SIMPLIFIED PREDICTION METHOD FOR SEISMIC RESPONSE OF ROCKING STRUCTURAL SYSTEMS WITH YIELDING BASE PLATES

Transcription

1 13 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August 1-6, 4 Paper No. 371 SIMPLIFIED PREDICTION METHOD FOR SEISMIC RESPONSE OF ROCKING STRUCTURAL SYSTEMS WITH YIELDING BASE PLATES Tatsua AZUHATA 1, Mtsumasa MIDORIKAWA, Tadash ISHIHARA 3 and Akra WADA 4 SUMMARY The rockng structural sstems wth eldng base plates, whch cause rockng vbraton durng strong earthquakes to reduce sesmc responses of steel buldngs, are now under development. In ths paper, smplfed methods are proposed to predct the sesmc responses of these sstems based on the energ conservaton law and so on. These methods are verfed b shakng table tests usng a half scale steel frame wth three stores and one ba. The predcton values for uplft dsplacement, base share, lateral roof dsplacement and vertcal compressve force at column bases of the rockng structural sstems are n good agreement wth the correspondng test results. INTRODUCTION To reduce sesmc responses of steel buldng structures, we are now developng structural sstems whch can cause rockng vbraton under approprate control durng sever earthquakes [1]-[7]. One of these sstems has base plates eldng n uplft moton at the column bases as shown n Fg.1. The base plates eld before eldng of the super structure. In ths paper, we propose smplfed predcton methods for sesmc responses of ths rockng structural sstem [BPY sstem], and verf the adequac of these predcton methods comparng wth the results of shakng table tests on a half scale steel frame wth three stores and one ba[5][6]. SIMPLIFIED PREDICTION METHOD FOR SEISMIC RESPONSES OF BPY SYSTEMS Uplft, base shear and lateral roof dsplacement Basc concept To predct uplft at the column bases, base shear of the super structure and lateral roof dsplacement of the whole structure, the energ conservaton law [8] whch s generall used for practcal sesmc desgn of buldng structures s appled. Ths method evaluates the energ whch s dsspated b the sstem untl 1 Internatonal Insttute of Sesmolog and Earthquake Engneerng, BRI, Japan, azuhata@kenken.go.jp Buldng Research Insttute, Japan, mdor@kenken.go.jp 3 Natonal Insttute for Land and Infrastructure Management, Japan, shhara-t9hd@nlm.go.jp 4 Toko Insttute of Technolog, Japan, wada@serc.ttech.ac.jp

2 A base plate elds due to tenson of column. Fg.1 Basc concepton of rockng structural sstems wth eldng base plates Energ dsspaton b Eq.(7) Fg. Equvalent one mass model Fg.3 Uplft force-deformaton relatonshp of base plates Fg.4 Energ dsspaton b base plates the uplft at the column bases reaches the maxmum from, and predcts sesmc responses such as the maxmum uplft consderng the balance between the dsspated energ and the nput earthquake energ. It s assumed that the total energ dsspaton s composed of the energ dsspaton b elastc deformaton of the super structure E S, that b up-lftng of the super structure s gravt center E G and that b plastc deformaton of the eldng base plates E B. We gnore the energ dsspaton b compresson stress of columns brought b mpact force here. From the followng secton, the super structure s replaced to the equvalent one mass model as shown n Fg.. Accordng to ths fgure, uplft s expressed b the product of the buldng wdth B and the rgd rotatonal angle θ. And the restorng force characterstc of base plates s modeled nto the uplft-force relatonshp shown n Fg.3 based on tensle statc tests [7]. Energ dsspaton b elastc deformaton of the super structure Usng the equvalent one mass model, the base shear of the super structure can be evaluated b Eq.(1). (. B ) + n N B Q( H + δ ) = M g 5 δ (1) up h Where, Q: the base shear of the super structure, H: the frst effectve heght of the super structure, δ up : uplft dsplacement of the gravt center of the super structure, M: mass of the super structure, δ h : horzontal dsplacement of the super structure, n: the number of columns exstng n the uplft sde, N: tensle force at the column base. Because the uplft dsplacement δ up of the gravt center and the horzontal dsplacement δ h of the super structure are relatvel slght to the heght and the wdth of the super structure respectvel, we gnore them. The tensle force N at the column base can be evaluated b Eq.() based on the restorng force characterstc of the base plate shown n Fg..

