OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING- LEARNING-BASED OPTIMIZATION ALGORITHM

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1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 2013; 3(3): OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING- LEARNING-BASED OPTIMIZATION ALGORITHM M.H. Makabad, A. Baghlan *,, H. Rahnema and M.A. Hadanfard Faculty of Cvl and Envronmental Engneerng, Shraz Unversty of Technology, Shraz, Iran ABSTRACT In ths study, teachng-learnng-based optmzaton (TLBO) algorthm s employed for the frst tme for optmzaton of real world truss brdges. The objectve functon consdered s the weght of the structure subjected to desgn constrants ncludng nternal stress wthn bar elements and servceablty (deflecton). Two examples demonstrate the effectveness of TLBO algorthm n optmzaton of such structures. Varous desgn groups have been consdered for each problem and the results are compared. Both tensle and compressve stresses are taken nto account. The results show that TLBO has a great ntrnsc capablty n problems nvolvng nonlnear desgn crtera. Receved: 15 February 2013; Accepted: 22 July 2013 KEY WORDS: Brdge, truss, szng optmzaton, teachng-learnng-based optmzaton 1. INTRODUCTION Brdges are amazng structures usually regarded as landmarks. They play an mportant role n transportaton and development of countres. Desgn, fabrcaton and nstallaton of brdges are usually costly. Optmzaton methods can be used n order to reduce these expenses and hence such methods are of paramount mportance. Unfortunately, despte of ther effectveness n economcally desgn of real lfe structures, optmzaton technques are not practcally employed by engneers, especally n the area of brdge desgn. Optmal desgn can be performed based on szng, shape or topology of the structure. A * Correspondng author: A. Baghlan, Faculty of Cvl and Envronmental Engneerng, Shraz Unversty of Technology, Shraz, Iran E-mal address: baghlan@sutech.ac.r (A. Baghlan)

2 500 M.H. Makabad A. Baghlan, H. Rahnema and M.A. Hadanfard combnaton of these optmzaton approaches s also possble. In szng optmzaton of truss brdges, whch s the man concern of ths artcle, the cross sectonal areas of members are consdered as desgn varables and they should be optmzed such that the weght of structure s mnmzed. Moreover, some desgn constrants should be satsfed at the same tme. Generally, nternal stress wthn bar elements (strength) and servceablty of the structure (deflecton) are regarded as desgn constrants. The desgn procedure of structures usually nvolves prelmnary desgn, analyss of structure, controllng desgn constrants, re-analyss and re-desgn. On the other hand, procedure of fndng optmum structure s usually carred out by evolutonary algorthms because of ther robustness, effectveness and ease of applcaton. In populaton-based optmzaton algorthms several numbers of structures are generated randomly n the begnnng of the procedure and the desgn s mproved after evolutons. Some well-known and effcent populaton based algorthms such as Genetc Algorthm (GA), Ant Colony Optmzaton (ACO), Partcle Swarm Optmzaton (PSO), Frefly Algorthm (FA) and so on have been developed so far. In the last decade, several valuable studes have been carred out on the optmzaton of truss structures usng evolutonary and metaheurstc algorthms [1-14]. However, a few studes have been publshed concernng optmzaton of real lfe brdge structures. Hong Goun et al. [15] utlzed prncpal stress based evolutonary structural optmzaton method for optmzaton of arch, ted arch, cable-stayed and suspenson brdges wth stress, dsplacement and frequency constrants. By usng genetc algorthm, Cheng [16] nvestgated sze optmzaton of steel arch truss brdges. Chen [17] studed the shape optmzaton of brdge structures usng hybrd genetc algorthm. Hasanceb [18] nvestgated the applcaton of evolutonary strateges n sze, shape and topology optmzaton of truss brdges. Baldmor et al. [19] studed optmzaton problem of cable cross secton of a cable stayed brdge consderng cable stress and deck dsplacement as desgn constrants. In ths paper, the effectveness of a recently developed populaton-based optmzaton algorthms,.e. teachng-learnng-based optmzaton (TLBO) n szng optmzaton of real lfe brdge structures s nvestgated. The optmzaton problem s frst formulated for a general two-dmensonal steel truss arch brdge structure and then a teachng- learnng-base optmzaton algorthm s developed for the optmum desgn of steel truss arch brdges. Fnally, two numercal examples nvolvng detaled computatonal models of long span steel truss arch brdges wth man spans of 680 ft and ft are presented to demonstrate the applcablty and merts of the aforementoned optmzaton method. Both tensle and compressve stresses are consdered; several desgn groups are tested and the results are compared. 2. FORMULATION OF THE PROBLEM The problem of szng optmzaton of truss brdge structures nvolves optmzng cross sectons A of the bars such that the weght of the structure W s mnmzed and some constrants wth respect to desgn crtera are satsfed. The mathematcal formulaton of the

