link suny buffalo eight-story building with...

Software Verification PROGRAM NAME: SAP2000 REVISION NO.: 9

EXAMPLE 6-010 LINK – SUNY BUFFALO EIGHT-STORY BUILDING WITH RUBBER ISOLATORS

PROBLEM DESCRIPTION This example is presented in Section 2, pages 5 through 23, of Scheller and Constantinou 1999 (“the SUNY Buffalo report”). It is an eight-story building that is seismically isolated using rubber bearings. The model is subjected to a recorded pair of scaled horizontal ground acceleration histories from the 1971 San Fernando earthquake. SAP2000 results for superstructure displacements relative to the isolation system, superstructure accelerations, and isolator forces and deformations are compared with results obtained using the computer program 3D-BASIS-ME (see Tsopelas, Constantinou and Reinhorn 1994).

The SAP2000 model is shown in the figures on the following three pages. The superstructure is modeled as a stick using linear link elements. The superstructure stick connects joints 23 and 55 through 62. The floor masses are concentrated at the eccentric joints, joints 46 through 54. Diaphragm constraints are used at each floor level above the isolation system to connect the mass to the superstructure. Only the Ux, Uy and Rz degrees of freedom are active for the analysis. The superstructure is assumed to have 3% modal damping and the isolation system to have 0% modal damping.

Joints 1 through 45 define the location of the 45 rubber isolators in the model. Those joints are constrained to joint 46 using a body constraint. Joints 101 to 145 are in the same location as joints 1 through 45, respectively, and are fully restrained (fixed to the ground). Zero-length link elements with rubber isolator properties connect joints 1 through 45 to joints 101 through 145. The properties for all of the link elements in the model are presented in the section titled “Link Element Properties” later in this example.

The SAP2000 model used in this verification example differs from that used in the Scheller and Constantinou 1999 report as follows. First, this verification example uses a linear link element for the stick superstructure rather than the damper element used in the report. The linear link is a simpler and more appropriate element to use, but it was not available when the report was written.

Second, this verification example uses the actual linear effective stiffness for the isolators, 6.55 kip/in, rather than the artificially small effective stiffness used in the report. This item is explained in more detail in the section titled “Linear Effective Stiffness of Rubber Isolator Elements” later in this example.

EXAMPLE 6-010 - 1


GEOMETRY AND PROPERTIES

1, 101

Y

X

5, 1052 3 4

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23, 123 24 25, 125

26 27 28 29 30

31 32 33 34 35

36 37 38 39 40

41, 141 42 43 44 45, 145

4 @ 20' = 80'

8 @ 20

' = 16

0' CG

8'

46

Two joints at the same location with a rubber isolator link element connecting them, typical for joints 1 through 45 and 101 through 145

Note: Joints 1 through 46 are constrained using a body constraint

Plan View at Isolator Level

EXAMPLE 6-010 - 2


43,143 38,138 33,133 28,128 23,123 18,118 13,113 8,108 3,10346

47 55

48 56

49 57

50 58

51 59

52 60

53 61

54 62

Isolator Level

Level 1

Level 2

Level 3

Level 4

Level 5

Level 6

Level 7

Level 8

Y

Z

8'

8 @ 12

' = 96

'

8 @ 20' = 160'

Longitudinal Section

Joints constrained as diaphragm, typical at levels 1 through 8

Linear link element typical at each level for the stick representing the building superstructure

EXAMPLE 6-010 - 3


1, 101

23, 123

XY

Z

46

54

5, 105

41, 141

45, 145

62

2, 102

21, 121

25, 125

Active degrees of freedom are Ux, Uyand Rz

Linear link element typical at each level for the stick representing the building superstructure

Two joints at the same location with a rubber isolator link element connecting them, typical for joints 1 through 45 and 101 through 145.

