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Vibration of lightweight wooden floors : experimental and analytical evaluation Neumann, Greg A. 1992

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VIBRATION OF LIGHTWEIGHT WOODEN FLOORS:EXPERIMENTAL AND ANALYTICAL EVALUATIONbyGREG A. NEUMANNB.A.Sc., The University of British Columbia, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Civil Engineering)We accept this thesis as conformingTHE UNIVERSITY OF BRITISH COLUMBIAOCTOBER 1992©Greg A. Neumann, 1992In presenting this thesis in partial fuffihiment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make it freelyavailable for reference and study. I further agree that permission for extensive copyingof this thesis for scholarly purposes may be granted by the head of my department or byhis or her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of ,‘Li;/The University of British ColumbiaVancouver, CanadaDate /5 /9•AbstractAn experimental study of two lightweight wooden floors’ dynamic response was performed. The objective of this study was to evaluate the applicability of the computerprograms, NAFFAP and DYFAP, for the case of lightweight wooden floor vibration. Theprogram NAFFAP solves for the floor system’s natural frequencies and mode shapeswhile DYFAP performs a time domain integration of the equations of motion in responseto a specified loading on the floor. DYFAP also has the capacity to model the coupledresponse of oscillators upon the floor system. These programs employ a T—beam finitestrip analysis. The test data of the floors’ dynamic response to various impacts werecompared with the programs’ simulations. One floor used 2x8 sawn lumber joists whilethe other used composite wood I—Joists. The transient dynamic response of the two floorsto three types of excitation were recorded. The three types of excitation were created bya hammer tap, releasing a sand filled bag and a standard human heel drop. The modelling of a human occupant with a single degree of freedom oscillator, the InternationalStandards Organization’s two degree of freedom oscillator model and a forcing functionwere investigated with DYFAP. The oscillators were composed of lumped masses, springsand viscous dashpots.The program NAFFAP was sufficiently successful in predicting the floors’ naturalfrequencies. The floor parameters that were used with NAFFAP were then used withDYFAP to produce displacement time histories that compared well with the test data.DYFAP’s modelling of occupants with simple oscillators rather than a forcing functionproved to be appropriate.IITable of ContentsAbstract iiList of Tables viList of Figures viiAcknowledgement xi1 Introduction 11.1 Human Perception of Floor Vibrations 11.2 Objective and Scope 72 Material Properties 92.1 Joists 92.2 Plywood 132.3 Sheathing—joist Connections 153 Experimental Setup 233.1 Floor Parameter Study 233.2 Floor Design 263.3 The Floor’s Support System 273.4 Data Acquisition 304 Testing 334.1 Hammer Impact Tests 331114.2 Sandbag Impact Tests 344.3 Heel Drop Tests 395 Dynamic Response of Test Floors 425.1 Evaluation of Damping Ratios 425.2 Identifying the Floors’ Natural Frequencies 466 NAFFAP 566.1 Introduction 566.2 Data File 566.3 NAFFAP’s Results for the Test Floors 597 DYFAP 637.1 Introduction 637.2 Modelling of a Hammer Impulse 637.3 The Bagdrop Impulse 657.4 Comparison of Hammer and Bagdrop Tests with DYFAP Results 657.5 Excitation by an Oscillator 707.5.1 DYFAP Results using the ISO Model 727.5.2 Modelling a Heel Drop with a Single Mass Oscillator 787.5.3 DYFAP Results using the Calibrated Single DOF Model 807.5.4 Comparison of Results for the Case of Two Occupants 848 Summary and Conclusion 908.1 Summary 908.2 Concluding Remarks 908.3 Further Areas of Study 92ivBibliography 93VList of Tables2.1 Joist E—moduli obtained from 4—point bending tests 112.2 Joist mass density at time of floor construction 122.3 Assumptions required by the composite plate property equations 152.4 Moduli values obtained from the composite plate property equations. 