OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS

Transcription

1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING It. J. Optm. Cvl Eg., 2017; 7(1): Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS M. Rezaee-Pajad *, ad H. Afsharmoghadam Cvl Egeerg Departmet, Ferdows Uversty of Mashhad ABSTRACT I ths paper, the effect of agle betwee predctor ad corrector surfaces o the structural aalyss s vestgated. Two objectve fuctos are formulated based o ths agle ad also the load factor. Optmzg these fuctos, ad usg the structural equlbrum path s geometry, lead to two ew costrats for the olear solver. Besdes, oe more formula s acheved, whch was prevously foud by other researchers, va a dfferet mathematcal process. Several bechmark structures, whch have geometrc olear behavor, are aalyzed wth the proposed methods. The fte elemet method s utlzed to aalyze these problems. The abltes of suggested schemes are evaluated tracg the complex equlbrum paths. Moreover, comparso study for the requred umber of cremets ad teratos s performed. Results reflect the robustess of the authors formulatos. Keywords: optmzato; path-followg aalyss; geometrc olear behavor; costrat equalty; load-dsplacemet curve; lmt pots. Receved: 2 Aprl 2016; Accepted: 12 Jue INTRODUCTION Nolear aalyss expresses the real behavor of structures uder dfferet types of loadg. I other words, to fd actual structural performaces, materal or geometrc olear behavor should be vestgated. So far, a lot of dfferet strateges for olear structural aalyss have bee suggested. These techques have ther ow merts ad demerts, ad yet; o perfect procedure s avalable. As otfed the lterature revew, a accurate method to trace the whole equlbrum paths of all structures has ot bee proposed utl ow. * Correspodg author: Cvl Egeerg Departmet, Ferdows Uversty of Mashhad E-mal address: rezaee@um.ac.r (M. Rezaee-Pajad)

2 110 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Newto-Raphso algorthm s oe of the most popular foudatos of teratve methods [1]. Due to the falure of Newto-Raphso techques o crossg sap-through regos, the dsplacemet cotrol procedure was troduced [2]. Ths soluto was ot able to passg through sap-back regos. I the 70 s, Wemper [3] ad Rks [4] veted arc legth methods. They defed the arc legth parameter as the dstace of the last statc pot to the supposed terato path. The approach of fdg arc legth parameter led frst to the ormal plae method [5] ad the, to the updated ormal plae techque [6]. I the costrat equato of the cyldrcal arc legth method, Crsfeld gored the force compoet [7]. He amed hs proposed solver, modfed Rks-Wemper procedure. Subsequetly, Tsa et al. used arc legth algorthm fte elemet method for aalyzg of a composte cyldrcal shell-lke structure [8]. Some researchers mmzed effectve varables of the olear aalyss. Reduced resdual load by Berga [9], resdual dsplacemet by Cha [10] ad resdual legth, permeter ad area the research of Rezaee-Pajad et al. [11], were mmzed. I 1990, Yag et al. troduced geeralzed dsplacemet cotrol (GDC) method that could catch both load, ad dsplacemet lmt pots [12]. Cardoso et al. detected ths effcet strategy, GDC method, as a orthogoal cyldrcal arc legth method [13]. I ormal flow scheme, sequetal teratve aalyses o les, whch were perpedcular to Davdeko s flow, were mplemeted to reach the structural equlbrum path [14]. Afterward, Saffar et al. formulated mproved ormal flow scheme [15]. O the other had, the dyamc relaxato (DR) method was performed for post-bucklg aalyss of trusses [16]. Rezaee-Pajad et al. appled DR techque the olear aalyss for varous structures [17, 18]. Ths capable algorthm set up a fcttous dyamc system to solve the olear system of equatos goverg the structural behavor. Recetly, Rezaee-Pajad et al. created a effcet strategy by combg DR method wth load factor ad dsplacemet cremets [19]. From a strctly mathematcal vewpot, t could be mpled to mult-pot procedures wth dfferet covergece [20, 21]. I a extesve research, the geometrc olear aalyss methods of structures were vestgated ad compared wth each other [22]. Geometrc olear behavor s due to the structural large deformatos. To perform ths aalyss, researchers foud dverse costrat relatos wth dfferet assumptos. Accordg to the related lterature, o accurate method to trace the whole equlbrum paths of all structures has ot bee proposed utl ow. I the preset study, the geometres of predctor ad corrector steps path are surveyed, ad two objectve fuctos are formed. Each of them has two depedet varables. By ths mathematcal base, two ovel costrat equatos are created. Furthermore, oe more costrat formula s cocluded, whch was foud by other vestgators a dfferet way. Ths outcome clearly demostrates the valdty of assumptos ad formulatos process. It has bee show that the geometrc olear behavor ca lead to varous complex structural statc paths. To explore these performaces, ad show the abltes of the authors techques, several two ad three-dmesoal trusses ad plaar frames are olearly aalyzed by exertg proposed costrats.

