problem 6 007

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R Software VerificationPROGRAM NAME: SAP2000

REVISION NO.: 0

EXAMPLE 6-007 - 1

EXAMPLE 6-007

LINK SUNY BUFFALO DAMPER WITH NONLINEAR VELOCITY EXPONENT

PROBLEM DESCRIPTION

This example comes from Section 5 of Scheller and Constantinou 1999 (the

SUNY Buffalo report). It is a two-dimensional, three-story moment frame with

diagonal fluid viscous dampers that have nonlinear force versus velocitybehavior. The model is subjected to horizontal seismic excitation using a scaledversion of the S00E component of the 1940 El Centro record (see the section

titled Earthquake Record later in this example for more information). TheSAP2000 results for modal periods, interstory drift and interstory force-deformation are compared with experimental results obtained using shake table

tests. The experimental results are documented in the SUNY Buffalo report.

The SAP2000 model is shown in the figure on the following page. Masses

representing the weight at each floor level, including the tributary weight from

beams and columns, are concentrated at the beam-column joints. Those masses,2.39 N-sec2/cm at each joint, act only in the X direction. In addition, small

masses, 0.002 N-sec2/cm, are assigned to the damper elements. The small masses

help the nonlinear time history analyses solutions converge. Diaphragmconstraints are assigned at each of the three floor levels.

Beams and columns are modeled as frame elements with specified end lengthoffsets and rigid-end factors. The rigid-end factor is typically 0.6 and the end

length offsets vary as shown in the figure. The frame elements connecting the

lower end of the dampers to the Level 1 and Level 2 beams are assumed to berigid. This is achieved in SAP2000 by giving those elements section properties

that are several orders of magnitude larger than other elements in the model. See

the section titled Frame Element Properties later in this example for additionalinformation.

The dampers are modeled using two-joint, damper-type link elements. Both

linear and nonlinear properties are provided for the dampers because thisexample uses both linear and nonlinear analyses. See the section titled Damper

Properties and the section titled Discussion of Nonlinear Damper StiffnessUsed in SUNY Buffalo Report later in this example for additional information.

This problem is solved using a nonlinear modal time history analysis and also

using a nonlinear direct integration time history analysis. See the section titled

Analysis Cases Used later in this example for additional information.

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EXAMPLE 6-007 - 2

GEOMETRY AND PROPERTIES

Level 3

Level 2

Level 1

Base

1ST

COL

10cm

10cm

10cm

18.8cm

1

Stiff

15cm

40.25 cm

40.25 cm

10 cm 10 cm

120.5 cm

26cm

100.5

cm

76.2cm

76.2

cm

3

5

7

2

4

6

8

9

8 cm

10

11

12

13

26cm

X

Y

Damp

er

Damp

er

Damper

Stiff

2XST2X3

2XST2X3

2XST2X3

ST2

X385

ST2X385

1ST

COL

ST2X385

ST2X385

10cm

Joints constrained asdiaphragm, typical at

Levels 1, 2, and 3

Frame element endlength offsets, typical.Rigid-end factor is 0.6

2.39 N-sec2/cm mass at

joints 3, 4, 5, 6, 7 and 8acting in X direction only

20cm

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EXAMPLE 6-007 - 3

FRAME ELEMENT PROPERTIES

The frame elements in the SAP2000 model have the following material

properties.

E = 21,000,000 N/cm2

= 0.3

The frame elements in the SAP2000 model have the following section properties.

1STCOL

A = 9.01 cm2

I = 14.614 cm4

Av = 4.42 cm2

ST2X385

A = 6.61 cm2

I = 5.95 cm4

Av = 2.02 cm2

2XST2X3

A = 13.22 cm2

I = 11.9 cm4Av = 2.02 cm

2

STIFF

A = 10,000 cm2

I = 100,000 cm4

Av = 0 cm2 (shear deformations not included)

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EXAMPLE 6-007 - 4

DAMPER PROPERTIES

The damper elements in the SAP2000 model have the following properties.

Linear (k is in parallel with c)

k = 0 N/mmc = 0 N-sec/mm

Nonlinear (k is in series with c)k = 2,000 N/mm

c = 220 N-sec/mm at Level 3

= 235 N-sec/mm at Level 2= 300 N-sec/mm at Level 1

exp= 0.5

As described in Scheller and Constantinou 1999, the c values were determined bytest. See the following section titled Nonlinear Damper Stiffness Used in the

SUNY Buffalo Report for additional information.

NONLINEAR DAMPER STIFFNESS USED IN THE SUNY BUFFALO REPORT

The results for the SAP2000 model used in the SUNY Buffalo report (Scheller

and Constantinou 1999) significantly underestimated the interstorydisplacements. We believe this significant difference occurred because thedamper modeled in the SUNY Buffalo SAP2000 model did not match that used

in the experiment. We believe that an inappropriate nonlinear stiffness, k, was

used in the SUNY Buffalo SAP model.