152.5 2x8 joist floor connection’s test results 162.6 I—Joist floor connection’s test results 202.7 Material properties for 3M wood adhesive 215.1 A selection of proportional damping constants for the I—Joist floor . . 455.2 A selection of proportional damping constants for the 2x8 joist floor . 455.3 Natural frequencies for the test floors 506.1 Percentage difference between 2x8 joist floor tests’ and NAFFAP frequencies 596.2 NAFFAP results compared with frequencies identified from floor tests . . 607.1 Parameter values for ISO human model 737.2 Grid of model parameters for oscillator calibration 797.3 Frequency weighting factors used for calculation of RMS acceleration 88viList of Figures1.1 Human subjective tolerance to whole body sinusoidal vertical vibrations,standing position (Grether, 1971) 41.2 Human response to various RMS levels of floor vibration induced by heeldrop tests (Chui, 1986) 41.3 Human response to steady—state vibration, vibration vertical (Wiss andParmelee, 1974) 51.4 Curves of mathematical model for evaluation of transient vertical vibrations (Wiss and Parmelee, 1974) 51.5 Annoyance criteria for floor vibration: residences, offices, and schoolrooms(Allen and Rainer, 1976) 61.6 Human response to transient vertical vibration induced by dropping a 701bsteel weight onto the floor centroid (Polensek,1970) 62.1 Joist cross—sections with coordinate systems 102.2 Test setup for testing the joist/plywood connection in shear; front view 172.3 Test setup for testing the joist/plywood connection in shear; side view 182.4 Cut away samples of typical joist to plywood glue connection 192.5 Load deflection curve for a drawn polyamide fibre ( Pechhold, 1973 ) 223.1 Connection details at joist ends for the I—Joist joist floor 283.2 A view of the floor framing and the steel support system 293.3 Anchoring system for the W—shape frame 313.4 Real spectral “Energy” density of a noise trace 32vii4.1 Impact and recording locations used on the 2x8 joist floor4.2 Impact and recording locations used on the I—Joist floor4.3 The hammer impact test4.4 The bagdrop impact test4.5 A closeup of the bag, load cell and accelerometers ( Photograph shows adrop height approximately twice that of the 27mm test height)4.6 The starting position of a typical heel drop test with an observer standingon the same joist as the tester4.7 Averaged plot of force versus time for heel drop impact (Folz and Foschi,1991)5.1 The decay curve from an acceleration trace filtered to isolate 30.4Hz, IJH6-2 445.2 Fast Fourier Transform of a six second acceleration record 475.3 Fast Fourier Transform of a bounded acceleration record, time bounds of4849SLH5-2 49SLH6-1 51SLH6-2 51SLH7-l 52SLH7-2 52IJH2-1 53IJH2-2 54IJH4-l 54IJH6-1 55IJH8-l 55343536373840415.35 — 9.45 seconds5.4 Fourier spectrum for hammer test, SLH5-l5. for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,spectrum for hammer test,VII’6.1 A T—beam finite strip 576.2 NAFFAP mode shapes for the 2x8 joist floor 626.3 NAFFAP mode shapes for the I—Joist floor 627.1 Modelling of a hammer impulse 647.2 Typical recorded bagdrop time history compared with that used with DYFAP 667.3 Comparison of the 2x8 floor’s test and DYFAP’s response to a hammerimpact 677.4 Comparison of the I—Joist floor’s test and DYFAP’s response to a hammerimpact 687.5 Fourier spectrum of the 2x8 floor’s acceleration response to a bagdrop impact 697.6 Fourier spectrum of the I—Joist floor’s acceleration response to a bagdropimpact 697.7 The 2x8 floor’s acceleration response to a bagdrop impact 707.8 The I—Joist floor’s acceleration response to a bagdrop impact 717.9 The 2x8 floor’s displacement response to a bagdrop impact 717.10 The I—Joist floor’s displacement response to a bagdrop impact 727.11 ISO, idealized lumped parameter vibratory human model 737.12 Fourier spectrum of the 2x8 floor’s test response to a heel drop 747.13 Fourier spectrum of the I—Joist floor’s test response to a heel drop . . . 