Problem 6 011

Software Verification

PROGRAM NAME: SAP2000 REVISION NO.: 8

EXAMPLE 6-011 - 1

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

PROBLEM DESCRIPTION This example is presented in Section 4, pages 43 through 59, of Scheller and Constantinou 1999 (“the SUNY Buffalo report”). It is a seven-story building that is seismically isolated using a friction pendulum isolation system. The model is subjected to a recorded, scaled horizontal ground acceleration history from the 1940 El Centro earthquake. See the section titled “Earthquake Record” later in this example for more information. The SAP2000 results for base shear versus Level 1 displacement and isolator force-deformation are compared with experimental 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 and columns, is estimated to be 47.5 kips. The weight of each floor is estimated to be 7.6 kips at Level 1, 6.7 kips at Levels 2 through 6 and 6.4 kips at Level 7. The gravity load associated with the total building weight is applied at the top joint of the friction pendulum isolator elements. The gravity loads applied are 7.92 kips at the exterior isolators and 15.83 kips at the interior isolators.

Masses representing the weight at each floor level are concentrated throughout the height of the structure at the beam-column joints. One-sixth of the floor mass is lumped at the exterior joints at that level and one-third is lumped at the interior joints. The mass is active in the Ux and Uz directions. In addition, small masses are applied directly to the isolator elements. The isolator masses are set to 0.0002 k-sec2/in. This mass is chosen to be about two orders of magnitude smaller than the typical joints masses. Thus it has essentially no effect on the overall dynamics, of the structure but it does provide modes associated with the isolators that help the convergence of the modal time history analysis.

Diaphragm constraints are assigned at each of the seven floor levels. A diaphragm 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 frame elements with specified end length offsets and rigid-end factors. The rigid-end factor is 0.45 for all beams and columns. All beams and columns have a 4.5 inch end offset at each end, except for the Level 1 columns, which have a 4.5 inch end offset at their lower ends (just above the isolators) and a 5.5 inch end offset at



EXAMPLE 6-011 - 2

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

The friction pendulum isolators are modeled using two-joint, zero-length, link elements. Both linear and nonlinear properties are provided for the isolators. The linear properties are used for the linear modal load case and the nonlinear properties are used for the nonlinear time history load cases. See the section titled “Friction Pendulum Isolator Properties” later in this example for additional information.

The analysis results for models using friction pendulum isolators sometimes exhibit high frequency fluctuations in the response. Typically those high frequency fluctuations have not been observed in experimental results. This is the case in this example. It appears that the high frequency fluctuations in the model are a result of the instantaneous opening and closing of the vertical gap element inherent in the friction pendulum and, to a lesser degree, a result of the instantaneous stick/slip friction behavior in the horizontal direction.

The high frequency fluctuations can be damped out in the analysis either by specifying appropriate damping in the time history load case or by including vertical dampers in the model at the isolator level. Both methods are considered in this example.

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

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



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



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



EXAMPLE 6-011 - 5

LOAD CASES USED The following two tables describe the load cases used in this example for each model.

MODEL A

Load Case Description

RITZ Modal load case for Ritz vectors. Ninety-nine modes are requested. The program will automatically determine that a maximum of forty-three modes are possible and thus reduce the number of modes to forty-three. The starting vectors are Ux acceleration, Uz acceleration, and all link element nonlinear degrees of freedom.

MGRAV Nonlinear modal time history load case that applies the gravity load to the isolators using a ramp function. The NLMHIST1A and NLMHIST2A modal time history load cases are started from the final condition of this load case.

DGRAV Nonlinear static load case used to apply the gravity load to the isolators. The NLDHIST1A direct integration time history load case is started from the final condition of this load case.

NLMHIST1A Nonlinear modal time history load case that uses the modes in the RITZ load case and starts from the final conditions of load case MGRAV. This case includes proportional damping that is defined to provide damping similar to, but not exactly the same as, the 0.59% modal damping used in Scheller and Constantinou 1999. It is the same damping specification as that used in load case NLDHIST1 for Model A. See the section titled “Proportional Damping for Time Histories in Model A” later in this example for more information.