3 N ( B θ δ ) = N + K () Where, N : tensle eld strength of the base plate, K : secondar stffness of the base plate, δ : eld uplft deformaton of the base plate. Substtutng Eq.() nto Eq.(1), we can derve the followng equaton. Q n B K H B H (.5 M g + n N n K δ ) = θ + (3) The energ dsspaton b elastc deformaton of the super structure s evaluated b Eq.(4). Substtutng Eq.(3) nto ths equaton, the energ dsspaton can be expressed as the functon of the rgd rotatonal angle θ. E ( ) =.5 Q K =.5 Q ω (4) U M U Where, ω: the frst fundamental frequenc, M u : the frst effectve mass. Energ dsspaton b uplft of the super structure s gravt center Energ dsspaton b uplft of the super structure s gravt center s evaluated b Eq.(5) based on the geometrc relatonshp shown n Fg.. (. B H θ ) dθ E G = M g 5 ( B ) =.5 M g θ H θ (5) Because the member Hθ, whch means horzontal dsplacement s relatvel slght to the wdth of the structure, we gnore t as shown b Eq.(6). E G =. 5 M g B θ (6) Energ dsspaton b plastc deformaton of the base plates The bold lne n Fg. 4 llustrates the uplft response of the base plate. In ths fgure, the tensle stffness of the base plate s estmated to be degraded wth ts eldng to several waves untl the uplft reaches the maxmum. However, t s dffcult to evaluate the degraded tensle stffness precsel because the tme hstor of the uplft response wth eldng depends on the earthquake characterstc. Thus we evaluate energ dsspaton b plastc deformaton of the base plates usng the force - deformaton relatonshp shown b the dotted lne n ths fgure. Ths dea provdes the most conservatve evaluaton. Ths energ dsspaton s evaluated b Eq.(7). E B ( B θ ) =.5 n B θ δ K (7) Predcton usng the energ conservatve law The energ conservatve law of the BPY sstem s expressed b Eq.(8). E = E + E + E (8) I U G B Suppose the member E I s nearl equal to the elastc potental energ of the super structure whose bases are fxed, t s evaluated b Eq.(9).

4 E I E ( M ) =.5 Q ω (9) U Where, Q E : Base shear of the super structure whose bases are fxed Fnall, b substtutng Eq.(4), (6), (7) and (9) nto Eq.(8), we can predct the rgd rotatonal angle θ. Furthermore, we can predct the uplft b multplng the θ b the buldng wdth B, the base shear Q of the super structure b substtutng the θ nto Eq.(3) and the horzontal roof dsplacement of the whole structure b addng the elastc deformaton of the super structure whch corresponds to the base shear Q to the rgd deformaton whch s the product of θ and the buldng heght h. Vertcal compressve force at column bases The past shakng table tests cleared that mpact effect on the structure s landng amplfes vertcal compressve force at the column bases [5][6]. The model shown n Fg. 5 whch s composed of concentrated masses and vertcal sprngs s used to evaluate ths effect. Ths mass-sprng model expresses vertcal structural characterstcs of the landng sde of the super structure. Each mass s equal to the half of mass at the correspondng floor level. Rgdt of each vertcal sprng s equal to vertcal rgdt of the correspondng column. It s supposed that the mass-sprng model starts oscllatng vertcall wth the mpact veloct v of column bases as the ntal veloct from the moment when the column bases land. The varable and vertcal compressve force N C at the column base on each floor s predcted b the followng equaton. Ths equaton regards that IMP, whch s the force caused b the mpact effect on the -th floor, s equal to vertcal force on each sprng of the mass-sprng model. C T N = N + IMP (1) Where, N T : Varable and vertcal tensle force at the column base on the -the floor Suppose that there s no mpact effect, the absolute value of the varable and vertcal force n the compressve sde s equal to the one n the tensle sde. Then t s consdered that N C s equal to N T n Eq.(1). The ntal veloct v n Fg. 5 used to predct IMP s calculated b Eq.(11). v π δ = max uplft (11) T uplft Where, T uplft : response vbraton perod of uplft and max δ uplft :the maxmum uplft The response vbraton perod of uplft s obvousl equal to that of the whole sstem. Ths perod can be calculated usng the equvalent lnear stffness of the whole sstem. Fg.5 Mass-sprng model for evaluaton of mpact effect on column bases