757.14 Comparison of the I—Joist floor’s response to a heel drop with DYFAP’sresults using the ISO model or a forcing function, IJF6-2 767.15 Comparison of the 2x8 floor’s response to a heel drop with DYFAP’s resultsusing the ISO model or a forcing function, SLF6-2 777.16 Displacement time histories for composite floor (2% damping) 77ix7.17 Single degree of freedom model 787.18 Influence of varying the stiffness of the oscillator 81.7.19 Influence of varying the viscous damping of the oscillator 827.20 Comparison of the 2x8 floor’s response to a heel drop with DYFAP’s resultsusing the 1 DOF model or a forcing function, SLF6-1 837.21 Comparison of the I—Joist floor’s response to a heel drop with DYFAP’sresults using the 1 DOF model or a forcing function, IJF6-2 837.22 The 2x8 floor’s response two joists away from the heel drop, SLF6-1 . . . 847.23 The I—Joist floor’s response two joists away from the heel drop, IJF6-1 . 857.24 Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on the same joist as the heel dropper,2x8 floor 867.25 Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on a joist adjacent to the heel dropper,2x8 floor 867.26 Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on the same joist as the heel dropper,I—Joist floor 877.27 Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on a joist adjacent to the heel dropper,I—Joist floor 877.28 Series #4 test data, a:SLN6, b:SLS6, c:IJN6, d:IJS6, plotted on Wiss andParmalee’s mathematical model 897.29 Series #4 test data, a:SLN6, b:SLS6, c:IJN6, d:IJS6, plotted on Chui’sRMS scale 89xAcknowledgementI wish to express gratitude to my supervisor Ricardo 0. Foschi and to Bryan Folz fortheir advice and assistance throughout this project. Thanks to the Council of ForestIndustries of B.C. for their donation of building material. The continuous support offamily and friends is appreciated for it was important for the successful completion ofthis thesis.xiChapter 1IntroductionExcessive vibrations of lightweight floor systems have become a common design problem now that higher strength materials are becoming available. They allow designs toemploy lighter and longer spanning structural members. Lightweight wooden floors constructed for the North American residential market have received an increasing numberof consumer complaints. This has prompted a number of studies and has resulted inthe NBCC199O adopting a new, more stringent vibration criteria. Allowable floor joistspans have been reduced in an effort to minimize the number of floors that have naturalfrequencies which fall within or near the human’s primary natural frequencies that liebetween 4Hz and 20Hz.1.1 Human Perception of Floor VibrationsMuch work has been done to identify human vibration threshhold levels. Relationshipsbetween acceptable vibrational performance and acceleration, deflection, velocity, or frequency have been explored. A subjective testing procedure has generally been used.Subjects have been asked to evaluate floor vibrations via various scales. Most humansenses are believed to be of a logarithmic rather than linear nature, thus Ohisson believesthat vibration perception ought to behave similarly [Ohisson, 1982]. Y.H. Chui listed asmall literature review concerning this area [Chui, 1986]. A short summary of his listingsfollows:1Chapter 1. Introduction 2Onysko and Bellosillo (1978) made a comprehensive literature review;Russel (1954) and Hanson (1960) correlated discomfort level with deflection under concentrated load;Polensek (1970) studied human response to impulsive vibrations in terms of their frequency and amplitude;Lenzen (1962) had human subjects evaluate floor vibrations using the Reiher and Meister Scale (1946);Leuzen (1962), Polensek (1975), Rainer and Pernica (1981) considered the added damping capacity to the floor system provided by a human