A new algorithm for size optimization of the truss structures using finite element method

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS A new algorthm for sze optmzaton of the truss structures usng fnte element method To cte ths artcle: Vu Th Bch Quyen et al 2018 IOP Conf. Ser.: Mater. Sc. Eng Vew the artcle onlne for updates and enhancements. Related content - Weght optmzaton of plane truss usng genetc algorthm D Neeraja, Thejesh Kamreddy, Potnuru Santosh Kumar et al. - Multobjectve Smultaneous Topology, Shape and Szng Optmzaton of Trusses Usng Evolutonary Optmzers Teerapol Techasen, Kttnan Wansasueb, Natee Panagant et al. - Montorng of progressve collapse of skeletal structures A Swercz, P Kolakowsk and J Holnck- Szulc Ths content was downloaded from IP address on 24/12/2018 at 16:37

2 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ A new algorthm for sze optmzaton of the truss structures usng fnte element method Vu Th Bch Quyen 1, Tran Th Thuy Van 2, Cao Quoc Khanh 3 1,2 Lecturer, Hano Archtectural Unversty, Vetnam 3 Lecturer, Men Tay Constructon Unversty, Vetnam Emal: bquyen1312@gmal.com, ttthvan.hau@gmal.com, khanhcf@gmal.com Abstract. The paper presents a new algorthm for sze optmzaton of the trusses usng fnte element method. The objectve functon was establshed based on weght mnmzaton of trusses. The constrant condtons were formulated based on the condtons that the structure satsfes the strength, stffness and compatblty requrements, and equlbrum equatons. Determnaton of nternal forces, dsplacements and establshment of the equatons are based on fnte element method. In order to solve optmzaton problems usng fnte element method, the authors proposed a new method for seekng optmal solutons. For truss weght optmzaton, the authors proposed a coeffcent and called ths coeffcent as the correlaton coeffcent of nternal forces among truss elements. Ths coeffcent shows the relatonshp between the nternal forces of truss elements, whch was used as the bass for re-selectng the dmensons of cross secton of truss elements n teratve process. The paper ntroduces the calculaton procedure and analyss algorthm for weght optmzaton of planar and space trusses. The examples for weght optmzaton of planar and space trusses were mplemented n a subroutne wrtten n Matlab software. The calculaton results usng the method proposed by the authors matched wth the calculaton results usng other methods such as Hybrdzed Genetc Algorthm, Harmony search method, HyperWork, etc. However, usng the method proposed by the authors gves less teratve crcles. 1. Introducton Trusses are common structures n constructon due to ther outstandng advantages n materal savng and maxmum utlzaton of structure load capacty. Optmzaton of trusses structure s a problem of seekng best soluton n Prelmnary Desgn stage. Dependng on desgn varables optmzaton problems of trusses are classfed nto three categores: sze optmzaton, shape optmzaton and structure optmzaton. In order to solve the optmzaton problems, the mathematcal and mechancal analyses are requred, n whch: Use the mechancs theores of deformable bodes to establsh equlbrum equatons (n the case of that dsplacements are varable functons) or compatblty equatons (n the case of that stresses are varable functons). From there, problems of structure analyss are determnaton of stresses, strans, dsplacements and nternal forces. Use the mathematcal optmzaton theores to solve structural optmzaton problems wth constrants determned from the equlbrum or compatblty equatons of structural analyss problems. Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by IOP Publshng Ltd 1

3 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ In the early study perod, solvng problems of truss structure optmzaton was manly based on the analytcal method, so that only smple problems could be solved. The development of nformaton technology helps the solvng optmzaton problems to acheve hgh effcency. Usng numercal methods, especally fnte element method allows solvng problems of structural analyss wth any structure szes and shapes. Programmng languages, mathematcal software are powerful tools for mplementaton to solve optmzaton problems. The authors studed to solve weght optmzaton problems of plane trusses usng fnte element method to determne nternal forces, dsplacements and to establsh constrant condtons [2]. For solvng optmzaton problem, the authors proposed a new teratve algorthm for seekng optmal soluton and establshment of calculaton program of sze optmzaton of plane trusses by Matlab software. Ths artcle presents proposed algorthm whch apples n solvng problem of sze optmzaton of plane and space trusses wth dfferent selectons of desgn varables. 2. Establshment of sze optmzaton problems trusses based on Fnte Element Method [1, 3] 2.1. Optmzaton objectve functon In sze optmzaton problems, optmzaton objectve functon s choosen to fnd mnmum of truss weght G. N G = γla mn (2.1) =1 Where N s number of desgn varable groups. Desgn varable s cross-secton area A, for convenence n fabrcaton truss elements are dvded nto several groups of element havng the same cross-secton area and dfferent lengths; L, A s the total length and cross-secton area of th element group; γ s specfc weght of materal of group. In the case of homogeneous structures optmzaton objectve functon s the volume of the structure as follows N V L A (2.2) 2.2. Constrants Strength of materals constrants 1 ten com max mn Where ; are allowable stresses n tenson and compresson. Stresses and nternal forces of elements are determned by fnte element method ten com E 1 1 T e L 1 1 f EA 1 1 T e L 1 1 e e (2.3) (2.4) In whch, E s elastc modulus; A s cross-secton area; L s element length; T e s coordnate transformaton matrx; δ e s element nodal dsplacement vector n global coordnate system. Dsplacements constrants yp y (2.5) Where [y] s allowable dsplacement wth dfferent postons of loads; 2