5 In order to estmate structural safet of columns, we need the maxmum value of the N C b Eq.(1). The maxmum value of N T s estmated at the summaton of tensle strength of base plates correspondng to ther uplft deformaton and the self weght supported b the column. On the other hand, the maxmum value of IMP n Eq.(1) can be evaluated usng the maxmum vertcal deformaton of the equvalent onemass sstem of the mult-mass-sprng model shown n Fg. 5 b Eq.(1). T v π v max = (1) Where, T v : the frst natural perod of the mass-sprng model shown n Fg.5 After calculatng max and dstrbutng t to each sprng of the orgnal mult-mass-sprng model, the maxmum compressve force of each stor can be predcted b multplng the deformaton b the rgdt of each vertcal sprng. In Eq.(1), t s assumed that all knetc energ whch the mult-mass-sprng model possesses on the landng s replaced to the potental energ of columns wth no energ dsspated b vscous dampng. Fnall, the maxmum value of the maxmum IMP n Eq.(1). N C s evaluated b substtutng the maxmum N T and VERIFICATION OF THE PROPOSED METHOD The adequac of the sesmc predcton methods proposed n the prevous chapter s verfed usng results of the past shakng table tests [5][6]. Photo.1 shows a steel frame wth three stores and one ba used for the shakng table tests. Yeldng base plates shown n photo. and Fg.6 are nstalled at each column base on the ground floor. The shakng table s oscllated onl n one lateral drecton. Photo. Base plate Table 1 Cross secton Column Beam H-148x1x6x9 Brace φ11 (PC steel bar) Fg.6 Plan of base plate Table Mass at floor level Floor RF 3F F Mass[t] Table 3 Characterstcs values of base plates of BP6 model Q (kn) δ (mm) K1 (kn/mm) K (kn/mm) K/K Photo.1 Test steel frame Table 4 Characterstcs values of base plates of BP9- model Q (kn) δ (mm) K1 (kn/mm) K (kn/mm) K/K

6 Table 5 Sstem parameters used for sesmc response predcton Frst-mode effectve heght H Frst-mode effectve mass Mu Frst-mode natural crcular frequenc of the super-structure ω Relaton between base shear and the maxmum nput acceleraton a max (from test results) Relaton between frame roof deformaton and base shear (from test results) 4.13 m 1.8 t 33.7 rad/s Q E = 55.5 a δ s =. 715 max Q E The heght of the frst stor s 1.7 m and that of the other stores s 1.8 m. The total heght of the frame s 5.3 m. The frame wdth n the oscllaton drecton s 3. m. Braces made of PC steel bar are nstalled n each stor. Each brace s stressed prevousl so that the tensle stran s 1 µ. Mass at each floor level and cross secton of structural members are lsted n Table 1 and Table respectvel. Ever column s arranged so that ts weak axs drecton concdes wth the oscllaton drecton. The frst natural perod and vscous dampng rato are.18 s and.5% respectvel. The are derved b whte nose oscllaton tests. Table 3 and 4 show eld strength Q, eld deformaton δ, elastc stffness K1 and secondar stffness K of the base plates n the uplft moton. The BP6 model of Table 3 has the base plates of whch thckness s 6 mm. The BP9- model of Table 4 has the ones of whch thckness s 9 mm. Ever base plate has four wngs as shown n photo. and Fg.6. But n the case of BP9- model, the bolts fxng wngs to base beams whch are arranged n the vertcal drecton to the oscllaton drecton are removed. Thus the base plates of the BP9- model have onl two wngs. The eld strength of the base plates s calculated regardng ther wngs as beam models. The other values n Table 3 and 4 are derved usng the curves envelopng the uplft force-deformaton relatonshps whch are obtaned b statc tensle tests of the column bases wth the base plates [7]. The nput ground moton for the shakng table tests s 194 El Centro NS of whch tme axs s shrunk to (1/).5. Each test model s oscllated several tmes amplfng nput level graduall wthout changng eldng base plates. To predct uplft, base shear and roof dsplacement of each model b the proposed method, sstem parameters lsted n Table 5 and restorng force characterstcs of base plates lsted n Table 3 and 4 are used. The frst natural mode shape, whch s used to calculate the frst effectve mass and heght shown n Table 5, s estmated to be reversed trangle shape. Elastc base shear and elastc roof dsplacement of the super structure lsted n Table 5 are derved based on the test results on the fxed base models [5][6]. Fg.7 shows test results of uplft, base shear and roof dsplacement comparng wth the correspondng predcted values. The predcted values are approxmatel n good agreement wth the correspondng test results. But the predcted values for uplft and roof dsplacement of the BP9- model overestmate test results a lttle when PGA becomes larger. The reason of these errors s probabl that energ dsspaton b base plates s evaluated too conservatvel b Eq.(7) and Fg.4. As the another reason, t s ponted out

7 Uplft dsp.(mm) Predcton BP6 4 6 Input acceleraton (m/s/s) (a) Uplft Uplft dsp.(mm) Predcton BP9-4 6 Input acceleraton (m/s/s) Base shear (kn) Predcton BP Input acceleraton (m/s/s) Base shear (kn) (b) Base shear Predcton 5 BP9-4 6 Input acceleraton (m/s/s) 3 5 Predcton 5 4 Predcton Roof dsp. (mm) BP Input acceleraton (m/s/s) Roof dsp. (mm) 3 1 BP9-4 6 Input acceleraton (m/s/s) (c) Roof dsplacement Fg.7 Comparson of between predcted values and test results: uplft, base shear and roof dsplacement that the dfference between the predcted EI b Eq.(9), whch s based on the elastc frst perod, and the real one mght become large consderabl after the frame uplfts. Fg.8 shows test results of vertcal compressve force at the column base on the frst floor comparng wth the correspondng predcted values, whch are the summaton of the calculaton result b Eq.(1) and the self weght supported b the column. Predcted values are n good agreement wth test results or estmate test results conservatvel a lttle.

