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MATERIJALI I KONSTRUKCIJE53 (2010) 3 (14-31)14

SPEKTRALNA MODALNA ANALIZA ZGRADA SA POLUKRUTIM I EKSCENTRINIMVEZAMA

SPECTRAL MODAL ANALYSIS OF BUILDINGS WITH SEMI-RIGID AND ECCENTRICCONNECTIONS

piro GOP EVI Stanko BR I Ljiljana UGI

PREGLEDNI RD

UDK: 69.057:517.962 = 861

1 UVOD

Skeletne zgrade predstavljaju najee primenjivanekonstrukcije u zgradarstvu. Jedan deo skeletnih zgradasu zgrade sa elinom konstrukcijom. Veza greda-stubkod zgrada je medijum koji prenosi odgovarajue sile imomente sa elementa na element. Prilikom idealizacijeveza u vorovima, polazi se od pretpostavke da su vezeidealne: krute ili zglobne. Veliki broj ispitivanja realnihveza pokazao je da veina krutih veza nije apsolutnokruta, kao i da veina zglobnih veza nije idealna. Kruteveze pri optereenju dozvoljavaju izvesnu relativnurotaciju na mestu veze, dok zglobne veze pri optere-enju pokazuju odreen stepen rotacione krutosti. Vezekoje po svome ponaanju predstavljaju prelaz izmedjuzglobnih i krutih veza nazivaju se polukrute veze. Kaoto su eline veze vie ili manje fleksibilne, tako su onetakoe vie ili manje ekscentrine. Najee se eks-centricitet veze zanemaruje, meutim u nekim slu-ajevima to nema opravdanja. To je sluaj kada su vezeostvarene preko vornog lima, tako da odnos ekscen-triciteta i duine linijskog elementa nije mali. Kod ree-tkastih nosaa odnos ekscentriciteta i duine tapamoe da iznosi i do 20%, dok je kod ramovskih sistemaon znaajno manji i iznosi oko 5%. Zbog velikog znaaja

polukrutih i ekscentrinih veza na konane rezultateprorauna, poslednjih godina definisanje ponaanja veze je predmet mnogobrojnih naunih radova [2,5,6,7].

Dr piro Gopevi, dipl.in.gra.JP eleznice Srbije, Nemanjina 6, 11000 Beograd, Srbija;e-mail:[email protected] Prof. dr Stanko Bri, dipl.in.gra.Univerzitet u Beogradu, Graevinski fakultet, Bulevar kraljaAleksandra 73, 11000 Beograd; e-mail:[email protected] Dr. Ljiljanaugi , dipl.in.gra.Univerzitet Crne Gore, Graevinski fakultet, Cetinjski putbb, 81000 Podgorica, Crna Gora; e-mail:[email protected]

1 INTRODUCTION

Framework buildings represent one of the veryfrequent structural systems of buildings. Among framedbuildings, steel structures are an important part.Connection beam-to-column is the medium to transferthe corresponding forces and moments from element toelement. Numerical idealization of joint connectionsusually assumes an ideal connection: either rigid orpinned. A large number of investigations of the realconnections show that the majority of rigid connectionsare not absolutely rigid, and also that the majority ofpinned connections are not ideally hinged. Rigidconnections, when loaded, allow some relative rotationat a joint, while pinned connections exhibit somerotational stiffness under loads. The connections that intheir behavior under loads represent an intermissionbetween the ideally pinned and rigid connections arecalled semi-rigid or flexible connections. In the sameway as the connections in steel frames are more or lessflexible, they are also, more or less, eccentric. Usually,the joint eccentricity is disregarded; however, in somecases it is not justified. It is the case of joints with nodalplates, when the ratio between eccentricity and elementlength is not small. In steel trusses the ratio of ec-

centricity and the bar length may be up to 20%, while inframed systems that ratio is substantially smaller and isabout 5%. Due to substantial effect of semi-rigid conec-

Dr piro Gopevi, dipl.in.gra.JP eleznice Srbije, Nemanjina 6, 11000 Beograd, Srbija;e-mail:[email protected] Prof. dr Stanko Bri, dipl.in.gra.Univerzitet u Beogradu, Graevinski fakultet, Bulevar kraljaAleksandra 73, 11000 Beograd; e-mail:[email protected] Dr. Ljiljanaugi , dipl.in.gra.Univerzitet Crne Gore, Graevinski fakultet, Cetinjski putbb, 81000 Podgorica, Crna Gora; e-mail:[email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]

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MATERIJALI I KONSTRUKCIJE53 (2010) 3 (14-31) 15

U odnosu na proraun u statikoj analizi, dinamikiproraun zgrada je ne samo obimniji, ve i znatnokomplikovaniji. Razlog za to je vei broj potrebnihparametara, ukljuujui i vremensku dimenziju, kao i,inenjerski gledano, mnogo neizvesnija procena ulaznihveliina. Dinamiko optereenje zgrade moe da budedato kao: optereenje nekom proizvoljnom dinamikomsilom, seizmikim optereenjem datim preko dinamikog

pomeranja oslonaca ili seizmikim optere

enjem datimpreko krive spektra pseudoubrzanja. Pri optereenju

zgrade spektrom pseudoubrzanja, u definisanom pravcu,dobijaju se pribline vrednosti maksimalnog odgovora jerkombinacija modalnih odgovora uzima u obzir samomaksimalne vrednosti odgovora za pojedine oblike nevezujui se za vremenski trenutak u kojem sumaksimalne vrednosti nastale. Meutim, za svaki uticajmoe da se odredi pravac spektra pseudoubrzanja zakoji posmatrani uticaj ima ekstremnu vrednost i da setada nau ekstremne vrednosti uticaja usled datogspektra pseudoubrzanja [8].

Zgrade su kontinualni trodimenzionalni sistemi sakompleksnom raspodelom krutosti, mase i optereenja.Pri njihovom matematikom modeliranju obino seusvaja linearno elastino ponaanje koje dozvoljavaprincip superpozicije.

2 MATEMATIKI MODEL TAPA SA POLUKRUTIMI EKSCENTRINIM VEZAMA

Na osnovu principa superpozicije, opti sluajprostornog naponskog stanja tapa u okviru linearneanalize, moe da se razdvoji na: aksijalno naprezanje,torziju i savijanje u dve ortogonalne ravni i predstavljaetiri nezavisna problema. Korektivna matrica, prekokoje se uzima u obzir uticaj polukrutih i ekscentrinihveza, ima uticaja samo na lanove matrice krutostielementa koji se odnose na savijanje. Posle odreivanjamatrice krutosti tapa na savijanje, na osnovu principasuperpozicije, odreuje se kombinovana matrica krutostitapa usled savijanja, torzije i normalnih sila.

