link – suny buffalo seven-story building with friction...

C OMP UTE R S &S TRU CTU R ES

IN C.

R Software VerificationPROGRAM NAME: SAP2000REVISION NO.: 0

EXAMPLE 6-011 - 1

EXAMPLE 6-011LINK – SUNY BUFFALO SEVEN-STORY BUILDING WITH FRICTION PENDULUM ISOLATORS

PROBLEM DESCRIPTION

This example is presented in Section 4, pages 43 through 59, of Scheller andConstantinou 1999 (“the SUNY Buffalo report”). It is a seven-story building thatis seismically isolated using a friction pendulum isolation system. The model issubjected to a recorded, scaled horizontal ground acceleration history from the1940 El Centro earthquake. See the section titled “Earthquake Record” later inthis example for more information. The SAP2000 results for base shear versusLevel 1 displacement and isolator force-deformation are compared withexperimental results obtained using shake table tests.

The SAP2000 model is shown in the figures on pages 3 and 4 of this example.The total building weight, including the tributary weight from beams andcolumns, is estimated to be 47.5 kips. The weight of each floor is estimated to be7.6 kips at Level 1, 6.7 kips at Levels 2 through 6 and 6.4 kips at Level 7. Thegravity load associated with the total building weight is applied at the top joint ofthe friction pendulum isolator elements. The gravity loads applied are 7.92 kipsat the exterior isolators and 15.83 kips at the interior isolators.

Masses representing the weight at each floor level are concentrated throughoutthe height of the structure at the beam-column joints. One-sixth of the floor massis lumped at the exterior joints at that level and one-third is lumped at the interiorjoints. The mass is active in the Ux and Uz directions. In addition, small massesare applied directly to the isolator elements. The isolator masses are set to0.0002 k-sec2/in. This mass is chosen to be about two orders of magnitudesmaller than the typical joints masses. Thus it has essentially no effect on theoverall dynamics, of the structure but it does provide modes associated with theisolators that help the convergence of the modal time history analysis.

Diaphragm constraints are assigned at each of the seven floor levels. Adiaphragm constraint is not provided at the top of the isolators.

As shown in the figure on the page 3, beams and columns are modeled as frameelements with specified end length offsets and rigid-end factors. The rigid-endfactor is 0.45 for all beams and columns. All beams and columns have a 4.5 inchend offset at each end, except for the Level 1 columns, which have a 4.5 inch endoffset at their lower ends (just above the isolators) and a 5.5 inch end offset at


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EXAMPLE 6-011 - 2

there upper ends (at Level 1). The frame section properties are shown in thefigure on page 4 of this example.

The friction pendulum isolators are modeled using two-joint, zero-length, linkelements. Both linear and nonlinear properties are provided for the isolators. Thelinear properties are used for the linear modal analysis case and the nonlinearproperties are used for the nonlinear time history analysis cases. See the sectiontitled “Friction Pendulum Isolator Properties” later in this example for additionalinformation.

The analysis results for models using friction pendulum isolators sometimesexhibit high frequency fluctuations in the response. Typically those highfrequency fluctuations have not been observed in experimental results. This is thecase in this example. It appears that the high frequency fluctuations in the modelare a result of the instantaneous opening and closing of the vertical gap elementinherent in the friction pendulum and, to a lesser degree, a result of theinstantaneous stick/slip friction behavior in the horizontal direction.

The high frequency fluctuations can be damped out in the analysis either byspecifying appropriate damping in the time history analysis case or by includingvertical dampers in the model at the isolator level. Both methods are consideredin this example.

Two models are created for this example. The models are identical, except thatModel A does not have vertical dampers included at the isolator level and ModelB does have vertical isolators at the damper level. The damper element nonlinearproperties used in Model B are the same as those used in the SUNY Buffaloreport. See the section titled “Vertical Damper Properties” later in this examplefor additional information.

Both a nonlinear modal time history analysis case and a direct integration timehistory analysis case are considered in this example. See the section titled“Analysis Cases Used” later in this example for additional information.