3, 103

43, 143

EXAMPLE 6-010 - 4


LOAD CASES USED Three different load cases are run for this example. They are described in the following table. It is important to note that the 3D-BASIS-ME model uses 3% modal damping for all modes associated with the superstructure. No modal damping is associated with the isolation system in 3D-BASIS-ME.

Load Case Description

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

NLMHIST1 Nonlinear modal time history load case that uses the modes in the RITZ load case. This case includes 3% modal damping in all modes, except that modes 1, 2 and 3, which are the modes associated with the isolation system, are assigned 0% modal damping. See the section titled “Linear Effective Stiffness of Rubber Isolator Elements” later in this example for more information.

NLDHIST1 Nonlinear direct integration time history load case. This case includes proportional damping, which is defined to provide damping similar to, but not exactly the same as, the damping for the modal time history. See the section titled “Proportional Damping for Direct Integration Time History” later in this example for more information.

Note that the inherent viscous damping in the superstructure, which is specified to be 3% of critical damping, is accounted for differently in 3D-BASIS-ME, the nonlinear modal time history in SAP2000, and the nonlinear direct integration time history in SAP2000. Thus, slight differences in the results for each of the three time history analyses (one in 3D-BASIS-ME and two in SAP2000) are expected.

EXAMPLE 6-010 - 5

Software Verification PROGRAM NAME: SAP2000 REVISION NO.: 9 EARTHQUAKE RECORD

The following figures show the earthquake records used in this example. They are the recorded pair of horizontal ground acceleration time histories from the 1971 San Fernando earthquake at station number 211. The earthquake records are provided in files named EQ6-010-trans.txt and EQ6-010-long.txt. Those files have one acceleration value per line, in g. The acceleration values are provided at an equal spacing of 0.02 second.

Inside SAP2000 each of the two components is multiplied by a factor of 2.345, as described in the SUNY Buffalo report, and also by a factor of 386.22 to convert from g to in/sec2. The recorded north and west components are applied in the transverse and longitudinal directions of the model, respectively.

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 5 10 15 20 25 30 35 40 45

Time (sec)

Gro

und

Acc

eler

atio

n (g

)

Ground Acceleration for Transverse (X) Direction

-0.10

-0.05

0.00

0.05

0.10

0.15

0 5 10 15 20 25 30 35 40 45

Time (sec)

Gro

und

Acc

eler

atio

n (g

)

Ground Acceleration for Longitudinal (Y) Direction

EXAMPLE 6-010 - 6

Software Verification PROGRAM NAME: SAP2000 REVISION NO.: 9 LINK ELEMENT PROPERTIES

This section presents the properties used for all of the 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 local 3 axis is parallel to the positive global Y axis.

The superstructure linear link elements have an effective stiffness, ke, and for the shear degrees of freedom, a distance from the J-end to the shear spring, DJ. Properties are specified for the U2, U3 and R1 degrees of freedom.

Between the Isolator Level and Level 3 (Property Name LINST123) ke U2 = 3401.8 k/in DJ U2 = 72 in ke U3 = 3401.8 k/in DJ U3 = 72 in ke R1 = 3.996 E+09 k-in/radian Between the Level 3 and Level 6 (Property Name LINST456) ke U2 = 2551.3 k/in DJ U2 = 72 in ke U3 = 2551.3 k/in DJ U3 = 72 in ke R1 = 2.997 E+09 k-in/radian Between the Level 7 and Level 8 (Property Name LINST78) ke U2 = 1700.9 k/in DJ U2 = 72 in ke U3 = 1700.9 k/in DJ U3 = 72 in ke R1 = 1.998 E+09 k-in/radian