Na slici 1 prikazan je obostrano ukljeteni tap uravni, sa polukrutim i ekscentrinim vezama, i sausvojenim generalisanim pomeranjima.

tions upon the final results of calculation, during the lastyears many scientific research are devoted to analysis of joint connections, see [2,5,6,7].

With regard to static analysis, dynamic analysis ofbuildings is not only more extensive, but it is also morecomplicated. The reason for that is the larger number ofnecessary parameters, including the time dimension,and also, in engineering sense, much more uncertain

estimate of the input values. Dynamic loading ofbuildings may be given as: loading defined by sometime-dependent force, seismic loading given by dynamicmotion of supports, or seismic loading given by thespectral pseudo-acceleration curve. When the seismicloading of a building is defined by the spectral pseudo-acceleration in a given direction, approximate values ofthe maximum response are obtained, since thecombination of modal responses is taking into accountonly the maximum values for various modes withoutregard of the time instances when the maximumoccured. However, for every effect one can also obtaindirection of the spectrum of pseudo-acceleration forwhich that effect has the maximum value and then toobtain the extreme values of the effect due to a givenspectrum of pseudo-acceleration, [8].Buildings are continuous three-dimensional systemsusually with complex distribution of stiffness, mass andloading. In their mathematical modeling one usuallyassumes the linear behavior which allows the principle ofsuperposition.

2 MATHEMATICAL MODEL OF A BEAM ELEMENTWITH SEMI-RIGID AND ECCENTRIC CONNECTIONS

Due to the principle of superposition, a general caseof the spatial state of stress of a beam, within the linearanalysis, may be partitioned into the axial stresses,torsion and bending in the two orthogonal planes, thusrepresenting the four independent problems. Thecorrective matrix, which takes care of the effect of semi-rigid and eccentric joints, has influence only upon theelements of the stiffness matrix that correspond tobending. After obtaining the stiffness matrix of a beamwith respect to bending, due to the principle ofsuperposition, the combined stiffness matrix of a beamconsidering combined bending, torsion and axial forcesis determined.

Slika 1 tap u ravni sa polukrutim i ekscentri nim vezama Fig. 1 Beam in a plane with semi-rigid and eccentric

connections

Slika 2 Uglovi obrtanja deformisanog tapa Fig. 2 Angles of rotation of deformed beam

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Ponaanje polukrute veze, koje je definisano relaci- jom izmeu momenta M i rotacije na kraju elementakoji je polukruto vezan, usvaja se da je linearno.

Polukruta veza krajeva tapa modelirana je pomourotacionih opruga na krajevima, a ekscentrinost vezepredstavljena je kratkim beskonano krutim elementima.Formulacija elementa izvedena je tako da se moerazdvojiti uticaj usled polukrute veze i uticaj ekscentrine

veze.

2.1 Uticaj polukrutih veza na savijanje tapa u ravni

Razmatra se linearna polukruta veza. Veza izmeuvertikalnog pomeranja ose tapa v(x) i vektorageneralisanih pomeranjaq , na krajevima tapa, moeda se prikae preko interpolacionih funkcija kao

Fig. 1 represents the both and fixed beam in a plane,with semi-rigid and eccentric joints, displaying theadopted generalized displacements.

The behavior of a semi-rigid connection, defined bythe bending moment M and rotation, is assumed aslinear. Semi-rigid connection at beam's ends isrepresented by the rotational springs at ends, while theeccentric connection is modeled by infinitely rigid

elements. Formulation of the finite element is derived insuch a way to be able to separate effects of semi-rigidand eccentric connections.

2.1 Effect of semi-rigid connections upon bendingof a planar beam

Linear semi-rigid connection is considered. Therelation between the lateral displacement of a beam axisand the vector of the generalized displacementsq atbeam's ends may be presented by the interpolationfunctions as

qN )( xv(x) = [ ])()()()()( 4321 x N x N x N x N x =N [ ]2211 vvT

=q (1)pri emu se za interpolacione funkcijeN i (x) (i = 1,2,3,4)usvajaju Hermite-ovi polinomi prve vrste.

Obrtanjevorova sistema i jednako zbiru obrtanjatapa

ii dodatnog obrtanja i kraja tapa nastalog

kao posledica polukrute veze (slika 2):

The interpolation functionsN i (x) (i = 1,2,3,4) areassumed as the Hermite's polynomials of the first kind.Rotations of joints i are equal to the sum of beam rota-tion i and the additional rotation i of beam's end, as aconsequence of the semi-rigid connection, see Fig. 2:

iii += 1, 2i = (2)

Jednaina (1), vodei rauna o jednaini (2), moese napisati kao

Equation (1), due to Eq. (2), may be written in theform

(3)

Vektor, u jednaini (3), moe da se izrazi kao Vector, in Eq. (3), may be expressed as 1 2

1 2

0 0T M M k k

=

i

i

k

M =i 2,1=i (4)

gde je k i rotaciona krutost opruge, a i M momenat uvoru i tapa. Veza sila i pomeranja na krajevima tapa je

where k i represents the rotational spring stiffness, while

i M is the moment at jointi of the beam. Therelationship between forces and displacements atbeam's ends is given by

( )

1 1

1 1 1

2 2

2 2 2

0

( ) ( ) ( ) ( ) ( )0

v v

v x x x x xv v

= = = =

N N N q N q%

[ ]1 1 2 2T v v =q% [ ]1 20 0T =

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12 2

103

22 2

2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

l lT

l l l l M EI l llT

l l l l M

= =

q K q (5)

gde je: 0K matrica krutosti na savijanje obostranoukljetenog tapa, E Young-ov moduo elastinosti i I moment inercije poprenog preseka. Momenti nakrajevima tapa u jednaini (5) mogu da se izraze ufunkciji vektoraq~ . Iz jednaine (5), vodei rauna o jednainama (3) i (4), dobijaju se momenti na krajevimatapa kao

where: 0K is the bending stiffness matrix of both andfixed beam,E is the Young's modulus of elasticity andI is the moment of inertia of the cross section. Moments atbeam's ends in Eq.(5) may be expressed as a functionof the vectorq~ . From Eq.(5), taking care about Eqs.(3)and (4), one obtains the moments at beam's ends as

(6)

gde je g bezdimenzionalna rotaciona krutost opruge.Vektor rotacije, dat jednainom (4), vodei rauna o jednaini (6), moe sada da se napie u obliku

where g is the non-dimensional rotational springstiffness. Vector of rotation, given by Eq.(4) and takingcare about Eq. (6), may be written in the form

1 12 1 2 1 2 1

1

221 2 2 1 2 1

2

0 0 0 0 0

6 61 2 4 1 3 1 2 2

1 0 0 0 0 0

6 61 2 2 1 2 4 1 3

M g( g ) g ( g ) g ( g ) gk l l

g M ( g ) g g ( g ) g ( g )l lk

+ + + = = = + + +

q Gq% (7)

gde je G korektivna matrica tapa sa polukrutim vezamana oba kraja.

