SOME ASPECTS OF THE STIC SYSTEM STABILITY CALCULATION 1 UDC : (045)

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1 FACTA UNIVERSITATIS Series: Architecture ad Civil Egieerig Vol. 7, N o 1, 9, pp DOI: 1.98/FUACE9135B SOME ASPECTS OF THE STIC SYSTEM STABIITY CACUATION 1 UDC 64.46: (45) Emra Bujar 1, Dragoslav Stojić 1 Departmet of Civil Egieerig, Uiversity of Prištia, Serbia Departmet of Civil Egieerig ad Achitecture, Uiversity of Niš, Serbia stojicd@gaf.i.ac.rs Abstract. The problem preseted i this paper has bee treated itesively by various researchers. However, cosiderig that by its ature the problem of structure stability belogs to the area of oliear aalysis, it represets a iexhaustible source for ivestigatios. Thus, we propose a simple method to determie the elemets of geometrical matrix of member rigidity by eergetic method through fiite elemet method. Key words: Stability, oliear aalysis 1. INTRODUCTION Structures deform depedig o the magitude ad the dispositio of exteral forces, physical ad mechaical characteristics of the material of which they were made, ad their geometrical characteristics. Uder the ifluece of loads the geometry of a structure, or its elemets, experieces chages such as dilatatios ad mutual displacemets of cross-sectios ad poits. Determiatio of these chages, i the process of examiatio, eables the estimatio of the strai coditio, deformatio ad load carryig capacity. The deformatio of the structure elemets or its particular parts could be defied through liear displacemet of the poits or the group of poits. Tesio or compressio or both occurs that umerical shift (of poits, rotatio agles, cross sectio axes or ay other plae ad logitudial deformatio). Desig of egieerig structures requires ot oly the stregth calculatio i order to get the complete picture of the structure safety, but also a aalysis of the deformable system balace stability, that represets a essetial task i civil egieerig. Received Jue 3, 9 1 This paper is dedicated to our teacher Prof. PhD Milić Milićević, Departmet of Civil Egieerig Uiversity of Niš, Serbia

2 36 E. BUJAR, D. STOJIĆ Static method, deformatio method, eergetic method could be used for determiig the critical forces of the liear elastic plae systems cosistig of members which are iflueced by coplaar logitudial forces ad trasversal loadig:. FORCES METHOD I solvig the stability problem usig the force method (statical method), a give system that is time statically idefiite is reduced to the so called basic system that is statically defiite ad geometrically uchageable. The system of the liear algebra equatio is used for fidig the ukow forces X k (k = 1,,3,... ). If the ukow values of X k are uequal to zero, i respect to the observed beded elemet of the system, it is a critical coditio i this case oly if the determiat of the liear equatio system is equal to zero. where: δ ik = δ ik ϕi (ν) δ11 δ1... δ1 D = δ1 δ... δ δ1 δ... δ The critical forces or their critical parameter ca be calculated from the equatio (1). While observig the equatio (1) as a equatio o ν we ca fid all its values (of ν) that are equal to all critical magitudes of kot loadigs. Usually the calculatio of the equatio (1) is very difficult. To accomplish that, a certai value for ν is adopted, ad the the fuctio ϕ i (ν) is calculated usig the table [1] ad oly afterwards the value of the determiat is calculated. (1) 3. DEFORMATION METHOD This method yields the basic system by implatig certai " brakes " i the system. With these "brakes" we are stoppig the rotatios ad liear displacemets of kots. The rotatio agles ad liear movemets of the kots are cosidered as ukow values Z k, (k = 1,,3,...). Sice i this paper the static systems with such kid of kot loadig are observed, that actually oly pressures or tesios i some elemets of the system are caused util the momet of losig the stability, the reactios i fictitious coectios are equal to zero, so the whole system of fiite equatios will be homogeeous. Z r + Z r Z r = = Z r Z r Z r () Z r + Z r Z r = 1 1

3 Some Aspects of the Stic System Stability Calculatio 37 With the symbols r ik = r ik (i,k = 1,,3,...) the reactios i the basic system are labeled, which appears i a itroduced i-th coectio of the uit displacemet Z k = 1. The loss of stability correspods to the iequality of the ukow values to zero, which is possible oly whe the determiat, cosistig of coefficiets alog with the ukow values equals to zero: r11 r1... r1 r1 r... r r 1 r... r = (3) The equatio (3) is equatio of stability by the calculatio of the deformatio method. 4. ENERGETIC METHOD IN THE FEM (FINITE EEMENT METHOD) FORM I the calculatio of the stability of the member systems by usig the eergetic method, as of the equatio (4), which characterize the deformatio of the system, movemets ad rotatios of the kot system are adopted as Z k, ad also some of the member cross-sectios. y = f Z + f Z f Z ( x) 1( x) 1 ( x) ( x) (4) If a certai form of the member deformatio ϕ i is give it is possible to avoid the applicatio of the special trascedet fuctio ad to more simply defie P kr ad the form of stability loss Z i a algebraic solutio such as (7) or (8), which ca be obtaied from the system of equatios (5). a11 P b11) Z1 + ( a1 P b1 ) Z + K + ( a1 P b ) Z = ( 1 (5) where: a ij a 1 P b 1) Z1 + ( a P b ) Z + K + ( a P b ) Z = ( d f d f df df i j i j = E I x dx bij N x dx dx dx = dx ( ) ; ( ) dx (6) If the equatio (5) is represeted i a metrical form we have: {A P B} {Z} = {} (7) Characteristic equatio of the stability that will give us the value of the critical force P kr will be: [A P B] = (8) If the members are of the costat rigidity, it is possible to get for elemets a ij ad b ij (6) simple mechaical ad geometrical iterpolatio as i the static problems.