The rubber isolator link elements have a linear effective stiffness, ke, a nonlinear initial stiffness, k, a nonlinear yield strength, Fy, and a post yield stiffness ratio, r. See the following section titled “Linear Effective Stiffness of Rubber Isolator Elements” for more information about ke. Properties are specified for the U2, and U3 degrees of freedom and the properties are the same for the two degrees of freedom. The rubber isolator property name is BILIN and its properties are:

ke = 6.55 k/in Fy = 12.8 k k = 25.6k/in r = 0.1887

EXAMPLE 6-010 - 7


LINEAR EFFECTIVE STIFFNESS OF RUBBER ISOLATOR ELEMENTS This verification example uses the calculated linear effective stiffness, ke, of 6.55 kip/in for the isolators. The SUNY Buffalo report used an artificially small effective stiffness of 0.0001 kip/in in their SAP2000 model to match the 3D-BASIS-ME results. The report further shows that when they used the actual isolator effective stiffness of 6.55 kip/in in their SAP2000 model, their SAP2000 results underestimated the 3D-BASIS-ME results.

The calculated isolator effective stiffness of 6.55 kips/in is the appropriate value to use and, as shown in this verification example, leads to results that match the 3D-BASIS-ME results when the SAP2000 model is made essentially equivalent to the 3D-BASIS-ME model.

It is important to recognize that the 3D-BASIS-ME model has 3% modal damping in the superstructure and no modal damping in the isolation system. Thus, for the SAP2000 model to be essentially equivalent to the 3D-BASIS-ME model, it must have 3% damping in all modes, except for modes 1, 2 and 3, which are dominated by the isolation system behavior. Thus, to be equivalent to the 3D-BASIS-ME model, modes 1, 2 and 3 in the SAP2000 model must be assigned 0% damping with all other modes assigned 3% damping.

The SUNY Buffalo report SAP2000 model underestimated the 3D-BASIS-ME results when using a linear effective stiffness of 6.55 kip/in for the isolators because the SAP2000 model had 3% damping in all modes, including modes 1, 2 and 3. Thus, the SAP2000 model was not equivalent to the 3D-BASIS-ME model.

When the SUNY Buffalo report SAP2000 model used a linear effective stiffness of 0.0001 kip/in for the isolators, the results matched the 3D-BASIS-ME results. Using 0.0001 kip/in effective stiffness for the isolators made the periods of the isolated modes (modes 1, 2 and 3) 528, 512 and 435 seconds, respectively. Noting that the entire earthquake duration is approximately 44 seconds, it is apparent that very little energy can be absorbed by modes 1, 2 and 3, thus making the damping associated with them approximately 0%, which is consistent with the 3D-BASIS-ME model.

Again, we recommend using the actual effective stiffness and adjusting the modal damping associated with the isolated modes, rather than using an artificially small effective stiffness.

EXAMPLE 6-010 - 8


Note that the preceding explanation is only relevant for the nonlinear modal time history load case. It is not relevant for the nonlinear direct integration time history load case. The linear effective stiffness of the isolators is only used in linear load cases. In this verification example, the only linear load case is the modal load case, RITZ. Both the nonlinear modal time history and the nonlinear direct integration time history load cases use the nonlinear properties of the isolator, not the linear properties. However, the nonlinear modal time history uses modes from the modal load case RITZ that are based on the linear effective stiffness of the isolators. Thus, the nonlinear modal time history load case is indirectly affected by the linear effective stiffness of the isolators, whereas the nonlinear direct integration time history load case is unaffected by the linear isolator properties.

PROPORTIONAL DAMPING FOR DIRECT INTEGRATION TIME HISTORY The nonlinear direct integration time history load case NLDHIST1 uses mass and stiffness proportional damping. For this load case the challenge is to designate appropriate proportional damping that approximates 3% damping in all modes, except the isolator modes (modes 1, 2 and 3), which are to have 0% damping. The isolator modes have periods of approximately 2 seconds, and the superstructure periods range from approximately 0.06 to 0.60 second.

