UDK ; Linearna in nelinearna ravnovesna enačba v računalniško podprtem konstruiranju A Linear and a Nonlinear Equilibrium Equation i

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1 UDK ; Linearna in nelinearna ravnovesna enačba v računalniško podprtem konstruiranju A Linear and a Nonlinear Equilibrium Equation in CAD of Structures MAKS OBLAK - MARKO KEGL - BRANKO BUTINAR V prispevku obravnavamo vpeljavo nelinearne ravnovesne enačbe v postopek optimalnega projektiranja konstrukcij, pri katerem se je doslej v glavnem uporabljala linearna enačba. Če v nelinearni enačbi upoštevamo končne pomike in zasuke, naletimo pri tem na težave v zvezi z nestabilno ravnovesno krivuljo konstrukcije. Standardno optimalno projektiranje je zaradi tega praktično neuporabno. Nov prijem, ki je predstavljen v prispevku, temelji na modificiranem problemu optimalnega projektiranja in spremenljivi ravni obremenitve. Ponazorjen je na dveh zgledih, na katerih je tudi narejena primerjava med uporabo linearne in nelinearne enačbe. The paper discusses the introduction of a nonlinear equilibrium equation into the optimal design of structures where a linear equation has been mostly used to date. If kinematic nonlinear effects enter the equilibrium equation, difficulties can arise owing to an unstable equilibrium path of the structure. Because of that, the classic approach to optimal design becomes practically unusable. The new approach presented in this paper is based on a modified problem of optimal design and a variable load level. It is illustrated with two examples where the use of linear and nonlinear equilibrium equations is also compared. 0. UVOD Optimalno izbiranje konstrukcijskih parametrov pomeni dandanes že sestavni del računalniško podprtega projektiranja (CAD) konstrukcij. Gre za avtomatično določanje optimalnih vrednosti nekaterih konstrukcijskih parametrov, ki do te faze projektiranja še niso izbrane, na primer dimenzije in oblike robov posameznih delov konstrukcije, karakteristike geometrijske oblike prerezov, vrste materiala, smeri podprtja konstrukcije in podobno. V računalniško podprtem projektiranju konstrukcij predpostavljamo, da imamo na voljo ravnovesno enačbo, iz katere lahko pri danih obremenitvah izračunamo odziv, ki dovolj dobro ustreza dejanskemu. Rutinsko nastajanje takšne enačbe omogoča za večino konstrukcij metoda končnih elementov (MKE). Če so izpolnjene zahteve: a) pomiki vozlišč so zanemarljivo majhni, b) robni pogoji se med obremenjevanjem ne spreminjajo, c) material je linearno elastičen, je ravnovesna enačba linearna. V nasprotnem primeru je linearna enačba neustrezna 111 in uporabiti moramo nelinearno. Glede na izpolnjevanje pogojev a) do c) delimo vzroke za nelinearnost enačbe konstrukcije v kinematične in materialne. Prvi so posledica neizpolnjenih pogojev a) ali b), drugi pa neizpolnjenega pogoja c). 0. INTRODUCTION Optimal selection of design parameters - optimal design - is today an integral part of computer aided design (CAD) of structures. It involves the automatic determination of optimal values of some design parameters which are still undetermined in the current stage of design, for example, dimensions and shapes of structural parts, geometric properties of cross-sections, material properties, directions of supports and so on. In CAD of structures, we assume that we have an equilibrium equation at our disposal from which we can calculate the structural response for given loads with satisfactory accuracy. For most structures, this equation can be successfully generated using the finite element method (FEM). If the following requirements are fulfilled: a) nodal displacements are negligibly small, b) boundary conditions do not change during loading. c) the material is linear elastic, then the equilibrium equation can be linear. In the opposite case, a linear equation is not appropriate 111 and one has to use a nonlinear one. With regard to the fulfillment of conditions a) to c), the nonlinearities are divided into kinematic and material nonlinear efects. The first emerge if conditions a) or b) are not fulfilled, while the second correspond to unfulfilled condition c).

