Demonstration of Lateral-Torsional Coupling in Building Structures

Transcription

1 Demonstration of Lateral-Torsional Coupling in Building Structures A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES ucist.cive.wustl.edu/ Required Equipment: Shake Table One-Stor Pleiglas Model -Pound Circular Weights Four Single-Channel Accelerometers Dnamic Data Acquisition Board Computer Matlab Data Acqusition Program SAP Developed B: Undergraduate Student: Carol Choi and Jack Rosenfeld Facult Advisor: Dr. Joel P. Conte Universit of California, Los Angeles This Project is supported in part b the National Science Foundation Grant No. DUE-99534

2 Instructor s Manual Universit of California, Los Angeles. REMARKS TO THE INSTRUCTORS This is the instructor s supplement to the eperiments described in the Student s Manual. It contains epected eperimental results, sample solutions to eercises, and further details on certain analtical developments. There are man topics that relate to these eperiments which are not eplored in this manual, which the instructor ma wish to bring out. This manual contains a number of tests to be performed using a phsical model. The instructor ma ask the students to perform all tests in several lab sessions, or choose particular tests that pertain to the emphasis and curriculum of the class. The instructor ma use equipments and data acquisition sstems of his/her choice as long as the can accomplish the tasks specified in the eperiments. The main two software packages used are MATLAB for data reduction analsis and SAP for numerical simulation of the eperiment. Students should be familiar with both of these programs before starting to work on the eperiments. Objectives: This eperiment serves two main objectives: To demonstrate the dnamic lateral-torsional coupling phenomenon ehibited b baseisolated building structures due to non-coincident center of mass and center of stiffness. To compare eperimental simulation with numerical simulation of a specialized structural dnamic eperiment.. INTRODUCTION Base-isolated buildings can be idealized as single-stor structures with clindrical columns. In this eperiment, ou will be directed to build a single-stor, two-ba b two-ba, Pleiglas model with adjustable level of mass eccentricit. Upon completion of the model, ou will perform static stiffness tests and free vibration tests to determine the actual properties of the model such as stiffness, damping ratio, and natural vibration frequencies. You will then compare the sstem properties identified eperimentall with those predicted b the theor or simulated numericall. Net, ou will operate the shaking table to simulate earthquake ecitations and observe the seismic response of the phsical model for different levels of eccentricit between the center of mass and the center of stiffness. To further enrich our background in the use of finite element programs, ou will simulate numericall the seismic response of the phsical model using a finite element model created in SAP. You will then compare the seismic response observed eperimentall with that predicted numericall and eplain the possible sources of discrepanc.. BACKGROUND Before beginning the eperiment, it is necessar to understand the concepts and rationale behind the model idealizations.

3 What is an eccentric sstem? An eccentric sstem is defined as a sstem with non-coincident center of mass and center of stiffness. When such a sstem is subjected to dnamic ecitations e.g., earthquakes, wind, ocean waves, 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, resulting into a torsional effect coupled with the lateral motion. An eccentric sstem is depicted in Figure. center of stiffness center of mass Figure : Plan View of an Eccentric Sstem In real life, eccentricit in a building is impossible to eliminate. Even if the design is nominall smmetric, accidental eccentricit is unavoidable due to imperfection in construction and uncertaint about the spatial distribution of dead and live loads. Therefore, structural engineers must understand its effects on the dnamic response of the building and account for these effects at the design stage. How does a single-stor building structure represent a multi-stor base-isolated building? Base isolators are used as structural devices to lessen the damage done to a building during earthquakes. Base isolators are tpicall clindrical laminated-rubber bearings designed to take large shear deformations. The are interposed between the base of the building and its foundation. During an earthquake, the base isolators act essentiall as shock absorbers, so that the building above undergoes less deformation. Consequentl, the building suffers much less damage and the risk of injur to the occupants is decreased. Figure : Tpical Base Isolator 3