Obzirom da je vektor rotacije odreen i dat jednainom (7), moe se eliminisati iz jednaine (3), tetransverzalno pomeranje proizvoljne take ose tapaiznosi

where G is the corrective matrix of both and fixed beamwith semi-rigid connections at both ends.

Since the rotation vector is determined by Eq.(7), itmay be eliminated from Eq.(3), so the lateraldisplacement of an arbitrary point along the beamelement is given by

( ) ( )( )v x x= N I G q (8)

Za sluaj polukrutih centrinih veza je 1 1v v= i2 2v v= , a vektor =q q% , pa je vektor interpolacionih

funkcija za tap sa polukrutim centrinom vezama jednak

In the case of semi-rigid connections one has

1 1v v= i 2 2v v= , and also vector =q q% , so thevector of interpolation functions for a beam with semi-rigid and centric connections is given as

( ) ( )( ) x x= N N I G (9)

q~)31(4)21(62)21(6

2)21(6)31(4)21(6

111

222

22

1

++++++

= glglglgglg

l EI

M

M

2121 12441 gggg +++=

ii lk

EI g = 2,1=i

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0

1 1 ( ) ( )2 2

lT T T A EI x x dx

= + = Tq N N G SG q q Kq

= +G G(I E) =

2

1

000

0000

000

0000

k

k S

(15)

U jednaini (15)K predstavlja matricu krutosti tapa

sa ekscentrinim i polukrutim vezama na savijanje. IzrazK iz jednaine (15), vodei rauna o izrazu za ( ) xN u jednaini (13), moe da se napie kao

In Eq. (15)K represents the bending stiffness matrixof a beam with eccentric and semi-rigid connections.Expression for K in Eq.(15), having in mind theexpression for ( ) xN in Eq.(13), may be written as

GSG)G(IK)G(IK T1T

1

0 +++=

[ ] [ ]dx EI l

T ""0

0 NNK = (16)

gde je 0K matrica krutosti tapa sa krutim centrinim

vezama.

3 MODALNA ANALIZA

Ova metoda je primjenjiva ako je vremenskazavisnost sila pobude svih masa ista ili srazmerno ista,to u sluaju seizmikog optereenja zadovoljava uslov.

Pretpostavlja se da je seizmiko optereenje datopreko vektora generalisanog dinamikog pomeranjaoslonaca a(t). Ovaj vektor sastoji se od vektoradinamikog pomeranja oslonaca ad(t) i vektoradinamikog obrtanja oslonaca a(t). U proraunu sepretpostavlja da je vektor dinamikog obrtanja oslonaca jednak nula vektoru. Vektor dinamikog pomeranjaoslonaca ad(t) ima proizvoljan pravac u prostoru (slika4). Vrh vektora ad(t) jedne take na povrini zemlje zavreme zemljotresa opisuje proizvoljnu krivu u prostoru.Vektor ad(t) moe da se razloi u pogodnomkoordinatnom sistemu 123 na tri komponente, te vektorgeneralisanog dinamikog pomeranja oslonaca iznosi

where 0K represents the stiffness matrix of a beam

with rigid and centric connections.

3 MODAL ANALYSIS

This method may be applied if the time dependanceof excitation forces of all masses is the same orrelatively the same, which, in the case of an earthquakeis satisfied.

It is assumed that the seismic loading is given by thevector of generalized dynamic displacement of supporta(t ). This vector consists of a vector of dynamic supportdisplacements ad (t ) and a vector of dynamic supportrotationsa(t ). It is assumed that the vector of supportrotations is equal to a zero vector. Vector of dynamicsupport displacementsad (t ) has an arbitrary direction inspace (Fig. 4). The tip of the vectorad (t ) of a point on theearth's surface during earhquake is inscribing anarbitrary curve in space. Vectorad (t ) may be projectedinto three orthogonal components with respect to aconvenient coordinate system123, so the vector ofgeneralized dynamic support displacements may bepresented as

[ ] [ ]0aaaa )()()()( t t t t T d T T d T == [ ]321)( aaat T d =a [ ] 0a == 000)(t T

(17)

Kretanje konstrukcije usled seizmikog optereenja

tretira se kao sloeno kretanje. Ukupni vektorpomeranja , j absq svake mase j ( j =1,2,...,N ), sastoji seod vektora prenosnog pomeranjaq j,k (t ) koje je jednakoseizmikom pomeranju tla i vektora relativnogpomeranjaq j (t ) (slika 3), i iznosi

The motion of the structure due to seismic excitationis considered as the compound motion. The absolutedisplacement vector , j absq of each mass j ( j =1,2,...,N ),consists of the vector of imposed displacementq j,k (t )which is equal to the seismic soil displacement at thebase of the building, and the vector of relativedisplacementq j (t ) (Fig. 3), so, it is

, , j abs j k j= +q q q (18)

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Neka osa 1 koordinatnog sistema123 u kojem je datvektor dinamikog pomeranja oslonaca zaklapa ugaosa globalnom osomX , a osa 3 je u pravcu oseZ (slika4). Ako se pretpostavi da je broj generalisanihpomeranja vora est, vektor prenosnog pomeranjavora j u pravcu osa globalnog koordinatnog sistemaiznosie

Let the axis1 of the coordinate system123, which isused as the reference frame for support displacement,forms the angle with the globalX axis, and the axis3 is in direction of the vertical Z axis (Fig. 4). If oneassumes that the number of generalized displacementsof a joint is six, the vector of imposed displacements of joint j with respect to the axes of the global coordinatesystem may be given as

,d

j k j

= = 0 a

q B a0 0 0

=

1

cossin

sincos

(19)

gde je matrica transformacije prenosnog pomeranjavora iz koordinatnog sistema123 u koordinatni sistemXYZ . Vektor ukupnog (apsolutnog) pomeranja sistemasada iznosi

where represents the transformation matrix of imposedmotion at considered joint from the coordinate system123 into the coordinate systemXYZ . The vector of thetotal (i.e. absolute) displacement is given as

( ) ( ) ( )abs t t t = +q Ba q 1, , ,T T T T abs abs j abs N abs = q q q qK KT N T jT T BBBB KK1=

1T T T T

j N = q q q qK K (20)