2 Razvoj optimalnega projektiranja v okviru CAD je v zadnjih dvajsetih letih temeljil na uporabi linearne ravnovesne enačbe, kar je tudi razvidno iz ponudbe komercialno dosegljivih računalniških programov. Uporaba nelinearne enačbe se v inženirski praksi še ni uveljavila predvsem zaradi težav, ki preprečujejo, da bi v že razvitem postopku optimalnega projektiranja linearno enačbo konstrukcije zamenjali z nelinearno. Te se lahko pojavijo pri neizpolnjevanju pogoja a) in se nanašajo na stabilnostne lastnosti mehanskega sistema. Nekaj poskusov odpravljanja teh težav smo v zadnjih letih sicer lahko opazili 121, vendar so bili rezultati precej skromni. V Laboratoriju za optimiranje mehanskih sistemov na Tehniški fakulteti v Mariboru smo razvili nov način optimalnega projektiranja statično obremenjenih konstrukcij, ki omogoča učinkovito uporabo nelinearne ravnovesne enačbe tudi pri upoštevanju velikih pomikov. Na tej podlagi smo razvili programsko opremo za optimalno projektiranje paličnih konstrukcij in na zgledih potrdili uporabnost novega načina. 1. RAVNOVESNA ENAČBA KONSTRUKCIJE IN RAVNOVESNA KRIVULJA Če predpostavimo uporabo metode končnih elementov in se omejimo na statično obremenjene konstrukcije brez učinkov lezenja, lahko ravnovesno enačbo konstrukcije v splošni obliki zapišemo 111: 0(1 n n kjer je Q( u) e M, u e IR pa je vektor vozliščnih pomikov in zasukov. Enačbo (1) imenujemo linearna ravnovesna enačba, če je funkcija Q linearna glede na spremenljivko u. V tem primeru (1) zapišemo v običajni obliki: K u kjer sta /ftogostna matrika konstrukcije, Rn pa vektor zunanjih vozliščnih obremenitev. Togostna matrika K in vektor obremenitev R sta konstantna in neodvisna od pomikov u. Reševanje ravnovesne enačbe je z matematičnega vidika neproblematično, saj je treba rešiti sistem linearnih enačb, čigar matrika je simetrična, pasovna in vedno dobro pogojena. Kadar je funkcija Q nelinearna glede na spremenljivko u, imamo nelinearno ravnovesno enačbo. V tem primeru togostna matrika konstrukcije ali vektor zunanjih obremenitev nista konstantna in sta odvisna od pomikov u. Reševanje nelinearne enačbe nadomestimo 131 z reševanjem zaporedja enačb: Development in the field of optimum design in CAD was based on the use of a linear equilibrium equation, which is also evident from the supply of commercially available computer programs. The use of a nonlinear equation could not be established in practical engineering tasks owing to difficulties in replacing a linear equation with a nonlinear one in a well-formed procedure of optimal design. These difficulties, which concern structural stability properties, can appear if condition a) is not fulfilled. Some attempts to solve these difficulties have already been made 121, but the results have been rather modest. In the Laboratory for optimum design of mechanical systems at the Technical Faculty in Maribor, a new approach to optimum design of statically loaded structures has been developed, which enables effective use of a nonlinear equilibrium equation accounting also for large displacements. On this basis, software for the optimum design of truss structures has been developed and its usefulness has been confirmed on several examples. 1. THE EQUILIBRIUM EQUATION OF THE STRUCTURE AND THE EQUILIBRIUM PATH If we suppose the use of the finite element method and if we restrict ourselves to statically loaded structures, we can write the equilibrium equation of the structure in a general form 111 as: = 0 (1), where Q( u) e R n, u e R " represents the vector of nodal displacements and rotations. Equation (1) is termed to be a linear equilibrium equation, if the function Q is linear with respect to the variable u. In this case (1) is commonly written as: R where K is the structural stiffness matrix and R e [R71 is the vector of external nodal loads. From the mathematical point of view, solving the linear equilibrium equation causes no difficulties since it is necessary to solve a linear system of equations where the coefficient matrix is symmetric, banded and always well-conditioned. When the function Q is nonlinear with respect to the variable u, the equilibrium equation is nonlinear. In that case, the structural stiffness matrix or the external load vector are not constant but depend on displacements u. The usual procedure for solving a nonlinear equilibrium equation is to solve a sequence of equations 131.