3 OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING-LEARNING problem can be stated as follows: Mnmze k 1 1 ng mk W A A ρl (1) k Subject to: low up, 1,2,..., nm (2) b 0, 1,2,..., ncm (3) low up, 1,2,..., nn (4) A A A, 1,2,..., n (5) low up g n whch A s the vector contanng the desgn varables (.e. cross sectons A A1, A2,..., Ang ), W A s the weght of the truss structure, s the densty of member, L s the length of member, nm s the number of members n the structure, ncm s the number of compresson members, nn s the number of nodes, ng s the total number of member groups (.e. desgn varables), A s the cross sectonal area of the members k belongng to group k, mk s the total number of members n group k, s the stress of the b th member, s the allowable bucklng stress for the th member, s the dsplacement of the th node, and low and up are the lower and upper bounds for stress, dsplacement and cross-sectonal area. 3. TEACHING-LEARNING-BASED- OPTIMIZATION (TLBO) ALGORITHM In 2011 Rao et al. [20] presented a new metaheurstcs called teachng-learnng-basedoptmzaton (TLBO). TLBO s a populaton-based algorthm whch tres to smulate the process of teachng and learnng n a classroom. The optmzaton process nvolves two stages ncludng teacher phase and learner Phase. In teacher phase, learners frst get nformaton from a teacher and then from other classmates n learner phase. The best soluton s regarded as the teacher (X teacher ) n the populaton. In the teacher phase, learners learn from the teacher and 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. Two randomly-generated parameters r n the range of 0 and 1 and T F are appled n update formula for the soluton X for stochastc purposes as follows: X new X r.( X teacher T F.X mean ) (6) where X new and X are the new and exstng soluton of, and T F s a teachng factor whch can be ether 1 or 2 [21,22]. In second phase,.e. the learner phase, the learners ncrease ther knowledge by

4 502 M.H. Makabad A. Baghlan, H. Rahnema and M.A. Hadanfard communcatng wth other students n the classroom. Therefore, an ndvdual wll learn new knowledge f the other ndvduals have more knowledge than hm/her. Durng ths stage, the student X nteracts randomly wth another student X j ( j ) n order to develop hs/her knowledge. In the case that X j s better than X (.e. f ( X j ) f ( X ) for mnmzaton problems), X s moved toward X j. Otherwse t s moved away from X j : X new X r.( X j X ) f f ( X ) f ( X j ) (7) X new X r.( X X j ) f f ( X ) f ( X j ) (8) If the new soluton X new s better, t s accepted n the populaton. The algorthm wll contnue untl the termnaton condton s met. For more detals about the algorthm, the nterested reader s referred to relevant references [21,22]. 4. DESIGN EXAMPLES In order to nvestgate the effectveness of TLBO algorthm n szng optmzaton of truss brdge structures, two real lfe truss brdges are optmzed. These brdges were selected because farly complete nformaton about the geometry, loadng and desgn crtera of these structures are avalable. Snce there are no publshed artcles n the lterature regardng optmzaton of these brdge structures, the results are compared wth the actual weght of structures and other results obtaned by re-groupng of the desgn varables. Therefore, the current study can be regarded as a benchmark problem for further nvestgatons and comparson wth our results n the future. A fnte element code n MATLAB s used for analyss of structures combned wth a code for the process of optmzaton based on TLBO. To explore usefulness of the optmzaton technque n solvng problems nvolvng nonlnear desgn crtera, both tensle and compressve stresses are taken nto account. For both examples, allowable tensle and compressve stresses are consdered accordng to AISC ASD (1989) [23] code as follows: b 0.6F for 0 up y for 0 (9) b 2 12 E 2 for C c Fy for Cc Cc Cc Cc (10) where Fy s the yeld stress of steel; E s Young's modulus of elastcty of steel; s

5 OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING-LEARNING slenderness rato kl r ; k s the effectve length factor, L s the length of each member ; r s the radus of gyraton of member ; and Cc s defned as: C c 2 F 2 E (11) y For these structures, the Young's modulus of elastcty was lb/ft 2 ; the materal densty was 495 lb/ft 3. The radus of gyraton was expressed n terms of cross-sectonal areas b as r aa [24]. Here, a and b are the constants dependng on the types of sectons adopted for the members such as ppes, angles, and tees. For ppe sectons consdered n ths study a= and b= were adopted. Allowable dsplacement s determned based on recommendatons of the Australan Brdge Code [25] where the deflecton allowance under the servce load should not exceed 1/800 of the man span of the brdge. 4.1 Burro Creek Brdge Burro Creek Brdge s located n Arzona U.S. Hghway 93 runs north to south through central Arzona and s the prmary transportaton corrdor between Phoenx and Las Vegas. The Burro Creek Brdge, whch carred two-way auto traffc, s a truss arch structure wth spandrel columns supportng the roadway deck and plate grder approach spans. Two vews of ths brdge are shown n Fgure 1. Fgure 1. Burro Creek Brdge The man span of the brdge s 680 ft whch conssts of 34 panels of 20 ft n length. Both upper and lower chords shapes are quadratc parabola. The elevaton vew of the brdge s shown n Fgure 2. The averaged dead loads for varous parts of the structure are summarzed n Table 1 [26-27]. Equvalent lve load plus mpact loadng on each arch for fully loadng structure s consdered as 1420 lb/ft. Accordng to Australan Brdge Code [25], allowable dsplacement s 0.85ft. Moreover, the mnmum cross-sectonal area was