U sluaju seizmikog optereenja, inercijalne sile

zavise od apsolutnog ubrzanja, sile priguenja odrelativne brzine i restitucione sile od relativnogpomeranja, a spoljanje dinamike sile u vorovimasistema su jednake nuli. Dinamika jednaina ravnoteesistema, odn. diferencijalna jednaina kretanja, usledseizmikog optereenja, glasi

In the case of a seismic loading, the inertial forcesdepend upon the absolute acceleration, viscous dis-sipative forces upon the relative velocity and the resti-tution forces upon the relative displacement, while theexternal dynamic nodal forces are equal to zero.Dynamic equilibrium equations, i.e. differential equationsof motion, due to seismic loading, are given in the matrixform as

abs + + =Mq Cq Kq 0&& & (21)gde je: Mmatrica masa, C matrica viskoznog priguenjai K matrica krutosti sistema. Jednaina (21), vodeirauna o izrazu (20), glasi

where: M is the mass matrix, C the matrix of viscousdamping andK the stiffness matrix. Eq. (21), consideringexpression (20), becomes

Slika 3 Pomeranje objekta pri zemljotresu Fig. 3 Displacement of a structure due to an earthquake

Slika 4 Razlaganje vektora dinami kog pomeranja oslonaca a d

Fig. 4 Decomposition of the vector of dynamic support displacement a d

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+ + = Mq Cq Kq MBa&& & & (22)

Matrica priguenjaC usvojena je, na uobiajennain, kao linearna kombinacija matrice masa i matricekrutosti:

Damping matrixC is assumed in the usual way asthe linear combination of the mass and stiffnessmatrices:

KMC += 1

1

2 n

n

= + 1

2

n

= + (23)

U jednaini (23) se koeficijenti i obino rau-

naju tako to se za dve svojstvene frekvencije1 i n ,za dva razliita svojstvena oblika, usvaja da je relativnopriguenje isto: 1 n = = .

Sistem simultanih diferencijalnih jednaina (22), pri-menom metode modalne analize, moe se transformi-sati u sistem meusobno nezavisnih jednaina od kojih je svaka sa jednim stepenom slobode. Da bi se izvrilaova transformacija, prvo je potrebno uraditi linearnutransformaciju vektora nepoznatih generalisanih pome-ranja q(t ), koristei glavne forme sopstvenih neprigu-enih oscilacija sistema, preko modalne matrice , koja je nezavisna od vremena, te se dobija da je

The coefficients and in Eq.(23) are usuallydetermined in such a way that for the two natural freque-ncies 1 and n , corresponding to two different naturalmodes, an equal relative damping is adopted:

1 n = = .The system of simultaneous equations (22), using

the modal analysis, may be transformed into the systemof mutually independent equations, each onecorresponding to one degree of freedom. In order topreform the modal analysis, the corresponding modaltransformation is done, that is the generalized displace-ments q(t ) are expressed as the linear combination ofthe modal matrix , which is independent of time, andthe new modal, or normal, coordinatesi(t):

( ) ( )t t =q (24)gde je ( )t vektor normalnih koordinata. Modalnamatrica jednaka je

where ( )t is the vector of normal (or modal)coordinates. Modal matrix is given as

[ ]1 i n= K K 1, 2,...,i n= (25)

gde je i svojstveni (karakteristini, modalni) vektorza svojstvenu frekvencijui . Unosei izraze (24) u jedna-inu (22) i mnoei je sa leve strane sa T dobija se

where i is the natural vector (eigen vector, or modalvector) corresponding to the eigen-frequencyi .Inserting expressions (24) into Eq. (22) andpremultiplying by T one obtains

( ) ( ) ( )T T T T t t t + + = M C K MBa&& & & (26)Imajui u vidu proporcionalnost (23) i ortogonalnost

svojstvenih vektora, dobija se da jeHaving in mind the linear combination (23) and

orthogonality of eigen-vectors, one obtains

( )2T i idiag = C (27)pri emu su modalni vektori jo i ortonormirani u odnosuna matricu mase, tako da je

while the modal vectors are also orthonormalized withrespect to the mass matrix, so

IM =T 2( )T idiag = K (28)

Sistem jednaina (26), vodei rauna o jedainama(27) i (28), prelazi u skup nezavisnih jednaina ponormalnim koordinatama (i =1,2,...,n):

The system of equations (26), having in mindrelations (27) and (28), becomes the set of indepenedentmodal (or normal) equations (i =1,2,...,n):

22 T i i i i i i i + + = MBa&& & (29)

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Reenje ove jednaine moe da bude prikazanopreko integrala konvolucije

The solution of each modal equation may be givenwith use of the convolution integral in the form

( )

0

) sin ( )i iT t

t ii i

i

( e t d

= MB

a&& (30)

U jednaini (30), vodei rauna o jednainama (23),(27) i (28), priguenjei -tog tona oscilovanja iznosi

Keeping in mind equations (23), (27) and (28), therelative damping of the modei in Eq.(30) is given by

1

1

i ni

in

+=

+(31)

4 SPEKTRALNA ANALIZA

Spektar odgovora moe se upotrebiti samo samodalnom analizom. Detaljniji prikaz spektralne analize,ali i drugih savremenih postupaka analize uticaja zemljo-tresa, dat je u radu [9], dok je analiza vrednovanjaaseizmikog projektovanja data u [10]. O pristupuseizmike analize u skladu sa odredbama Evrokoda 8,dato je u [12], dok rad [11] posebno posmatra analizuvremenskog odgovora za zadati akcelerogram. Priizvoenju izraza koji slede, pretpostavljeno je dapobuivanje konstrukcije moe da bude samo uhorizontalnoj ravni i to u pravacu samo jedne od osa1 ili2 . Ovo ogranienje pojednostavljuje izvoenje, ali neograniava metodu. Poto se radi u elastinompodruiju, vai zakon superpozicije i rezultati proraunaza pojedine pravce pobuivanja mogu da se sabiraju.Pretpostavlja se da je pobuda (odn. ubrzanje) u pravcuose 1 i da iznosi ( )a t && . Izraz (19) se transformie u

4 SPECTRAL ANALYSIS

The response spectrum may be used only in thecontext of the modal analysis. More detailed presen-tation of the spectral analysis, and also other con-temporary methods of seismic analysis, is given in [9],while the analysis of evaluation of aseismic designsolutions is given in [10]. Seismic analysis according toEurocode 8 provisions is given in [12], while [11] isaddressing the time history response for a givenaccelerograms. In derivation of the following expressionsit is assumed that the seismic excitation of the structureis acting only in the horizontal plane and in direction ofonly one axis,1 or 2 . This restriction simplifies deriva-tion, but does not restrict the method itself. Since theelastic behavior is assumed, the principle of super-position is valid, so the results obtained separetely forvarious directions may be added. It is assumed that theexcitation (i.e. acceleration) is acting in direction of theaxis 1 and is given as ( )a t && . Expression (19) is trans-

formed into

) j ja(t =B a(t) b&& && [ ]cos sin 0 0 0 0T j =b

(32)