3 Q{u,t) =O, t= t0, t i...,t j, ć0 = 0,fj = l (2), kjer t pomeni neodvisno spremenljivko, ki ima pri statičnih problemih brez učinkov lezenja vlogo označevanja intenzivnosti obremenitve (inkrementni postopek). Vsaka enačba iz zaporedja (2) je sicer nelinearna, vendar lahko njeno rešitev tu izračunamo preprosteje, če je obremenitveni inkrement ( - tj ) majhen. Postopek reševanja je iteracijski in ga lahko z označbami r = J+, in s = tj zapišemo 111 : r (o) s II = II ; where t represents an independent variable, which in static analysis without creep effects merely denotes the imposed load level (incremental procedure). Each equation from the sequence (2) is still nonlinear, but its solution tl can be more easily obtained if the load increment ( tj +1 - tj ) is small. The solution procedure is iterative and can be written 111 with the symbols r = tj+i and s = tj as: r ( i +1 ) r ( i ) II = II + kjer je N0 množica pozitivnih celih števil. Pri tem izračunamo Au(i+1) iz enačbe: where is a set of non-negative integer numbers. The increment A u 1+ is calculated from: v o) kjer je qk tangentna togostna matrika 111 konstrukcije pri ravni obremenitve q t Itj, J + 11, ki je odvisen od izbrane tehnike reševanja, J5*e 1R pa je vektor notranjih sil, izračunan na podlagi napetostne porazdelitve, pri nivoju obremenitve r. Reševanje (3) zahteva invertiranje tangentne togostne matrike. Če so zunanje obremenitve neodvisne od pomikov, je ta matrika sicer še simetrična in pasovna, ni pa več vedno dobro pogojena. Z vpeljavo neodvisne spremenljivke t lahko definiramo množico: = > where qk is the structural tangent stiffness matrix 111 at load level q? 1.-, J + 11, q depend on the chosen solution technique, and 'F e IRn is a vector of internal forces (calculated from the stress distribtion) at load level r. To solve (3) it is necessary to invert the tangent stiffness matrix. If the external loads are displacement independent, this matrix is still symmetric and banded, but it is no longer always well- -conditioned. By introducing the independent variable t one can define the set: ki jo imenujemo ravnovesna krivulja konstrukcije. Ravnovesna krivulja je v točki tu stabilna 141, če je d etc /f) > O, oziroma nestabilna, če je det(t Ä') < 0. Točko, v kateri ni ravnovesna krivulja niti stabilna niti nestabilna, imenujemo kritično točko. V kritični točki, ki jo bomo označili z cu, je izpolnjen pogoj: t/= { V 0(o, t) = 0, Os f s 1}, det (CK) = 0, which is termed the equilibrium path of the structure. An equilibrium path is said to be stable 141 at the point *11 if det Č /f) > 0, and unstable if det^/f) < 0 respectively. The point at which the equilibrium path is neither stable nor unstable is called a critical or stability point. At the critical point, which will be denoted by cu, the following condition is fulfilled: kjer je CK tangentna togostna matrika pri nivoju tc. Obremenitveni nivo tc, ki ustreza točki cu imenujemo kritični nivo obremenitve. V grobem delimo kritične točke na mejne in bifurkacijske 141. Pri povečanju obremenitve prek tc postane pri mejnih točkah ravnovesna krivulja nestabilna, medtem ko se pri bifurkacijskih razcepi na primarno in sekundarne veje. Natančnejša razdelitev kritičnih točk zahteva precej računskega dela in je podrobno opisana v 141. where CK is the tangent stiffness matrix at load level tc. The load level tc corresponding to the point Ln is called the critical load level. Critical points are classified into limit and bifurcation points 141. At a limit point, the equilibrium path becomes unstable when the load level is increased over tc while at a bifurcation point, the equilibrium path bifurcates into primary and secondary paths. A more precise classification of critical points requires much computational work and described in detail in 141.

4 2. VPELJAVA NELINEARNE RAVNOVESNE ENAČBE V OPTIMALNO PROJEKTIRANJE KONSTRUKCIJ Namen optimalnega projektiranja je določitev najboljših vrednosti nekaterih konstrukcijskih parametrov, ki so do te stopnje projektiranja še nedoločeni. Označimo te parametre za «IRm. Razumljivo je, da vrednost a vpliva na odziv konstrukcije pri dani obremenitvi, kar pomeni, da se vektor a pojavlja v ravnovesni enačbi konstrukcije (1). Ker pri optimalnem projektiranju konstrukcijske parametre obravnavamo kot spremenljivke, moramo torej ravnovesno enačbo v tem primeru zapisati nekoliko drugače: 2. INTRODUCTION OF A NONLINEAR EQUILIBRIUM EQUATION INTO THE OPTIMUM DESIGN OF STRUCTURES The purpose of optimum design is to evaluate the optimum values of some design parameters which are still undetermined in the current stage of design. Let us denote these parameters with a e [Rm. It is clear that the value of a influences the response of the structure at given loads, which means that a is included in the equilibrium equation (1) of the structure. Since in optimum design the design parameters are treated as design variables, we have to write the equilibrium equation in a slightly different way: C (u,a) = 0 (4), kjer je Giu, a) e IRn. Ravnovesno enačbo zapisano v obliki (4) imenujemo tudi enačba stanja. Optimalne vrednosti konstrukcijskih parametrov ustrezajo zahtevam projektanta, ki jih mora v matematični obliki izraziti z uporabo namenske funkcije in omejitvenih pogojev. Tako lahko problem optimalnega projektiranja zapišemo v obliki problema matematičnega programiranja: where Glu, a) e R n An equilibrium equation of the form (4) is also termed a state equation. Optimaum values of design parameters correspond to the requirements of the designer which has to be expressed in a mathematical form by an objective function and constraints. In this manner, the problem of optimum design can be written in - the form of a problem of mathematical programming: ob upoštevanju pogojev: min fla) subject to constraints (5), gj (u,a) š (6) in enačbe stanja: and the state equation G(u kjer so f namenska funkcija, gj skalama funkcija (pomik vozlišča, napetost itn.), katere vrednost želimo omejiti, gj~ pa je njena največja dopustna vrednost. V problemu (5) (7) se pojavlja poleg namenske funkcije v (5) in omejitvenih pogojev (6) tudi enačba stanja (7). Razlog je v tem, da se v (6) odvisna spremenljivka u ne da izraziti eksplicitno v odvisnosti od spremenljivke a. Njuna povezava je zato implicitno podana s (7). Problem (5) (7) je praktično vedno nelinearen. Rešitev a* zato običajno izračunamo kot limito zaporedja (a(l)), ki ga naredimo takole: a(o) izberemo; a(1 + 1) določimo kot rešitev problema P (1), ki je definiran na podlagi problema (5) (7) ter točke acl) in, i \l0. i) = 0 (7) where f is the objective function, gj is a scalar function (nodal displacement, effective stress, etc.), whose value has to be restricted and g, is its limiting value. Besides of the objective function (5) in the constraints (6), the problem (5) (7) also includes the state equation (7). The reason is that in (6) the dependent variable u cannot be expressed explicitly in terms of the independent variable a Therefore, the relationship between u and a is implicitly given by (7). The problem (5) (7) is almost always nonlinear. Its solution a is therefore usually calculated as a limit of the sequence (a(1)), which is generated in the following way a(o) has to be chosen; a(l + 1) is obtained as a solution of the problem P (l), which is defined from problem (5) (7) at the point a ll) in, / e W0.