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EXAMPLE 6-011 - 3

GEOMETRY AND PROPERTIES

1, 33

5

2, 34 3, 35 4, 36

6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24

25 26 27 28

29 30 31 32

3 @ 4' = 12 '

7 @

3' =

21'

Z

X

Level 7

Level 6

Level 5

Level 4

Level 3

Level 2

Level 1

Base andIsolator Level

Base level has two joints in the same location at the bottom of each column. Zero length friction pendulum elements (and in Model B also vertical damper elements) connect joints 1 to 33, 2 to 34, 3 to 35 and 4 to 36. Joints 1, 2, 3 and 4 are connected to the bottoms of the columns. Joints 33, 34, 35 and 36 are connected to ground, that is, restrained.

15.83 k 7.92 k7.92 k 15.83 kBuilding weight is applied directly to the top of the isolators

Floor joints are constrained as a diaphragm, typical for Levels 1 through 7

Ux and Uz mass equal to 1/6 of floor mass at exterior joints and 1/3 of floor mass at interior joints typical at Levels 1 through 7

Active degrees of freedom for model are Ux, Uz and Ry

End offsets typical for all frame members at all joints.

Joint numbers, typical


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EXAMPLE 6-011 - 4

Z

X

Level 7

Level 6

Level 5

Level 4

Level 3

Level 2

Level 1

Base andIsolator Level

Frame element number

FSEC

1 29

1

FSEC

25

FSEC

29

FSEC

213

FSEC

217

FSEC

221

FSEC

225

FSEC

12

FSEC

26

FSEC

210

FSEC

214

FSEC

218

FSEC

222

FSEC

226

FSEC

13

FSEC

27

FSEC

211

FSEC

215

FSEC

219

FSEC

223

FSEC

227

FSEC

14

FSEC

28

FSEC

212

FSEC

216

FSEC

220

FSEC

224

FSEC

228

FSEC330

FSEC331

FSEC3

32

FSEC233

FSEC234

FSEC2

35

FSEC236

FSEC237

FSEC2

38

FSEC239

FSEC240

FSEC2

41

FSEC242

FSEC243

FSEC2

44

FSEC245

FSEC246

FSEC2

47

FSEC248

FSEC249

FSEC2

Frame section name

SectionName

AreaA (in2)

Moment of InertiaI (in4)

Shear AreaAv (in2)

FSEC1 7.46 12.18 4.375

FSEC2 3.34 5.04 1.02

FSEC3 5.58 13.58 2.608


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EXAMPLE 6-011 - 5

ANALYSIS CASES USED

The following two tables describe the analysis cases used in this example foreach model.

MODEL A

Analysis Case Description

RITZ Modal analysis case for Ritz vectors. Ninety-nine modesare requested. The program will automatically determinethat a maximum of forty-three modes are possible and thusreduce the number of modes to forty-three. The startingvectors are Ux acceleration, Uz acceleration, and all linkelement nonlinear degrees of freedom.

MGRAV Nonlinear modal time history analysis case that applies thegravity load to the isolators using a ramp function. TheNLMHIST1A and NLMHIST2A modal time historyanalysis cases are started from the final condition of thisanalysis case.

DGRAV Nonlinear static analysis case used to apply the gravity loadto the isolators. The NLDHIST1A direct integration timehistory analysis case is started from the final condition ofthis analysis case.

NLMHIST1A Nonlinear modal time history analysis case that uses themodes in the RITZ analysis case and starts from the finalconditions of analysis case MGRAV. This case includesproportional damping that is defined to provide dampingsimilar to, but not exactly the same as, the 0.59% modaldamping used in Scheller and Constantinou 1999. It is thesame damping specification as that used in analysis caseNLDHIST1 for Model A. See the section titled“Proportional Damping for Time Histories in Model A”later in this example for more information.


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EXAMPLE 6-011 - 6

MODEL A


NLMHIST2A Nonlinear modal time history analysis case that uses themodes in the RITZ analysis case and starts from the finalconditions of analysis case MGRAV. This case includes0.59% modal damping in all modes, except modes 40, 41,42 and 43 (the modes associated with the vertical excitationof the isolators) are assigned 99.9% modal damping.

NLMHIST3A Nonlinear modal time history analysis case that uses themodes in the RITZ analysis case and starts from the finalconditions of analysis case MGRAV. This case includes0.59% modal damping in all modes with no modal dampingoverwrites.