4 Since deformations localize mainl in the base-isolators, the structure above can be assumed to displace as a rigid unit. This eplains wh a multi-stor base-isolated building structure can be idealized as a single stor structure. In the phsical model used here, the base-isolators are represented b fleible, clindrical columns supporting the roof. 3. THEORY 3. Multiple-Degree-of-Freedom Sstem In this eperiment, the model under investigation is a one-stor structure composed of a rigid square top plate supported b nine columns. The coordinate sstem of the structure is illustrated in Figure 3. The -ais along the lateral direction crosses the -ais along longitudinal direction at the center of mass of the superstructure top plate + added weights. The columns of the model are assumed to be inetensible and their mass negligible compared to the mass of the whole sstem. The top plate is quasi-rigid in and out of plane. Thus, the sstem has three degrees of freedom, namel u, u, and u θ as illustrated in Figure 3. Columns are placed smmetricall in both the and directions so that the center of rigidit is located at the geometric center of the top plate. Different levels of eccentricit are introduced into the structure b placing additional concentrated masses at various pre-determined locations on the top plate. As a result, the center of mass of the whole sstem varies according to the location of the added masses. For the purpose of this eperiment, the model is ecited onl in the -direction and eccentricit is imposed onl along the -ais. The distance between the two centers center of mass, c.m., and center of stiffness, c.s. along the - and -direction is denoted as eccentricit along the -direction, E, and eccentricit along the -direction, E Y, respectivel., u, p E Longitudinal Direction u, p θ θ c.m. Lateral Direction E, u, p c.s. Figure 3: Coordinate Sstem The general equation of motion for a multiple-degree-of-freedom sstem is derived from Newton s second law and takes the form 4

5 mu t + cu t + ku t p t 3 4 where ut is the displacement vector of the sstem as a function of time and a superimposed dot denotes a single differentiation with respect to time; m, c, and k are the mass, damping, and stiffness matrices, respectivel; pt is the dnamic forcing vector function. Equation epresses the equilibrium between the sstem s inertial forces, damping forces, elastic restoring forces 3, and the eternal dnamic forces 4. The dnamic response of the eccentric sstem is derived under two hpotheses: The lateral stiffness of each base isolator column is uniform in all directions of deformation. The rotational response u θ developed under dnamic ecitation is small so that u θ sinu θ tanu θ. Hence, the dnamic response is governed b the following set of coupled differential equations of motion. Note that based on the assumption stated above, E p p θ. m m u t u t + m ρ u t θ [ C' ] u u u t k t + t k E θ k k E k E u k E u k θθ u θ t p t p t p θ t t t where u t, u t, u θ t m I p translation along the - and - directions and rotation along the z-ais, respectivel, of the base-isolated sstem; total mass of the superstructure top plate + added weights; polar mass moment of inertia of the superstructure with respect to the z-ais which passes through the center of mass; I p ρ mass radius of gration of the superstructure with respect to the m z-ais; [C ] i, i k i damping matri; - and - coordinates of the i th base isolator; lateral stiffness of the i th base isolator in an direction; N Total number of base isolators ; k N k i i lateral stiffness of the total base isolation sstem in an direction; 5

6 k k E k k E N i i i N i i i, eccentricit positive as shown in Figure 3 in the - and - direction, respectivel, of the center of stiffness of the total base isolation sstem relative to the center of mass; + i N i i i k k θθ rotational stiffness about the z-ais of the total base isolation sstem; p t, p t, p θ t eternal dnamic forces/moment applied in the -, -, and z- direction, respectivel. To identif the dimensionless parameters governing the coupled lateral-torsional response of the sstem, Equation can be rewritten as note that E p p θ in our application, [ ] + + t p t p t p t u t u t u e e e e m t u t u t u C t u t u t u m L θ θ θ θ ρ ρ γ ω ρ ρ 3 where [ ] [ ] ' C C ρ damping matri; k uncoupled lateral longitudinal or transversal natural circular frequenc L ω of vibration; m k natural circular frequenc of rotational vibration of a fictitious non- θθ θ ω p I eccentric structure having the same rotational stiffness and mass moment of inertia with respect to the z-ais as the eccentric sstem considered; ratio of rotational uncoupled natural frequenc ω θ to the lateral γ θθ θ uncoupled natural frequenc ω L as defined above; L ρ ω k k ω 6

7 D e ρ equivalent diagonal of the eccentric sstem which, for rectangular shape and uniform mass distribution, coincides with the actual length of the diagonal of the sstem; e E D e relative eccentricit in the -direction, recalling that in our case the eccentricit in the -direction is zero, i.e., E e. Special Case of Support Ecitation In the case of support earthquake ecitation in the -direction, the effective earthquake load vector is given as the product of the mass matri, a load influence vector, and the ground acceleration [Chopra, ]: m p eff t - m 4 u g t m ρ Through modal analsis, it is possible to uncouple the equations of motion of a linear sstem b making use of the natural vibration mode shapes [Chopra, ]. The solution of the eigenvalue problem [ k m] φ 5 ω n which governs the undamped free vibration response of a linear sstem provides the following frequencies for the first three natural modes of vibration of the sstem, ω, ω, and ω3 ωn and φ n denote the circular frequenc in rad/sec and mode shape, respectivel, of the n-th natural mode of vibration: Ω ω ω L n e { + γ γ + 48e } + Θ 6a Ω ω ω L 6b Ω 3 ω 3 ω L e { + γ + γ + 48e } + Θ3 6c where e e + e ; e 7 7