5 Pri tem je lahko izbira a(o) precej poljubna, problem matematičnega programiranja P (l) pa najpogosteje definiramo, kot aproksimacijo problema (5) (7) v točki a(i). Nastajanje problema - P (1) terja izračun mnogih vrednosti, vendar je najpomembnejši in računsko najzahtevnejši izračun odziva u in odvoda du/da v točkia(i) [51. Oba izračuna opravimo na temelju enačbe stanja (7). Pri uporabi linearne ravnovesne enačbe je funkcija G v (7) linearna glede na u. Omenili smo že, da lahko takšno enačbo uporabimo le ob nekaterih predpostavkah, ki sicer niso preprosto določljive. K temu je treba dodati, da je veljavnost linearne enačbe odvisna tudi od vrednosti konstrukcijskih spremenljivk. Ker se pri optimalnem projektiranju te vrednosti iz iteracije v iteracijo spreminjajo, se lahko znajdemo v še posebej težavnem položaju, tako da postane vpeljava nelinearne ravnovesne enačbe nujno potrebna. Preprosto obravnavanje problematike bi nas lahko pripeljalo do sklepa, da terja vpeljava nelinearne ravnovesne enačbe le uporabo nelinearne funkcije G glede na spremenljivko u. To je celo res pri upoštevanju nekaterih nelinearnosti, kakor je na primer materialna. Toda ob pozornejšem pregledu lahko hitro ugotovimo, da bi tako naleteli na nepremagljive ovire, če ne bi upoštevali vseh mehanskih in matematičnih vidikov problema. Pri upoštevanju kinematičnih nelinearnosti obstaja namreč možnost, da bo v poljubni iteraciji (v točki a (1)) ravnovesna krivulja konstrukcije odsekoma nestabilna. Takšne razmere so običajno z inženirskega vidika nesprejemljive, saj to pomeni, da odziva konstrukcije ni mogoče popolnoma napovedati. Pri povečanju obremenitve čez mejno točko pride namreč do»preskoka«konstrukcije v močno deformirano stanje, medtem ko lahko pri bifurkacijski točki odziv ob povečanju obremenitve ustreza različnim vejam ravnovesne krivulje. V obeh primerih lahko pride do porušitve konstrukcije. Zaradi tega moramo problemu (5) (7) pri upoštevanju kinematičnih nelinearnosti obvezno dodati stabilnostni pogoj: The choice of a (o)can be quite arbitrary while the problem of mathematical programming P (1)is mostly defined as an approximation of the problem (5) (7) at the point a (1). Generation of the problem P (i) requires the evaluation of many quantities. However, the most important and most difficult is the calculation of the response uand the derivative du/daat the point a(1) [51. Both calculations are performed by using the state equation (7). If a linear equilibrium equation is used, then the function Gin (7) is linear with respect to the variable u As it has already been mentioned, such an equation can only be used on some assumptions which are not simple to predict. In addition, the validity of the linear equation depends also on the values of design variables. Since in optimum design these values change from iteration to iteration, we are confronted with a quite difficult situation, making the introduction of a nonlinear equation necessary. A naive treatment of this subject could yield the conclusion that the introduction of a nonlinear equilibrium equation only requires the use of a nonlinear function Gwith respect to the variable if. This is even true when some of the nonlinear effects are considered, for example, the material nonlinearity. However, it can be seen that considering kinematic nonlinearities in this case, we would soon meet some insuperable problems if all of the mechanical and mathematical aspects of the problem were not considered. In other words, in considering kinematic nonlinearities, there exists the possibility that in an arbitrary iteration (at the point a(i)) the equilibrium path of the structure is partially unstable. From an engineering point of view, such conditions are usually unacceptable since in this case the structural response is not completely predictable. Increasing the load level over the critical value leads to snap-through at the limit points while at the bifurcation points, the response may correspond to different branches of the equilibrium path. In both cases, the structure may collapse. Because of that, when kinematic nonlinearities are considered, we have to add to the problem (5) (7) the