3 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS SOLVING NONLINEAR EQUATIONS Goverg equato of the olear behavor of structures s as follows: Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 R ( u, ) P F( u) (1) I the curret equalty, dsplacemet vector, load factor, resdual load vector, exteral force vector ad teral force vector are show by u,, Ru (, ), P ad Fu ( ), respectvely. Resdual load vector depeds o the dsplacemet ad load factor varables. It has bee emphaszed that the load factor has a mportat role the olear structural aalyss. If the dsplacemet vector has m arrays, the the system of Equatos (1) wll have m+1 ukows. The extra ukow for a structure wth m degrees of freedom s because of. Hece, for calculatg the ukows, oe more relato s requred addto to Equatos (1). Based o Fg. 1, the -th cremetal step, the process of aalyss s accomplshed to fd the structural equlbrum curve betwee -1-th ad -th pots. I the predctor step, to dscover dsplacemet cremet Equato (2), a sutable cremetal load should be supposed. u ( K ) P (2) I ths relato, K s the tagetal stffess matrx at the -1-th pot of the structural statc curve. I the -th cremetal step, the coordates of the frst terato pot s accessed by Equalty (2). Cosecutve teratos are doe to get beyod the pot wth a defed tolerace. The load factor of each terato s computed by costrat relato. It should be metoed, dsplacemet cremet of the successve terato steps s calculated by followg lear equalty [23]: u (3) Superscrpt ad subscrpt represet the cremet ad terato umber of aalyss, u respectvely. I the recet equato, ad are the dsplacemet cremets due to resdual load ad exteral force, correspodgly. These vectors are defed the followg relatos: ( K ) R (4) 1 1 u ( K ) P (5) I fact, the values of forces R ad P are kow at the begg of each terato. Therefore, ther related dsplacemets are avalable. Accordg to Equato (3), fdg u

4 112 M. Rezaee-Pajad ad H. Afsharmoghadam oly depeds o the load factor. Based o the succeedg relatos, the load factor ad dsplacemet cremets of structure are gaed by addg ad u to ther prevous cremets, correspodgly: Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 (6) 1 u u u (7) 1 It s worth metog that the former proposed methods, the postos of the predctor s path ad the corrector s path were assumed arbtrary ad the costrat equalty was obtaed [5-7]. O the cotrary, ths study optmal agle betwee the predctor s path ad the corrector s path s acheved by usg mathematcal tools alog wth the geometry of the equlbrum curve. Based o the optmzed crtera, the relatos requred for the structural aalyss are formulated. 2.1 Frst ew method Fgure 1. The process of advaced cremetal-teratve methods I ths procedure, the agle, betwee the taget of the statc pots o the structural equlbrum path ad the trajectory whch passes through the sequetal teratve aalyses, s

5 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 113 assumed as. Outle of the proposed scheme for the j-th degree of freedom of structure s show Fg. 2. Based o ths fgure, the ext relatos betwee the specfed agles are avalable: Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th ta( ) ta( ( )) ta( ) (8) where, α ad β are the agles betwee the taget of the statc pots ad ts relevat teratve aalyses path wth the dsplacemet axs, correspodgly. I accordace wth ths defto, the taget of these agles ca be determed the below forms: P ta( ) (9) u K ta( ) (10) Utlzg the trgoometry commads for Equato (8) ad substtutg Equato (9) ad (10) t, the succeedg equalty s foud: P K ta( ) K ta( ) 1 u Isertg Equatos (3) ad (5) to (11) ad smplfyg t, coclude the followg relatoshp: ta( ) P P 2 P P ta( ) T T T T T ta( ) 0 (11) (12) It s possble to defe the left had of the curret equato as a goal fucto. Cosequetly, the ext fucto, G, terms of two depedet varables s establshed: G (, ) ta( ) P P 2 P P ta( ) T T T T ta( ) T (13) The, the process of optmzg Fucto (13), wth respect to the agle ad the load factor, has the comg shapes: G ta ( ) P T P 2 P T ta ( ) T 0 (14)