Jednaina (30) moe, vodei rauna o izrazu (32),

da se napie kaoEq. (30), having in mind expression (32), may be

written as

( )

0

( )) sin ( )i i

T t t i i

i i ii i

D t a( e t d

= = Mb &&

1

T T T T j N = b b b bK K

(33)

Sa D i je oznaen integral. i je faktor participacije

definisan izrazomThe integral in Eq. (33) is denoted asD i while i is

the participation factor defined by

T i i = Mb (34)

U sluaju optereenja konstrukcije spektrom odgovo-ra, kod sistema sa jednim stepenom slobode pomeranja,vrede izrazi

When the structure is loaded by the responsespectrum, for the system with one degree of freedom,the folowing relations hold

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2max( ) pv pad

S Sq t S

= = =

(35)

gde je S d spektar relativnih pomeranja,S pv spektarpseudobrzine iS pa spektar pseudoubrzanja. Svi spektriS su funkcije krune frekvencije i priguenjaS=S( , ),ali, alternativno, mogu da budu izraeni i u funkcijisvojstvenog perioda i priguenja.Maksimalna veliina kolinikaD i (t)/ i po analogiji sasistemom sa jednim stepenom slobode, jednaka jeveliini u spektru pomeranja, za krunu frekvencijui iza priguenjei

where S d is the spectrum of relative displacements,S pv the spectrum of pseudovelocity andS pa the spectrum ofpseudoacceleration. All spectrumsS are the functions ofthe circular frequency and the relative dampingS=S( , ), but, alternatively, may be expressed as thefunctions of the natural period and damping.

The maximum value of the ratioD i (t)/ i in analogywith the system with one degree of freedom, is equal tothe corresponding value in the displacement spectrum,for a given modal circular frequancyi and for a modaldampingi

( )max

( , ) idi di i ii

D t S S

= = (36)

Na taj nain, maksimalna veliina i data izrazom

(33), vodei rauna o izrazu (36), moe da se napie uobliku

Therefore, the maximum value of i given by

expression (33), and having in mind expression (36),may be written in the form

,maxi i di S = (37)Vektor maksimalnih generalisanih pomeranja i vektor

maksimalnih unutranjih sila (doprinos vibracija u tonui )u osnovnom koordinatnom sistemu su, imajui u vidu(24), dati sa

Due to transformation (24), the vector of maximumgeneralized displacements and the vector of maximuminternal forces (contribution of the natural modei ), in theinitial generalized coordinates, are given, having in mindtransformation (24), as

, ,i max i i max i i diS= = q (38)2

,max ,maxT

i i i i i i paiS= = =S Kq M q M Mb (39)Veliina S pai je veliina u spektru pseudoubrzanja.

Indeks i odnosi se na ton vibracijai . Vektor unutranjihsila Si koristi se kao ekvivalentno statiko optereenje(seizmike sile) statikog modela konstrukcije.

Kao krajnji rezultat analize, potrebno je poznavatiukupan uticaj (presene sile, pomeranja) usled svihtonova vibracija. Kombinacije uticaja pojedinih tonova,kod primene spektra odgovora, mogue je nainiti samopriblino. Kao jedna od metoda kombinacije upotrebljavase kompletna kvadratna kombinacija (eng . CompleteQuadratic Combination) ili skraeno CQC. Ukupni vektoruticaja u (presene sile, pomeranja) u konstrukcijiusled svih modova oscilovanja, iznosi

The quantity S pai is the value in the spectrum ofpseudoaccelerations and the indexi corresponds to thenatural mode numberi . The vector of internal forcesSi isused as the equivalent static loading (i.e. seismic forces)in the static model of the structure.

As the final result of the analysis, it is necessary toobtain the complete effects (internal forces, displace-ments) due to all natural modes. The proper combina-tions of the contributions of particular natural modes inthe overall response, using the spectral approach, maybe determined only approximately. As one of the modalcombinations, the Complete Quadratic Combination (orCQC), is used here. The complete vector of some effect

in the structure (internal forces, displacements), denotedas u, as the corresponding contribution of all naturalmodes, is given as

1 11

T

T k k k

T nn n

u

u

u

= =

u u

u u u

u u

MM

M M

, ,T k 1,k i k n k u u u = u K K

(40)

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gde je k u vektor k -tog uticaja u konstrukciji usled svih

modova oscilovanja, alan ,i k u vektora k u je vrednost

uticajak za i -ti ton oscilovanja ilan k u vektorau jeukupnik -ti uticaj usled svih modova oscilovanja. jematrica korelacije, prema Der Kiureghian-u [3], i iznosi

where k u is the vector of thek -th effect in the structure

due to all modal contributions, and the term ,i k u of the

vector k u is the value of the effectk for thei -th mode of

oscillation and the term k u of the vectoru is the totalk -th effect due to all modes of vibrations. Finally, is the

corelation matrix, according to Der Kiureghian [3], and isgiven by

=mmm

ij

m

L

MM

L

1

111

222222

23

4141

8

)r ( )r r( )r (

)r r (

ji ji

/ j ji ji

ij +++++

=

0 1ij

i

j

r =

(41)

Izraz (41) nije upotrebljv kada je 0, 1r = = i tada se

uzima da je 1ij = .

5 EKSTREMNE VREDNOSTI UTICAJAPretpostavie se da spektralno optereenje, dato

preko krivihS 1 i S 2 , deluje na konstrukciju u dvaortogonalna pravca, u horizontalnoj ravni, istovremeno.

Vektor rezultujuih uticaja u , ije su komponentevrednosti uticaja 1u usled optereenja S 1 koje deluje

pod uglom i 2u usled optereenja S 2 koje deluje poduglom 90 + , iznosi

Expression (41) may not be used when 0, 1r = = ,so in that case, one uses 1ij = .

5 EXSTREME VALUES OF EFFECTSIt is assumed that the spectral loading, defined by

curves S 1 and S 2 , is acting upon the structure in thehorizontal plane, along the two orthogonal directions,simultaneously.