occupant;Lenzen (1966) established that damping strongly influences perception of transient vibrations;Shaver (1976) attempted to correlate acceleration and displacement with human responseParks (1962), Wiss and Parmalee (1974) studied the response of subjects to transientvibration tests, and their evaluation of the vibrations via a four or five point scalerespectively.Many threshhold and performance curves have been proposed by a number of theabove authors. Grether performed a study to determine to what degree vibration wouldaffect human performance [Grether, 1971]. The results indicate that humans are sensitive to frequencies in the range of 4— 8Hz. This conclusion was supported by exposure durations from short time (impulse/transient) to continuous vibrations. The curvesof equal tolerance of Fig. 1.1 show a marked dip over this range. The InternationalChapter 1. Introduction 3Standards Organization (ISO), noted and recognized this and recommended that frequencies be weighted to reflect this sensitivity prior to analysis. Y.H. Chui utilized theRMS of the acceleration to develop the graph of tolerable levels shown in Fig. 1.2. Hefound that a RMS acceleration of 0.375m/s2 resulting from a heel drop impact testdefined a boundary between acceptable and unacceptable floor vibration for a typicalperson [Chui, 1986]. The Reiher and Meister scale for steady—state vibration is shownin Fig. 1.3. Wiss and Parmalee worked with transient vibrations. Shown in Fig. 1.4 is aplot of their mathematical model that they derived from their transient vibration studies[Wiss and Parmelee, 1974]. Allen and Rainer performed tests on long span floors. Shownin Fig. 1.5 is their plot of peak acceleration versus frequency with damping considered.The lines mark acceptability threshholds [Allen and Rainer, 1976]. Shown in Fig. 1.6 isPolensek’s plot, which is similar to that of Reiher and Meister’s shown in Fig. 1.3. Hewas interested in the maximum peak—to—peak displacement of the floor response. This ismeasured by taking the absolute value of the displacement between a positive peak andthe following negative peak [Polensek, 1970].Chapter 1. Introduction 4/0— i i ‘ I I I I I I J ‘0 2 4 6 8 10 12 14 16 18 20Frequency (Hz)Figure 1.1: Human subjective tolerance to whole body sinusoidal vertical vibrations,standing position (Grether, 1971)0.800N0,SDISTURBING0.500 Unacceptable to all occupants0PERCEPTIBLE0.375 Unacceptable to most occupantsQ)SLUGHTLY PERCEPTIBLE0.200 Acceptable to many occupantsCl)S NOT PERCEPTIBLE0.100 Acceptable to nearly all occupantsFigure 1.2: Human response to various RMS levels of floor vibration induced by heeldrop tests (Chui, 1986)Chapter 1. Introduction 50.1 -DISFURBINGC:Frequency (cps)Figure 1.3: Human response to steady—state vibration, vibration vertical (Wiss andParmelee, 1974)1—R=20.01R Classification—1 Imperceptible2 Barely perceptible3 Distinctly perceptible4 Strongly perceptible5 Severe0.001—i i i , i1 4 I I0.01 0.1Damping, Re:Critical DampingFigure 1.4: Curves of mathematical model for evaluation of transient vertical vibrations(Wiss and Parmelee, 1974)Chapter 1. Introduction 6100 -DAMPING RATIO 12%C.Y 1ODAMPING RATIO 6%HEEL DROP IMPACTDAMPING RATIO 3%1CONTINUOUS WBRATIONI I I I‘‘I1 10Frequency (Hz)Figure 1.5: Annoyance criteria for floor vibration: residences, offices, and schoolrooms(Allen and Rainer, 1976)cv0I.2-DISTURBINGPERCEPTIBLE0— I I I I I I I I I I I I I I10 11 12 13 14 15 16 17 18 19 20 21Frequency (Hz)Figure 1.6: Human response to transient vertical vibration induced by dropping a 701bsteel weight onto the floor centroid (Polensek,1970)Chapter 1. Introduction 71.2 Objective and ScopeAs previously mentioned, much work has been done to address the floor vibration problem from a subjective testing point of view. From these studies several vibration acceptance criteria have been produced. In an effort to analytically model a floor’s dynamicresponse, two computer programs, ( NAFFAP