Problem 6 006gdfgdfgdfggghhhhhhhhhhhhhh

Software Verification PROGRAM NAME: SAP2000 REVISION NO.: 0

EXAMPLE 6-006 - 1

EXAMPLE 6-006 LINK SUNY BUFFALO DAMPER WITH LINEAR 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 linear force versus velocity behavior. The model is subjected to horizontal seismic excitation using a scaled version of the S00E component of the 1940 El Centro record (see the section titled Earthquake Record later in this example for more information). The SAP2000 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.

Diaphragm constraints are assigned at each of the three floor levels.

Beams and columns are modeled as frame elements with specified end length offsets 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 be rigid. 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 additional information.

The dampers are modeled using two-joint, damper-type link elements. Both linear and nonlinear properties are provided for the dampers because this example uses both linear and nonlinear analyses. See the section titled Damper Properties and the section titled Load Cases Used later in this example for additional information.


EXAMPLE 6-006 - 2

GEOMETRY AND PROPERTIES

Level 3

Level 2

Level 1

Base

1STC

OL

10 cm

10 cm

10 cm

18.8 cm1

Stiff

15 cm

40.25 cm

40.25 cm

10 cm 10 cm120.5 cm

26 cm

100.5

cm76

.2 cm

76.2

cm

3

5

7

2

4

6

8

98 cm

10

11

12

13

26 cm

X

Y

Damper

Damper

Damper

Stiff

2XST2X3

2XST2X3

2XST2X3

ST2X

385

ST2X

385

1STC

OLST

2X38

5ST

2X38

5 10 c

m

Joints constrained as diaphragm, typical at Levels 1, 2, and 3

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

2.39 N-sec2/cm mass at joints 3, 4, 5, 6, 7 and 8 acting in X direction only

20 cm


EXAMPLE 6-006 - 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 cm4 Av = 2.02 cm2 STIFF A = 10,000 cm2 I = 100,000 cm4 Av = 0 cm2 (shear deformations not included)


EXAMPLE 6-006 - 4

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

Linear (k is in parallel with c) k = 0 N/cm c = 160 N-sec/cm Nonlinear (k is in series with c) k = 1,000,000 N/cm c = 160 N-sec/cm exp = 1

The damping coefficient used for the dampers for both the linear and nonlinear analyses is c = 160 N-sec/cm. This value was determined using the average value from a series of experimental tests. As described in Scheller and Constantinou 1999, the tested values of the damping coefficient ranged from 135 to 185 N-sec/cm. The average value of 160 N-sec/cm was used for all dampers in the SAP2000 model

LINEAR AND NONLINEAR ANALYSIS USING DAMPERS This example uses both linear and nonlinear load cases. It is important to understand that there are differences in the damper element behavior for linear and nonlinear analysis.

For nonlinear analysis the damper acts as a spring in series with a dashpot and uses the specified nonlinear spring stiffness and damping coefficient for the damper. In contrast, for linear analyses the damper element acts as a spring in parallel with a dashpot and uses the specified linear spring stiffness and damping coefficient for the damper. This is illustrated in the figure to the right.

Damper Propertiesfor

Nonlinear Analyses

knonlinear

cnonlinear

clinearklinear

Damper Propertiesfor

Linear Analyses


EXAMPLE 6-006 - 5

In this example, for the linear analysis, the linear effective stiffness, klinear, is set to zero so that pure damping behavior is achieved. For nonlinear analysis the nonlinear stiffness, knonlinear, is set to an approximation of the stiffness of the brace with the damper.

If pure damping behavior is desired from the damper element for nonlinear analysis with dampers, as is the case in this example, the effect of the spring can be made negligible by making its stiffness, knonlinear, sufficiently stiff. The spring stiffness should be large enough so that the characteristic time of the spring-dashpot damper element, given by = c/ knonlinear, is approximately one to two orders of magnitude smaller than the size of the load steps. Care must be taken not to make knonlinear excessively large because numerical sensitivity may result.

For this example:

00016.0000,000,1

160 nonlinearkc seconds

Thus is approximately two orders of magnitude less than the 0.01 second load steps and the 1,000,000 N/cm seems to be a reasonable value to obtain pure damping behavior.