The SUNY Buffalo report ran the SAP2000 analysis multiple times using k

values of 100,000 N/mm and 25,000 N/mm. Those damper stiffnesses areapproximately 10 to 50 times larger than the 2,000 N/mm stiffness used in this

verification problem. This section describes why we believe that the 2,000 N/mm

stiffness is a more appropriate value.

The SUNY Buffalo report makes several references to the damper force-velocity

relationship. It indicates that the dampers were tested and found to exhibit abehavior described by

)(5.0

vsignvCF =

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EXAMPLE 6-007 - 5

except that for velocities below approximately 15 mm/sec, the force-velocitybehavior was essentially linear. In the preceding equation sign(v) = -1 if v < 0,

sign(v) = +1 if v > 0 and sign(v) = 0 if v = 0.

The figure below plots the force-velocity characteristics for a damper with

c = 220 N-(sec/mm)0.5

, a velocity exponent of 0.5 and various values of k, the

damper stiffness, in N/mm units. Similar plots can be obtained for dampers withc = 235 N-(sec/mm)0.5and c = 300 N-(sec/mm)0.5.

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40 45 50 55 60

Velocity (mm/sec)

k=100 N/mm

k=1000 N/mm

k=2000 N/mm

k=10000 N/mm

F=CV^0.5 (k=infinity)

Suny Buffalo report used k=100,000 N/mm and k=25,000 N/mm

c=220 N-(sec/mm)^0.5, exp=0.5

The plots in the figure were obtained by subjecting a damper to a linearlyincreasing velocity. This was achieved using a displacement time history. The

time history had a load that was a unit displacement at one end of the damper anda function that specified the displacement value as proportional to the square of

the time value. The SAP2000 model named Example 6-007 Damper Study was

used to obtain the data for the figure.

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EXAMPLE 6-007 - 6

In the figure the F=cv0.5line is equivalent to a k of infinity. The k=10,000 N/mmline and the F=cv0.5 line are essentially identical. Thus for stiffnesses of 10,000

N/mm and above, the damper will be well described by F=cv0.5. The SUNY

Buffalo report SAP2000 models used k values of 100,000 N/mm and 25,000N/mm. Thus both of those models were inconsistent with the properties of the

dampers observed in experimental testing because their force-velocity

relationship did not deviate from the F line at velocities below 15 mm/sec.

For this verification problem we have chosen a damper stiffness, k, of 2,000

N/mm. This value of k provides a force-velocity relationship that deviates fromthe F=cv0.5line at velocities below 15 mm/sec and matches the F=cv0.5line well

at velocities above 15 mm/sec.

See the section titled Study of the Sensitivity of Results to the Damper

Stiffness later in this example for more information.

ANALYSIS CASES USED

Three different analysis cases are run for this example. They are described in the

following table.

Analysis Case Description

MODAL Modal analysis case for ritz vectors. Ninety-nine modes are

requested. The program will automatically determine that a

maximum of ten modes are possible and thus reduce thenumber of modes to ten. The starting vectors are Uxacceleration and all link element nonlinear degrees of

freedom.

NLMHIST1 Nonlinear modal time history analysis case that uses the

modes in the MODAL analysis case. This case includes

modal damping in modes 1, 2 and 3.

NLDHIST1 Nonlinear direct integration time history analysis case. Thiscase includes proportional damping.

The modal time history analysis used 2.71%, 1.02% and 1.04% modal damping

for modes 1, 2 and 3, respectively. As described in Scheller and Constantinou

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EXAMPLE 6-007 - 7

1999, those modal damping values were determined by experiment for the framewithout dampers.

The direct integration time

history used mass and

stiffness proportional damping

that is specified to have 2.71%damping at the period of the

first mode and 1.02%

damping at the period of thesecond mode. The solid line in

the figure to the right showsthe proportional damping usedin this example.

EARTHQUAKE RECORD

The following figure shows the earthquake record used in this example. Asdescribed in Scheller and Constantinou 1999, it is the S00E component of the

1940 El Centro record compressed in time by a factor of two. It is compressed to

satisfy the similitude requirements of the quarter length scale model used in the

shake table tests.

The earthquake record is provided in a file named EQ6-007.txt. This file has oneacceleration value per line, in g. The acceleration values are provided at an equal

spacing of 0.01 second.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 5 10 15 20 25 30 35 40

Time (sec)

Acceleration

(cm/sec

2)

0

0.01

0.02

0.03

0.04

0.05

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Period (sec)

Mass

Stiffness

Rayleigh

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EXAMPLE 6-007 - 8

TECHNICAL FEATURES OF SAP2000 TESTED

Damper links with nonlinear velocity exponents Frame end length offsets Joint mass assignments Modal analysis for ritz vectors Nonlinear modal time history analysis Nonlinear direct integration time history analysis Generalized displacements

RESULTS COMPARISON

Independent results are experimental results from shake table testing presented in

Section 5, pages 61 through 73, of Scheller and Constantinou 1999.