and DYFAP ), have been developed atthe University of British Columbia [Filiatrault and Folz, 1989] [Filiatrault et al, 1990][Folz and Foschi, 1991]. The main objective of the investigation reported in this thesiswas to study experimentally the applicability of the programs NAFFAP and DYFAP forthe case of lightweight wooden floor vibration.Two wooden floors were constructed as per the NBC199O guidelines. One used traditional sawn lumber joists while the other used composite wood I—Joists. The I—joistis becoming particularly popular due to its ability to span greater distances with lessmaterial than available sawn lumber products.Currently, the data from in—situ testing of floors takes the form of acceleration timehistories. The acceleration time history can be processed and integrated to producevelocity and displacement time histories and frequency spectra. Three types of impacttesting were performed on each of the floors: tapping the floors with a common hammer,releasing a sand bag from a predetermined height and a human heel drop.The three forms of floor excitation were modelled by DYFAP. The hammer impulsewas estimated and entered as a discretized forcing function. The bagdrop impulse wasrecorded and therefore was available for discretization. The human heel drop, however,posed the most difficult problem for DYFAP. Previous research by Folz and Foschi showedthat for the case where the occupants’ mass is a significant percentage of the floor system’smass, an assumed forcing function was inadequate. An oscillator model of the humanwas placed upon the floor and was given an initial velocity to simulate the heel dropChapter 1. Introduction 8action.The natural frequencies of the floors were determined experimentally. NAFFAP’spredicted frequencies and DYFAP ‘s time histories were then compared with the experimental data. In the time domain, such features as peak acceleration, damping and peakdisplacement were used to evaluate the performance of DYFAP.Chapter 2Material PropertiesThe primary components of the floor systems, the joists and sheathing, were testedbefore the floors were constructed. The geometrical and mechanical properties that arerequired by NAFFAP and DYFAP were evaluated. Each joist was tested in flexure,weighed and its physical dimensions measured. Sixteen small plywood samples, assumedto be representative of the plywood stock used for the two floors, were similarly tested.2.1 JoistsOne floor was built with ten SPF No.2 2x8 joists spaced at 400mm while the other usednine TTS (Jager Industries) MSR2100 wood I—Joists spaced at 600mm. The programsrequired five geometrical properties and two moduli for each joist.The joists were tested in flexure under 4—point loading. They were cut to length priorto testing so that the test and floor span would be the same. Therefore the ten 2x8 joistsspanned 3100mm and the nine Jager I—Joists spanned 4200mm. The loading span wasset at 600mm for both types of joists. The Jager and 2x8 joists were loaded to 400lbs(1780N) and 2401bs ( 1070N ) respectively. The defiections at these loads were recorded.The bending stiffness was then derived from the standard beam formula of Eq. 2.1.El— Pa(3L2 — 4a2)2 124 (.)The elastic modulus for each of the 2x8 joists was calculated directly from Eq. 2.1.The inertia and the torsional constant were calculated from the physical dimensions of9Chapter 2. Material Properties 10Figure 2.1: Joist cross—sections with coordinate systemsthe joists’ cross—section. The shear modulus was derived from the elastic modulus. Itwas assumed that the ratio of elastic to shear modulus was 17. Shown in Table 2.1 arethe 2x8 joists’ elastic moduli that were used in the programs’ data files.The elastic moduli for the I—Joists were not immediately available from Eq. 2.1. TheI—Joists are composite structures. Their bending stiffness can be expressed as(EI) =2EFIFY + EIy (2.2)There are two unknowns in Eq. 2.2. The elastic moduli for