NLDHIST1A Nonlinear direct integration time history analysis case thatstarts from the final conditions of analysis case DGRAV.This case includes proportional damping that is defined toprovide damping similar to, but not exactly the same as, the0.59% modal damping used in Scheller and Constantinou1999. It is the same damping specification as that used inanalysis case NLMHIST1 for Model A. See the sectiontitled “Proportional Damping for Time Histories in ModelA” later in this example for more information.


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EXAMPLE 6-011 - 7

MODEL B


RITZ Same as Model A.

MGRAV Same as Model A. This is a nonlinear modal time historyanalysis case that applies the gravity load to the isolatorsusing a ramp function. The NLMHIST1B andNLMHIST2B modal time history analysis cases are startedfrom the final condition of this analysis case.

DGRAV Same as Model A. This is a nonlinear static analysis caseused to apply the gravity load to the isolators. TheNLDHIST1B direct integration time history analysis case isstarted from the final condition of this analysis case.

NLMHIST1B Nonlinear modal time history analysis case that uses themodes in the RITZ analysis case and starts from the finalconditions of analysis case MGRAV. This case includesproportional damping that is defined to provide dampingsimilar to, but not exactly the same as, the 0.59% modaldamping used in Scheller and Constantinou 1999. It is thesame damping specification as that used in analysis caseNLDHIST1 for Model B. See the section titled“Proportional Damping for Time Histories in Model B”later in this example for more information.

NLMHIST2B Nonlinear modal time history analysis case that uses themodes in the RITZ analysis case and starts from the finalconditions of analysis case MGRAV. This case includes0.59% modal damping in all modes with no modal dampingoverwrites.


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EXAMPLE 6-011 - 8

MODEL B


NLDHIST1B Nonlinear direct integration time history analysis case thatstarts from the final conditions of analysis case DGRAV.This case includes proportional damping that is defined toprovide damping similar to, but not exactly the same as, the0.59% modal damping used in Scheller and Constantinou1999. It is the same damping specification as that used inanalysis case NLMHIST1 for Model B. See the sectiontitled “Proportional Damping for Time Histories in ModelB” later in this example for more information.

In Model A the damping is set high for modes associated with the verticalexcitation of the isolators. This is not the case in Model B, which includesvertical damper elements at the isolator level.

In the nonlinear direct integration time history analysis cases, a maximumsubstep size of 0.0005 second is used and the Hilber-Hughes-Taylor integrationfactor, alpha, is set to -1/3.


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EXAMPLE 6-011 - 9

PROPORTIONAL DAMPING FOR TIME HISTORIES IN MODEL AIn Model A the nonlinear direct integration time history analysis caseNLDHIST1A and the nonlinear modal time history analysis case NLMHIST1Ause mass and stiffness proportional damping. The proportional damping for thoseanalysis cases should approximate 0.59% modal damping for all periods, exceptthat the higher frequencies (lower periods) should be more highly damped. Formodel A the proportionaldamping is selected bysetting the damping atperiods of 1 second and 0.1second to 0.59%. This yieldsa mass proportionalcoefficient of 0.0674 and astiffness proportionalcoefficient of 1.707E-04.The resulting damping isdisplayed in the figure to theright.

PROPORTIONAL DAMPING FOR TIME HISTORIES IN MODEL BIn Model B the nonlinear direct integration time history analysis caseNLDHIST1A and the nonlinear modal time history analysis case NLMHIST1Ause mass and stiffness proportional damping. The proportional damping for thoseanalysis cases should approximate 0.59% modal damping for all periods, exceptthe higher frequencies (lower periods) should not be more highly damped. Formodel B the proportional damping is selected by setting the damping at a periodof 1 second to 0.59% andthe damping at a period of0 second to 0%. Thisyields a mass proportionalcoefficient of 0.0741 and astiffness proportionalcoefficient of 0. Theresulting massproportional damping isdisplayed in the figure tothe right.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Period (sec)

Dam

pin

g R

atio

Mass

Stiffness

Rayleigh

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Period (sec)

Dam

pin

g R

atio

Mass

Stiffness

Rayleigh


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EXAMPLE 6-011 - 10

EARTHQUAKE RECORD

The following figure shows the earthquake record used in this example. It is theS00E component of the 1940 El Centro earthquake record scaled up to a peakacceleration of 0.57g. This is twice the recorded level of the earthquake. Thetime scale is also compressed by a factor of two to satisfy the similituderequirements of the experiment.

