the truss after desgnng wth TLBO has been depcted n Fg. 3. Desgn varable Table 5. Optmum desgn for the smply supported 37 bar truss from varous methods Y coordnates (m) and areas (cm 2 ) Intal Wang et al. [7] et al. [5] et al. [22] Y3, Y Y5,Y Y7,Y Y9,Y Y A1,A A2,A A3,A A4,A A5,A A6,A A7,A A8,A A9,A A10,A A11,A A12,A A13,A A Mass (kg) Table 6. Optmum desgn of natural frequences (HZ) for the 37 bar truss from varous methods Frequency No. Natural frequences (HZ) Intal Wang et al. [7] et al. [5] et al. [22] Table 7. Statstcal results for fve ndependent runs of TLBO for the smply supported 37 bar truss Mean mass usng (kg) Standard devaton (kg) Coeffcent of varaton (%) Mean no. of searches Fg. 3. Fnal confguraton desgn optmzed for the 37-bar truss structure by the present work c) 52 bar dome structure The next benchmark example s 52 bar spatal truss wth the ntal confguraton shown n Fg. 4 (top vew) and Fg. 5 (lateral vew). Ths example s a hghly non-lnear dynamc optmzaton problem wth December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +

8 416 A. Baghlan and M. H. Makabad multple frequency prohbted band constrants. A nonstructural mass of 50 kg s attached to each free node (nodes 1-13) of the structure. The dome s made of steel wth modulus of elastcty of 10 3 E 2110 paand materal densty of 7800 kg / m. Shape and sze optmzaton of ths truss to mnmze ts mass wth multple natural frequency constrants s consdered. Nodal coordnates and cross sectonal areas of the members are consdered as desgn varables. In order to mantan the structural symmetry n the desgn, the 52 bars are lnked nto eght groups as shown n Table 8. Thus, there are 13 ndependent desgn varables ncludng fve shape and eght szng varables. The natural frequency constrants are Hz and Hz. The cross sectonal area of each bar s ntally equal to 2 10 m, and s permtted to vary between 110 m and 1 10 m. The three coordnates (x, y, z) of each movable node are taken as ndependent varables and the movable range of each coordnate s 2m. Addtonal masses Fg. 4. Intal confguraton for the 52-bar dome structure wth added masses (top vew) Addtonal masses Fg. 5. Intal confguraton for the 52-bar dome structure wth added masses (lateral vew) IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

9 Teachng-learnng-based optmzaton algorthm for 417 Table 8. Member lnkng detal for the 52 bar space truss Group number Members 1 1-2, 1-3, 1-4, , 3-8, 4-10, , 3-7, 3-9, 4-9, 4-11, 5-11, 5-13, , 3-4, 4-5, , 7-8, 8-9, 9-10, 10-11, 11-12, 12-13, , 7-15, 8-16, 9-17, 10-18, 11-19, 12-20, , 8-15, 8-17, 10-17, 10-19, 12-19, 12-21, , 7-16, 9-16, 9-18, 11-18, 11-20, 13-20, Ths problem has already been nvestgated by Ln et al. [10] usng Optmal Crtera (OC), et al. [5] usng Nche Hybrd Genetc Algorthm (NHGA), et al. [22] usng parallel genetc algorthm, usng PSO, Zuo et al. [23] usng adaptve egenvalue re-analyss methods, and usng both Harmony search () and Frefly algorthm () methods. Thus, t s a sutable example for comparng TLBO wth other approaches. The optmal desgn varables, natural frequences and statstcal results for fve ndependent runs are shown n Tables 9-11, respectvely. Desgn varable Table 9. Optmum desgn for the 52 bar space truss from varous methods Coordnates (m) and areas (cm 2 ) Intal Ln et al. [10] et al. [5] et al. [22] Z X Z X Z A A A A A A A A Mass (kg) Table 10. Optmum desgn of natural frequences (HZ) for the 52 bar truss from varous methods Frequency No. Natural frequences (HZ) Intal Ln et al. [10] et al. [5] et al. [22] Table 11. Statstcal results for fve ndependent runs of TLBO for the 52bar space truss Mean mass usng (kg) Standard devaton (kg) Coeffcent of varaton (%) Mean no. of searches It s clear from Table 9 that the results obtaned by the present work are better than all results reported n the lterature. It s worth pontng out that none of the natural frequency constrants were volated usng TLBO as can be seen n Table 10. The slght value of standard devaton from mean value for fve ndependent runs gven n Table 11 agan shows the usefulness of the present for solvng ths knd of complex problems. It should be mentoned that the results presented for comparson are the best results obtaned among the runs. Fgure 6 depcts the fnal desgn of the structure optmzed by TLBO. December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +

10 418 A. Baghlan and M. H. Makabad Fg. 6. Fnal confguraton desgn optmzed for the 52-bar dome structure by the present work d) 72 bar space truss The fnal benchmark problem s devoted to 72 bar space truss shown n Fg. 7. Ths problem has been studed by varous researchers and t s approprate for comparson [11, 17, 23, 24, 25]. Fg bar space truss structure wth added masses (dmensons n m) The desgn varables are the cross sectonal areas of the members whch are lnked nto 16 groups n order to mantan the structural symmetry, as shown n Table 12. A nonstructural mass of 2268 kg (5000 lb ) s attached to four nodes on the top of the structure (nodes 1-4). The structure s made of 7 alumnum wth modulus of elastcty E Gpa (10 ps ) and materal IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

11 Teachng-learnng-based optmzaton algorthm for densty kg m (0.1 lb n ). The natural frequency constrants are 1 4Hzand Hz. The allowable mnmum area of the cross sectonal area s m (0.1 n ). The optmal desgn varables, natural frequences and statstcal results for fve ndependent runs are shown n Tables 13-15, respectvely. Desgn varable Table 12. Member lnkng detal for the 72 bar space truss Group number Members Group number Members Table 13. Optmum desgn for the 72 bar space truss from varous methods Areas(cm 2 ) Konzelman [24] Sedaghat [25] Zuo et al. [23] Mass (kg) Table 14. Optmum desgn of natural frequences (HZ) for the 72 bar truss from varous methods Frequency no Natural frequences (HZ) Konzelman Sedaghat [24] [25] Zuo et al. [23] Table 15. Statstcal results for fve ndependent runs of TLBO for the 72 bar space truss Mean mass usng (kg) Standard devaton (kg) Coeffcent of varaton (%) Mean no. of searches As Table 13 shows, Konzelman [24] usng Dual Method (DM) and Sedaghat [25] usng the Force Method (FM) found the same values for the desgn varables, resultng n optmal mass of kg. Further, usng PSO found a heaver structure and Zuo et al. [23] usng adaptve egenvalue reanalyss methods found the optmal soluton of whch volates the frequency constrants (see Table 14). Recently, by usng two metaheurstcs,.e. and, found feasble solutons for ths problem wth optmal weghts of kg and kg, usng and, respectvely. Table 13 shows that the optmal feasble soluton found by TLBO wth the weght of December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +

12 420 A. Baghlan and M. H. Makabad kg s the best feasble soluton among the feasble solutons obtaned by other methods, and s very close to optmal solutons found by Konzelman [24] and Sedaghat [25] but wth dfferent cross sectonal areas. It s mportant to note that from Table 14, the soluton found by TLBO satsfes all frequency constrants, unlke the soluton found by Zuo et al. [23]. 5. CONCLUSION Shape and sze optmzaton of truss structures wth multple frequency constrants whch s a hghly nonlnear problem was nvestgated n ths paper. A recently developed approach,.e. teachng-learnng-based optmzaton (TLBO) was used for ths purpose. The method s smple to mplement snce no tunng parameter should be calbrated n the algorthm. Some benchmark problems were solved va the proposed approach and the results were compared wth other methods ncludng other metaheurstc approaches such as PSO, and. In all examples TLBO gves very satsfactory results whch satsfy all frequency constrants. The results of ths paper show that TLBO s an outstandng approach sutable for solvng complcated optmzaton problems wth hghly non-lnear behavor. REFERENCES 1. Grandh, R. V. & Venkayya, V. B. (1988). Structural optmzaton wth frequency constrants, AIAA Journal, Vol. 26, pp Khot, N. S. (1985). Optmzaton of structures wth multple frequency constrants. Computers and Structures, Vol. 20, pp Sedaghat, R., Suleman, A. & Tabarrok, B. (2002). Structural optmzaton wth frequency constrants usng fnte element force method. AIAA Journal, Vol. 40, pp Tong, W.H. & Lu, G.R. (2001). An optmzaton procedure for truss structure wth dscrete desgn varables and dynamc constrants. Computers and Structures. Vol. 79, pp , W., Me, Z., Guangmng, W. & Guang, M. (2005). Truss optmzaton on shape and szng wth frequency constrants based on genetc algorthm. Computatonal Mechancs, Vol. 25, pp Rozvany, G. I. N., Bendsoe, M. P. & Krsh, U. (1995). Layout optmzaton of structures. Appled Mechancs Revews, Vol. 48, pp Wang, D., Zhang, W. H. & Jang, J. S. (2004). Truss optmzaton on shape and szng wth frequency constrants. AIAA Journal, Vol. 42, pp Grandh, R. V. (1993). Structural optmzaton wth frequency constrants A revew. AIAA Journal, Vol. 31, pp Bellagamba, L. & Yang, T. Y. (1981). Mnmum-mass truss structures wth constrants on fundamental natural frequency. AIAA Journal, Vol. 19, pp Ln, J. H., Chen, W. Y. &. Yu, Y. S. (1982). Structural optmzaton on geometrcal confguraton and element szng wth statc and dynamc constrants. Computers and Structures, Vol. 15, pp , H. M. (2011). Truss optmzaton wth dynamc constrants usng a partcle swarm algorthm. Expert Systems wth Applcatons, Vol. 38, pp Eskandar, H., Sadolla, A., Bahrennejad, A. & Hamd, M. (2012). Water cycle algorthm A novel metaheurstc optmzaton method for solvng constraned engneerng optmzaton problems. Computers and Structures, Vol , pp Sadolla, A., Bahrennejad, A., Eskandar, H. & Hamd, M. (2012). Mne blast algorthm: A populaton based algorthm for solvng constraned engneerng optmzaton problems. Appled Soft Computng, IJST, Transactons of Cvl Engneerng, Volume 37, Number C + December 2013