8 Vertcal compressve force (kn BP6 model Predcton for BP6 4 6 F model Input acceleraton (m/s/s) BP9- model Predcton for BP9- Fg.8 Comparson of between predcted values and test results: maxmum vertcal compressve force Usng the proposed predcton methods, prelmnar structural desgn of the BPY sstem can be performed accordng to the followng procedure. But pror to ths procedure, the BPY sstem should be ensured not to cause rockng vbraton under moderate earthquakes. Step 1: Set the crtera, δ and Q, for uplft and base shear of the super structure. Step : Calculate the crteron N for tensle force of base plates b Eq.(13). N = Q H B. 5Mg (13) If the crteron N s, the structure needs no base plate. If t s smaller than, the BPY sstem can not be appled to ths structure. Step 3: Choose the specfcaton of base plates so that tensle force of base plates becomes lower than or equal to ther crteron when the uplft becomes equal to ts crteron. Step 4: Confrm that the uplft predcted b the proposed method does not surpass ts crteron. Step 5: Confrm that the vertcal compressve force at the column bases does not surpass the correspondng eld force. On the Step 4, f t s judged that uplft surpasses ts crteron, the crteron of uplft or base shear must be changed nto larger one on the Step 1. If the crteron of base shear s changed, the crteron of tensle force of base plates s also changed nto larger one on the Step. Then harder base plates can be chosen. Usng base plates of whch secondar stffness s smaller s also effectve to decrease uplft, because the have superor energ dsspaton capact. On the Step 5, f t s judged that vertcal compressve force at the column bases surpass the correspondng eld force, the column must be changed nto one whch possesses hgher strength. Decreasng the

9 crteron of base shear mght be also effectve to decrease the vertcal compressve force at the column bases. However uplft and the mpact effect due to uplft become larger n ths case. CONCLUSION [1] To predct uplft, base shear and roof dsplacement of the BPY sstems, the smplfed method was proposed based on the energ conservaton law. [] To predct vertcal compressve force at the column bases on the frst floor, whch s affected b mpact force on the landng, the smplfed method was proposed. A mass-sprng model whch expresses structural propertes of the landng sde of the structure s used n ths method. [3] The adequac of these predcton methods was verfed comparng wth the results of shakng table tests usng a half scale steel frame wth three stores and one ba. ACKNOWLEDGEMENT Ths work was carred out under the US-Japan cooperatve structural research project on Smart Structure Sstems (Charperson of Japan sde: Prof. S. Otan, Unverst of Toko). And ths work was partall supported b grants-n-ad for scentfc research 3 b MEXT (No ). The authors would lke to acknowledge all project members for ther useful advce and suggestons and the support b MEXT. REFERENCES 1. Mdorkawa, M., Azuhata, T., Ishhara, T., Matsuba, Y., Matsushma, Y. and Wada, A.: Earthquake Response Reducton of Buldngs b Rockng Structural Sstems, SPIE s 9th Annual Internatonal Smposum on Smart Structures and Materals, Proc. of SPIE, vol.4696, pp.65-7,.3.. Azuhata, T., Mdorkawa, M., Ishhara, T. and Wada, A.: Shakng Table Tests on Rockng Structural Sstems wth Base Plate Yeldng, Proc. of SEWC, T-7-a-, Mdorkawa, M., Azuhata, T., Ishhara, T., Matsuba, Y. and Wada, A.: Shakng table tests on earthquake response reducton effects of rockng structural sstems, SPIE s 1th Annual Internatonal Smposum on Smart Structures and Materals, pp.7-86, Azuhata, T., Mdorkawa, M. and Wada, A.: Stud on applcablt of rockng structural sstems to buldng structures, SPIE s 1th Annual Internatonal Smposum on Smart Structures and Materals, pp.87-96, Mdorkawa, M., Azuhata, T., Ishhara, T. and Wada, A.: Shakng table tests on rockng structural sstems nstalled eldng base plates n steel frames, proc. of STESSA3, Mdorkawa, M., Azuhata, T., Ishhara, T. and Wada, A.: Dnamc Behavor of Steel Frames wth Yeldng Base Plates n Uplft Moton for Sesmc Response Reducton, J. Struct. Const. Eng., AIJ, No.57, pp.97-14, Ishhara, T., Mdorkawa, M., Azuhata, T. and Wada, A.: Hsteress Characterstcs of Column Base for Rockng Structural Sstems wth Base Plate Yeldng, Journal of Constructon Steel, Vol.11, pp Penzen, J., Elasto-Plastc Response of Idealzed Mult-Stor Structures Subjected to a strong Moton Earthquake, II WCEE, Toko, 196.