The earthquake record is provided in a file named EQ6-011.txt. This file has oneacceleration value per line, in g. The acceleration values are provided at an equalspacing of 0.01 second.

Inside SAP2000 the earthquake record is multiplied by a factor of 386.22 toconvert from g to in/sec2.

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35 40

Time (sec)

Gro

un

d A

ccel

erat

ion

(g

)


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EXAMPLE 6-011 - 11

FRICTION PENDULUM ISOLATOR PROPERTIES

This section presents the properties used for the friction pendulum link elementsin the model. All link elements in the model are oriented such that the positivelocal 1 axis is parallel to the positive global Z axis, the positive local 2 axis isparallel to the positive global X axis and the positive local 3 axis is parallel to thepositive global Y axis. Different properties are specified for the interior andexterior link elements.

The properties for the exterior friction pendulum link are:

Linear analysis propertieske U1 = 20,000 k/inke U2 = 1.05 k/inke R3 = 10,000 k-in/radian

Nonlinear analysis propertiesk U1 = 20,000 k/ink U2 = 31.6667 k/inFriction coefficient, slow U2 = 0.04Friction coefficient, fast U2 = 0.06Rate parameter U2 = 1.0897 sec/inRadius of sliding surface U2 = 9.75 in

The properties for the interior friction pendulum link are:

Linear analysis propertieske U1 = 20,000 k/inke U2 = 2.10 k/inke R3 = 10,000 k-in/radian

Nonlinear analysis propertiesk U1 = 20,000 k/ink U2 = 63.3333 k/inFriction coefficient, slow U2 = 0.04Friction coefficient, fast U2 = 0.06Rate parameter U2 = 1.0897 sec/inRadius of sliding surface U2 = 9.75 in

The ke U1 property of 20,000 k/in used in this example is different from that usedin the Scheller and Constantinou 1999 SAP2000 model where a value of 0.0001k/in was used.


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EXAMPLE 6-011 - 12

VERTICAL DAMPER PROPERTIES

The damper element nonlinear properties used in Model B are the same as thoseused in Scheller and Constantinou 1999. The damping coefficient, c, is selectedon the basis of providing a damping ratio of 0.10 for the total building weight of47.5 kips and the total vertical stiffness of 80,000 kip/in (four isolators, each at20,000 kip/in). Thus,

g

kWkmm

m

kmc ξξξξω 22224 ====

===386

5.47*000,80

2

10.0

2 g

kWc

ξ5 kip-sec/in

The damper stiffness, k, is set to 10,000 kip/in to achieve pure damping behaviorin the damper. This means that the characteristic time of the spring-dashpotsystem, given by τ = c / k = 5 / 10000 = 0.0005 sec, is approximately one to twoorders of magnitude smaller than the size of the load steps, which is 0.01 secondin this case. This characteristic time should give pure damping behavior.

The linear properties of the damper are set to zero so that the damper has noeffect on the modal analysis.

TECHNICAL FEATURES OF SAP2000 TESTED

Friction pendulum link elements Damper link elements Zero-length, two-joint link elements Diaphragm constraints Frame end length offsets Modal analysis for ritz vectors Nonlinear modal time history analysis Nonlinear direct integration time history analysis Joint masses


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EXAMPLE 6-011 - 13

RESULTS COMPARISON

Independent results are experimental results from shake table testing presented inSection 4, pages 43 through 59, of Scheller and Constantinou 1999.

The figures on page 14 of this example plot base shear versus Level 1displacement for the four time history cases in Model A, which has no addeddamper elements, and for the three time history cases in Model B, which doeshave added damper elements.

The plot shown at the bottom center of page 14 is for Model A, analysis caseNLMHIST3A. Recall that Model A does not have vertical dampers at the isolatorlevel and that analysis case NLMHIST3A has 0.59% modal damping for allmodes with no increased damping in the higher frequencies. This plot showssubstantial high frequency fluctuations in the response. Note that the other plots,all of which have some increased damping for the higher frequencies (as modaldamping, mass and stiffness proportional damping, or added vertical damperelements), show significantly fewer of those high frequency fluctuations. In allcases the peak response values compare well with the experimental values. Thiscomparison is tabulated in the table on page 15.

