th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 2004 Paper No. 3256

Transcription

1 h World Conference on Earhquake Engineering Vancouver, B.C., Canada Augus 1-6, 2004 Paper No. 256 TOUCHING ANALYSIS OF TWO BUILDINGS USING FINITE ELEMENT METHOD Mircea IEREMIA 1, Silviu GINJU 1, Adrian Ivan 2 SUMMARY The presen paper focuses on he conac-impac analysis of wo buildings during he 1977 Vrancea Earhquake, Romania. The srengh srucure, for boh of hem, is made up of seel frame. The buildings have differen heighs, laeral siffness and masses. A Time hisory analysis in elaso-plasic range of seel behaviour has been accomplished, in order o model he impac. Thus, for modeling he conac of hese wo buildings Gap finie elemens are being used. So, his analysis shows he individual vibraion characerisics of each building bu also, he deformaions and forces resuled from he conac in he srucures. This paper poins ou boh he necessiy of a cerain deerminaion for compuing he gap beween wo new buildings, as well as he mehod of analysis for he old buildings wih unsuiable gaps beween hem. Figure 1. The discreizaion in finie elemens of wo buildings srucure 1 Prof. dr., Technical Universiy of Civil Engineering Buchares, Romania. mieremia@mail.ucb.ro 2 Prof. dr., Poliechnical Universiy of Timisoara, Romania. aivan@cef.u.ro

2 INTRODUCTION The paper presens he modal and nonlinear dynamic analysis of wo buildings in earhquake ineracion (figure 1). The higher building (figure 2) is a saion for coal disribuion, nirogen and oxygen srengh srucure, afferen o F2 furnace, belonging o S.C. SIDEX S.A. Galai, Romania. The Disribuion Saion has 4 echnological levels. The srengh srucure is made up of many-sored bracing seel frames, being assembled by welding. The building has 4 levels of 6m heigh, one span of 6m and wo bays of 6m. In he gable, he building has wo consoles abou.20m and 1.95m respecively. The Saion is parallel wih he furnace house. The building is covered wih iron pleaed. The building is suppored by foundaion plaes up o he 2.20m deph and by drilling piles of 800 mm diameer beyond ha. Figure 2. The high srucure The access in he Saion is done hrough a gangway being suppored by he furnace house and he sairs beween he floors. Beween he furnace house and he Saion here are some pipes which penerae in he Saion from where he connecions wih he furnace are made. The lower building (figure ) is an indusrial hall wih ground floor and parial firs floor. The ground floor space is used as sorage space and he firs floor houses offices. The srengh srucure is made of meallic farms wih siffness bars verically as well as horizonally. The pillars ransmi he loads o he foundaion land hrough some isolaed foundaions. Beween he wo buildings here is an earhquake disance of cm. The buildings have he C caegory of imporance in accordance wih H.G. 766/1997, and III in accordance wih P sandard. The buildings are placed in C seismic zone wih Tc=1.5 period of corner and Ks=0.20, seismic zone coefficien.

3 Figure. The lower srucure CONSIDERATIONS REGARDING THE MODELLING The srengh of boh srucures has been analysed using a finie elemen mehod. For his purpose he high srucure has been divided ino 214 BEAM finie elemens conneced in 90 nodes. For defining he secions of he elemens hree consans ses have been necessary. The columns have I cross-secion composed of welding seel (flange: 50x20, web: 500x15), he bracings are made up of wo U20 laminaed channel secion and he beams have I cross-secion composed of welding seel (flange: 240x15, web: 400x8). The lower srucure has been divided ino 554 BEAM finie elemens conneced in 22 nodes. The columns has wide flange cross-secion composed of welding seel (flange: 180x20, web: 450x12), he verical bracings are made up of pipe 140/8 secion, he horizonal bracings are made up of angle 100x100x10 secion, he purling are made up of wo U20 laminaed channel secion and he frame girders have flange composed of wo 140x140x40 angle secion and he russ of frame girder is composed of wo 140x140x40 angle secion. All he seel elemens are made of OL7. For he simulaion of he seel elemens in elaso-plasic range, BEAM finie elemen wih von Mises consiuive law has been used. The ineracion beween he srucures was modelled wih Gap finie elemens. CONSIDERATIONS REGARDING THE DYNAMIC ANALYSIS Linear modal analysis The moving equaion for an undamped dynamic sysem, expressed in marix noaion is: [M] {ü} + [K] {u} = {0} (1) [M] he srucure mass marix; [K] he srucure siffness marix in elasic range. For a linear sysem, free vibraions will be harmonic having he form: {u} = {φ} i cos ω i (2) {φ} i - eigenvecor represening he mode shape of he i-h naural frequency; ω i - i-h naural circular frequency (radians per uni ime); - ime.