EXAMPLE 6-011 - 6

MODEL A


NLMHIST2A Nonlinear modal time history load case that uses the modes in the RITZ load case and starts from the final conditions of load case MGRAV. This case includes 0.59% modal damping in all modes, except modes 40, 41, 42 and 43 (the modes associated with the vertical excitation of the isolators) are assigned 99.9% modal damping.

NLMHIST3A Nonlinear modal time history load case that uses the modes in the RITZ load case and starts from the final conditions of load case MGRAV. This case includes 0.59% modal damping in all modes with no modal damping overwrites.

NLDHIST1A Nonlinear direct integration time history load case that starts from the final conditions of load case DGRAV. This case includes proportional damping that is defined to provide damping similar to, but not exactly the same as, the 0.59% modal damping used in Scheller and Constantinou 1999. It is the same damping specification as that used in load case NLMHIST1 for Model A. See the section titled “Proportional Damping for Time Histories in Model A” later in this example for more information.



EXAMPLE 6-011 - 7

MODEL B


RITZ Same as Model A.

MGRAV Same as Model A. This is a nonlinear modal time history load case that applies the gravity load to the isolators using a ramp function. The NLMHIST1B and NLMHIST2B modal time history load cases are started from the final condition of this load case.

DGRAV Same as Model A. This is a nonlinear static load case used to apply the gravity load to the isolators. The NLDHIST1B direct integration time history load case is started from the final condition of this load case.

NLMHIST1B Nonlinear modal time history load case that uses the modes in the RITZ load case and starts from the final conditions of load case MGRAV. This case includes proportional damping that is defined to provide damping similar to, but not exactly the same as, the 0.59% modal damping used in Scheller and Constantinou 1999. It is the same damping specification as that used in load case NLDHIST1 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 load case that uses the modes in the RITZ load case and starts from the final conditions of load case MGRAV. This case includes 0.59% modal damping in all modes with no modal damping overwrites.



EXAMPLE 6-011 - 8

MODEL B


NLDHIST1B Nonlinear direct integration time history load case that starts from the final conditions of load case DGRAV. This case includes proportional damping that is defined to provide damping similar to, but not exactly the same as, the 0.59% modal damping used in Scheller and Constantinou 1999. It is the same damping specification as that used in load case NLMHIST1 for Model B. See the section titled “Proportional Damping for Time Histories in Model B” later in this example for more information.

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

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



EXAMPLE 6-011 - 9

PROPORTIONAL DAMPING FOR TIME HISTORIES IN MODEL A In Model A the nonlinear direct integration time history load case NLDHIST1A and the nonlinear modal time history load case NLMHIST1A use mass and stiffness proportional damping. The proportional damping for those load cases should approximate 0.59% modal damping for all periods, except that the higher frequencies (lower periods) should be more highly damped. For model A the proportional damping is selected by setting the damping at periods of 1 second and 0.1 second to 0.59%. This yields a mass proportional coefficient of 0.0674 and a stiffness proportional coefficient of 1.707E-04. The resulting damping is displayed in the figure to the right.

PROPORTIONAL DAMPING FOR TIME

HISTORIES IN MODEL B In Model B the nonlinear direct integration time history load case NLDHIST1A and the nonlinear modal time history load case NLMHIST1A use mass and stiffness proportional damping. The proportional damping for those load cases should approximate 0.59% modal damping for all periods, except the higher frequencies (lower periods) should not be more highly damped. For model B the proportional damping is selected by setting the damping at a period of 1 second to 0.59% and the damping at a period of 0 second to 0%. This yields a mass proportional coefficient of 0.0741 and a stiffness proportional coefficient of 0. The resulting mass proportional damping is displayed in the figure to the 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)

Da

mp

ing

Ra

tio

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



EXAMPLE 6-011 - 10

EARTHQUAKE RECORD The following figure shows the earthquake record used in this example. It is the S00E component of the 1940 El Centro earthquake record scaled up to a peak acceleration of 0.57g. This is twice the recorded level of the earthquake. The time scale is also compressed by a factor of two to satisfy the similitude requirements of the experiment.