6 114 M. Rezaee-Pajad ad H. Afsharmoghadam G 2 (1 ta ( )) ( T T T P P u u u u ) 0 (15) Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 The costrat equalty of the frst suggested method s obtaed by solvg the system of two Equatos (14) ad (15). As a result, the followg load factor s calculated: T T T P P Ths costrat s used for the geometrc olear aalyss of the bechmark structures the ext secto. If the load compoet of Equalty (16) s eglected, the famlar relato s attaed, as follows: T T The last metoed approxmato leads to the outcome of the well-kow procedure, called the mmum resdual dsplacemet method [10]. I other words, the authors formulato provdes alteratve proof of the former scheme, ad t shows the geeralty ad rghtess of the suggested techque. (16) (17) Fgure 2: Frst proposed scheme

7 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 115 Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th Secod ew method 1 I ths way, the tagetal stffess matrx the -1-th pot of the equlbrum path ( K ) s utlzed. From the geometrc vewpot, ths stffess matrx s deoted Fg. 1. I fact, t 1 s assumed that the stffess matrx of teratve steps ( K ) s equal to K. By replacg Equato (3) to Equato (11), the equalty s rewrtte the succeedg form: ta( ) K P P K u K u ta( ) u ta( ) 0 Accordg to the curret relato, the ext two-varable goal fucto, H, s proposed: H K P P K u K u u (, ) ta( ) ta( ) ta( ) To optmze ths fucto, ts dfferetato wth respect to the depedet varables should be take, as follows: H 1 1 ta ( ) K P P K ta ( ) 0 H 2 1 (1 ta ( )) ( K P ) 0 Solvg the last system of equatos results the followg load factor: 1 K P (18) (19) (20) (21) (22) By substtutg Equato (5) the recet equalty, the last relato ca be rewrtte the below form: K K K P P (23) The subsequet vector form of the secod proposed costrat s acheved by sertg Equato (2) to the curret relato: P T 1 1T 1 T ( K K 1) P u1 (24)

8 116 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 To perform the aalyss process, stffess matrx must be tured to vector form. To acheve ths goal, oly ma dagoal arrays of the stffess matrx are utlzed. By ths acto, the stffess matrx chages to a vector wth m compoets. I ths paper, Costrat (24) s also used for the aalyss of bechmark structures. To summarze the outcomes, the ovel costrats are wrtte Table 1. Method Frst method Secod method Table 1: Proposed costrats Costrat T T T P P P T 1 1T 1 T ( K K 1) P u1 3. NUMERICAL SAMPLES Based o the proposed Equatos (16) ad (24), a olear geometrc aalyss computer program s provded. To be sure, the school program was utlzed to fd the accurate aswers, as well. For years, ths olear fte elemet program has bee used the authors egeerg school, ad t has bee proved to be free of errors. Several bechmark problems wth geometrc olear behavor are vestgated. Wth respect to the umber of cremets ad teratos eeded to fd the structural equlbrum paths; the abltes of the proposed techques are evaluated. It should be emphaszed; the aalyss mportat propertes of each sample are otfed ts subsecto. 3.1 Seve-member truss As demostrated Fg. 3, the plaar truss structure s uder cocetrated load P. Ths structure has seve degrees of freedom. The cross-sectoal area of the horzotal members s cm 2 ad for the others s cm 2, correspodgly. The members modulus of elastcty s kn/cm 2. Ths truss was used to verfy hgher-order stffess matrx the predcto of structural behavor [24]. Furthermore, t was employed for vestgatg the elastc bucklg of members [25]. I ths research, the referece load, the arc legth of the frst loadg step, the maxmum umber of teratos each cremet ad the resdual error are assumed to be 1 kn, 0.01, 10 ad 10-4, respectvely. The load-deformato curves of ode 1 the vertcal drecto, ad for the ode 2 the horzotal drecto are draw Fg. 4 ad 5, respectvely. Accordg to the results, both structural equlbrum paths have the load ad dsplacemet lmt pots. I spte of the fact that these odes have complex behavoral curves, both proposed methods are able to trace etrely the relevat load-dsplacemet dagram.