For this example, the proportional damping is specified as stiffness proportional damping only. The mass coefficient for the damping is set equal to zero and the stiffness coefficient is set equal to 0.0030. The solid line in the chart to the right plots the resulting proportional damping used. The dashed line shows a constant 3% damping.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Period (sec)

Dam

ping

Rat

io

EXAMPLE 6-010 - 9


TECHNICAL FEATURES OF SAP2000 TESTED Rubber isolator links Linear links Zero-length, two-joint link elements Diaphragm constraints Modal analysis for ritz vectors Nonlinear modal time history analysis Nonlinear direct integration time history analysis Generalized displacements

RESULTS COMPARISON Independent results are obtained using the computer program 3D-BASIS-ME (see Tsopelas, Constantinou and Reinhorn 1994).

The eight figures shown on the following four pages plot results from 3D-BASIS-ME and from the SAP2000 load case NLMHIST1 (nonlinear modal time history). The results for SAP2000 load case NLDHIST1 (nonlinear direct integration time history) are similar. The following plots are shown:

Level 8 X direction displacement relative to the isolation system Level 8 Y direction displacement relative to the isolation system Level 8 rotation about Z relative to the isolation system Base shear in the X direction Level 3 absolute acceleration in the X direction Level 3 absolute acceleration in the Y direction Link 23 force-deformation in the X direction Link 23 force-deformation in the Y direction

In SAP2000, the level 8 displacements and rotations relative to the isolation system are determined using generalized displacements. The generalized displacements are defined to subtract the displacement or rotation at joint 23 from that at joint 62.

EXAMPLE 6-010 - 10


-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 U

x Dis

plac

emen

t Rel

ativ

e to

Isol

atio

n Sy

stem

(in)

3D-BASIS-ME SAP2000 NLMHIST1

Level 8 Ux Displacement Relative to Isolation System

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 U

x Dis

plac

emen

t Rel

ativ

e to

Isol

atio

n Sy

stem

(in)


Level 8 Ux Displacement Relative to Isolation System

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 U

y Dis

plac

emen

t Rel

ativ

e to

Isol

atio

n Sy

stem

(in)


Level 8 Uy Displacement Relative to Isolation System

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 U

y Dis

plac

emen

t Rel

ativ

e to

Isol

atio

n Sy

stem

(in)


Level 8 Uy Displacement Relative to Isolation System

EXAMPLE 6-010 - 11


-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 R

z Rot

atio

n R

elat

ive

to Is

olat

ion

Syst

em (r

adia

ns)


Level 8 Rz Rotation Relative to Isolation System

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0 5 10 15 20 25 30

Time (sec)

Leve

l 8 R

z Rot

atio

n R

elat

ive

to Is

olat

ion

Syst

em (r

adia

ns)


Level 8 Rz Rotation Relative to Isolation System

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 5 10 15 20 25 30

Time (sec)

Bas

e F x

(ki

p)


Base Shear Fx

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 5 10 15 20 25 30

Time (sec)

Bas

e F x

(ki

p)


Base Shear Fx

EXAMPLE 6-010 - 12


-80

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30

Time (sec)

Leve

l 3 U

x Abs

olut

e A

ccel

erat

ion

(in/s

ec2 )


Level 3 Ux Absolute Acceleration

-80

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30

Time (sec)

Leve

l 3 U

x Abs

olut

e A

ccel

erat

ion

(in/s

ec2 )


Level 3 Ux Absolute Acceleration

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30

Time (sec)

Leve

l 3 U

y Abs

olut

e A

ccel

erat

ion

(in/s

ec2 )


Level 3 Uy Absolute Acceleration

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30

Time (sec)

Leve

l 3 U

y Abs

olut

e A

ccel

erat

ion

(in/s

ec2 )


Level 3 Uy Absolute Acceleration

EXAMPLE 6-010 - 13


-50

-40

-30

-20

-10

0

10

20

30

40

50

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Isolator 23 Ux Deformation (in)

Isol

ator

23

F x F

orce

(kip

)


Isolator 23 Force-Deformation in the X Direction

-50

-40

-30

-20

-10

0

10

20

30

40

50

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Isolator 23 Ux Deformation (in)

Isol

ator

23

F x F

orce

(kip

)