The vector of the resulting effectu , whosecomponents are denoted as 1u due to loadingS 1 whichis acting along the angle (with respect to X axis) and

2u due to loading S 2 which is acting along theangle 90 + , is given as

0190

1 10

2 2902

cos sin

cos sin

uu u

u u

u

= =

u (42)

gde je iu vrednost uticaja u usled spektralnog

optereenja S i (i =1,2) koje deluje pod uglom( 0,90 = ) i rauna se prema izrazima datim upoglavlju 4. Predznak u jednaini (42) je neohodanpoto vrednosti iu

gube predznake u modalnojkombinaciji. Poto suS 1 i S 2 statiki nezavisni, ukupnavrednost uticajau je

where iu is the value of the effectu due to spectral

loading S i (i =1,2) which is acting along theangle ( 0,90 = ) and is determined according toexpressions given in the section 4. The sign in Eq. (42)is necessary since the values iu

are losing their signsin the modal combination. SinceS 1 and S 2 are staticallyindependent, the total value of the effectu is given as

( ) T u = u u (43)

Ugao za koji e neki uticaj u imati ekstremnuvrednost, nalazi se iz uslova [8]

The angle that corresponds to the case when someeffect u has an extreme value may be obtained from thecondition, see [8],

( )0

u

=(44)

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Iz izraza (44) dobija se, vodei rauna o izrazima(42) (ispred sinusa i kosinusa ugla uzima se predznak+) i (43), izraz iz kojega se rauna kritini ugao cr zakoji uticaju ima ekstremnu vrednost i iznosi

Using expression (44), taking care about expressions(42) (in front of sine and cosine terms the sign + isassumed) and also (43), one obtains the expressionfrom which one might determine the critical anglecr that corresponds to the extreme value of the effectu andis given by

0 90 0 901 1 2 2

0 2 90 2 0 2 90 21 1 2 2

2( )tan(2 )(( ) ( ) ) (( ) ( ) )cr

u u u uu u u u

+= + (45)

Svaki uticaj moe da ima razliitu vrednost kritinogugla usled spektralnog optereenja. Zamenom ugla cr u izrazu (42), iz izraza (43) moe da se izraunaekstremna vrednost uticaja.

6 PARAMETARSKA ANALIZA

Za parametarsku analizu korien je razvijeniprogram ELAN [2] koji je napisan u jeziku C++ iomoguava linearnu i nelinearnu analizu konstrukcijausled statikog i dinamikog optereenja.

Da se u analizi uticaja (pomeranja i presene sile),usled promene fleksibilnosti i ekscentrinosti veza naseizmiki odgovor konstrukcije, ne bi koristile stvarnenumerike vrednosti uticaja, uvode se bezdimenzionalneveliine: koeficijenat krutosti, koeficijenat ekscentrinostii normalizovani uticaj. Pri tome se definiu dve vrstedijagrama [8]:

Dijagram A: Koeficijent krutosti (K k ) - Normalizovani uticaj usled promene koeficijenta krutost ( ,k u N ). Uticaji u konstrukciji su dati kao funkcije koeficijenta krutosti veze koja je definisana kao

Every effect may have the different value of thecorresponding critical angle due to spectral loading.Substituting the angle cr in expression (42), using (43),one might determine the corresponding extreme value ofconsidered effect.

6 PARAMETRIC ANALYSIS

For the parametric numerical analysis, the cor-responding computer code ELAN [2] is used. The code,developed using C++ language, enables the linear andnon-linear static and dynamic analysis of frameworkbuilding structures.

When analyzing effects (displacements and cross-sectional forces) due to change of flexibility andeccentricity of joints under seismic loading, in order toavoid the real numerical values, the corresponding non-dimensional values are introduced: coefficient of jointrigidity, coefficient of joint eccentricity and normalizedconsidered effect. Two types of diagrams are defined,according to [8]:

Diagram A:Coefficient of rigidity (K k ) - Normalized effect due to change of coefficient of rigidity ( ,k u N ).Structural effects are given as functions of the coefficientof joint rigidity, which is defined as

13

1k K EI

lk

=+

(46)

gde je k krutost veze. Uticaji kod ovog dijagrama senormalizuju deljenjem njihovih vrednosti sa vrednostimadobijenim za konstrukciju sa krutim vezama. Normali-zovani uticaj je dat kao

where k is the joint rigidity. Effects are normalized bydividing their values by the value of the same effect inthe case of a structure with rigid connections.Normalized effect is given as

,1

Kk k ut

Kk

u N u =

= (47)

gde je u Kk posmatrani uticaj (oznaka ut= U zapomeranja,ut= F za sile) za koeficijenat krutost vezeK k ,a u Kk=1 je isti taj uticaj zaK k =1.

Dijagram B: Koeficijent ekscentri nosti (K e ) - Normalizovani uticaj usled promene koeficijenta ekscentri nosti ( ,e u N ). Uticaji u konstrukciji su dati kaofunkcije koeficijenta ekscentrinosti veze. Koeficijentekscentrinosti veze dat je kao

where u Kk is considered effect (notationut= U fordisplacements,ut= F for forces) for the coefficient of jointrigidityK k , whileu Kk=1 is the same effect forK k =1.

Diagram B: Coefficient of eccentricity (K e ) - Normalized effect due to change of the coefficient o eccentricity ( ,e u N ). Structural effects are given asfunctions of the coefficient of joint eccentricity. Thecoefficient of joint eccentricity is given as

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k e

lK

l=

(48)

gde je l k duina krute zone uvoru, a l duina tapa.Uticaji kod ovog dijagrama se normalizuju deljenjemnjihovih vrednosti sa vrednostima dobijenim zakonstrukciju bez ekscentriciteta. Normalizovani uticajiznosi

where l k is the length of the rigid zone in a joint, whileis the length of a beam element. The effects in thisdiagram are normalized by dividing their values with thecorresponding values obtained for the structure withouteccentricity. The normalized effect is given as

,0

Kee ut

Ke

u N

u == (49)

gde je Keu posmatrani uticaj za koeficijenat ekscentri-nostiK e , a 0Keu = isti taj uticaj zaK e =0.

U parametarskoj analizi se, kao uticaji, analizirajuekstremni uticaji u konstrukciji, koji se raunaju premapostupku datom u poglavlju 5. Normalizovane vrednostiekstremnih uticaja dobijaju se iz izraza (47) odnosno (49).

Kao primer, na kojem je uraena parametarskaanaliza, razmatrana je nesimetrina zgrada sa tri kulerazliite spratnosti. Glavna kula ima deset spratova, dok

sporedne kule imajuetiri, odnosno est spratova (slika5). Osovinski razmak stubova uX i Y pravcu je 8 metara,a meuspratna visina je 4 metra. Karakteristike popre-nih preseka su: grede: povrinaF =0.306m 2, momenatinercije oko lokalne ose paralelne ravniXOY I =0.002569 m 4; stubovi: povrinaF =0.1224m 2 , momenatinercije oko obe lokalne oseI = 0.001798m 4. Young -ovmoduo elastinosti materijala je 8 22.1 10 E x kNm=(elik). Ekscentricitet i krutost veza uzimana je samo naspoju greda i stubova.

