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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 Faculty Advisor: Joel P. Conte 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 and added circular weights. 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. 6 Added Mass: 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 Coordinates of Added Mass and Corresponding Parameters TABLE 1: Coordinates of Added Mass and Resulting Parameters ex x coordinate y coordinate Ex Position (relative of added masses of added masses center of mass Alpha 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 Acceleration (rad/sec ) 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 Acceleration (irad/sec ) 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