6 504 M.H. Makabad A. Baghlan, H. Rahnema and M.A. Hadanfard consdered to be 0.2 ft 2 and F y as lb/ft 2. Fgure 2. Elevaton vew of Burro Creek Brdge Table 1: Average dead load on Burro Creek Brdge Average dead load lb per ft Deck slab and surfacng for roadway 3140 Slabs for sdewalks 704 Ralngs and parapets 470 Floor steel for roadway 800 Floor bracng 203 Arch trusses 2082 Arch bracng 580 Arch posts and bracng 608 Total 8587 For smplfcaton, a total unform load of lb/ft for both dead and lve loads s consdered on the deck. Because of symmetry, half of the structure s consdered n the analyss whch s shown n Fgure 3 ncludng numberng of all bars. Fgure 3. 2D fnte element model and element numberng of Burro Creek Brdge (for one half of the brdge)

7 OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING-LEARNING Optmzaton of the structure s accomplshed consderng three dfferent groups of varables ncludng 4, 8 and 12 varables n the desgn. Table 2 demonstrates the cross sectons consdered for these three dfferent cases. Table 3 reports the results found after optmzaton of the structure for three aforementoned cases. Table 2: Desgn varables for Burro Creek Brdge for three dfferent cases Desgn Member number varables Case (4 varables) Case (8 varables) Case (12 varables) 1 67, 63, 59, 55, 51,47, 43, 39, 35, 67, 63, 59, 55, 51,47, 31, 27, 23, 19, 15, 11, 7, 3 43, 39, 35, 31, 27 67, 63, 59, 55, 51,47, , 62, 58, 54, 50, 46, 42, 38, 34, 66, 62, 58, 54, 50, 46, 30, 26, 22, 18, 14, 10, 6, 2 42, 38, 34, 30, 26 66, 62, 58, 54, 50, 46, , 65, 61, 57, 53, 49, 45, 41, 37, 69, 65, 61, 57, 53, 49, 33, 29, 25, 21, 17, 13, 9, 5, 1 45, 41, 37, 33, 29 69, 65, 61, 57, 53, 49, , 64, 60, 56, 52, 48, 44, 40, 36, 68, 64, 60, 56, 52, 48, 32, 28, 24, 20, 16, 12, 8, 4 44, 40, 36, 32, 28 68, 64, 60, 56, 52, 48, , 19, 15, 11, 7, 3 39, 35, 31, 27, 23, , 18, 14, 10, 6, 2 38, 34, 30, 26, 22, , 21, 17, 13, 9, 5, 1 41, 37, 33, 29, 25, , 20, 16, 12, 8, 4 40, 36, 32, 28, 24, , 15, 11, 7, , 10, 6, , 13, 9, 5, , 12, 8, 4 Table 3: Comparson of optmal desgn for Burro Creek Brdge for three dfferent cases Optmal cross-sectonal areas (ft 2 ) Desgn Case Case Case varables (4 varable) (8 varable) (12 varable) A A A A A A A A A A A A Weght (lb) The optmum weght of lb s found when 4 group of varables s consdered

8 506 M.H. Makabad A. Baghlan, H. Rahnema and M.A. Hadanfard and the optmum weght of lb f found n the case of 12 varables. As expected, ncludng more desgn varables results n flexblty n the optmzaton procedure and fndng lghter structures. It s worth pontng out that the actual weght of the structure s approxmately lb. In Fgure 4 a comparson among convergence rates n TLBO for three cases s presented Case I Case II Case III Weght (lb) Number of analyses Fgure 4. Comparson of the convergence rates for Burro Creek Brdge for three dfferent cases 4.2 West End-North Sde Brdge The West End-North Sde Brdge s a steel bowstrng arch brdge over the Oho Rver n Pttsburgh, Pennsylvana, approxmately one mle below the confluence of the Allegheny and Monongahela Rvers. A vew of the brdge s depcted n Fgure 5. Fgure 5. The West End-North Sde Brdge