The top left plot on page 14 shows the base shear versus Level 1 displacementfor analysis case NLMHIST1A which is a nonlinear modal time history withproportional damping. The plot third down on the left shows the same base shearversus Level 1 displacement plot for analysis case NLDHIST1A which is anonlinear direct integration time history with proportional damping. Theproportional damping specified for these two analysis cases is identical. The plotfor NLDHIST1A has much less high frequency fluctuation than that shown in theplot for NLMHIST3A (bottom center), and more high frequency fluctuation thanthat shown in the plot for NLMHIST1A (top left). The difference between theplots for NLMHIST1A and NLDHIST1A is caused by the differences in howproportional damping is handled in the nonlinear modal and direct integrationtime history analysis cases.


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EXAMPLE 6-011 - 14

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Level 1 Displacement (in)

Bas

e S

hea

r / W

eig

ht

Experimental SAP2000

Model ANonlinear modal time historyProportional dampingAnalysis case NLMHIST1A

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model BNonlinear modal time historyIncludes vertical damper elementsAnalysis case NLMHIST1B

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model ANonlinear modal time historyModal damping w/ overwritesAnalysis case NLMHIST2A

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model BNonlinear modal time historyIncludes vertical damper elementsAnalysis case NLMHIST2B

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model ANonlinear direct integration time historyProportional dampingAnalysis case NLDHIST1A

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model BNonlinear direct integration time historyIncludes vertical damper elementsAnalysis case NLDHIST1B

Weight = 47.5 kips

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5


Bas

e S

hea

r / W

eig

ht


Model ANonlinear modal time historyModal damping w/o overwritesAnalysis case NLMHIST3A

Weight = 47.5 kips


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EXAMPLE 6-011 - 15

OutputParameter Model

AnalysisCase SAP2000

IndependentExperimental

PercentDifference

NLMHIST1A -2.020 -2%

NLMHIST2A -2.034 -1%

NLMHIST3A -2.147 +5%A

NLDHIST1A -2.020 -2%

NLMHIST1B -2.017 -2%

NLMHIST2B -2.081 +1%

MinimumLevel 1

Displacement(in)

B

NLDHIST1B -1.988

-2.053

-3%

NLMHIST1A 1.982 -3%

NLMHIST2A 1.981 -3%

NLMHIST3A 2.000 -2%A

NLDHIST1A 1.968 -4%

NLMHIST1B 1.996 -2%

NLMHIST2B 2.021 -1%

MaximumLevel 1

Displacement(in)

B

NLDHIST1B 1.967

2.043

-4%

NLMHIST1A -0.250 +2%



NLDHIST1A -0.245 0%



MinimumBase

Shear/Weight

B

NLDHIST1B -0.253

-0.245

+3%

NLMHIST1A 0.253 +2%

NLMHIST2A 0.258 +4%

NLMHIST3A 0.270 +9%A

NLDHIST1A 0.266 +7%

NLMHIST1B 0.237 -4%

NLMHIST2B 0.240 -3%

MaximumBase

Shear/Weight

B

NLDHIST1B 0.235

0.248

-5%


IN C.


EXAMPLE 6-011 - 16

In nonlinear modal time history analysis cases with proportional damping, theproportional damping is converted to modal damping based on the initialstiffness of the analysis. This damping does not change as the analysis proceeds.

In nonlinear direct integration time history cases with proportional damping, thestiffness proportional component of the damping can change during the course ofthe analysis as the stiffness of the structure changes. If the stiffness goes to zeroduring a portion of the analysis, the associated stiffness proportional componentof the damping also goes to zero.

In this example, analysis case NLMHIST1A has its damping based on the initialconditions of the analysis. For those conditions, the isolator is under axialcompression and it is not sliding. Thus, nonzero vertical and horizontal stiffnessis present at the isolators. Therefore, vertical and horizontal stiffness proportionaldamping is present at the isolators throughout the entire analysis.

Analysis case NLDHIST1A has damping that changes as the analysis proceeds.When the isolator is under axial compression and it is not sliding, vertical andhorizontal stiffness proportional damping is present at the isolators. When theisolators begin to slide, the horizontal stiffness proportional damping disappears.When the isolator uplifts (as it is sliding), both the vertical and horizontalstiffness proportional damping at the isolators disappears.