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

Inside SAP2000 the earthquake record is multiplied by a factor of 386.22 to convert 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

cce

lera

tio

n (

g)



EXAMPLE 6-011 - 11

FRICTION PENDULUM ISOLATOR PROPERTIES This section presents the properties used for the friction pendulum link elements in the model. All link elements in the model are oriented such that the positive local 1 axis is parallel to the positive global Z axis, the positive local 2 axis is parallel to the positive global X axis and the positive local 3 axis is parallel to the positive global Y axis. Different properties are specified for the interior and exterior link elements.

The properties for the exterior friction pendulum link are:

Linear analysis properties ke U1 = 20,000 k/in ke U2 = 1.05 k/in ke R3 = 10,000 k-in/radian Nonlinear analysis properties k U1 = 20,000 k/in k U2 = 31.6667 k/in Friction coefficient, slow U2 = 0.04 Friction coefficient, fast U2 = 0.06 Rate parameter U2 = 1.0897 sec/in Radius of sliding surface U2 = 9.75 in

The properties for the interior friction pendulum link are:

Linear analysis properties ke U1 = 20,000 k/in ke U2 = 2.10 k/in ke R3 = 10,000 k-in/radian Nonlinear analysis properties k U1 = 20,000 k/in k U2 = 63.3333 k/in Friction coefficient, slow U2 = 0.04 Friction coefficient, fast U2 = 0.06 Rate parameter U2 = 1.0897 sec/in Radius of sliding surface U2 = 9.75 in

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



EXAMPLE 6-011 - 12

VERTICAL DAMPER PROPERTIES The damper element nonlinear properties used in Model B are the same as those used in Scheller and Constantinou 1999. The damping coefficient, c, is selected on the basis of providing a damping ratio of 0.10 for the total building weight of 47.5 kips and the total vertical stiffness of 80,000 kip/in (four isolators, each at 20,000 kip/in). Thus,

g

kWkmm

m

kmc 22224

386

5.47*000,80

2

10.0

2 g

kWc

5 kip-sec/in

The damper stiffness, k, is set to 10,000 kip/in to achieve pure damping behavior in the damper. This means that the characteristic time of the spring-dashpot system, given by τ = c / k = 5 / 10000 = 0.0005 sec, is approximately one to two orders of magnitude smaller than the size of the load steps, which is 0.01 second in 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 no effect 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



EXAMPLE 6-011 - 13

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

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

The plot shown at the bottom center of page 14 is for Model A, load case NLMHIST3A. Recall that Model A does not have vertical dampers at the isolator level and that load case NLMHIST3A has 0.59% modal damping for all modes with no increased damping in the higher frequencies. This plot shows substantial high frequency fluctuations in the response. Note that the other plots, all of which have some increased damping for the higher frequencies (as modal damping, mass and stiffness proportional damping, or added vertical damper elements), show significantly fewer of those high frequency fluctuations. In all cases the peak response values compare well with the experimental values. This comparison is tabulated in the table on page 15.

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



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


Ba

se S

hea

r / W

eig

ht


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

Weight = 47.5 kips



EXAMPLE 6-011 - 15

Output Parameter Model Load Case SAP2000

Independent Experimental

Percent Difference

NLMHIST1A -2.02 -2%

NLMHIST2A -2.034 -1%

NLMHIST3A -2.145 4% A

NLDHIST1A -1.883 -8%

NLMHIST1B -2.016 -2%

NLMHIST2B -2.081 1%

Minimum Level 1

Displacement (in)