4 38 E. BUJAR, D. STOJIĆ The eergy equatio of the beded member i the Fig. 1 from the revolutio of the left fixed ed for a small agle Z, 3 3 x 1 x f( x)= x + Fig. 1 is of the form: 1 Ee = r Z (9) 11 For the member bedig caused by the ifluece of the ormal ad trasversal forces we have: r 11 i ϕ 1 (ν); so r 11 = r 11 ϕ1 (ν); where, E I N r11 = 3 ; ν = (1) E I I this case the eergy equatio will be preseted as: Ee = U + Π = U + W (11) For the deflectios of the rotatios Z = 1 of the same fuctio f (x), as i the static problems without the ormal force N: 3 3 x 1 x y( x) = f( x) = x + I accordace with (6): 1 U = r11 Z where: 11 r d f E I = E I dx 3 =. dx ad: with: N W = N Δ = 1 P γ 11 Z γ 11 = df N N = ; P - basic loadig parameter P N dx = dx 5 (1)

5 Some Aspects of the Stic System Stability Calculatio 39 The coefficiet of the rigidity i the equatio (9) ca be preseted: r = r P γ (13) where: r 11 rigidity without cosideratio of iflueces of the ormal ad trasversal forces. γ 11 coefficiet that is proportioal to the movemets Coefficiet γ 11 will be called geometrical coefficiet ad it will be calculated from the equatio (1) ad (15). Equally for ay other coefficiet r ik the deformatio methods will be calculated by the equatio: rik = rik P γ ik (14) where: d f ij d f kj rik = E I dx i 1 dx dx = 11 γ ik j dϕ ij dϕ = N j dx dx i= 1 kj dx (15) I these equatios the coefficiets o the members are multiplyig ad the very same are deformig i the basic system from the positio Z = 1 for each oe of the members whe E I = cost. ad N = cost. Coefficiets r j ik ad geometrical coefficiets could be calculated for ay of the members. 5. NUMERICA EXAMPE Now we will show the way of defiig the rigidity matrix for the member with oe fixed ad oe freely supported poit Fig.. Fig.

6 4 E. BUJAR, D. STOJIĆ γ x 1 x f( x ) = x + ; df( x) x 3 x = + dx 1 3 d f ( x ) 3 x = 1 ; dx 11 r = E I d f ( x ) E I dx 3 dx = ( ) = P df x dx = P 3 E I P ; rik = rik γ ik ; r11 = r11 γ 11 = dx E I P r11 = 1 15 E I With: E I k = ; K11 = r11 we have: P K11 = 3 k 1 15 k From the coditio of balace we have: 3 E I P A= B= 5 or: 3 k P A= B= 5 or: A = B = P k 3 5 k 6. CONCUSIONS The goal of the paper is to determie the elemets of geometrical matrix of member rigidity by eergetic method through fiite elemet method. A very simple method procedure for determiatio of relevat factors of this problem has bee demostrated. REFERENCES 1. A. R. Ržaici: Stroitelaja mehaika, Moskva, 198 god.. I. M. Rabiović: Osovi stroiteloj mehaiki sterževih sistem, Moskva, A. A. Umaskovo: Sprovočik projektirovščika promišleih,žilih i občestveih zdaij i sooružeij, Moskva, 1973 god. 4. A. F. Smirov, A.V. Aleksadrov, B. J. ašeikov, N. N. Šapošikov: Stroitelaja mehaika, Moskva, 1984 god. 5. A. B. Aleksadrov, B. J. ašeikov, N. N. Šapošikov: Stroitelaja mehaika, Moskva, 1983 god. 6. Mila Đurić: Statika kostrukcija. IRO Građeviska kjiga Beograd, 1983 god. 7. Miodrag Sekulović: Metod koačih elemeata. IRO Građeviska kjiga Beograd, 1984 god. 8. Milić Milićević, Brako Popović, Slavko Zdravković: Statika kostrukcija. Građeviski fakultet Uiverziteta u Nišu. Niš,199 god.

7 Some Aspects of the Stic System Stability Calculatio 41 OSVRT NA PRORAČUN STABINOSTI INIJSKIH SISTEMA Emra Bujar, Dragoslav Stojić Problem koji se tretira u ovom radu u svetu i kod as je dosta razrađiva. Međutim, imajući u vidu da problemi stabilosti kostrukcija, po svojoj prirodi spadaju u područje elieare aalize sa komplikovaim algoritmom rešavaja, ovde se izosi jeda sasvim jedostava postupak rešavaja tog problema. U sistemu jedačia ravoteža, geometrijska matrica krutosti je epozata, pošto je zavisa od aprezaja, koji su takođe epozate, pa se javlja problem eliearosti. U ovom radu izlože je jedostava postupak određivaja člaova geometrijske matrice krutosti štapa, eergetskom metodom u obliku MKE (Metod koačih elemeata). Ključe reči: stabilost, elieara aaliza