As a consequence, over the full course of the analysis, analysis caseNLDHIST1A is less damped than analysis case NLMHIST1A. This is why morehigh frequency fluctuations are evident in the plot for NLDHIST1A than that forNLMHIST1A.

The plot for NLMHIST2A shows some small high frequency fluctuations that arenot present for NLMHIST1A. Recall that NLMHIST1A uses mass and stiffnessproportional damping previously described in the section titled “ProportionalDamping for Time Histories in Model A.” NLMHIST2A uses constant 0.59%modal damping, with the damping overwritten to 99.9% for the four highestfrequency modes, which all have periods of approximately 0.0004 second. Theproportional damping used in NLMHIST1A provides 0.59% damping at a periodof 0.1 second and increases to approximately 134% damping as the period isdecreased to 0.0004 second. The damping is increased over the entire range from0.1 second to 0.0004 second rather than just at 0.0004 second as is the case inNLMHIST2A. Thus, more high frequency damping is present in NLMHIST1Athan in NLMHIST2A. This explains why the plot for NLMHIST2A shows somesmall high frequency fluctuations that are not present for NLMHIST1A. If


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EXAMPLE 6-011 - 17

increased damping were provided for the modes between 0.1 second and 0.0004second in NLMHIST2A, the results for NLMHIST2A would appear more similarto those for NLMHIST1A.

The following figure compares the Level 1 displacement versus time for analysiscase NLMHIST1A to the experimental results. The comparison is similar for theother analysis cases.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

Time (sec)

Lev

el 1

Dis

pla

cem

ent

(in

)



The following figures show isolator force-deformation plots for an exterior andan interior isolator for analysis case NLMHIST1A. The exterior isolator islocated at joints 1 and 33. The interior isolator is located at joints 2 and 34.

As described in Scheller and Constantinou 1999, “The gravity loads on thebearings [during the experiment] were not exactly known and they could verywell have been different than assumed in the [SAP2000] analysis.” This couldcontribute to the difference in the experimental and SAP2000 results for theforce-deformation response of the exterior isolator.


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EXAMPLE 6-011 - 18

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Isolator Displacement (in)

Iso

lato

r S

hea

r F

orc

e (k

ip)



Exterior isolator at joints 1 and 33

-4

-3

-2

-1

0

1

2

3

4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Isolator Displacement (in)

Iso

lato

r S

hea

r F

orc

e (k

ip)



Interior Isolator at Joints 2 and 34

The following table compares the peak values of the isolator force anddeformation with the experimental values for the NLMHIST1A analysis case.Similar results are obtained for other time history analysis cases with damping atthe high frequencies.


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EXAMPLE 6-011 - 19




PercentDifference




NLDHIST1A -1.796 +7%


NLMHIST2B -1.855 +10%

Exterior Isolator(Joints 1 and 33)

Minimum Deformation

(in) B

NLDHIST1B -1.772

-1.686

+5%

NLMHIST1A 1.961 +3%

NLMHIST2A 1.976 +4%


NLDHIST1A 1.946 +2%

NLMHIST1B 1.981 +4%

NLMHIST2B 2.004 +5%


Maximum Deformation

(in) B

NLDHIST1B 1.950

1.909

+2%

NLMHIST1A -5.672 -17%

NLMHIST2A -5.726 -17%

NLMHIST3A -5.922 -14%A

NLDHIST1A -5.782 -16%

NLMHIST1B -5.683 -17%

NLMHIST2B -5.834 -15%


Minimum Shear Force

(kip) B

NLDHIST1B -5.543

-6.872

-19%

NLMHIST1A 0.911 -23%

NLMHIST2A 0.904 -24%


NLDHIST1A 1.064 -10%

NLMHIST1B 0.933 -21%

NLMHIST2B 0.919 -22%


Maximum Shear Force

(kip) B

NLDHIST1B 0.894

1.183

-24%


IN C.