4 Thus, equaion (1) becomes: Figure 4. High srucure. The firs mode of vibraion ( -ω 2 [M] +[K] ) {φ} i = {0} () This equaion is saisfied if eiher {φ} i is zero or he deerminan of (-ω 2 [M]+[K]) is zero. The firs opion is he rivial one and, herefore, is no of ineres. Thus, he second one gives he soluion: [K] -ω 2 [M] = 0 (4) This is he sandard form of eigenvalues equaion which mus be solved for up o n values of ω 2 and n eigenvecors {φ} i which saisfy equaion (), where n is he number of DOF s. Figure 5. High srucure. The 2-h mode of vibraion

5 The eigenvalue and eigenvecor problem needs o be solved for mode-frequency analysis. I has he form of: [K] {φ i }= λ i [M] {φ i } (5) λ i - he eigenvalue. By applying his ype of analysis, he eigenvalues and eigenvecors of he firs 20 modes of vibraoins have been obained. The rae of modal mass paricipaion obained for horizonal vibraions is ~90%, and for verical vibraions is ~80%. Due o he obained firs period of T 1 = 0.87 s, he srucure is considered a semi-flexible srucure. Figure 6. High srucure. The -h mode of vibraion Table1. High srucure. Eigenvalue and mass paricipaion facors Mode ω (rad/s) ν (herz) T (s) Fpx (%) Fpy (%)

6 Figure 7. Lower srucure. Firs mode of vibraion Figure 8. Lower srucure. The 2-h mode of vibraion Figure 9. Lower srucure. The -h mode of vibraion

7 Table 2. Lower srucure. Eigenvalue and mass paricipaion facors Mode ω (rad/s) ν (herz) T (s) Fpx (%) Fpy (%) Time hisory analysis The srengh srucures have been dynamically analyzed using he direcly inegraion mehod of he differenial moion equaion, considering he damping influence in srucure s dynamic response. When using he finie elemen mehod for srucures discreizaion, he moving equaions sysem becomes: [M]{ U & ( )} + [ C]{ U& ( )} + [ K]{ U ( )} = [ M ]{ u& ( )} (8) Using he Newmark [1] inegraion scheme, we can make he following assumpions: U& U& + 1 δ U&& + δ * U&& (9) [( ) ] = U = U + U& + + α U&& + α * U&& + 2, (10) α si β are he parameers which give he sabiliy and accuracy of he inegraion process. For solving he displacemens, speeds and acceleraions a ime, we considered he equaions (9), (10) and also he equilibraion equaions a ime: MU & + + CU& + KU = R (11) + Solving he equaion (10) for U & varying wih + U + and hen placing he obained relaion in (9), we obain he equaions for U & si + U& +. Boh of hem varying only wih he unknown displacemens U +. These wo equaions for U & si + U& + are placed in (11) o find ou U +. Then, using he equaions (9) and (10) i obains acceleraion U & and speeds + U& +. The complee algorihm using he Newmark inegraion scheme is made of: The firs compuaions: i forms he marix K (of siffness), M (of masses) and C (of damping); i gives o he displacemen, speed and respecively acceleraion, one firs value: U 0, U& 0siU & 0 i selecs sep of ime, α and δ parameers, and also, he inegraion consans are compued: 1 δ 1 a0 = ; a α 2 1 = ; a2 = ; α 2 α a = 1 2α ; δ a 1 4 = ; α δ a 5 = 2 ; a 6 = (1 δ ) ; a7 = δ ; 2 α

8 δ 0.50; α 0.25 (0.50+δ) 2 (12) i forms he siffness marix: Kˆ : Kˆ =K+a 0 M+a 1 C (1) Figure 10. The 1977 Vrancea earhquake For each ime-sep: i calculaes loading for sep of ime: R ˆ R M ( a U a U& a U&& ) C( a U a U& + a U& + = ) (14) i calculaes he displacemen for sep of ime: Kˆ ˆ (15) U + = R + i calculaes he acceleraions and speeds for he sep of ime: U&& = a ( U U ) a U& a U&& 0 2 U& = U& + a U&& + a U&& 6 7 Figure 11. High srucure response

9 For simulaing he dynamic acion of he earhquake, he 1977 Vrancea recordings (Figure 10) have been used. The earhquake has a magniude of M=7.4 and he following characerisics: - ground acceleraions 0.20g; - he maximum displacemen a he ground level.7 cm; - he lengh of acion 42 sec. The recorded acceleraion graph have been divided in 200 seps having he firs case of loading: dead loadings (he weigh of he srucure) and live loadings placed on he floors. The response of high srucure in elaso-plasic range is presened in figure 11. The low srucure, during he earhquake acion, is in elasic range (figure 12). Figure 12. Lower srucure response THE CONSTITUTIVE RULE OF THE MATERIAL The consiuive Von Misses [2] rule of he seel wih cinemaic consolidaion has been used in ime-hisory analysis for modelling he behaviour in elaso-plasic range. The expression of he equivalen sress is : σ e = [ 2 ({s} {a}) T [M]({s} {a})] 1/2 (17) {s} he deviaoric sress vecor; {s} - {σ} - σ m [ ] T (18) 1 σ m = (σx + σ y + σ z ) (19) σ m he mean of hydrosaic sress; {a} he yield surface ranslaion vecor. Noe ha since he equaion (17) is dependen on he deviaoric sress, yielding is independen of he hydrosaic sress. When equivalen sress σ e is equal o he uniaxial yield sress, σ y, he maerial is assumed o yield.