8 e F 8 γ Θ + 48F F Θ F F 9 Equations 6a-c indicate that the modal frequencies of the subject eccentric sstem are functions of structural parameters γ and e onl. Another important structural parameter, the alpha parameter, α, is defined as the product of the mass radius of gration of the structure and the ratio of the maimum rotational to the maimum longitudinal displacement response developed b an undamped one-stor eccentric sstem in free vibration induced b an initial displacement in the longitudinal direction Trombetti and Conte. uθ ma α ρ u ma 48F 48F + The γ, e, and α parameters reveal fundamental characteristics of the coupled lateral-torsional response of eccentric sstems. The are used in designing the phsical model. 3. Stiffness Determined From Simplified Analtical Model Figure 4 shows a prismatic column that is fied at both ends. It is displaced b one unit in the horizontal direction. From Bernoulli-Euler linear beam theor, the lateral stiffness, k i, of a single column of height h, with material Young s modulus E, and moment of inertia I of the crosssection can be determined b Equation. All columns in the model are assumed to have the same lateral stiffness. Multipling k i from Equation b the total number of columns N in the structure gives the total stiffness k of the sstem. k i EI 3 h Figure 4: Stiffness Coefficients of a Prismatic Column 8

9 3.3 Damping Ratio Determined b Logarithmic Decrement of Amplitude Deca The damping ratio characterizes the rate of deca of motion of the sstem. Therefore, the damping ratio can be obtained eperimentall from free vibration records of a sstem. For lightl damped sstems, the damping ratio can be determined from the following equation [Chopra, ]: u i ζ ln 3 j u π i+ j where j denotes the number of ccles during which successive peaks ü i, ü i+,, ü i +j are used in the calculation as shown in Figure 5. Acceleration Record of Free Vibration Test 5 ü i ü i+ Acceleration [in/s ] ü i+ ü i+3 ü i+4 ü i T d T d Time [sec] Figure 5: Damping Ratio Determined b Logarithmic Decrement 4. PHYSICAL MODEL 4. Target Prototpe The target prototpe structure is a three-stor base-isolated structure with planar dimensions 67 ft 67 ft and an average floor height of ft. The prototpe structure is assumed to have an π uncoupled lateral natural period of TL seconds, which is ver realistic. The ratio of the ω L rotational uncoupled natural frequenc to the lateral uncoupled natural frequenc, γ, ranges from. to.8, while the relative eccentricit, e e, ranges from. to.. Similitude Requirements Scaling factors for the various geometrical and phsical quantities are selected to produce a reasonabl sized model and to accommodate the capacit of the small shaking table available. 9

10 Tprototpe The time scale factor of prototpe to model is λ T 5and the length scale factor of T Lprototpe prototpe to model is λ L 4. Therefore, the phsical model is designed to have an Lmodel uncoupled lateral natural period of vibration of.4 second, and plan dimensions of the top plate of in in. 4. Design of the Model To maimize the visual effects of the model's dnamic behavior, the material for the columns must be fleible enough, et resistant enough against buckling. Using the MTS uni-aial testing machine, tension and compression tests were performed on both Pleiglas and Lean samples. Pleiglas was selected as the basic material for the model because it possesses more desirable properties than other materials considered, such as light weight, fleibilit in the elastic range, and workabilit. The Young's modulus and strength of Pleiglas were measured to be approimatel 4, psi and, psi, respectivel. The densit of Pleiglas was measured to be.46 lb/in 3. Nine columns of.5 inch in diameter are set 8 inches apart. As stated before, the model must show visual effects of torsional response when ecited; in other words, the "alpha parameter", defined in Equation, of the sstem is to be maimized. From the equations in Section 3., the "alpha parameter" is a function of structural parameters γ and e, which in turn are functions of man phsical and geometrical variables such as columns stiffness, amount of added masses, and position of added masses. A program written in MATLAB see masspos.m in the Matlab folder on the CD-ROM is used to calculate the "alpha parameter" from basic geometric and material model parameters. Several ccles of iterations produced an optimized model with a top plate thickness of.375 inch, 9 columns of 9. inches in length measured from the top of the base plate to the middle of the top plate placed smmetricall with 8 inches spacing, added circular weights of 6. lbf each, and γ.. To facilitate referencing the location of the added masses, a secondar coordinate sstem is defined with its origin set at the geometric center of the top plate. The coordinates of the added masses and the resulting locations of the center of mass of the superstructure and the relative eccentricities are tabulated in Table. Figure 6 illustrates the pre-determined positions of added masses at two inch intervals from to 9 inches awa from the geometric center of the plate along the -ais. model TABLE : Coordinates of Added Mass and Resulting Parameters Position reference coordinate of added masses [inches] coordinate of added masses [inches] E center of mass [inches] e relative eccentricit [%] α Alpha parameter