Important Note: In linear modal time history analysis (and response spectrum analysis) of systems with damper elements, only the diagonal terms of the damping matrix are used; the off-diagonal, cross-coupling terms are ignored. All other analyses of systems with damper elements use all terms in the damping matrix. Thus linear modal time history analysis (and response spectrum analysis) of systems with damper elements should be used with great care and should typically be considered as only an approximation of the solution. In general, nonlinear analysis should be used for final design of systems with damper elements.


EXAMPLE 6-006 - 6

LOAD CASES USED Five different load cases are run for this example. They are described in the following table.

Load Case Description

MODAL Modal load 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 the number of modes to ten. The starting vectors are Ux acceleration and all link element nonlinear degrees of freedom.

MHIST1 Linear modal time history load case that uses the modes in the MODAL load case. This case includes modal damping in modes 1, 2 and 3.

NLMHIST1 Nonlinear modal time history load case that uses the modes in the MODAL load case. This case includes modal damping in modes 1, 2 and 3.

DHIST1 Linear direct integration time history load case. This case includes proportional damping.

NLDHIST1 Nonlinear direct integration time history load case. This case includes proportional damping.

The modal time history analyses use 2.71%, 1.02% and 1.04% modal damping for modes 1, 2 and 3, respectively. As described in Scheller and Constantinou 1999, those modal damping values were determined by experiment for the frame without dampers.


EXAMPLE 6-006 - 7

The direct integration time histories use mass and stiffness proportional damping that is specified to have 2.71% damping at a the period of the first mode and 1.02% damping at the period of the second mode. The solid line in the figure to the right shows the proportional damping used in this exampl

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)D

ampi

ng R

atio

Mass Stiffness Rayleigh

e.

EARTHQUAKE RECORD The following figure shows the earthquake record used in this example. As described 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 EQ6-006.txt. This file has one acceleration 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)

Acc

eler

atio

n (c

m/s

ec2 )


EXAMPLE 6-006 - 8

TECHNICAL FEATURES OF SAP2000 TESTED Damper links with linear velocity exponents Frame end length offsets Joint mass assignments Modal analysis for ritz vectors Linear modal time history analysis Nonlinear modal time history analysis Linear direct integration 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 the experimental results.

Modal Period Load Case SAP2000 Independent Experimental

Percent Difference

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 the experimental results for the story drift versus time for each of the three story levels for the NLDHIST1 load case. Similar results are obtained for the other time history load cases.

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 is calculated 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. The story 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.


EXAMPLE 6-006 - 9

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 2

Time (sec)

Leve

l 3 S

tory

Drif

t (%

)

0

NLDHIST1 Experimental

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18

Time (sec)

Leve

l 2 S

tory

Drif

t (%

)

20


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 2

Time (sec)

Leve

l 1 S

tory

Drif

t (%

)

0



EXAMPLE 6-006 - 10

The following table compares the maximum and minimum values of story drift obtained from SAP2000 and the experimental results at each story level for each of the four time history load cases.

Output Parameter Load Case Story Level SAP2000

Independent Experimental

Percent Difference

Level 1 0.734 0.750 -2% Level 2 0.877 0.947 -7% MHIST1 Level 3 0.538 0.608 -12% Level 1 0.764 0.750 +2% Level 2 0.879 0.947 -7% NLMHIST1 Level 3 0.524 0.608 -14% Level 1 0.764 0.750 +2% Level 2 0.879 0.947 -7% DHIST1 Level 3 0.524 0.608 -14% Level 1 0.764 0.750 +2% Level 2 0.879 0.947 -7%

Maximum Story Drift

NLDHIST1 Level 3 0.524 0.608 -14% Level 1 -0.589 -0.615 -4% Level 2 -0.789 -0.878 -10% MHIST1 Level 3 -0.551 -0.629 -12% Level 1 -0.639 -0.615 +4% Level 2 -0.807 -0.878 -8% NLMHIST1 Level 3 -0.526 -0.629 -16% Level 1 -0.638 -0.615 +4% Level 2 -0.806 -0.878 -8% DHIST1 Level 3 -0.526 -0.629 -16% Level 1 -0.638 -0.615 +4% Level 2 -0.806 -0.878 -8%

Minimum Story Drift

NLDHIST1 Level 3 -0.526 -0.629 -16%

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 load case. Similar results are obtained for the other time history load cases. The SAP2000 story shears are normalized by dividing them by 14,070 N.


