Dynamics of Structures Project

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1、同济大学DynamicsofStructuresProjectMiguelAngelHidalgoCalvoID:1375464th7January2014DynamicsofStructuresProjectGeneralStatementOurproblemistheclassicalmulti-storiedbuildingbutwithalead-corerubber-bearingisolationsystemplacedinthebasement.Inadditionwehavetoconsidereran

2、on-linearrestoringforceinoursystemstiffnessmatrix.??̈+??̇+??=?(?)HystereticcomponentIfweconsiderthatthetimederivativeofthehystereticcomponentisonlytimedependentwiththetimederivativeofthebasementdisplacementwecanconcludethefollowingrelation:?̇?=???−1[???̇?−??

3、?̇?

4、

5、

6、??

7、??−1??−???̇?

8、??

9、??]??≈???−1[????−??

10、??

11、

12、??

13、??−1??−????

14、??

15、??]Substitutingtheparametersvalues:??=20[??−2

16、??

17、

18、??

19、??−??

20、??

21、2]=??(??)Inordertosolvetheimplicitrelationthefollowingstaticinitialconditionsaresupposed:??(0)=0?????(0)=0Bouc-WenmodelOnceconsideredtheh

22、ystereticcomponentapproximationwehavedeterminedthestiffnessrestoringforceofthelead-corerubber-bearing.Fortheinitialconditionsourstiffnessmustbekb.???=??????+(1−??)???????Substitutingtheparametersvalues:???=0.6∗180.5∗??+(1−0.6)∗180.5∗0.05?????=108.3??+3.61??Basem

23、entmodelWhenevertheearthquakemotionstartsthestiffnessofourbasementbecomesnon-linearandwehavetouseatime-stepmethodfornon-linearsystems.m?̈?+??̇?+????=0????m?̈?+??̇?+??=−??̈?(?)???SystemMatricesThesystemmassanddampingmatriceshavetobemodelledforafive-storiedbuildin

24、g,whichmeansa5x5matrixwithanon-linearcomponentduetotheBouc-Wenmodel.Ontheotherhandwehavethenon-linearstiffnessmatrixthatwillbeachievedusingtheNewmark’smethod.?100004500000⎡⎤⎡⎤⎢0?2000⎥⎢0345.6000⎥?=⎢00?300⎥=⎢00345.600⎥⎢000?40⎥⎢000345.60⎥⎣0000?5⎦⎣0000345.6⎦?1+?2−?2

25、000⎡−?2?2+?3−?300⎤?=⎢0−?3?+?−?40⎥=⎢34⎥⎢00−?4?4+?5−?5⎥⎣000−?5?5⎦26.17+490−490000⎡⎤−490490+467−46700⎢⎥=⎢0−467467+410−4100⎥=⎢00−410410+386−386⎥⎣000−386386⎦516.17−490000⎡⎤−490957−46700⎢⎥=⎢0−467877−4100⎥⎢00−410796−386⎥⎣000−386386⎦?1+?2−?2000⎡⎤⎢−?2?2+?3−?300⎥?=⎢0−?3?3

26、+?4−?40⎥=⎢00−?4?4+?5−?5⎥⎣000−?5?5⎦?1+340.4−340.4000⎡⎤−340.4340.4+325.7−325.700⎢⎥=⎢0−325.7325.7+284.9−284.90⎥=⎢00−284.9284.9+268.6−268.6⎥⎣000−268.6268.6⎦?1+340.4−340.4