4 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ Dsplacement y s determned by fnte element method 1 y K.P (2.6) In whch, K s stffness matrx of truss; P s nodal load vector of truss Equlbrum constrants satsfes the equaton (2.6) n solvng problem usng fnte element method 3. The method of correlaton between nternal forces of optmum elements for weght optmzaton of trusses Results of determnaton of dsplacements, stresses, nternal forces of truss elements can be expressed as functons of desgn varables A as follows: -1-1 y = a.a ; σ = b.a ; f = c (3.1) Where a, b, c are constants determned from equatons usng fnte element method Accordng to strength of materal constrants (elements wth the same strength of materal) t s receved: f1 f2 f... (3.2) A1 A2 A From (3.1) and (3.2) can derve the relatonshp: f A c n = = (3.3) f 1 A 1 c 1 Accordng to (3.3) the relatonshp between desgn varables A can be calculated from the correlaton of elements nternal forces. Thereof, the authors proposed a new method to solve problems of weght mnmzaton of trusses wth constrants establshed usng fnte element method. The bass content of the method can be expressed n algorthm of seekng of optmum values of desgn varables by correlaton between elements nternal forces. The authors call ths method as the method of correlaton coeffcent between element nternal forces. The order of solvng problems of weght optmzaton of trusses by method of correlaton between element nternal forces ncludes followng steps: Step 1: Prelmnary selecton of cross-secton szes for desgn varable groups. Dsplacements y, stresses σ, elements nternal forces f are determned usng fnte element method n the case of that the elements have the same desgn varables A. Determne A (j) mn for desgn varable groups accordng to the constrants of strength materal and dsplacement of all elements n groups as follows: y y a (j) b => A = max ; mn (3.4) σ y σ σ Prelmnary determnaton of truss volume: V N ( j) 1 LA 1 Step 2: Select cross-secton szes one more tme Aj=njA1 of desgn varable group j accordng to correlaton coeffcent between elements nternal forces of desgn varable groups n j = f N.f1 Where f s total element nternal force n the same group; N s number of elements n the same group. In prncple, nternal forces can be selected correspondng to A (j) mn of any element as standard value f 1. Groups of elements whch have value approxmately null should not selected to avod errors Determne truss volume: V N ( j) 2 LA 1 Determne the percentage of decrease of cross-secton area. (3.5) 3

5 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ V1 - V2 100% (3.6) V1 These steps are mplemented teratvely to n crcle untl archevng requred percentage of decrease of cross-secton area. The teratve method accordng to correlaton between element nternal forces can be represented n the form of a dagram as n Fgure 1. Fg. 1. Iteratve dagram usng correlaton coeffcent between element nternal forces 4. Establshment of calculaton programme for weght optmzaton of trusses usng Matlab software Based on the proposed above method of correlaton coeffcent between element nternal forces the block dagram of algorthm s establshed (Fg.2) for weght optmzaton of plane and space trusses wth any desgn varable group usng Matlab software. 4

6 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ Fgure 2. Block dagram of algorthm for weght optmzaton usng method of correlaton coeffcent between element nternal forces 5. Examples The authors used the above establshed programme for optmzaton to solve several examples of determnng optmum weght of plane and space trusses wth any number of desgn varable group. Caculated results were compared wth results mplemented by other method n publshed studes [4-13]. 5