Uticaji u konstrukciji su analizirani za optereenjadata preko krivih spektra pseudoubrzanja (slika 6a):kriva S Ec1 prema [1], sa parametrima: 21.2a ms =&& ,srednje tlo (tipB ), faktor ponaanjaq =1, =0%.; kriva

S Ec2 prema [1], sa parametrima: 21.2a ms =&& , srednje tlo(tip B ), faktor ponaanjaq =1, =5%.; krivaS Yu prema[4], sa parametrima: II kategorija tla, II kategorija objekta,tip konstrukcije 1 i IX seizmiko podruije. Poto su, prianalizi preko dijagramaK k - ,k ut N i K e - ,e ut N , od interesasamo normalizovane vrednosti uticaja, kao optereenjemogu da se koriste normalizovane vrednosti spektarapseudoubrzanja (slika 6b). Ova normalizacija je izvrenau odnosu na maksimalnu vrednost pseudoubrzanja zasvaki dijagram posebno. Osim normalizovanih krivihspektara pseudoubrzanja S Ec1, S Ec2 i S Yu uveden je i jedinini spektar pseudoubrzanjaS C , da bi se iz prora-una eliminisao uticaj oblika krive spektra pseudo-ubrzanja, i na taj nain dobile granine vrednost norma-lizovanih uticaja u konstrukciji sa fleksibilnim i ekscen-trinim vezama.

Uticaji u konstrukciji trae se zaetiri sluaja optere-enja, data preko etiri spektralne krive. Prilikom odre-ivanja uticaja, za svaki sluaj optereenja posebno, istispektar pseudoubrzanja deluje u pravcu osa1 i 2 istovremeno.

Tavanice konstrukcije su krute betonske ploe.Obrtanjevorova eline konstrukcije nije spreeno. Zaproraun uticaja u konstrukciji, u radu je korien modelzgrade sa krutim tavanicama - pseudo trodimenzionalnimodel, opisan u radu [2]. Mase konstrukcije su koncen-trisane u teitima tavanica i vrednosti su date u tabeli 1.

whereKeu is considered effect for the coefficient of

eccentricityK e , while 0Keu = is the same effect forK e =0.The effects calculated in the parametric analysis are

the extreme effects in the structure, determinedaccording to the procedure outlined in section 5.Normalized values of extreme effects are obtainedaccording to expressions (47), or (49).

As an example structure, used in the parametricanalysis, a non-symmetric framed building with three

towers of different number of stories is considered. Themain tower has ten stories, while the other two towershave four and six stories, (Fig. 5). The bay distance ofcolumns in X and Y directions is 8m, and the story heightis 4m. Cross-sectional properties are, for beams: areaF =0.306m 2, moments of inertiaI = 0.002569m 4 and forcolumns: area F =0.1224m 2 moments of inertia in bothdirectionsI = 0.001798m 4. Young's modulus of elasticityof material is 8 22.1 10 E x kNm= (steel). Eccentricity andrigidity of joint connections is considered only at beam-to-column connections.

The structure is analyzed for loadings that aredefined as given curves of pseudo-acceleration (Fig. 6a):the curve S Ec1 according to [1], with parameters:

2

1.2a ms

=&& , soil type B (middle soil), behavior factorq =1, =0%.; curveS Ec2 according to [1], with parameters:21.2a ms =&& , soil type B (middle soil), behavior factor

q =1, =5%.; and curve S Yu according to [4], withparameters: soil type II, category of structure II,structural type 1 and the seismic zone IX. Since theresults of parametric analysis are presented by thenormalized diagramsK k - ,k ut N and K e - ,e ut N i.e. onlynormalized effects are of interest, it is possible to use thenormalized values of the pseudo-acceleration spectrums(Fig. 6b). The normalization is performed with respect tothe maximum value of pseudo-acceleration for eachcurve separately. Besides the normalized curves of

pseudo-acceleration spectrumsS Ec1, S Ec2 and S Yu theunit pseudo-acceleration spectrumS C is introduced too,in order to eliminate the influence of the shape of thespectral curve in the calculation and to obtain the limitvalues of normalized effects in the structure with theflexible and eccentric joint connections.

Effects in the structure are determined for the fourloading cases, given by four spectral curves. Deter-mination of extreme effects was done for simultaneouspseudo-acceleration spectrum in both axes1 and 2 , foreach loading case.

Floor slabs of the structure are rigid concrete plates.Rotations of joints of the steel part of the structure is notprohibited. Numerical model of a building was assumed

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Na slici 7 prikazan je uticaj promene koeficijenta kru-tosti na normalizovana ekstremna pomeranjavorova 1,11 i 19 u pravcu oseX . Normalizovana ekstremna po-meranja rastu sa opadanjem koeficijenta krutosti. Za isti

Fig. 7 is presenting the influence of the change of thecoefficient of rigidity upon the normalized extreme dis-placements of nodes 1, 11 and 19 in direction of theglobal X axis. The normalized extreme displacements are

a) a)

b) b)

c) c)

d) d)

Slika 7 Uticaj krutosti veze na normalizovane ekstremne vrednosti pomeranja vorova u pravcu X

usled a) S Ec1 b) S Ec2 c) S Yu d) S C Fig. 7 Influence of the joint rigidity upon the

normalized extreme displacements in X direction due to: a) SEc1 b) SEc2 c) SYu d) SC

Slika 8 Uticaj ekscentriciteta veze na nor - malizovane ekstremne vrednosti pomeranja vora 1

usled a) S Ec1 b) S Ec2 c) S Yu d) S C Fig. 8 Influence of the joint eccentricity upon the

normalized extreme displacements of node 1 due to: a) SEc1 b) SEc2 c) SYu d) SC

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koeficijenat krutosti, normalizovano ekstremno pomera-nje se uveava sa uveanjem sprata. Normalizovanaekstremna pomeranja, data na slikama 7a-c, ne prelazeodgovarajue granine vrednosti pomeranja date na slici7d.

Na slici 8 prikazan je uticaj promene koeficijentaekscentrinosti, a za tri vrednosti koeficijenta krutostiK k =0.1, 0.5, 1.0, na normalizovano ekstremno pomera-

njavora 1 u pravcu oseX . Sa slike se vidi, da to jevei koeficijent ekscentrinosti, za bilo koji koeficijenat

krutosti, normalizovano ekstremno pomeranje sesmanjuje priblino linearno. Za istu vrednost koeficijentaekscentrinosti, a za razliite koeficijente krutosti,normalizovana ektremna pomeranja su priblino ista.