B

NLDHIST1B -1.988

-2.053

-3%

NLMHIST1A 1.982 -3%

NLMHIST2A 1.981 -3%

NLMHIST3A 2 -2% A

NLDHIST1A 1.899 -7%

NLMHIST1B 1.996 -2%

NLMHIST2B 2.021 -1%

Maximum Level 1

Displacement (in)

B

NLDHIST1B 1.963

2.043

-4%

NLMHIST1A -0.25 2%

NLMHIST2A -0.251 2%


NLDHIST1A -0.23 -6%

NLMHIST1B -0.26 6%

NLMHIST2B -0.263 7%

Minimum Base

Shear/Weight

B

NLDHIST1B -0.253

-0.245

3%

NLMHIST1A 0.253 2%

NLMHIST2A 0.258 4%

NLMHIST3A 0.269 8% A

NLDHIST1A 0.255 3%

NLMHIST1B 0.237 -4%

NLMHIST2B 0.24 -3%

Maximum Base

Shear/Weight

B

NLDHIST1B 0.235

0.248

-5%



EXAMPLE 6-011 - 16

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

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

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

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

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

The plot for NLMHIST2A shows some small high frequency fluctuations that are not present for NLMHIST1A. Recall that NLMHIST1A uses mass and stiffness proportional damping previously described in the section titled “Proportional Damping for Time Histories in Model A.” NLMHIST2A uses constant 0.59% modal damping, with the damping overwritten to 99.9% for the four highest frequency modes, which all have periods of approximately 0.0004 second. The proportional damping used in NLMHIST1A provides 0.59% damping at a period of 0.1 second and increases to approximately 134% damping as the period is decreased to 0.0004 second. The damping is increased over the entire range from 0.1 second to 0.0004 second rather than just at 0.0004 second as is the case in NLMHIST2A. Thus, more high frequency damping is present in NLMHIST1A than in NLMHIST2A. This explains why the plot for NLMHIST2A shows some small high frequency fluctuations that are not present for NLMHIST1A. If increased damping were provided for the modes between 0.1 second and 0.0004



EXAMPLE 6-011 - 17

second in NLMHIST2A, the results for NLMHIST2A would appear more similar to those for NLMHIST1A.

The following figure compares the Level 1 displacement versus time for load case NLMHIST1A to the experimental results. The comparison is similar for the other load 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)

Le

vel 1

Dis

pla

cem

ent

(in

)

Experimental

SAP2000


The following figures show isolator force-deformation plots for an exterior and an interior isolator for load case NLMHIST1A. The exterior isolator is located 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 the bearings [during the experiment] were not exactly known and they could very well have been different than assumed in the [SAP2000] analysis.” This could contribute to the difference in the experimental and SAP2000 results for the force-deformation response of the exterior isolator.



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 and deformation with the experimental values for the NLMHIST1A load case. Similar results are obtained for other time history load cases with damping at the high frequencies.