9 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 117 Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Fgure 3. Seve-member truss Fgure 4. The statc curve of seve-member truss of ode 1 at vertcal drecto Fgure 5. The statc curve of seve-member truss of ode 2 at horzotal drecto The umber of cremets ad teratos are lsted Table 2. It should be oted, these resposes for ode 1 are aalogous to ode 2. Accordg to the obtaed results, the frst techque ca termate the olear aalyss wth much fewer cremets ad teratos tha the secod soluto. I other words, although the secod procedure captures all lmt pots of truss aalyss, but t requres more tme for aalyzg the structure. Cosequetly, the frst proposed method s more powerful solvg ths problem.

10 118 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th Two-member truss Table 2: Aalyss results of seve-member truss for ode 1 or 2 Costrat Number of cremets Number of teratos Frst method Secod method A two-dmesoal truss wth two degrees of freedom uder a cocetrated load, s dsplayed Fg. 6. Youg s modulus of the members s kn/cm 2 ad ther crosssectoal areas are cm 2. Ths bechmark was umercally aalyzed by Papadrakaks [16]. I aother research, ths truss was aalyzed to check the accuracy of the hgher-order stffess matrx [24]. To solve ths two-member truss, the referece load, arc legth of the tal loadg step, the maxmum umber of teratos each cremet ad the tolerace of the respose covergece are cosdered to be 1 kn, 0.1, 10 ad 10-4, correspodgly. Fgure 6. Two-member truss Fg. 7 dcates the structural equlbrum path of ode 1 the vertcal drecto whch s obtaed by the authors schemes. Evdetly, both ew strateges ca pass through the load lmt pot. Based o the Fg. 7, the crtcal load value of ths truss s compatble wth the oe obtaed by Torkama et al. [24]. Fgure 7. The equlbrum curve of two-member truss s ode 1 at vertcal drecto

11 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 119 Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 I fact, the secod recommeded procedure s a approxmately method because of selectg oly the dagoal stffess matrx arrays for the aalyss process. Nevertheless, based o Table 3, ths procedure traces the equlbrum curve wth fewer umbers of cremets ad teratos tha the frst oe. By a lttle dfferece, aalyss of the twomember truss va the secod strategy takes the frst rak. 3.3 Fve-story frame Table 3: Aalyss results of two-member truss Costrat Number of cremets Number of teratos Frst method Secod method Fg. 8 depcts a plaar frame uder the horzotal cocetrated forces, ad the vertcal uform dstrbuted loads. The uform dstrbuted loads, appled to all beams, have the value of 10 kn/cm. Ths structure s modeled as a 29-elemet frame by supposg every beam or colum member as oe elemet. It should be added that the dyamc relaxato method was utlzed for aalyzg ths bechmark problem, as well [26]. The elastcty modulus of all members s kn/cm 2. Other specfcatos of structure s beams ad colums are regstered Table 4. It should be formed; the equvalet odal forces ad momets of the uform dstrbuted loads are computed ad used for the puts of the computer program. I ths paper, arc legth of the frst loadg step, maxmum teratos of each cremet ad resdual error equal to 0.1, 10 ad 10-4, respectvely. Fgure 8. Fve-story frame