stability constraint: tc (a) ž 1 (8), Čeprav bi lahko sklepali, da bodo s tem teoretične težave odpravljene, so v praksi stvari precej drugačne. Pogoj (8) bo namreč zanesljivo izpolnjen le v optimalni točki a*, medtem ko za poljubno točko a(1) tega ne moremo pričakovati. Torej se lahko zgodi, da bo v poljubni iteraciji tangentna togostna matrika na nekaterih nivojih obremenitve slabo pogojena ali celo singularna. Even if one could conclude that this would remove the theoretical difficulties, in practical tasks the situation is quite different. The constraint (8) will certainly be fulfilled only at the optimum point a* while one can not expect the same for an arbitrary point a(1 f It can thus happen that in an arbitrary iteration, the tangent stiffness matrix will be ill-conditioned or even singular at some load levels. Since the generation of problem

6 Ker terja nastajanje problema P (lj reševanje linearnih sistemov enačb, katerih sistemska matrika je tangentna togostna matrika, lahko pričakujemo numerične težave. V praksi velja, da je dejansko stanje še nekoliko slabše, saj se pojavijo tudi težave v zvezi s konvergenco zaporednja (a(1)) in veliko računsko neučinkovitostjo takega načina. Nov način, ki smo ga razvili v Laboratoriju za optimiranje mehanskih sistemov, zelo učinkovito odpravlja vse opisane težave. Bistvo novega načina je v tem, da P (i) določimo kot aproksimacijo problema: i P (1)requires the solution of a system oflinear equations with the tangent stiffness matrix as the coefficient matrix, one can expect numerical difficulties. Actually, in practical tasks, the conditions are even worse due to difficulties concerning the convergence of the sequence (a(n) and the extreme inefficiency of such approach. The new approach, which has been developed in the laboratory for optimum design of mechanical systems, avoids all the above mentioned difficulties in a very effective way. The essence of the new approach is that is determined as an approximation of the problem: min /Ta) (9), ob upoštevanju pogojev: subject to constraints: gj (ii,a) É (p(1)g ), j =1,..., k w (ti, a) ž 1, (10) (11) in enačbe stanja: anc^ the state equation: G(u,a, p (i)) = 0 (12), v točki a(i), kjer je w(u, a) ocena kritične obremenitvene ravni, tako da je w (cu,a) = = /c (a), s p li) pa smo označili fiksno raven obremenitve, ki jo določimo med analizo odziva (reševanje enačb (2)) s preverjanjem naslednjih pogojev: 1) at the point a(i), where w(u,a) is an estimation of the critical load level, so that w (cu,a) = = /c(a), and p (1) is a fixed load level which is determined during response analysis (while solving equations (2)) by checking the following conditions: t = 1; (13) 2) /=Zc (a(1)); (14) 3) max {gj (*11, a(1)) / (t gj) - 1 ; j= 1,..., A' } ž 1 - t (15). Če je katerikoli izmed pogojev (13) (15) izpolnjen, vrednost parametra t priredimo parametru p (1) in analizo odziva končamo. Bistveni razliki med problemoma (5) (7) in (9) (15) sta dodana ocena stabilnostnega pogoja in prilagodljiva raven obremenitve, za katero zaradi ( 13) ( 15) veljap(i) é min {l, /c(atn)}. Dodan ocenjen stabilnostni pogoj zagotavlja pravilnost rešitve a* prilagodljiva raven obremenitve pa učinkovit in numerično zanesljiv numerični postopek optimalnega projektiranja. Pokažemo lahko (6), da zaporednje ravni p (i\ ki ustreza zaporedju a(i), konvergira k 1. Zaradi tega sklepamo, da lahko problem (5) (7) vedno nadomestimo z (9) (15). If any of the conditions (13) (15) is fulfilled, then the value of p (1) is set to be equal to the value of / and the response analysis is terminated. The essential differences between problems (5) (7) and (9) (15) are the added estimation of the stability constraint and an adjustable load level for which it holds that p (1)š min{ 1, Zc(aU))} because of (13) (15). The estimation of the stability constraint ensures the validity of the solution a, while the adjustable load level stabilizes the optimization process in an effective manner. It can be shown (61 that the sequence of load levels p (1), corresponding to the sequence a (l), converges to 1. Because of that, one can conclude that the problem (5) (7) can always be replaced with (9) (15).