7 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ Example of weght optmzaton of plane truss Optmzaton of plane truss wth followng parameters (Fg. 3): Specfc weght of materal -5 3 ρ = 2,77x10 (N / mm ); Elastc modulus 2 E = (N / mm ); Allowable 2 stress σ = ±172(N / mm ); : Allowable dsplacement d = ±50,8(mm); Loads P 2Y = P 3Y = (N). The problem was solved wth seven optons of dvdng desgn varables groups as below. Opton 0: A1; A2; A3; A4; A5; A6; A7; A8; A9; A10; Opton 1: A1=A5; A8=A9; A3=A4=A6=A10; A2; A7. Opton 2: A1=A5=A8=A9; A3=A4=A6=A10; A2; A7; Opton 3: A1=A2=A4 =A5; A3=A6; A7=A8=A9=A10. Opton 4: A1=A2; A4=A5; A3=A6; A7=A8; A9=A10; Opton 5: A1=A5; A6; A7; A8=A9; A3=A4=A10; A2. Opton 6: A1=A2=A4 =A5; A3=A6=A7=A8=A9=A10; \ Fgure 3. Plane truss under statc load Calculated results usng method of correlaton coeffcent between element nternal forces The results of weght optmzaton of plane truss usng method of correlaton coeffcent between element nternal forces accordng to optons of dvdng desgn varable groups are shown n Table 1. The dagram of weght relatonshp depends on the number of teratons accordng desgn varable groups s presented n Fgure 4. Order Calculated results of cross secton accordng to opton of desgn varable groups (n 2 ) Order Calculated results of cross secton accordng to opton of desgn varable groups (n 2 ) [0] [1] [2] [3] [4] [5] [6] [0] [1] [2] [3] [4] [5] [6] 1 28,9 28,78 23,94 22,08 25,4 29,34 23,72 7 8,24 8,55 9,99 18,74 18,78 5,34 15,76 2 0,01 14,34 14,51 22,08 25,4 14,67 23, ,43 20,33 23,94 18,74 18,78 23,92 15, ,35 0,05 0,01 3,19 3,47 0,01 15, ,43 20,33 23,94 18,74 18,78 23,92 15, ,44 0,05 0,01 22,08 18,86 0,01 23, ,02 0,05 0,01 18,74 18,78 0,01 15, ,78 23,94 22,08 18,86 29,34 23,72 Inter ,01 0,05 0,01 3,19 3, ,76 W Table 1. Caculated results of weght optmzaton of plane truss accordng to optons of desgn varable groups. 6

8 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ Fgure 4. Dagram of weght relatonshp of plane truss depends on the number of teratons usng varous optmzaton methods Example of weght optmzaton of space truss Optmzaton of space truss wth followng parameters (Fg. 5): 3 Specfc weght of materal ρ = 0,1(lb / n ); Elastc modulus E = 10000( ks); Allowable stress σ = ±40( ks ); Allowable dsplacement d = ±0,35(n); Loads P 1X = 1(kp); P = P = -10(kp); 1Y 1Z P = P = -10(kp); P = 0,6(kp); P = 0,5(kp);P = 0,6(kp) 2Y 2Z 6X 3X 6X. Fgure 5. Space truss wth 25 elements The problem was solved wth fve optons of dvdng desgn varables groups as below. Opton 0: group not dvded, 4 dvson optons of groups are shown n Table 2. Optons of dvdng groups Order [Opton 1] [Opton 2] [Opton 3] [Opton 4] 1 A1 A1.A2,A3,A4,A5 A1 A1- A9 2 A2,A3,A4,A5 A6,A7,A8,A9 A2,A3,A4,A5 A10-A13 3 A6,A7,A8,A9 A10,A11 A6,A7,A8,A9 A14-A25 4 A10,A11 A12,A13 A10,A11,A12,A13 5 A12,A13 A14,A15,A16,A17 A14,A20,A22 7

9 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ A14,A15,A16,A17 A18,A19,A20,A21 A15,A18,A23 7 A18,A19,A20,A21 A22,A23,A24,A25 A16,A21,A25 8 A22,A23,A24,A25 A17,A19,A24 Table 2. Optons of dvdng groups of desgn varable of space truss Calculated results of weght optmzaton of space truss accordng to the optons and other studes are shown n Table 3. Dagram of weght relatonshp depends on the number of teratons s presented n Fg. 6-7 Order CROSS-SECTION AREA OF ELEMENTS BY STUDIES (n 2 ) [7] [4] [13] [Op0] [Op1] [Op2] [Op3] [Op4] 1 0,1 0,1 0, ,119 0,002 2, ,72 0,12 1,8 0,104 0,216 0,119 0,216 2, ,72 0,12 1,8 0,317 0,216 0,119 0,216 2, ,72 0,12 1,8 0,555 0,216 0,119 0,216 2, ,72 0,12 1,8 0,317 0,216 0,119 0,216 2, ,34 3,53 2,3 2,051 3,536 3,605 4,054 2, ,34 3,53 2,3 4,824 3,536 3,605 4,054 2, ,34 3,53 2,3 2,226 3,536 3,605 4,054 2, ,34 3,53 2,3 4,629 3,536 3,605 4,054 2, ,1 0,1 0, ,012 0, ,1 0,1 0, ,012 0, ,82 1,87 0,1 1,084 1,98 1,982 0,012 0, ,82 1,87 0,1 2,62 1,98 1,982 0,012 0, ,67 0,791 0,8 0,527 0,786 0,782 2,473 2, ,67 0,791 0,8 1,036 0,786 0,782 1,544 2, ,67 0,791 0,8 0,453 0,786 0,782 2,644 2, ,67 0,791 0,8 1,119 0,786 0,782 1,401 2, ,32 0,152 1,8 0,096 0,068 0,112 1,544 2, ,32 0,152 1,8 0,053 0,068 0,112 1,401 2, ,32 0,152 1,8 0,136 0,068 0,112 2,473 2, ,32 0,152 1,8 0,194 0,068 0,112 2,644 2, ,47 3,97 3 5,201 4,003 3,949 2,473 2, ,47 3,97 3 2,678 4,003 3,949 1,544 2, ,47 3,97 3 2,265 4,003 3,949 1,401 2, ,47 3,97 3 5,671 4,003 3,949 2,644 2,206 Inter W 466,8 467, ,1 467,6 466,4 584,5 675,8 Table 3. Results of weght optmzaton of space truss accordng to the varous optons and methods 8