13 Teachng-learnng-based optmzaton algorthm for Rajaboun, R. (2011). Cuckoo optmzaton algorthm. Appled Soft Computng, Vol. 11, pp Kaveh, A. & Laknejad, K. (2011). A hybrd mult-objectve optmzaton and decson makng procedure for optmal desgn truss structure. Iranan Journal of Scence and Technology, Transactons of Cvl Engneerng, Vol. 35, No. C2, pp Kaveh, A. & Malakout Rad, K. (2010). Hybrd genetc algorthm and partcle swarm optmzaton for the force method based smultaneous analyss and desgn. Iranan Journal of Scence & Technology, Transacton B: Engneerng, Vol. 34, No. B1, pp Mguel, L.F.F. & Mguel, L.F.F. (2012). Shape and sze optmzaton of truss structures consderng dynamc constrants through modern metaheurstc algorthms. Expert Systems wth Applcatons. Vol. 39, pp Kaveh, A. & Zolghadr, A. (2012). Truss optmzaton wth natural frequency constrants usng a hybrdzed CSS-BBBC algorthm wth trap recognton capablty. Computers and Structures, Vol. 103, pp Crespnsek, M., Lu, S. H. & Mernk, L. (2012). A note on teachng-learnng-based optmzaton algorthm. Informaton Scences, Vol. 212, pp Rao, R. V., Savsan, V. J. & Vakhara, D. P. (2012). Teachng-learnng-based optmzaton: An optmzaton method for contnuous non-lnear large scale problems. Informaton Scences, Vol. 183, pp Rao, R. V., Savsan, V. J. & Vakhara, D. P. (2011). Teachng learnng-based optmzaton: a novel method for constraned mechancal desgn optmzaton problems. Computer-Aded Desgn, Vol. 43, pp , W., Tanbng, T., Xanghong, X. & Wenje, S. (2011). Truss optmzaton on shape and szng wth frequency constrants based on parallel genetc algorthm. Struct Multdsc Opt, Vol. 43, pp Zuo, W., Xu, T., Zhang, H. & Xu, T. (2011). Fast structural optmzaton wth frequency constrants by genetc algorthm usng adaptve egenvalue reanalyss methods. Struct Multdsc Opt, Vol. 43, pp Konzelman, C. J. (1986). Dual methods and approxmaton concepts for structural optmzaton. M.A.Sc. thess, Department of Mechancal Engneerng, Unversty of Toronto. 25. Sedaghat, R. (2005). Benchmark case studes n structural desgn optmzaton usng the force method. Internatonal Journal of Solds and Structures, Vol. 42, pp December 2013 IJST, Transactons of Cvl Engneerng, Volume 37, Number C +