7 Učinkovita numerična uporaba novega načina zahteva seveda precej več ko zgolj opisani postopek, vendar bomo, zaradi preglednosti, podrobnosti v tem prispevku izpustili. Omenimo le, da je novi način v celoti opisan v 161 in ZGLEDI Slabosti uporabe linearne ravnovesne enačbe si oglejmo na dveh reprezentativnih zgledih. V obeh primerih gre za palične konstrukcije, pri katerih so že določene koordinate vozlišč, topologija, gradivo, podprtje in obremenitve. Nedoločeni so le še prerezi posameznih elementov. Te bi radi izbrali tako, da bo prostornina porabljene snovi minimalna. Pri tem napetosti v posameznih elementih ne smejo preseči predpisane vrednosti. Izbrani material lahko v pričakovanem območju obremenjevanja (dopustne napetosti) obravnavamo kot linearno elastičen. Izmed zahtev a) do c), ki upravičujejo uporabo linearne enačbe, sta izpolnjena pogoja b) in c). Glede pogoja a) za zdaj še ne moremo reči ničesar, saj je odvisen od deformacij konstrukcije pri polni obremenitvi. Na dveh računskih zgledih bomo pokazali, da brez uporabe nelinearne enačbe pravzaprav ne vemo, ali je pogoj a) izpolnjen ali ne. Torej je uporaba nelinearne enačbe mnogokrat - nujno potrebna, če se želimo izogniti neprijetnim posledicam. Za zgled smo uporabili dve izmed standardnih konstrukcij, ki se pojavljajo v specializirani literaturi v zvezi z razvojem postopkov optimalnega projektiranja. Zaradi tega bomo tudi uporabili izvirne številčne podatke. Za njihov preračun v sistem SI je te številčne vrednosti treba pomnožiti s konstantami in enotami, ki so navedene v 151. Oba zgleda sta rešena z aproksimacijsko metodo matematičnega programiranja (aproksimacija s tremi parametri) I8I. Zaradi jasnosti prikaza rezultatov se še dogovorimo, da bomo uporabljali naslednji označbi: optimalna konstrukcija, dobljena z uporabo linearne enačbe; optimalna konstrukcija, dobljena z uporabo nelinearne enačbe. Označbi in pomenita dve varianti iste konstrukcije. Ustreznost ali neustreznost variante bomo preverili z analizo odziva konstrukcije z uporabo nelinearne ravnovesne enačbe, ki natančneje opisuje dejanski odziv. An effective numerical implementation of the new approach requires of course some further discussion, but for the sake of brevity in this paper the details will be omitted. It should just be noted that a detailed description of the new approach can be found in 16] and EXAMPLES The drawback of using a linear equilibrium equation will be illustrated with two examples. Both cases involve truss structures with known nodal coordinates, topology, material properties, as well as support and loading conditions. Still undetermined are the cross-sections of individual elements. These should be chosen in such a way that the volume of the material used will be minimum. At the same time, the stresses in individual members may not exceed the prescribed values. The chosen material can be treated as linear elastic in the expected range of loading (allowable stresses). Among the conditions a) to c) which justify the use of a linear equation, conditions b) and c) are fulfilled. However, nothing can be concluded yet about condition a) since it depends on the amount of deformation at full loading. On two numerical examples, it will be shown that without using a nonlinear equation we actually do not know whether condition a) is fulfilled or not. The use of a nonlinear equation is thus necessary if we wish to avoid unpleasant consequences. In the examples, we used two standard structures which frequently appear in the specialized literature in connection with the development of optimum design procedures.. We will therefore also use the original numeric data. To express this data in the SI system these numbers should be multiplied by the constants and units given in 151. Both examples have been solved by using an approximation method of mathematical programming (three-parameters controlled approximation) I8I. For the sake of clarity, we will use the following notations: the optimum structure obtained with the use of a linear equation the optimum structure obtained with the the use of a nonlinear equation. The notations and represent two options of the same structure. Whether the option is adequate or not will be verified by analyzing the response of the structure by using a nonlinear equilibrium equation which describes the actual response more precisely.