11 8 6 4 Positions of Added Masses and Corresponding Sstem Center of Mass Model period.4 sec, γ., e, ma 8.8% Location of added masses at Position 9. The -coordinate of the added masses is 9. from the center, resulting in a relative eccentricit, e 8.8%. -position in position in Locations of added masses influence the position of the center of mass of the whole sstem. When masses are at Position 9, the resulting center of mass of the sstem is at 5.96 from the center. Figure 6: Position of Added Masses 4.3 Model Construction Figure 7: The Completed Model Material and Equipment Pleiglas plates of size.375 top plate and.375 bottom plate; 9-pieces of.5 diameter Pleiglas solid rod 9.9 in length; make sure that both end surfaces form a 9 degrees angle with the ais of the rod;

12 Liquid Pleiglas welding agent; 3 small C-clamps; Various sizes of washers, nuts, and fasteners; Circular weights with eternal diameter of approimatel 4 inches; Electric scale; Punching bit or nail and hammer for marking top plate; Milling machine at machine shop; G drill bit, or drill bit slightl larger than.5 diameter for columns to fit through plate; 3/8 drill bit. Construction Procedure The Pleiglas model must be fabricated carefull and precisel to minimize an construction discrepanc e.g., accidental eccentricit that could effect significantl the eperimental results. Referring to Table, mark the eact positions of the added masses on a piece of paper of size. On the same paper, mark the eact positions of the columns using a different color pen. 3 Tape the paper securel on the top Pleiglas plate. 4 Hold the punching bit straight up and firml at one of the marks on the paper; hammer the punching bit lightl to make an indentation on the Pleiglas under the paper. Repeat for all the columns and added mass locations. 5 Clamp both pieces of the Pleiglas plates together, make sure that the plates are centered relative to one another and the edges are parallel. 6 Using the milling machine and a G drill bit, drill the holes for the nine columns through both plates. Before ou turn on the milling machine, lower the drill bit to line up the tip of the drill bit with the indentation. You might want to practice drilling on some scrap Pleiglas plates first!. 7 Unclamp the plates be sure that ou can re-align the holes on the top plate with the holes on the bottom plate later, ou ma want to make marks as a reminder. Using a 3/8 drill bit, drill the holes at the positions of the added masses through the TOP PLATE ONLY. 8 Drill two etra holes at coordinates., 9. and.,-9. for the non-eccentric case. 9 Flip the top plate over, place the nine Pleiglas rods into the column holes. Make sure that the ends flush with the bottom side of the plate. Using the welding agent, place a few drops around the column and in the hole of the plate. Allow for 3 minutes to dr. Weigh the assembl of top plate and columns. Record this value. After the whole assembl has dried, flip it over it should look like a table with 9 legs, insert the legs into the corresponding holes in the bottom plate. As before place a few drops of the welding agent around the column and in the hole of the plate. Using a combination of circular weights, washers, and fasteners, make up two weights of 6. lb each. 3 Allow for 4 hours to dr before subjecting the model to the static and dnamic tests described in the net section.

13 5. EXPERIMENTS 5. Stiffness Test The first test performed on the phsical model of a one-stor eccentric sstem is to eperimentall determine the lateral stiffness of the model which represents the lateral stiffness of the total base isolation sstem. According to the equation of static equilibrium, F ku, the lateral stiffness of the structure k is obtained b simpl measuring the applied lateral force F it takes to displace the structure through a displacement u in an horizontal direction. Eperimental Set-Up and Equipment The equipment required to perform the static test consists of Load cell or potentiometer calibrated; Voltmeter; Dial gage; String; C-clamps. Testing Procedure Securel clamp the base plate of the model to the edge of a large table to prevent sliding. Place the two added masses each consisting of several circular weights on the model at the zero eccentricit position. 3 Place the dial gage in the position indicated in Figure 8, note the orientation of the model in the figure which has as the vertical ais and as the horizontal ais. Make sure that the spring of the dial gage is in contact with the top plate with a zero reading. 4 Tie the string to the top plate through a hole at location., -9. and to the hook at the end of the load cell. 5 Slowl pull the load cell horizontall in the longitudinal direction; for each increment of.5 in measured b the dial gage, read the corresponding voltage on the voltmeter measuring the applied force. 6 To sta awa from the deformation capacit of the columns, do not displace the top plate b more than.5 in. 7 Convert the voltage reading from the voltmeter to force in pounds lbf using the calibration constant of the load cell, i.e., number of Volts per pound. Graph the Force in pounds lbf versus the displacement in inches in as discrete data points. 8 Find the slope of the best-fit line through the data points using linear least-squares fit to obtain the total lateral stiffness k in lbf/in. 3