EXAMPLE 6-006 - 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 (%)

Leve

l 3 S

tory

She

ar /

Wei

ght

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


Leve

l 2 S

tory

She

ar /

Wei

ght




-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

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


Leve

l 1 S

tory

She

ar /

Wei

ght





EXAMPLE 6-006 - 12

The following table compares the maximum and minimum values of normalized story shear obtained from SAP2000 and the experimental results at each story level for each of the four time history load cases.

Output Parameter Load Case Story Level SAP2000

Independent Experimental

Percent Difference

Level 1 0.256 0.324 -21% Level 2 0.204 0.248 -18% MHIST1 Level 3 0.112 0.136 -18% Level 1 0.291 0.324 -10% Level 2 0.223 0.248 -10% NLMHIST1 Level 3 0.121 0.136 -11% Level 1 0.276 0.324 -15% Level 2 0.223 0.248 -10% DHIST1 Level 3 0.121 0.136 -11% Level 1 0.291 0.324 -10% Level 2 0.223 0.248 -10%

Maximum Normalized Story Shear

NLDHIST1 Level 3 0.121 0.136 -11% Level 1 -0.208 -0.322 -35% Level 2 -0.190 -0.280 -32% MHIST1 Level 3 -0.122 -0.174 -30% Level 1 -0.271 -0.322 -16% Level 2 -0.239 -0.280 -15% NLMHIST1 Level 3 -0.144 -0.174 -17% Level 1 -0.234 -0.322 -27% Level 2 -0.238 -0.280 -15% DHIST1 Level 3 -0.143 -0.174 -18% Level 1 -0.271 -0.322 -16% Level 2 -0.239 -0.280 -15%

Minimum Normalized Story Shear

NLDHIST1 Level 3 -0.144 -0.174 -17%

The inaccuracies associated with ignoring the off-diagonal, cross-coupling terms in the damping matrix for the linear modal time history load case MHIST1 are more significant in the story shear results than they have been in other results displayed in this example.


EXAMPLE 6-006 - 13

The Level 1 story shear results shown for the MHIST1 load case do not include the force in the Level 1 damper. This damper force is not reported because, for the linear modal time history, the damping associated with the dampers is converted to modal damping and added to any other modal damping that may be specified. If a stiff frame element was included in the model below the Level 1 damper, similar to the stiff element at the other levels, the Level 1 story shear could be cut through three frame elements and all of the shear would be accounted for. However, the inaccuracies caused by ignoring the off-diagonal terms in the damping matrix would still be present.

COMPUTER FILE: Example 6-006

CONCLUSION The SAP2000 results show an acceptable comparison with the independent results. The clearest comparison of results is evident in the graphical comparisons.

The results using linear modal time history analysis (load case MHIST1) are slightly different from the other analyses because the linear modal time history analysis uses only the diagonal terms in the damping matrix, ignoring any off-diagonal, cross-coupling terms. The other analyses use all terms in the damping matrix. For this example load case MHIST1 shows a good approximation of the other solutions. The error introduced by ignoring the cross-coupling terms tends to improve the comparison with experimental results in some items and to make it worse in other items.

In general we recommend that linear modal time history analysis of models with damper elements only be used for quick, preliminary checks, and that another type of analysis be used for final analysis.

EXAMPLE 6-006Link SUNY Buffalo Damper with Linear Velocity ExponentProblem DescriptionGeometry and PropertiesFrame Element PropertiesDamper PropertiesLinear and Nonlinear Analysis Using DampersLoad Cases UsedEarthquake RecordTechnical Features of SAP2000 TestedResults ComparisonComputer File: Example 6-006Conclusion

Problem 6 006gdfgdfgdfggghhhhhhhhhhhhhh

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