EXAMPLE 6-011 - 20




PercentDifference




NLDHIST1A -1.923 +7%


NLMHIST2B -1.983 +10%

Interior Isolator(Joints 2 and 34)

Minimum Deformation

(in) B

NLDHIST1B -1.892

-1.796

+5%

NLMHIST1A 1.854 +4%

NLMHIST2A 1.853 +4%


NLDHIST1A 1.836 +3%

NLMHIST1B 1.871 +5%

NLMHIST2B 1.896 +6%


Maximum Deformation

(in) B

NLDHIST1B 1.842

1.786

+3%

NLMHIST1A -3.493 0%



NLDHIST1A -3.429 -2%


NLMHIST2B -3.504 0%


Minimum Shear Force

(kip) B

NLDHIST1B -3.384

-3.498

-3%

NLMHIST1A 3.909 +17%

NLMHIST2A 3.973 +19%


NLDHIST1A 4.036 +21%

NLMHIST1B 3.859 +15%

NLMHIST2B 3.811 +14%


Maximum Shear Force

(kip) B

NLDHIST1B 3.795

3.346

+13%


IN C.


EXAMPLE 6-011 - 21

COMMENTS ON SUNY BUFFALO REPORT

The SUNY Buffalo report (Scheller and Constantinou 1999) indicates the use ofan extremely small value for the axial linear effective stiffness, ke U1, of thefriction pendulum isolators to achieve acceptable results. The SUNY Buffaloreport used a value of 0.0001 kip/in for ke U1. This verification example uses arealistic value of 20,000 kip/in for ke U1 and achieves acceptable results.

The comparisons of SAP2000 isolation system displacement with experimentalresults appear better in this verification example than they do in the SUNYBuffalo report. In Section 4-4 on page 50 of that report, the isolation systemdisplacement is defined as the displacement of the first floor with respect to theground; that is, the isolator displacement plus the displacement in the first levelcolumn. This displacement is called the Level 1 Displacement in the plots shownin this verification example.

When the SAP2000 isolator displacement is plotted versus the base shear, theresulting plot is very similar to that shown in the SUNY Buffalo report. Thus, itappears that the report may in some instances be making comparisons where theexperimental displacement is for Level 1 and the SAP2000 displacement is forthe Isolator Level. This would explain why the comparisons appear better in thisverification example.

SOLUTION PARAMETERS FOR DIRECT INTEGRATION TIME HISTORY

The nonlinear direct integration time histories in this example were run using amaximum substep size of 0.0005 second. A larger maximum substep size of0.005 second was tried and found to yield larger displacements than the 0.0005second step size. A smaller maximum substep size of 0.00005 second was triedand found to yield the same solution as the 0.0005 second step size. Thus, it wasconcluded that a 0.0005 second maximum step size was appropriate for thisexample.

Similarly, the nonlinear direct integration time histories in this example were runusing a relative iteration convergence tolerance of 1E-4. A smaller relativeiteration convergence tolerance was tried and found to yield the same results.Thus the 1E-4 tolerance was deemed to be sufficient.

In general, parameter studies, such as described herein, should be performed fornonlinear analyses. This helps to build confidence that appropriate results havebeen obtained.


IN C.


EXAMPLE 6-011 - 22

COMPUTER FILES: Example 6-011a, Example 6-011b

CONCLUSION

In general, the SAP2000 results show an acceptable comparison with theindependent results. For analysis case NLMHIST3A, which has no damperelements and no additional damping at the higher frequencies, the comparison ofpeak values for the isolator force-deformation curves is poor. Additionaldamping associated with the high frequencies improves the comparison.

For nonlinear modal time history analysis cases, modal damping, proportionaldamping and added dampers can all be used to significantly reduce the highfrequency fluctuations that can occur in the models with friction pendulumisolators.

For nonlinear direct integration time history analysis cases, proportionaldamping or added dampers can both be used to significantly reduce the highfrequency fluctuations that can occur in the models with friction pendulumisolators. However, it is important to realize that proportional damping will notcompletely eliminate the fluctuations because the stiffness proportionalcomponent of the damping will be zero when the isolators are uplifted, and it isthe stiffness proportional component of the damping that is effective in dampingout the high frequency behavior. Thus, if nonlinear direct integration timehistories are used, added damper elements may be a better alternative thanproportional damping in the analysis case to reduce the high frequencyfluctuations in the results.

link – suny buffalo seven-story building with friction...

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