EXAMPLE 6-011 - 19



Percent Difference

NLMHIST1A -1.799 7%

NLMHIST2A -1.814 8%


NLDHIST1A -1.681 0%

NLMHIST1B -1.807 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 1.872 -2%

NLMHIST1B 1.982 4%

NLMHIST2B 2.003 5%


Maximum Deformation

(in) B

NLDHIST1B 1.947

1.909

2%

NLMHIST1A -5.672 -17%

NLMHIST2A -5.724 -17%

NLMHIST3A -5.939 -14% A

NLDHIST1A -5.238 -24%

NLMHIST1B -5.671 -17%

NLMHIST2B -5.834 -15%


Minimum Shear Force

(kip) B

NLDHIST1B -5.535

-6.872

-19%

NLMHIST1A 0.911 -23%

NLMHIST2A 0.904 -24%


NLDHIST1A 1.046 -12%

NLMHIST1B 0.933 -21%

NLMHIST2B 0.919 -22%


Maximum Shear Force

(kip) B

NLDHIST1B 0.893

1.183

-25%



EXAMPLE 6-011 - 20



Percent Difference

NLMHIST1A -1.924 7%

NLMHIST2A -1.94 8%


NLDHIST1A -1.79 0%

NLMHIST1B -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 1.77 -1%

NLMHIST1B 1.871 5%

NLMHIST2B 1.896 6%


Maximum Deformation

(in) B

NLDHIST1B 1.838

1.786

3%

NLMHIST1A -3.493 0%

NLMHIST2A -3.564 2%


NLDHIST1A -3.292 -6%

NLMHIST1B -3.586 3%

NLMHIST2B -3.504 0%


Minimum Shear Force

(kip) B

NLDHIST1B -3.382

-3.498

-3%

NLMHIST1A 3.909 17%

NLMHIST2A 3.973 19%


NLDHIST1A 4.033 21%

NLMHIST1B 3.841 15%

NLMHIST2B 3.802 14%


Maximum Shear Force

(kip) B

NLDHIST1B 3.745

3.346

12%



EXAMPLE 6-011 - 21

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

The comparisons of SAP2000 isolation system displacement with experimental results appear better in this verification example than they do in the SUNY Buffalo report. In Section 4-4 on page 50 of that report, the isolation system displacement is defined as the displacement of the first floor with respect to the ground; that is, the isolator displacement plus the displacement in the first level column. This displacement is called the Level 1 Displacement in the plots shown in this verification example.

When the SAP2000 isolator displacement is plotted versus the base shear, the resulting plot is very similar to that shown in the SUNY Buffalo report. Thus, it appears that the report may in some instances be making comparisons where the experimental displacement is for Level 1 and the SAP2000 displacement is for the Isolator Level. This would explain why the comparisons appear better in this verification example.

SOLUTION PARAMETERS FOR DIRECT INTEGRATION TIME HISTORY The nonlinear direct integration time histories in this example were run using a maximum substep size of 0.0005 second. A larger maximum substep size of 0.005 second was tried and found to yield larger displacements than the 0.0005 second step size. A smaller maximum substep size of 0.00005 second was tried and found to yield the same solution as the 0.0005 second step size. Thus, it was concluded that a 0.0005 second maximum step size was appropriate for this example.

Similarly, the nonlinear direct integration time histories in this example were run using a relative iteration convergence tolerance of 1E-4. A smaller relative iteration 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 for nonlinear analyses. This helps to build confidence that appropriate results have been obtained.



EXAMPLE 6-011 - 22

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

CONCLUSION In general, the SAP2000 results show an acceptable comparison with the independent results. For load case NLMHIST3A, which has no damper elements and no additional damping at the higher frequencies, the comparison of peak values for the isolator force-deformation curves is poor. Additional damping associated with the high frequencies improves the comparison.

For nonlinear modal time history load cases, modal damping, proportional damping and added dampers can all be used to significantly reduce the high frequency fluctuations that can occur in the models with friction pendulum isolators.

For nonlinear direct-integration time history load cases, proportional damping or added dampers can both be used to significantly reduce the high frequency fluctuations that can occur in the models with friction pendulum isolators. However, it is important to realize that proportional damping will not completely eliminate the fluctuations because the stiffness proportional component of the damping will be zero when the isolators are uplifted, and it is the stiffness proportional component of the damping that is effective in damping out the high frequency behavior. Thus, if nonlinear direct integration time histories are used, added damper elements may be a better alternative than proportional damping in the load case to reduce the high frequency fluctuations in the results.

Note that this particular problem is very numerically sensitive to the order of operations, which may differ for different computers and/or different operating environments on the same computer. The SAP2000 analysis engine is separately optimized for different processors which could lead to a different order of operations for different computers. Furthermore, the multithreaded algorithms and vectorized operations may be performed in a different order on the same computer based on the memory alignment which can change each time the analysis is performed. Due to the nature of the nonlinear modal and direct-integration history analyses, a small difference in an earlier time step can accumulate, leading to slightly different results each time the model is analyzed. For these reasons, deviations from the published results are to be expected and do not indicate a problem with the software.

Problem 6 011

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