14 Voltmeter m m string dial gage load cell Direction of PULL Figure 8: Stiffness Test Apparatus Questions: a What is the eperimental lateral stiffness k of the structure? b Determine the lateral stiffness k of the structure analticall using Equation and compare with the eperimental stiffness obtained under a. Are the eperimental and analtical values of k identical? If not, wh? Answers: a See sample plot below: 7 Eperimental Determination of the Structure Lateral Stiffness 6 Load lbfs Displacement in b Using the best-fit line through the data, the eperimental stiffness is approimatel.56 lbf/in, which is.47% larger than the theoretical value of.98 lbs/in given b Equation. 5. Free Vibration Test You will perform free vibration tests on the phsical model under two conditions: 4

15 When the sstem has no eccentricit e e. When the sstem is at various degrees of eccentricit. With the non-eccentric sstem, ou can determine the damping ratio, ζ. In addition, ou can calculate the structure parameter γ from the eperimentall identified uncoupled lateral natural frequenc, f π ω, and uncoupled torsional natural frequenc, π ω. L 5.. Non-Eccentric Sstem Eperimental-Setup and Equipment L The following equipment is required to perform this free vibration test commonl referred to as a snap-back test: 3 single-channel accelerometers; Signal amplifier units for accelerometers; Terminal board; Multiple-channels dnamic data acquisition board; Data acquisition software LabView or equivalent; Spectrum Analzer. f θ θ m Amplifiers A B C accelerometers A, B, C m Terminal Board Data Acquisition Board LabView Figure 9: Free Vibration Test Apparatus The various components of the equipment required to perform the free vibration tests are connected according to Figure 9. The added masses are fied at the zero transversal eccentricit position e. The three accelerometers are attached at the following locations: accelerometer A at -8.,., accelerometer B at 8.,., and accelerometer C at 9.,.. Make sure that all accelerometers are placed in the right orientation see arrow engraved on each accelerometer so that accelerometers A and B measure the voltage proportional to the sstem s acceleration in the -direction, while accelerometer C measures the acceleration response in the -direction. The voltage signals are amplified before the are recorded b a data acquisition software here LabView through a dnamic data acquisition board. 5

16 Testing Procedure Figure : SAP Model Test : Pull the top plate slowl until ou displace it approimatel.5 in the -direction. Pull as straight as possible. Start the data acquisition program so it is read to record the data. 3 Release the top plate to set the model under free longitudinal vibration. Record and save to 5 sec of free vibration response data. Be aware of the sampling rate in our data acquisition program. In the present case, a sampling rate of Hz is recommended, although an sampling rate above 5 Hz is acceptable. Test : Place our hands at opposing corners of the top plate, pull-and-push in a twisting motion; caution: do not twist too much so as to keep the model in its safe domain! Start the data acquisition program so it is read to record the data. 3 Release the top plate to set the model under free rotation response. Record and save to 5 sec of free vibration response data. Task: Convert the voltage measured b the accelerometers in Test and Test to acceleration in in/s. Note that ou will also have to reduce the three measurements to accelerations ü, ü, and ü θ along the three degrees of freedom at the center of mass see Figure 3. The reader is referred to Appendi for instruction on how to accomplish this. 3 Graph the results obtained from Test and Test in the form of acceleration versus time. Observe the acceleration record in the -direction for Test, and determine the uncoupled natural lateral frequenc, f L, of the model. 4 Observe the acceleration record in the -direction for Test, and determine the damping ratio ζ of the model for its uncoupled lateral mode of vibration using the logarithmic decrement of amplitude deca according to Equation 3 in Section