12 120 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Table 4: The propertes of fve-story frame s beams ad colums Member Cross-sectoal area (cm 2 ) Momet of erta (cm 4 ) Beam Colum The load-dsplacemet resposes of the fve-story frame s roof level va ovel costrats are depcted Fg. 9. Accordg to ths chart, both algorthms have the ablty to fd the structural equlbrum path for ths frame. It should be added that the resulted curves are fully compatble wth the other studes [26]. Fgure 9. The load-dsplacemet path of fve-story frame s roof at horzotal drecto Comparg the fdgs of olear aalyss recorded Table 5, llustrates that the secod method eeds teratos to aalyze the structure. However, wth the same umber of cremets, the frst method completes the load-deflecto curve of the frame wth teratos. Thus, both ovel solutos aalyze ths two-dmesoal frame almost the same tme. Table 5: Aalyss results of fve-story frame Costrat Number of cremets Number of teratos Frst method Secod method Star shaped dome truss A three-dmesoal truss of Fg. 10 s uder the vertcal load P = 1 kn the ceter. Ths dome has 13 odes ad 24 members. Youg s modulus ad cross-sectoal area of members equal to N/cm 2 ad 3.17 cm 2, correspodgly. Ths bechmark structure s mostly cosdered for olear aalyss of three-dmesoal trusses. For stace, star shaped dome truss was used to vestgate the bucklg effect o the overall stablty of structure [27] ad also assessg the ablty of a automatc method to reveal the more precse

13 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 121 structural equlbrum path [28]. I addto, Rezaee-Pajad et al. utlzed the dyamc relaxato process for tracg the structural equlbrum curve of ths truss [29, 30]. I ths study, the tal loadg step s arc legth, maxmum teratos of each cremet ad the resdual error are assumed 0.1, 10 ad 10-4, respectvely. Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Fgure 10. Pla ad vew of star shaped dome truss At the frst stage, the vertcal dsplacemet of ode 1 s studed. The obtaed loaddsplacemet graph s exposed Fg. 11. It should be added that the vertcal axs of the graph s show dmesoless. Ths truss has sap-through behavor at ode 1. Based o Fg. 11, the frst proposed algorthm ca pass lmt pots ad yelds the whole equlbrum path. Whereas, the secod suggested solver s ot able to catch the frst lmt pot, ad the related aalyss was stopped there. Fgure 11. The load-deflecto curve of star shaped dome truss of ode 1 at vertcal drecto The total umber of aalyss cremets ad teratos are serted Table 6. The frst ew techque traverses the curve, whch s show Fg. 11, wth 500 cremets ad 1006 teratos. The secod ew procedure s capable of tracg small part of the structural equlbrum path by 160 cremets ad 350 teratos.

14 122 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Table 6: Aalyss results of star shaped dome truss for ode 1 or 2 Costrat Number of cremets Number of teratos Frst method Secod method Fgure 12. The load-deflecto curve of star shaped dome truss of ode 2 at vertcal drecto I aother aalyss of ths three-dmesoal structure, the vertcal dsplacemet of the ode 2 s checked. The related equlbrum curve has sap-through ad sap-back regos. I accordace wth Fg. 12 ad smlar to the results terpretato of ode 1, the frst ovel procedure crosses every lmt pots successfully, but the other preseted algorthm fals to catch the frst lmt pot. It should be poted out; the obtaed results are compatble wth predecessor fdgs [27]. The sgfcat results of ths ode are aalogous to the oes o Table 6. Evetually, t s cocluded that the frst strategy, whch ca accurately trace the structural equlbrum path for the specfed odes, s superor tha the secod oe. 3.5 Schwedler s dome truss Pla ad vew of the Schwedler s dome truss are dsplayed Fg. 13. Ths truss has 97 odes ad 264 members. All the perpheral odes are hged at the support. All axal stffess of the members are kn. Load P, whch s appled o the cetral ode, equals to 1 kn. To fd the geometrc olear behavor of the three-dmesoal trusses; Schwedler's truss was used [31]. Rezaee-Pajad et al. aalyzed ths bechmark problem to verfy orthogoal strateges ad also mmum resdual legth, permeter ad area methods [11]. Moreover, Saffar et al. gaed the structural equlbrum path of ths dome structure by applyg the two-pot strategy [20]. I ths artcle, puts of the program for the arc legth of tal loadg step, maxmum umber of teratos each cremet ad resdual error are 0.1, 10 ad 10-4, correspodgly.