8 3.1 Paličje z desetimi elementi 3.1 The 10-bar truss Konstrukcija je prikazana na sliki 1, kjer so tudi razvidne koordinate vozlišč, topologija in podprtje. Izbran material ima elastični modul E = = ksi, največ ja dopustna napetost pa je Iödopl = 20 ksi. Zunanja obremenitev/? = -10 kips deluje v navpični smeri v vozliščih 2 in 4. Prerez posameznega elementa ne sme biti manjši od mln The stucture is shown in figure 1, where the coordinates, the topology and the support conditions can also be seen. The chosen material has an elastic modulus E = ksi and the greatest allowable stress is ddop = 20ksi. The external load R = -10 kips acts in a vertical direction at nodes 2 and 4. The cross-section of an individual element may not be smaller than a min = 0.1 in Paličje smo optimirali z uporabo linearne in nelinearne ravnovesne enačbe. Rezultati optimiranja so podani v preglednicah 1 in 2. Sl. 1. Paličje z desetimi elementi. Fig. 1. The 10-bar truss. The truss has been optimized by using the linear as well as the nonlinear equilibrium equation. The results of the optimization are listed in tables 1 and 2. Preglednica 1: Optimalna prostornina. Table 1: Optimum volume 2071,8 2071,4 Preglednica 2: Optimalni prerezi Table 2: Optimum cross-sections a, 1,0621 1,0600 a2 0,4379 0,4370 a3 0,1000 0,1000 a4 0,1000 0,1000 as 0,9379 0,9382 a6 0,1000 0,1000 a7 0,6192 0,6202 3q 0,7950 0,7944 a9 0,1000 0,1000 a io 0,6192 0,6199 Iz preglednic 1 in 2 vidimo, da sta varianti From tables 1 and 2, it can be seen that in praktično identični. To pomeni, options and are practically idenda je v tem primeru linearna enačba dovolj dobra. tical. This means that in this case, the linear Potrditev te ugotovitve lahko najdemo tudi v equation is good enough. This can be confirmed preglednicah 3 in 4, kjer so podani vozliščni po- by inspecting tables 3 and 4, where nodal displamiki in napetosti obeh variant, izračunani z upo- cements and stresses calculated by using the nonlinear rabo nelinearne enačbe. equation are given.

9 Preglednica 3: Odziv konstrukcije (pomiki nelinearna enačba) Table 3: Sructural response (displacements nonlinear equation) «v U y ux U y v, 0,387-1,771 0,387-1,771 v 2-0,483-1,919-0,482-1,919 v3 0,239-0,720 0,239-0,720 v4-0,241-0,720-0,240-0,720 Preglednica 4: Odziv konstrukcije (napetosti nelinearna enačba) Table 4: Structural response (stresses nonlinear equation) 3.2 Pallčje s tridesetim i elementi Konstrukcija je prikazana na sliki 2, kjer so tudi razvidne koordinate vozlišč, topologija in podprtje. Zaradi simetrije je obravnavana le polovica konstrukcije. Elementi so razvrščeni v 4 skupine, in sicer: 1/(1, 2, 3, 4), 2/(5, 6, 7), 3/(8, 9, 10, 11) in 4/(12, 13, 14, 15, 16, 17) kar pomeni, -19,994-19,966-19,991-19,951 *3 12,424 12,437 12,416 12,429 19,998 19,997 0,044-0,018-19,995-20,009 19,997 19,996 *3-17,561-17,578 Ö,0 19,991 20, The 30-bar tru ss The structure is shown in figure 2, where the nodal coordinates, the topology and the support conditions can also be seen. Because of the symmetry, only half of the structure is considered. The elements are linked into 4 groups: 1/(1, 2, 3, 4), 2/(5, 6, 7), 3/(8, 9, 10, 11) and 4/(12, 13, 14, 15, 16, 17), which means that we have only four design variables and that each of SI. 2. Paličje s tridesetimi elementi. Fig. 2. The 30-bar truss.

10 da imamo le štiri konstrukcijske spremenljivke in vsaka pomeni prereze elementov, ki pripadajo ustrezni skupini. Izbrano gradivo ima elastični modul E = ksi, največja dopustna napetost pa jeiödopl = 20 ksi. Zunanje obremenitve: R i = = -10 kips deluje v navpični smeri v vozlišču 3; R2 = -1 kips deluje v navpični smeri v vseh drugih nepodprtih vozliščih. Prerez posameznega elementa ne sme biti manjši od amln = 0,1 in2. Tudi to paličje smo optimirali z uporabo linearne in nelinearne enačbe. Rezultati so podani v preglednicah 5 in 6. Iz preglednic 5 in 6 vidimo, da sta rezultata zelo različna. Iz tega izhaja, da uporaba linearne enačbe v tem primeru ni bila ustrezna. Preglednica 5: Optimalna prostornina these represents the cross-sections of the elements belonging to the corresponding group. The chosen material has an elastic modulus E = ksi and the greatest allowable stress is ddop = 20 ksi. External loads: /?, = -10 kips acts in a vertical direction at node 3; R 2 = -lkips acts in a vertical direction at all other unsupported nodes. The cross- -section of an individual element may not be smaller than amln = 0.1 in2. This truss also been optimized by using a linear as well as a nonlinear equilibrium equation. The results of the optimization are listed in tables 5 and 6. From tables 5 and 6, it can be seen that the two results are quite different. It follows