Isolator 23 Force-Deformation in the X Direction

-40

-30

-20

-10

0

10

20

30

40

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Isolator 23 Uy Deformation (in)

Isol

ator

23

F y F

orce

(kip

)


Isolator 23 Force-Deformation in the Y Direction

-40

-30

-20

-10

0

10

20

30

40

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Isolator 23 Uy Deformation (in)

Isol

ator

23

F y F

orce

(kip

)


Isolator 23 Force-Deformation in the Y Direction

EXAMPLE 6-010 - 14


The following table compares the maximum and minimum values of the output items shown in the charts on the previous four pages. Results are compared for both the modal time history load case, NLMHIST1, and the direct integration time history load case, NLDHIST1.

Output Parameter

Dir and Max/Min

Load Case SAP2000

Independent 3D-BASIS-ME

Percent Difference

Level 8 displacement

relative to isolation system

(in)

Ux Max

NLMHIST1 3.521 3.494

+1%

NLDHIST1 3.418 -2%

Ux Min

NLMHIST1 -2.875 -2.804

+3%

NLDHIST1 -2.760 -2%

Uy Max

NLMHIST1 2.642 2.538

+4%

NLDHIST1 2.695 +6%

Uy Min

NLMHIST1 -2.167 -2.029

+7%

NLDHIST1 -2.182 +8%

Level 8 rotation relative to

isolation system (rad)

Rz Max

NLMHIST1 0.00080 0.00075

+7%

NLDHIST1 0.00068 -9%

Rz Min

NLMHIST1 -0.00077 -0.00076

+1%

NLDHIST1 -0.00073 -4%

Base shear in the X direction

(kip)

Fx Max

NLMHIST1 1854 1818

+2%

NLDHIST1 1767 -3%

Fx Min

NLMHIST1 -2109 -2089

+1%

NLDHIST1 -2059 -1%

EXAMPLE 6-010 - 15


Output Parameter

Dir and Max/Min

Load Case SAP2000

Independent 3D-BASIS-ME

Percent Difference

Level 3 (jt 57) absolute

acceleration (in/sec2)

Ux Max

NLMHIST1 65.76 67.73

-3%

NLDHIST1 67.82 0%

Ux Min

NLMHIST1 -72.71 -71.47

-2%

NLDHIST1 -69.94 +2%

Uy Max

NLMHIST1 70.36 64.96

+8%

NLDHIST1 70.36 +8%

Uy Min

NLMHIST1 -58.35 -57.78

+1%

NLDHIST1 -56.99 -1%

Isolator 23 shear force

(kip)

Fx Max

NLMHIST1 48.66 48.17

+1%

NLDHIST1 47.14 -2%

Fx Min

NLMHIST1 -43.55 -42.68

+2%

NLDHIST1 -41.04 -4%

Fy Max

NLMHIST1 36.65 36.38

+1%

NLDHIST1 35.31 -3%

Fy Min

NLMHIST1 -30.91 -30.54

+1%

NLDHIST1 -30.41 +0%

Isolator 23 deformation

(in)

Ux Max

NLMHIST1 7.935 7.845

+1%

NLDHIST1 7.611 -3%

Ux Min

NLMHIST1 -6.916 -6.746

+3%

NLDHIST1 -6.386 -5%

Uy Max

NLMHIST1 6.247 6.150

+2%

NLDHIST1 5.93 -4%

Uy Min

NLMHIST1 -4.419 -4.304

+3%

NLDHIST1 -4.303 0%

EXAMPLE 6-010 - 16

Software Verification PROGRAM NAME: SAP2000 REVISION NO.: 9 COMPUTER FILE: Example 6-010

CONCLUSION The SAP2000 results show an acceptable comparison with the independent results considering that SAP2000 and 3D-BASIS-ME use different modeling and solution techniques for the isolated structure. The clearest comparison of results is evident in the graphical comparisons.

EXAMPLE 6-010 - 17

link suny buffalo eight-story building with...

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