10 The yield crierion is herefore: F = [ 2 ({s} {a}) T [M]({s} {a})] 1/2 - σ y = 0 (20) The associaed flow rule yields: Q F = = ({ s} { a}), (21) σ σ 2σ e so, he incremen in plasic srain is normal o he yield surface. The associaed flow rule wih he von Mises yield crierion is known as he Prandl Reuss flow equaion. σ 2 deviaoric plane ( σ = 0) ii υ ; ; σ σ 0 ({ }, ) = 0 σ σ 1 F σ c σ = 1 = σ 2 σ σ Figure 1. The von Mises yield crierion The yield surface ranslaion is defined as: {a} = 2G {ε sh } (22) G he ransversal shear modulus; The incremen deformaion is analogously compued wih (2) equaion: sh sh sh { } { ε n } + { } ε n ε = 1 (2) { ε } sh C = { } ε sh (24) 2G and: 2 EET C = (25) E ET E he longiudinal Young s modulus; E T he angen modulus from he bilinear uni-axial sress-srain curve. The yield surface ranslaion {ε sh } is iniially zero and changes wih subsequen plasic sraining.

11 The equivalen plasic srain is dependen on he loading hisory and is defined : pl pl pl ˆ ε = ˆ ε + n n 1 ˆ ε εˆ pl n pl εˆ n 1 - he equivalen plasic srain for his ime poin; - he equivalen plasic srain from he previous ime poin. The equivalen sress parameer is defined : (26) σˆ pl σˆ pl e EE = σ T e y + εˆ pl n E E T - he equivalen sress parameer. is equal o he σ e y yield sress. If he load were o be reversed afer plasic loading, he sress σ e would fall bellow yield, bu σˆ pl Noe ha when here is no plasic srain (εˆ pl = 0), σˆ pl would regiser above ( since εˆ pl is non zero). (27) e CONSIDERATIONS REGARDING THE TIME HISTORY ANALYSIS The seismic ineracion of he wo srucures has been dynamically analysed in four ypes of placemen. Thus, he following disances beween buildings have been considered: 0 cm, 1 cm, 2 cm, cm, he las choice represening he genuine placemen of he buildings. Following he performance of he 4 ime hisory analysis, he seismic forces dissipaed hrough collision have been rendered in graphs (figures 14 17). For he earhquake disance of 0 cm (figure 14), he ineracion beween he wo srucures is unfolded hrough ou he duraion of he seism. The second siuaion analysed, ha of he earhquake disance of 1cm (figure 15), shows ha he srucures inerac beween he momen of PGA (peak ground acceleraion) and he end of he acceleraion graph. For an earhquake disance of 2 cm (figure 16), 8 collisions beween he buildings are noiced. A single collision occurs for he earhquake disance of cm (figure 17). Figure 14. Gap forces for 0 cm Figure 15. Gap forces for 1 cm

12 Figure 16. Gap forces for 2 cm Figure 17. Gap forces for cm The movemen response of srucures in he four cases of ineracion is rendered by comparison wih he case when he srucures do no inerac. Figure 18. Lower srucure. Negaive increased displacemens Figure 18 shows he variaion of negaive maxims o he low srucure response for he 4 cases of earhquake disance. The variaion of posiive maxims of he low srucure response for he 4 cases of earhquake disance is rendered in figure 19. Similarly, in figure 20 are presened he negaive maxims variaion whereas in figure 21 he posiive maxims variaion in displacemens for he high srucure response, regarding he 4 ypes of earhquake disance. Figure 19. Low srucure. Posiive increased displacemens

13 Figure 20. High srucure. Negaive increased displacemens Figure 21 High srucure. Poziive increased displacemens CONCLUSIONS The independen analysis of he wo srengh srucures having differen masses and rigidiies, showed differen response o seismic acion. Thus, as a resul of he seismic acion, he low srucure remains in he elasic field of behaviour for seel, while he high srucure displays a mainly elaso-plasic behaviour. The wo buildings are differenly influenced by he size of he earhquake disance. Thus, he low building is influenced more by he 1cm earhquake disance, while a cm earhquake disance has major influences on he high building. Also, as a resul of collision, he lower srucure suffers minor local plasificaions, while for he higher building he dynamic amplificaion involves major elaso-plasic changes, and also in elemens locaed ouside he conac area. For he case when here is no gap beween he buildings, no major amplificaion of dynamic response upon seismic acion occurs for none of he buildings, he dissipaion of seismic energy hrough moderae collisions occurring hrough ou he duraion of he seismic movemen. This fac migh parially explain why buildings which do no comply wih seism safey sandards have no collapsed during major earhquakes. REFERENCECS [1] K.J.Bahe, Finie Elemen Procedures, Prenice Hall, Englewood Cliffs, New Jersey, U.S.A, [2] M.Ieremia, Elasiciy.Plasiciy.Nonlineariy., Ed.PRINTECH Buchares, Romania, 1998.