15 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 123 Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Fgure 13. Pla ad vew of Schwedler s dome truss To vestgate the ablty of ovel costrats gve Table 1, Schwedler s dome truss s aalyzed. The load-dsplacemet curve of the vertcal dsplacemet of the hghest ode s demostrated Fg. 14. It ca be see, there are two load lmt pots ths structural equlbrum curve. The frst solver captures the lmt pots ad etrely traverses the relevat equlbrum path. However, the secod strategy just represets a small part of the cetral ode s behavor. Fgure 14. The load-deformato curve of Schwedler s dome truss s crest at vertcal drecto I Table 7, two ma parameters of aalyss are regstered. The frst soluto thoroughly traces the structural load-dsplacemet curve wth 5209 cremets ad teratos. The secod solver completely passes the structural equlbrum curve. It s stopped the 576-th cremet ad wth 5760 teratos. Hece, the frst preseted solver s successful the olear aalyss of ths structure.

16 124 M. Rezaee-Pajad ad H. Afsharmoghadam Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th Arc frame Table 7: Aalyss results of Schwedler s dome truss Costrat Number of cremets Number of teratos Frst method Secod method Fg. 15 dsplays a two-dmesoal structure uder a asymmetrc loadg. Ths arc structure s desged by authors ad s modeled by 33 elemets. I addto, ths frame has two fxed supports. The elastcty modulus of all members s kn/cm 2. Furthermore, the crosssectoal area ad the momet of erta of the members are equal to cm 2 ad cm 4, correspodgly. It should be metoed, the referece load, the arc legth of the frst loadg step, maxmum teratos of each cremet ad the resdual error are cosdered 1 kn, 0.1, 10 ad 10-4, respectvely. The load-dsplacemet resposes of the arc frame for ode 1 are show Fg. 16. Accordg to ths dagram, both ew schemes ca represet the same structural equlbrum path. It should be oted that the obtaed curves are compatble wth the results of the other algorthms, as well. Fgure 15. Arc frame Fgure 16. The load-dsplacemet path of arc frame of ode 1 at vertcal drecto

17 OPTIMIZATION FORMULATION FOR NONLINEAR STRUCTURAL ANALYSIS 125 The fdgs of olear aalyss are recorded Table 8. As llustrated ths table, the secod ovel strategy aalyzes the frame wth fewer umbers of cremets ad teratos tha the frst suggested techque. I other words, usg the secod costrat of Table 1, for olear aalyss of ths structure, s more effcet tha aother soluto. Dowloaded from joce.ust.ac.r at 7:24 IRDT o Saturday July 7th 2018 Table 8: Aalyss results of arc frame Costrat Number of cremets Number of teratos Frst method Secod method CONCLUSION I the advaced cremetal-teratve methods, a load factor s assumed at the begg of each aalyss cremet. Ths part s called the predctor step. By performg the teratve process, the hypothetcal obtaed pot wll coverge towards the structural equlbrum path. Ths stage s called the corrector step. I ths paper, two objectve fuctos were establshed. These fuctos cosst of two depedet varables, amely, the load factor ad the agle betwee predctor ad corrector surfaces. By optmzg these goal fuctos, two ew costrat equaltes for the load factor cremet were obtaed. Sce the authors formulas were geeral, oe more costrat equato was also foud that was smlar to oe of the former olear solvers. By ths ew way, the prevous method was oce aga mathematcally verfed. To evaluate the ovel schemes, proposed costrats were appled the geometrc olear aalyss of several truss ad frame structures. The bechmark samples had sap-through or sap-back behavors. The outcomes of umercal tests llustrated that the frst recommeded techque could trace structural equlbrum paths sap-through ad sap-back regos ad had acceptable compatblty wth the prevous researches. O the other had, based o the umercal results, the frst suggested techque was more qualfed tha the other olear solver. It should be emphaszed; the secod proposed method s rooted the use of oly ma dagoal arrays of the structural stffess matrx formg the related costrat equato. REFERENCES 1. Che WF, Lu EM. Stablty Desg of Steel Frames, CRC press, Zekewcz OC. Icremetal dsplacemet o lear aalyss, It J Numer Meth Eg 1971; 3(4): Wemper GA. Dscrete approxmatos related to olear theores of solds, It J Solds Struct 1971; 7(11): Rks E. The applcato of Newto s method to the problem of elastc stablty, J Appl Mech 1972; 39(4):

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