17 Observe the acceleration record in the -direction for Test, and determine the uncoupled natural torsional frequenc, f θ, of the model. Run the SAP model from the model.sdb or model.$k file in the SAP folder on the CD-ROM. This model and analsis simulate numericall the phsical eperiments performed under Test and Test. Click on the heading Displa --> Show Time Histor Traces. At the Time Histor Case pull down menu, select FREEVIB, plot the time histories of response quantities,, and z the denote the translational acceleration in the - and - direction, respectivel, and the rotational acceleration at Joint center of mass. Click on Displa to actuall displa these acceleration response histories. Change the Time Histor Case to FREEROT, plot response quantities,, and z. the denote the translational acceleration in the - and - direction, respectivel, and the rotational acceleration at Joint center of mass. Click on Displa to actuall displa these acceleration response histories. Obtain the theoretical uncoupled natural lateral frequenc, f L, and uncoupled torsional frequenc, f θ, from the above SAP acceleration response histories. Questions: a b c a b What is the value of the damping ratio in percent obtained eperimentall? Complete the following chart: Natural lateral frequenc, f L [Hz] Natural torsional frequenc, f θ [Hz] Natural lateral period, T L [s] Natural torsional period, T θ [s] γ ω θ /ω L SAP Eperiment Do ou see a general agreement between the SAP results and eperimental results? What are the possible sources of discrepanc? Answers: The damping ratio is approimatel 3%. Theoretical Eperimental Natural lateral frequenc, f L [Hz].5.7 Natural torsional frequenc, f θ [Hz] Natural lateral period, T L [s].4.37 Natural torsional period, T θ [s].33.9 γ ω θ /ω L..7 c Figures a and b are free vibration response histories obtained from the eperiment performed under Test and the corresponding SAP analsis, respectivel. Both the eperimental lateral and torsional frequencies are larger than the ones predicted b the theor SAP. Possible sources of errors are overestimation of the model s stiffness and/or underestimation of the model s mass. Also, note that in Figure a, acceleration in the - direction and about the z ais are present, while the should be zero according to the theor. 7

18 It is ver likel that the model is not perfectl non-eccentric due to construction imperfections leading to some accidental eccentricit. 6 FILTERED ACCELERATION VS. TIME LATERAL a 4 a a θ acceleration in/s time s Figure a: Free Vibration Record from Eperiment Test THEORETICAL ACCELERATION VS. TIME LATERAL 6 4 acceleration in/sec time s Figure b: Free Vibration Record from SAP Simulation 8

19 5... Eccentric Sstem In this section, ou will perform free vibration tests on the phsical model with various degrees of transversal eccentricit e and find the corresponding modal frequencies through the use of a spectrometer and frequenc domain analsis discrete Fourier transform of the eperimental data. The eperimental results will be compared to theoretical results obtained from SAP and Equations 6a-c. Eperimental Set-up and Equipment Use the same apparatus described for the non-eccentric case, with the addition of a Spectrum Analzer. A spectrum analzer performs on the fl a Fourier transform of the data and displas the resonant frequencies in the form of ver narrow spectral peaks of the structure on a monitor. Testing Procedure Secure the two weights on the top plate at the holes -inch awa from the center of the plate. Displace the model to approimatel.5 in the negative -direction. 3 Release the top plate to set the model under free vibration. Record and save to 5 sec of free vibration response data. 4 Using the MATLAB FFT program fft.m in the Matlab folder on the CD-ROM, perform a Fast Fourier Transform of the three channels of saved data. You ma need to make some slight modifications in the program. A graph of transformed functions called Fourier amplitude spectra will appear on the screen. 5 Identif the natural frequencies from the Fourier amplitude spectra of the eperimental data. 6 Disconnect accelerometer A from the terminal board and connect it to the Spectrum Analzer. Tap the top plate using our finger. The frequencies corresponding to the obvious narrow spectral peaks appearing on the monitor of the Spectrum Analzer are the natural frequencies determined eperimentall. You ma need to tap the top plate in different directions in order to obtain good readings from the Spectrum Analzer for all three natural modes of vibration. 7 Open the SAP model from the model.sdb file in the SAP folder on the CD-ROM and assign concentrated or nodal translational masses to the locations defining the first level of eccentricit. You ma find the locations easil b clicking on Select Groups Mass; ou will see two nodes being selected that correspond to coordinates 7.46,. and -7.46,.. Remember to delete the masses originall assigned to the zero eccentricit position. 8 Run the modified SAP model, displa the first three natural mode shapes of vibration and the corresponding modal frequencies. click on Displa-> Displa Mode Shapes. Use the arrow kes at the lower right hand corner of the screen to show the net mode. 9 Observe the first three natural modes of vibration using the dnamic animation option b clicking the start button at the lower right hand corner of the screen. 9

20 Using Equations 6a-c, calculate the theoretical natural frequencies for the first three natural modes of vibration. Repeat steps through for eccentric mass positions 3, 5, 7, and 9. Complete the following chart: Position Name Relative Eccentricit [%] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Theoretical Equation 6 SAP FFT Spectrometer Questions: a b c a Compare the results obtained from each of the method used. Do the show a general agreement? Wh are we onl concerned about the first three modes of vibration? Wh does the frequenc of the second mode of vibration remain the same at different levels of eccentricit? Answers: Reasonable values are: Position Name Relative Eccentricit [%] Theoretical Equation 6 SAP FFT Spectrometer Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz] Mode [Hz] Mode [Hz] Mode3 [Hz]