that the: use of the linear equation was inadequate. Table 5: Optimum volume 648,1 1219,6 Preglednica 6: Optimalni prerezi Table 6: Optimum cross-sections «1-4 1,4142 2,4369 a5-7 0,6862 2,3089 a ,6853 0,6838 ai2-n 0,1000 0,1000 Bistveno manjša prostornina, ki smo jo dobili z uporabo linearne enačbe daje slutiti, da bodo pri varianti omejitve (dejanske napetosti) močno prekoračene. Iz preglednice 7 je razvidno, da je dejansko stanje še mnogo hujše od prvotnih pričakovanj. Kritični nivo obremenitve je namreč za enak tc =0, V praksi to pomeni, da bi pri tej obremenitvi prišlo do»preskoka«v močno deformirano stanje, kar lahko pripelje do porušitve. Preglednica 7: Odziv konstrukcije (pomiki nelinearna enačba) The essentially smaller volume which was obtained by using the linear equation gives rise to the suspicion that for the option, the constraints (actual stresses) will be seriously violated. From table 7, it is evident that the actual state is even worse than initially expected. The critical load level for OPT L is, namely, equal to tc = In practice, this means that the structure would collapse at this load level. Table 7: Structural response (displacements nonlinear equation) UX u y uz v\ -0,164 0,000-0,466 ^2-0,082 0,142-0,466 ^3 0,000 0,000-8,716 V4 0,082 0,142-0,466 ^5 0,164 0,000-0,466»preskok«(snap - through) tc = 0,

11 Preglednica 8: Odziv konstrukcije (napetosti nelinearna enačba) * ,958 ^5-7 13,680 ^ ,999 ^ \ ,293 Glede nato, da se prvi (paličje z 10 elementi) in drugi (paličje s 30 elementi) zgled bistveno ne razlikujeta, lahko sklepamo, da brez uporabe nelinearne ravnovesne enačbe nimamo preprostega merila, po katerem bi lahko ugotovili, ali je linearna enačba ustrezna ali ne. Seveda bi v drugem primeru, ki sicer daje slutiti, da linearna enačba ne bo primerna, lahko dodatno omejili vozliščne pomike. Izkaže se, da bi dobili zadovoljiv rezultat, če bi bil največji dopustni pomik vozlišč približno udop = 3-0 in>to(la to je precej slaba tolažba, saj brez nelinearne enačbe nimamo pravega kriterija za določitev udop, mimo tega pa da tak način rešitve, ki so daleč od optimalnih. Kot zgled navedimo, da bi z uporabo dodatnih pogojev u(a) = 3,0 indobili sicer stabilno konstrukcijo, vendar z optimalno prostornino 1687,7 in3, kar je za 38 odstotkov več ko prostornina. Table 8: Structural response (stresses nonlinear equation)»preskok«(snap - through) tc = 0, Since the first (10-bar truss) and the second (30-bar truss) example do not differ essentially one can conclude that we do not have a simple criterion to recognize whether a linear equilibrium equation is adequate or not. Of course, in the second example, for which one can suspect that the linear equation will not be appropriate, one could additionally constrain the nodal displacements. It turns out that a satisfactory result could be obtained by constraining the nodal displacements by the maximum value t/dop = 3.0 In. However, this is poor consolation since these is no adequate criterion for udop and, in this case, the results are far from optimum. As an example, let us state that the introduction of additional constraints u(a) = 3.0 would yield a stable structure, but with the optimum volume in2, which is 38 %more than the optimum volume of. 4. LITERATURA 4. REFERENCES (11 Bathe. K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall. Englewood Cliffs, Wu. C.C.-Arora. J.S.: Design Sensitivity Analysis and Optimization of Nonlinear Structural Response Using Incremental Procedure. AIAA J , Bathe. K.J.-Dvorkin. E.N.: On the Automatic Solution on Nonlinear Finite Element Equations. Computers and Structures. 17, Wriggers. P.-Simo. J.C.: A General Procedure for the Direct Computation of Turning and Bifurcation Points. Int. J. Numer. Methods Eng Butinar. B.: Analiza obnašanja mehanskega sistema pri spremembah nekaterih parametrov. Doktorska disertacija. TF Maribor Kegl, M.: Optimalno projektiranje konstrukcij na osnovi nelinearnega modela. Doktorska disertacija. TF. Maribor Oblak, M.M. Kegl, M. Butinar. B.J.: An approach to Optimal Design of Structures with Non-linear Response. Int. J. Numer. Methods Eng.. 36, , Kegl. M.S.-Butinar. B.J.-Oblak. M.M.: Optimization of Mechanical Systems: On Strategy of Non-linear First-Order Approximation, Int. J. Numer. Methods Eng., 33, Naslov avtorjev: prof. dr. Maks Oblak. dipl. inž. doc. dr. Marko Kegl, dipl. inž. doc. dr. Branko Butinar, dipl. mat. Tehniška fakulteta Univerze v Mariboru Smetanova 17, Maribor Prejeto: Received: Authors' Address: Prof. Dr. Maks Oblak, Dipl. Ing. Doc. Dr. Marko Kegl, Dipl. Ing Doc. Dr. Branko Butinar. Dipl. Mat. Faculty of Engineering University of Maribor Smetanova 17, Maribor, Slovenia Sprejeto: Accepted: