# Demonstration of Lateral-Torsional Coupling in Building Structures by tyndale

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```									Demonstration of Dynamic Lateral-
Torsional Coupling in Building
Structures

http:// ucist.cive.wustl.edu/

Developed By:

Undergraduate Students: Carol Choi and Jack Rosenfeld
at
University of California, Los Angeles

This Project is supported in part by the
National Science Foundation Grant No. DUE-9950340
Equipment Required:                    Software Required:
• Shaking Table                        • MatLAB (for model design and data analysis
• One-Story Plexiglas Building Model   • Data Acquisition Software (e.g. LabVIEW)
• 2-Pound Circular Weights             • SAP2000 (for numerical simulation of experiments)
• Four Single-Channel Accelerometers
• Power Supply
• Dynamic Data Acquisition Board
• Computer
Objectives
   To demonstrate dynamic lateral-torsional
coupling phenomenon exhibited by base-
isolated buildings due to non-coincident
center of mass and center of stiffness.

   To compare experimental simulation with
numerical simulation.
Introduction
   Base-isolation is widely applied for seismic retrofit.

   Based-isolated buildings can be idealized as one-story
structures with flexible cylindrical columns.

   In the real world all buildings are eccentric systems.

   Eccentric systems respond in laterally-torsionally coupled
motions during earthquakes.
Background
   Eccentric System
• A system with non-coincident center of stiffness and center of
mass.
• During dynamic excitation, the inertia forces can be modeled as
acting through the center of mass, while the resultants of the
resisting forces respond through the center of stiffness.
• This creates a moment between the two opposing forces.
• The torsional effect created is coupled with the lateral motion.
• It is impossible to eliminate eccentricity in a building because one
cannot predict the spatial distribution of the dead and live loads to
make the center of mass coincide with the center of stiffness of the
building.
Background (contd.)
   Base-isolation
• Typically cylindrical laminated-rubber bearings
designed to take large shear deformations.
• Used to lessen the damage done to buildings
during earthquakes.
• Essentially act as shock absorbers, so that the
building undergoes less deformation.
A Lead Rubber Bearing (LRB) Base-
Isolator
The Coordinate System
x,ux,px

Ex

y,uy,py
u , p
Ey
c.m.

c.s.

c.s. = center of stiffness
c.m. = center of mass
Theory
      Governing differential equation of motion:
m u(t )  c u(t )  ku(t )  p(t )
Where u (t) is the displacement vector of the system as a function of time; m, c, and k are the mass, damping, and stiffness matricies,
respectively; p (t) is the dynamic forcing function, which takes the special form
0 
p (t )   m  1       in the case of earthquake base excitation in the y-direction.
0 
 
                        
 u x (t )            u x (t )                1         0      (ey  12)   u (t )  1 0                         0   px (t ) 
 u (t )   C   u (t )   m                                            x                                      
m y                y                                         (ex  12)    u y (t )   0 1                   0    p y (t ) 
2
                        L          0         1                                                                         
   u (t )           u (t )                                            u (t )                          1   p (t ) 
                                       
 (ey  12) (ex  12)      2     
              0 0                                    

                      
m                  total mass of the superstructure
Ip                     polar mass moment of inertia of the superstructure with respect to the z-axis, which passes through the center of mass
Ip
                     mass radius of gyration of the superstructure with respect to the z-axis
m
C                    damping matrix
xi , yi                x- and y- coordinates of the i-th base isolator
ki                     lateral stiffness of the i-th base isolator
N
k   ki               lateral stiffness of the total base isolation system
i 1

 N                       N         
Ex    ki  xi  / k , E y    ki  yi  / k  eccentricity in the x- and y- direction, respectively, of the center of stiffness of the total
 i 1                    i 1      
base isolation system with respect to the center of mass
k   ki  ( xi2  yi2 )   = rotational stiffness (about the z-axis) of the total base isolation system

k
L                          = uncoupled lateral natural circular frequency of vibration
m

k
                          = natural circular frequency of rotational vibration of a fictitious
IP
non-eccentric structure having the same rotational stiffness and
mass moment of inertia (with respect to the z-axis) as the eccentric
system considered here

    k
=                          = ratio of  defined above to the lateral uncoupled natural frequency
L   2 k

De    12                  = "equivalent diagonal" of the system

e x  Ex / De , ey  E y / De = relative eccentricity in the x- and y- direction, respectively
   Normalized natural circular frequencies
                            
2
      1                                        e
1   1
       1   2  ( 2  1) 2  48  e 2  1   1

– 1st, 2nd, 3rd modes of vibration:                  L     2                                        2

   “Alpha” Parameter                                        
2

2   2
     1

–    mass radius of gyration                         L   
multiplied by the maximum
rotational to maximum lateral
                            
2
displacement response ratio in                        1                                        e
3   3
       1   2  ( 2  1) 2  48  e 2  1    3

undamped free vibration                          L     2                                        2
condition
–   Parameter “alpha” is a measure of
the intrinsic propensity of an
eccentric system to develop a                                         2       2      2         2
rotational-torsional response under
where:                e = ex + ey = ex              if ey = 0
free and forced vibration conditions
e
(e.g., earthquake excitations).
F
 2 1

1 
u                           48F 2                                 1       1  1  48 F 2 

         max
         1                                     F                 
uy   max undamped
48F  1
free vibration

 1  1  48F 2 
1
3                          
F                    
Physical Model
   Target Prototype
» A 3-story base-isolated building with dimensions of 67 ft x 67 ft plan
and an average floor height of 10 feet.
» The prototypes uncoupled lateral natural period of vibration,
TL, prototype = 2 sec.
   Scaling factors
» Time ratio of prototype to model is 5
T = Tprototype/Tmodel = 5.
» Length ratio of prototype to model is 40
L = Lprototype/Lmodel = 40.
   Model
» Nine columns of 0.25 inch diameter are set 8 inches apart.
» Top plate measures 20 in x 20 in plan with a thickness of 0.375 in.
» The model‟s uncoupled „lateral natural period of vibration,
TL, model = 0.40 sec.
Model Construction
   Print out on a 20” by 20” sheet of paper, the position of the columns
   Tape paper onto top plate.
   Using a milling machine and a 3/8” drill bit, drill holes at all the
locations for the weights.
   Carefully clamp the top and bottom plates together, make sure the
edges are parallel.
   Drill holes at the locations of the columns with a “G” bit, on the
milling machine.
   Place the Plexiglas rods in the locations drilled, place a few drops of
the adhesive in the space between the columns and the drilled holes
and let dry (make sure the columns are flush with the plate).
   After waiting 30 minutes for the adhesive to dry, flip the plate over and
place the rods so that they fit into the other plate.
   Use the adhesive once the columns are flush with this other plate.
Testing
   Static Stiffness Test
   Free Vibration Test
» Non-eccentric condition
» Eccentric condition

   Shaking Table Test
» Using a scaled down version of the El Centro 1940
earthquake record
» Non-eccentric condition
» Eccentric condition
Model Specifications
Positions of Added Mass (º) and Corresponding Center
   Material:
of Mass (•)                                                                Plexiglas, E = 422 ksi
   Dimensions:
10
•    top plate 20”x20”x3/8”
8                                                                 •    columns 1/4” dia., 9” ht.
4
0.0161 [lb-s2 /in]

2                                                               Total Mass:
y-position (in)

0.0486 [lb-s2 /in]
0

-2
   Lateral Stiffness:
k = 11.98 [lb/in]
-4
   Uncoupled Lateral Natural
-6
Period of vibration:
-8                                                               0.40 s
-10
-10   -8   -6   -4   -2       0        2   4   6   8   10
   Maximum Relative
x-position (in)
Eccentricity:
18.8%
    ratio:
1.2
Corresponding Parameters

TABLE 1: Coordinates of Added Mass and Resulting Parameters
ex
x coordinate          y coordinate              Ex                             
Position                                                                  (relative
reference                                                                eccentricity)
[inch]                [inch]              [inch]                       parameter
[%]
1              1           7.4573        -7.4573         .6627           2.48          .3636
3              3           7.5229        -7.5229         1.9880          7.29          .7539
5              5           7.6524        -7.6524         3.3133          11.70         .8788
7              7           7.8426        -7.8426         4.6386          15.54         .9257
9              9           8.0893        -8.0893         5.9639          18.79         .9473
Shaking Table Actuator Arm;
Plan View
El Centro 1940 Earthquake Record;
0% Relative Eccentricity;
SAP2000 and Experiment
El Centro 1940 Earthquake Record;
7.3% Relative Eccentricity;
SAP2000 and Experiment
El Centro 1940 Earthquake Record;
18.8% Relative Eccentricity;
SAP2000 and Experiment
El Centro 1940 Earthquake Record;
Eccentricity in Both X- and Y-Directions;
SAP2000 and Experiment
El Centro 1940 Earthquake Record;
18.8% Relative Eccentricity;
Side View
Acceleration Records from SAP2000 for scaled El
Centro 1940 Earthquake Record
SAP2000 El Centro magnitude 1/20: 7.3% relative eccentricity:
table measured acceleration
200

100
Acceleration (in/sec )
2

0

-100

-200
0   1        2        3         4        5         6         7          8          9       10
Time [sec]

SAP2000 Acceleration in ux, uy, u
150
________ Acceleration in ux
________ Acceleration in uy
100
________ Acceleration in u

2


50
Acceleration (in/sec )
2

0

-50

-100

-150
0   1        2        3         4        5         6         7          8          9       10
Time [sec]
Acceleration Records from Experiment for scaled El
Centro 1940 Earthquake Record
LabvVIEW El Centro magnitude 1/20: 7.3% Relative Eccentricity:
Table Measured Acceleration
200

100
Acceleration (in/sec )
2

0

-100

-200
0   1       2        3        4        5           6        7       8       9    10
Time [sec]
LabVIEW Acceleration in ux, uy, u
150
__________ Acceleration in ux
100                                                     __________ Acceleration in uy

2
Acceleration (in/sec )

__________ Acceleration in u
2

50                                                                                 

0

-50

-100

-150
0   1       2        3        4        5           6        7       8       9    10
Time [sec]
Natural Vibration Mode Shapes;
SAP2000;
0% Relative Eccentricity;
Plan View

1st mode   shape                                    2nd mode shape

3rd mode shape
Natural Vibration Mode Shapes;
SAP2000;
0% Relative Eccentricity;
3-D With Fill Elements

1st mode   shape                                    2nd mode shape

3rd mode shape
Natural Vibration Mode Shapes;
SAP2000;
0% Relative Eccentricity;
3-D Without Fill Elements

1st mode   shape                                    2nd mode shape

3rd mode shape
Natural Vibration Mode Shapes;
SAP2000;
18.8% Relative Eccentricity;
Plan View

1st mode   shape                                    2nd mode shape

3rd mode shape
Natural Vibration Mode Shapes;
SAP2000;
18.8% Relative Eccentricity;
3-D With Fill Elements

1st mode   shape                                    2nd mode shape

3rd mode shape
Natural Vibration Mode Shapes;
SAP2000;
18.8% Relative Eccentricity;
3-D Without Fill Elements

1st mode   shape                                    2nd mode shape

3rd mode shape
Acknowledgement

   This research is supported by:
– University Consortium on Instructional Shake Tables
– Pacific Earthquake Engineering Research Center
– University of California, Los Angeles

   Special thanks to:
– Professor Joel Conte, UCLA
– Mr. Jack Rosenfield, Research Project Partner
– Mr. Alberto Salamanca, Mr. Harold Kasper,
UCLA Machine Shop, UCLA Student Work Shop
– Ms. Gina Ring, UC Irvine

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