21 Mode [Hz] Mode [Hz] Mode3 [Hz] b c Our idealized model is assumed to have onl three degrees of freedom, since the capture the first three natural modes of vibration of the actual distributed mass sstem which contribute to most of the dnamic response of the sstem to the tpe of dnamic ecitation imposed in the above eperiments. Lateral-torsional coupling occurs onl between the displacement in the longitudinal - direction and the rotation about the z-ais. Mode is purel translational in nature and therefore the second mode frequenc is controlled onl b the total translational mass m and the total lateral stiffness k. 5.3 Shaking Table Test In this eperiment, ou will ecite the phsical building model on the shaking table with a scaled down version of the El Centro 94 earthquake record. You will observe how the model responds to seismic ecitation under two scenarios: When the sstem is non-eccentric e e. When the sstem is eccentric. Eperimental Set-Up and Equipment The equipment required to perform the shaking table test consists of Shaking table; Four single channel accelerometers; Signal amplifier units for accelerometers; Terminal board; Multiple-channels dnamic data acquisition board; Data acquisition software LabView or equivalent; Software: SAP and MATLAB. The test apparatus is similar to that for the Free Vibration Test with the addition of a fourth accelerometer unit, accelerometer D, to measure the shaking table acceleration as shown in Figure. You will need to make some small modifications in the data acquisition program in order to acquire the data from accelerometer D as well. Again, be sure to place Accelerometer D in the correct orientation so as to measure the table acceleration in the positive -direction.

22 m A Accelerometers A, B, C, and D m B C Amplifier Terminal Board Data Acquisition Board Shaking Table D LabView Figure : Shaking Table Test Apparatus Testing Procedure Clamp the Pleiglas model securel to the shaking table. Create a non-eccentric sstem b placing the two added masses at the zero transversal eccentricit positions as shown in Figure. 3 Cop the scaled El Centro 94 earthquake record from the CD-ROM file ELC4.prn in SAP folder. Use it as the input function for the shaking table test. The earthquake record is an acceleration record with 5 data points and a constant time step of.4 second. 4 Operate the shaking table to ecite the phsical model with the scaled El Centro earthquake record. 5 Record and save all measurements from all four accelerometers. 6 Convert the raw data in Volts from accelerometers A, B, C, and D to acceleration in in/s using the accelerometer calibration constant acceleration per Volt. Create a tet file *.tt with time s in the first column and the corresponding measured shaking table acceleration in/s, from accelerometer D, in the second column. 7 Reduce the measurements from accelerometers A, B, and C to the three components of acceleration, ü, ü, and ü θ, at the center of mass of the sstem according to the procedure outlined in Appendi. Plot the measured acceleration response histories ü t, ü t, and ü θ t, on three separate graphs versus time in seconds. 8 Use the above tet file as the Time Histor Function for the SAP model of the eperiment and perform the SAP simulation for the same level of eccentricit as in the eperiment. It is important that ou do not use the scaled El Centro 94 earthquake record provided in the CD-ROM as the SAP input function, since it is unlikel that the shaking table is capable of reproducing eactl this earthquake record. Save the response histories ü t, ü t, and ü θ t simulated using SAP.

23 9 Create a video of the building response simulated in SAP or open a SAP video file in the CD-ROM in the PowerPoint and Videos folder. Repeat steps 9 for the phsical model with two different levels of eccentricit e.g., e e 7% and 9%. Obtain the ground record for a second earthquake, such as Coalinga 989 from the 989 Loma Prieta earthquake. Scale the record using the time and length scaling factors mentioned in section 4.. Repeat steps - for the scaled Coalinga 989 earthquake record. Questions: a b c d e f a b c B comparing the response of the non-eccentric sstem to that of the eccentric sstem with two levels of eccentricit, describe the effects of eccentricit of the center of mass relative to the center of stiffness. Do ou observe an response in the transversal or - direction? If es, how does the amplitude of this response compare to that in the longitudinal or - direction? Is the response in the transversal or - direction epected from the theor? Refer to Equations and 4. How does the response simulated in SAP compare with the actual response? Plot each of the response histories ü t, ü t, and ü θ t simulated using SAP and observed eperimentall on separate graphs using the same scaling of aes and same grid for better comparison. What are the possible sources of discrepanc between analtical/numerical response predictions and eperimental results? Answer questions a through e for the numerical and eperimental data obtained using the scaled Coalinga 989 earthquake record. Answers: The rotational response of the sstem increases with increasing eccentricit. For zero eccentricit, there should be theoreticall no rotational response. Due to accidental transversal eccentricit in the phsical model introduced during construction and instrumentation of the model and due to imperfect positioning of the model on the shaking table, the model will ehibit some transversal response. However, the amplitude of the transversal response is small compared to that of the longitudinal response. Equations and 4, with p t p θ t, ield a zero transversal displacement and therefore acceleration response. 3

24 d See Figures a and b for sample graphs. Also see video files provided in the PowerPoint and Videos folder on the CD-ROM for simulated response using SAP. Note that the units of the angular acceleration response ü θ t are in radian/s as shown on the ais label on the right. LabvVIEW El Centro magnitude /: 7.3% Relative Eccentricit:Table Measured Acceleration Acceleration in/sec Time [sec] Acceleration in/sec LabVIEW Acceleration in u, u, u θ 5 Acceleration in u Acceleration in u 5 Acceleration in uθθ Time [sec] Acceleration irad/sec Figure a: Acceleration Records from Eperiment for Scaled El Centro 94 Earthquake - - c SAP El Centro magnitude /: 7.3% relative eccentricit: measured table acceleration Acceleration in/sec Time [sec] celeration in/sec SAP Acceleration in u, u, uθ Acceleration in u Acceleration in u Acceleration in uθ ccele ration rad/sec A - A Time [sec] Figure b: Acceleration Records from SAP for scaled El Centro Earthquake 4

25 e f Possible sources of discrepanc are: accidental eccentricit due to imperfections in model construction and instrumentation of the model; imperfect mathematical modeling due to idealized assumptions about geometr; boundar conditions, and material behavior and properties; 3 accelerometer sensor noise, digitization error in data acquisition, and filtering of data. The response of the model is ver much dependent on the input earthquake record. The model s response to the scaled Coalinga 989 earthquake record has a different pattern than that to the scaled El Centro 94 earthquake record. 6. SAP MODEL The numerical model, created using the finite element analsis software SAP, is developed with modeling assumptions and parameters representing as closel as possible the real phsical model. Most of the measurable parameters used in the SAP model, such as Young s modulus and mass/weight densit of Pleiglas, are all determined eperimentall. However, there is alwas some deviation between the real phsical model and its numerical counterpart. The reader is referred to the SAP model in the CD-ROM file model.sdb in the SAP folder. 6. Geometr and Joint Constraints The top plate with overall dimensions of.375 is modeled using quadrilateral shell elements. For the areas where the added masses are located, shell elements are meshed into a reduced size of to locate eactl the added masses. Column frame sections have a height of 9. and a cross sectional diameter of.5. The bottoms of the columns are defined as fied supports. 6. Material A new material, labeled as Plei, must be defined in SAP. Plei is used for both the top plate and the columns. The properties of Plei were determined through material uni-aial testing and information from the manufacturer. The following steps are followed to define the Plei material: Define Material: Plei Material Name: Plei Tpe of Material: Isotropic Mass per unit Volume:.E -4 [lbsec /in 4 ] Weight per unit Volume:.46 [lb/in 3 ] Modulus of Elasticit: 4, [psi] Poisson s Ratio:.35 Coefficient of Thermal Epansion: 3.3 Shear Modulus: 56,96.3 [psi] 5

26 6.3 Dnamic Analsis The model is analzed using Ritz-Vector Analsis with modes. The modal damping ratio identified eperimentall in Section 5. is used. 6

27 Appendi Reducing Accelerometer Measurements at the Center of Mass of the Sstem Referring to Figure below, in the general case, the center of mass c.m. and the center of stiffness c.s. of the sstem do not coincide. When the plate is displaced, the displacement at point A, B, and C are measured. The three accelerometers at points A, B, and C measure the acceleration response of the sstem along degrees of freedom s, s, and s 3, respectivel. Using simple linear kinematics, these three degrees of freedom can be reduced into the three degrees u, u, and u θ at the center of mass of the sstem. The distance between point A and the center of mass is denoted b d, while d represents the distance between points B and the center of mass. d d S u S A c.s. c.m. u θ u B C S 3 Figure. Reduction of Measurement DOF s s, s, and s 3 into Analsis DOF s u, u, and u θ B simple geometr, it follows that s u d u θ 4 s u + d u θ 5 s 3 u 6 Solving the above equations for u, u, and u θ ields u s 3 7 u θ s s / d + d 8 u s + d u θ 9 7

28 References: Chopra, A. K., Dnamics of Structures, nd Ed., Prentice Hall,. Trombetti, T. L. and Conte, J. P., New Insight Into and Simplified Approach to Analsis of Laterall-Torsionall Coupled One-Stor Sstems: Part I Formulation, submitted for possible publication in the Journal of Sound and Vibration, August. Conte, J. P. and Trombetti, T. L., New Insight Into and Simplified Approach to Analsis of Laterall-Torsionall Coupled One-Stor Sstems: Part II Numerical and Eperimental Verifications, submitted for possible publication in the Journal of Sound and Vibration, August. 8