9 OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING-LEARNING The man span of the brdge s ft whch conssts of 28 panels of ft n length. A elevaton vew of the brdge and ts geometry s shown n Fgure 6 wth more detals. Fgure 6. Elevaton vew of West End-North Sde Brdge The averaged dead loads for varous parts of the structure are reported n Table 4 [26-28]. Equvalent lve load plus mpact loadng on each arch for fully loadng structure s consdered as 1790 lb/ft. For ths structure, F y was consdered as lb/ft 2 and the mnmum cross-sectonal area was 0.15 ft 2. Accordng to Australan Brdge Code [25], allowable dsplacement s consdered to be 0.97 ft. Table 4: Average dead load on West End-North Sde Brdge Average dead load lb per ft Roadway, sdewalks, and ralngs 4870 Floor steel and Floor bracng 2360 Arch trusses 4300 Arch tes 2100 Arch bracng 550 Hangers 360 Utltes and excess 600 Total A total unform load of 9360 lb/ft for both deal and lve loads s consdered on the deck. Smlar to prevous problem, half of the structure s consdered n the fnte element analyss whch s depcted n Fgure 7, ncludng the bars numberng. Desgn varables for ths problem,,.e. cross sectonal areas, are categorzed n four and eght groups for Case I and Case II, respectvely. A lst of members consdered n each case s tabulated n Table 5. Table 6 shows the optmum cross sectonal areas found by TLBO and the optmum weght of structure n each case. The optmum weght of lb s found for Case I n whch 4 groups of varables are consdered, and lb for Case II when the varables are categorzed n 8 groups. Same as the prevous example, lghter structures can be found wth ncreasng the number of nvolved varables. In Fgure 8 a comparson between convergence rates n TLBO for two cases s depcted.

10 508 M.H. Makabad A. Baghlan, H. Rahnema and M.A. Hadanfard Fgure 7. 2D fnte element model and element numberng of West End-North Sde Brdge (for one half of the brdge). Table 5: Desgn varables for West End-North Sde Brdge for two dfferent cases Desgn Member number varables Case (4 varables) Case (8 varables) 1 53, 49, 45, 41, 37, 33, 29, 25, 21, 17, 13, 53,49,45,41,37,33,29, 25 9, 5, , 48, 44, 40, 36, 32, 28, 24, 20, 26, 12, 52, 48, 44, 40, 36, 32, 28, 8, 4, , 51, 47, 43, 39, 35, 31, 27, 23, 19, 15, 55, 51, 47, 43, 39, 35, 31, 11, 7, , 50, 46, 42, 38, 34, 30, 26, 22, 18, 14, 54, 50, 46, 42, 38, 34, 30, 10, , 17, 13, 9, 5, , 26, 12, 8, 4, , 19, 15, 11, 7, , 18, 14, 10, 6 Table 6: Comparson of optmal desgn for West End-North Sde Brdge for two dfferent cases Desgn Optmal cross-sectonal areas (ft 2 ) varables Case (4 varable) Case (8varable) A A A A A A A A Weght (lb)

11 OPTIMAL DESIGN OF TRUSS BRIDGES USING TEACHING-LEARNING Case I Case II Weght(lb) Number of analyses Fgure 8. Comparson of the convergence rates for West End-North Sde Brdge for two dfferent cases 5. CONCLUSIONS Applcaton and effectveness of one of the most recently developed optmzaton algorthms called teachng-learnng-based optmzaton (TLBO) n desgn optmzaton of real world steel truss arch brdges s nvestgated n ths paper. Two brdges are optmzed va TLBO, takng both tensle and compressve stresses nto account. Varous groups of varables are consdered and the results show that TLBO s very effectve n szng optmzaton of ths knd of structures wth nonlnear desgn crtera. REFERENCES 1. Kaveh A, Farahmand Azar B, Talatahar S. Ant colony optmzaton for desgn of space trusses, Int J Space Struct, 2008; 23: Lee K S, Geem Z W. A new structural optmzaton method based on the harmony search algorthm, Comput Struct, 2004; 82: Schmt L A, Farsh B. Some approxmaton concepts for structural synthess, AIAA J, 1974; 12: Kaveh A, Raham H. Analyss, desgn and optmzaton of structures usng force method and genetc algorthm, Int J Numer Methods Eng, 2006; 65: Adel H, Kumar S. Dstrbuted genetc algorthm for structural optmzaton, J Aerospace Eng, 1995; 8: Degertekn S O. Improved harmony search algorthms for szng optmzaton of truss structures, Comput Struct, 2012; 92: Adel H, Kamal O. Effcent optmzaton of plane trusses, Adv Eng Software, 1991; 13: Lee KS, Han SW, Geem ZW. Dscrete sze and dscrete-contnuous confguraton

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