The following table compares the modal periods obtained from SAP2000 and theexperimental results.

Modal PeriodAnalysis

Case SAP2000IndependentExperimental

PercentDifference

Mode 1 sec 0.438 0.439 0%

Mode 2 sec 0.135 0.133 +2%

Mode 3 sec

MODAL

0.074 0.070 +6%

The following three figures plot the SAP2000 analysis results and theexperimental results for the story drift versus time for each of the three story

levels for the NLDHIST1 analysis case. Similar results are obtained for the other

time history analysis case.

The story drift for Level 3 is calculated by subtracting the displacement at joint 5

from that at joint 7 and then dividing by the Level 3 story height of 76.2 cm and

multiplying by 100 to convert to percent. Similarly, the story drift for Level 2 iscalculated by subtracting the displacement at joint 3 from that at joint 5 and then

dividing by the Level 2 story height of 76.2 cm and multiplying by 100. Thestory drift for Level 1 is calculated by dividing the displacement at joint 3 by the

Level 1 non-rigid story height of 81.3 cm and multiplying by 100. The interstory

displacement results are obtained using SAP2000 generalized displacements.

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EXAMPLE 6-007 - 9

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

Level3StoryDrift(%)

NLDHIST1

Experimental

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

Level2StoryDrift(%)

NLDHIST1

Experimental

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

Level1S

toryDrift(%)

NLDHIST1

Experimental

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EXAMPLE 6-007 - 10

The following table compares the maximum and minimum values of story driftobtained from SAP2000 and the experimental results at each story level for each

of the two time history analysis cases.

OutputParameter

AnalysisCase Story Level SAP2000

IndependentExperimental

PercentDifference

Level 1 0.542 0.526 +3%

Level 2 0.589 0.631 -7%NLMHIST1

Level 3 0.313 0.323 -3%

Level 1 0.543 0.526 +3%

Level 2 0.588 0.631 -7%

Maximum

Story Drift

NLDHIST1

Level 3 0.312 0.323 -3%

Level 1 -0.610 -0.572 +7%

Level 2 -0.719 -0.746 -4%NLMHIST1

Level 3 -0.424 -0.488 -13%

Level 1 -0.610 -0.572 +7%

Level 2 -0.719 -0.746 -4%

MinimumStory Drift

NLDHIST1

Level 3 -0.424 -0.488 -13%

The three figures on the following page plot the SAP2000 analysis results and the

experimental results for the story drift versus normalized story shear for each of

the three story levels for the NLDHIST1 analysis case. Similar results are

obtained for the other time history analysis case. The SAP2000 story shears arenormalized by dividing them by 14,070 N.

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EXAMPLE 6-007 - 11

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Level 3 Story Drift (%)

Level3StoryShear/Weight

Experimental

NLDHIST1

Story Height for Drift = 76.2 cm

Structure Weight for Shear Normalization = 14,070 N

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


Level

2StoryShear/Weight

Experimental

NLDHIST1



-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


Level1Story

Shear/Weight

Experimental

NLDHIST1



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EXAMPLE 6-007 - 13

The results obtained for a k value of 1000 N/mm range from +13% to +31%different from (larger than) the experimental results. Using a k value of 1000

N/mm results in a structure that is underdamped compared to the experimental

structure.

The results obtained for a k value of 3000 N/mm range from -1% to -23%

different from (smaller than) the experimental results. Using a k value of 3000N/mm results in a structure that is overdamped compared to the experimental

structure.

The results obtained for a k value of 2000 N/mm, which was the chosen stiffness

for this verification problem, range from -13% to +7% different from (smallerthan) the experimental results.

The figure below shows the velocity across the Level 2 damper versus time.Similar plots are obtained for the other dampers. One reason this example is so

sensitive to the damper k value is that the velocities across the dampers are

typically in the range of +20 mm/sec to -20 mm/sec, except for a few peaks. Thek value affects the behavior of the damper at those low velocities as illustrated in

the figure and described in the section titled Nonlinear Damper Stiffness Used

in SUNY Buffalo Report earlier in this example.

-100

-80

-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

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EXAMPLE 6-007 - 14

COMPUTER FILES: Example 6-007, Example 6-007 Damper Study

CONCLUSION

The SAP2000 results show an acceptable comparison with the independent

results. The clearest comparison of results is evident in the graphicalcomparisons.

The nonlinear damper stiffness, k, used in the SUNY Buffalo SAP2000 model

gives a damper force-velocity relationship that did not appear to match the tested

force-velocity behavior of the dampers. This explains why the SUNY Buffalo

SAP2000 model significantly underestimates the displacements.

The results obtained for this example are sensitive to the value used for thedamper stiffness, k. For example, there is approximately a 20% difference in the

results obtained using a k value of 1,000 N/mm compared to using a k value of

2,000 N/mm. Thus when using dampers with nonlinear velocity exponents, itappears important to obtain accurate information on the force-velocity behavior

of the damper, particularly at low velocities

problem 6 007

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