Normalizovana ekstremna pomeranja, data na slika-ma 8a-c, ne prelaze odgovarajue granine vrednostidate slici 8d.

Na slici 9 prikazan je uticaj promene koeficijentakrutosti na normalizovane ekstremne vrednosti prese-nih sila u oslonakomvoru 21. Transverzalna silaT jedata u pravcu globalne oseY , a momenat savijanjaM jedat oko globalne oseX. Na dijagramima se vidi, da sasmanjenjem koeficijenta krutosti normalizovane ekstre-mne transverzalne sile ravnomerno opadaju. Normalizo-vani ekstremni momenti savijanja, sa smanjenjem koe-ficijenta krutosti, i u zavisnosti od primenjenog spektrapsudoubrzanja, mogu da opadaju pa da rastu (slika 9c).

increasing with decrease of the coefficent of rigidity. Fora given coefficient of rigidity normalized extremedisplacement is increasing with increase of floor level.Normalized extreme displacements, given in Figs. 7a-c,do not overcome the limit values of displacements givenin Fig. 7d.

Fig. 8 is presenting the influence of the change of thecoefficient of eccentricity, for the three values of the

coefficient of rigidityK k =0.1, 0.5, 1.0, upon thenormalized extreme displacement of node 1 in directionof the axis X. It may be seen from the figure that withincrease of the coefficient of eccentricity, for anycoefficent of rigidity, normalized extreme displacement isdecreasing approximately linearly. For the same value ofthe coefficent of eccentricity, normalized extremedisplacements are approximately the same.

Fig. 9 is presenting the influence of the change of thecoefficent of rigidity upon the values of the cross-sectional forces at the support joint 21. Transverse forceT is in direction of the global axis Y, while the bendingmoment M is about the global axis X. From the givendiagrams, it may be seen that the normalized extremetransverse forces are uniformly decreasing withdecrease of the coefficient of rigidity. Normalized extre-me bending moments, with decrease of the coefficient ofrigidity and depending on the applied spectrum ofpseudo-acceleration, may decrease and then increase

a) b)

c) d)

Slika 9 Uticaj krutosti veze na normalizovane ekstremne vrednosti prese nih sila u voru 21 usled a) S Ec1 b)S Ec2 c) S Yu d) S C

Fig. 9 Influence of joint rigidity upon the normalized extreme values of cross-sectional forces in joint 21 due to a) SEc1 b) SEc2 c) SYu d) SC

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Normalizovane ekstremne presene sile, date naslikama 9a-c, ne prelaze odgovarajue graninevrednosti date slici 9d.

Iz dijagrama na slikama 7 i 9 moe da se vidi dapromena krutosti veze ima dosta veeg uticaja napomeranja nego na presene sile.

7 ZAKLJUAKMatrica krutosti za gredne elemente sa polukrutim i

ekscentrinim vezama, kao i postupci modalne ispektralne analize, prikazani su u radu i implementiranisu u razvijenom programu ELAN, napisanom u jezikuC++. Program je namenjen za linearnu statiku idinamiku analizu prostornih ramovskih konstrukcija, asamim tim i za analizu prostornih i nesimetrinih zgrada,ukljuujii i seizmiku analizu datu preko spektralnogoptereenja.

U cilju ilustracije numerikog postupka razmatrana jenesimetrina zgrada, gde glavna kula zgrade ima desetspratova, dok sporedne kule imajuetiri, odnosno est

spratova. Za analizu uticaja u zgradi, korien je modelzgrade sa krutim tavanicama pseudo trodimenzionalnimodel. Uticaj polukrute i ekscentrine veze u proraun jeuveden preko korektivne matrice. Primenom korektivnematrice modifikovana je konvencionalna matrica krutostielementa sa krutim i centrinim vezama.

Analiziran je uticaj ekscentriciteta i krutosti veza naekstremne vrednosti uticaja (pomeranja i presene sile)usled optereenja zgrade spektralnim optereenjem.

Na osnovu analize dobijenih rezultata, moe da sezakljui da: promena koeficijenta krutosti veze ima dostaveeg uticaja na normalizovana ekstremna pomeranjanego na normalizovane ekstremne presene sile; sasmanjenjem koeficijenta krutosti: normalizovana eks-tremna pomeranja se poveavaju, normalizovanaekstremna transverzalna sila u osloncu ravnomernoopada, a normalizovani ekstremni momenat savijanja uosloncu moe na jednom delu dijagrama da opada azatim da raste u zavisnosti od primenjenog optereenja;sa poveanjem koeficijenta ekscentrinosti, za bilo kojikoeficijenat krutosti, normalizovana ekstremna pomera-nja linearno opadaju.

(Fig. 9c). Normalized extreme cross-sectional forces,given in Figs. 9a-c, do not overpass the correspondinglimit values given in Fig. 9d.

From diagrams given in Figs. 7 and 9 it may beconcluded that the change of joint rigidity has sub-stantially greater influence upon displacements thenupon the cross-sectional forces.

7 CONCLUSIONThe stiffness matrix for beam elements with semi-

rigid and eccentric connections, modal and spectralearthquake analyses, are presented in the paper, andalso implemented in the computer code called ELAN,see [2], which is developed in C++. The program isdevoted to linear static and dynamic analysis of spatialframes, therefore also to 3D and non-symmetricbuildings, including also seismic analysis defined byspectral loading.

As an illustration of the numerical procedure, a non-symmetric framed building is considered. The buildingconsists of three parts, or towers, of unequal heights: the

main one is 10 stories high, while the other two are 6and 4 stories. The building is modelled as a structurewith rigid floors - shear building pseudo tridimensionalmodel. The influence of semi-rigid and eccentric con-nections is introduced by the corresponding correctivematrix which is modifying the classical stiffness matrixfor beam elements with rigid and centric connections.

The influence of eccentricity and rigidity of jointconnections upon the extreme values of effects (displa-cements and cross-sectional forces) is analyzed, due toseismic loading of a building defined by the spectralcurves.

Based upon the analysis of obtained results, in maybe concluded that the change of coefficient of rigidity in joints has much larger influence upon the normalizedextreme displacements then upon the normalized ex-treme cross-sectional forces. When the coefficient of joint rigidity is decreased, normalized extreme displace-ments are increasing, while the normalized extremetransverse force (at considered support joint) is uniformlydecreasing. Also, the normalized extreme bendingmoment (at support joint) is partially decreasing and alsoincreasing, depending on applied loading. Finally, whenthe coefficient of joint eccentricity is increasing, for anycoefficient of joint rigidity, the normalized extremedisplacements are linearly decreasing.

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