10 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ Fgure 6. Dagram of weght relatonshp of space truss depends on the number of teratons wth the optons of desgn varable group Fgure 7. Dagram of weght relatonshp of space truss and number of teratons by varous methods 5.3 Comments On the bass of the results of weght optmzaton of plane and space trusses shown n paragraph 4, the followng comments can be gven: - The method of correlaton coeffcent between element nternal forces establshed by the authors gves the results of weght optmzaton of plane and space trusses wth a neglgble dfference (0.1%) compared wth the results of optmzaton by other methods. - Usng the method of correlaton coeffcent between element nternal forces requres sgnfcantly less number of teratons than other methods. Ths s a remarkable advantage n optmzaton problems of large-scale space trusses. The dvson of desgn varables groups affects the results and the convergence rate of the problem. In order to reduce the truss weght, t s necessary to select the element wth the same nternal force n the same group of desgn varables. The opton of the ndependent varables (opton 0) gves results n the smallest weght and the slowest convergence. 6. Conclusons The method of correlaton coeffcent between element nternal forces establshed by the authors s a smple and effectve method for weght optmzaton of plane and space trusses. In the next studes, the 9

11 IOP Publshng IOP Conf. Seres: Materals Scence and Engneerng (2018) do: / x/365/4/ authors wll contnue to develop the establshed algorthms to solve problems of shape and structure optmzaton of trusses. References [1] Lê Xuân Huỳnh, Tính toán kết cấu theo lý thuyết tố ưu, Nhà xuất bản khoa học và kỹ thuật, [2] Cao Quốc Khánh, Nghên cứu phương pháp tính tố ưu dàn phẳng, Luận văn thạc sĩ kỹ thuật xây dựng công trình dân dụng và công nghệp, Trường Đạ học Kến trúc Hà Nộ, [3] Võ Như Cầu, Tính kết cấu theo phương pháp phần tử hữu hạn, NXB Xây Dựng, Hà Nộ, [4] Al Kaveh & Omd Khadem Hossen, A hybrd HS-CSS algorthm for smultaneous analyss, desgn and optmzaton of trusses va force method, Perodca Polytechnca Cvl Engneerng 56/2 (2012). [5] Al Kaveh & Al Zolghadr, A mult-set charged system search for truss optmzaton wth varables of dfferent natures; element groupng, Perodca Polytechnca Cvl Engneerng (55/2 2011). [6] Al Alna-zazand Behrooz Farsh, Szng optmzaton of truss structures by method of centers and force formulaton, Internatonal journal of solds and structures 47 (2010). [7] A.Kaveh and M.Hassan,Smltaneous analyss, desgn and optmzaton of structures usng force method and ant colony algorthms, Asa journal of cvl engneerng vol 10, no 4 (2009). [8] Domnque Chamoret, Kepeng Qu and Maththeu Domaszewsk, Optmzaton of truss structures by a stochastc method, ASMDO (2009) [9] Hatham Farah Hanna Al Rabad, Truss sze and topology optmzaton usng harmony search method, Unversty of Iowa (2014). [10] J.N. Reddy, An Introducton to the Fnte Element Method Thrd Edton, Texas A&M Unversty College Staton, Texas, USA (2006). [11] Reza Najan Asl & Mohamad Aslan & Masoud Sharat Panah, Szng Optmzaton of Truss Structures usng a Hybrdzed Genetc Algorthm, (June 7, 2013). [12] S.B.Wade & K.N.Kadam (2015), Dscrete Sze Optmzaton of Indetermnate Truss Usng HyperWork, Journal of Cvl Engneerng and Envronmental Technology, (Volume 2, Number 10). [13] Shh CJ& Yang YC, Generalzed Hopfeld network based structural optmzaton usng sequental unconstraned mnmzaton technque wth addtonal penalty strategy, Advances n Engneerng Software 32 (2002), , DOI /S (02)