the web (Em), and flanges(EF), are unknown. The modulus for the web material was assumed. A value of 9653Mpa (1.4x106 psi) was taken for the 3/8” OSB web. A poor assumption ought notto have a large effect since the flange inertia (IFy) is substantially larger than the webinertia (Iwy). Since (EI) was measured, (EF) was calculated from Eq. 2.2. Equation 2.3was then used to derive a composite elastic modulus for the z—direction. The programsSPF 2x8Grade No.2x61/2[Jager TTS Wood I—Joistz SPF 2x4 3/8” OSB21 OOIvISRzChapter 2. Material Properties 11Table 2.1: Joist E—moduli obtained from 4—point bending testsE—Modulus “MFa)Joist 2x8 I—Joist1 14982 180162 19271 1.57343 15527 158104 16685 138865 17582 161896 14879 168927 14162 172718 16789 170029 15934 1494110 9425 —required that the elastic moduli for the y—direction and z—direction be equal. Therefore,from Eq. 2.3, the elastic moduli for the I—Joist became simply EF.(El)2 =2EFIFZ (2.3)Contribution from the web to Eq. 2.3 was neglected because the flange inertia wasvery much larger than the web’s. Also shown in Table 2.1 are the I—Joists’ elastic modulithat were used in the programs’ data files. The composite moment of inertia that theprograms require was then calculated from Eq. 2.4, using E = EF.1 = (EI)JE (2.4)The contribution from the web to the composite torsional constant was neglected sincethe flange’s constants were so much larger. The shear modulus was found by making thesame assumption as for the 2x8 joists. The ratio of the composite shear and elasticmoduli was assumed to be 17.Once the joist stiffnesses were determined, the mass density ( mass/volume ) of theChapter 2. Material Properties 12Table 2.2: Joist mass density at time of floor constructionMass Density (mNs2/mm4,10—7)Joist # 2x8 I—Joist1 4.27 5.422 5.31 5.763 4.28 5.394 4.62 5.315 5.01 5.106 5.03 5.737 4.50 5.238 4.71 5.369 5.36 5.4610 3.83 —joists were derived. An estimate of their volume via dimensions and their mass wasrequired. Moisture content readings for the 2x8 joists ranged from 12% to 19%. The 2x8joist floor was then constructed immediately but the mass of the Jager joists dropped2—3% from the time of flexure testing to floor construction. At the time of construction,the 2x8 and Jager joist floors weighed approximately 4901bs ( 2180N ) and 8201bs( 3650N ) respectively.Listed below are the joist parameters required by the programs.RIY ... .Moment of inertia of joist around Y-axis.RIZ ....Moment of inertia of joist around Z-axis.RIT .. ..Torsional inertia of joist.AJ ....Cross-sectional area of joist.HCJ . .. .Distance from centroidal axis of joist to bottom of cover.Chapter 2. Material Properties 13EJ . .. .Modulus of elasticity of joist.GJ .. . .Shear modulus of joist.RHOJ . ...Mass density of joist.2.2 PlywoodThe programs require a number of parameters to describe the plate action of the plywood sheathing. Eight specimens in each orientation, span parallel and perpendicularto the surface grain, were tested under 3-point bending. Each specimen had nominaldimensions of 275x1 188mm. Only the mass density and bending stiffnesses parallel andperpendicular to the surface grain were determined experimentally. The mass density ofthe plywood was determined in the same manner as for the joists. All other parameterswere derived from the composite plate property equations 2.5—2.12.EI 2 EItRKX = 22 11 + E 11 22 (2.5)1=1 — 1/121/21) i=1 — V121’21)E1 2 EtI2RKY = 11 11 + 22 22 (2.6)i=1 — 1221) i=1 — V1221)RKV = +E2Iv1 (2.7)i=1 (1 — 1/121/21) i=1 (1 — 1/121/21)RKG=E GI1 +2(2.8)‘ E t 2 E tDY = 11 11 + 22 22 (2.9)il— 1221) (1—11121/21)Chapter 2. Material Properties 142DX 22 11 + 22 (2.10):i — V12V21J j=1 (1 — -‘l2’2l)DG=Gd (2.11)DV =E1v2t+E2v1t (2.12)i=1 (1 —v1221) i=1 (1 v1221)where:RKX ....Bending stiffness of the cover in the direction parallel to the joists. For plywood,this is usually perpendicular to the face grain.RKY ... .Bending stiffness of the cover in the direction perpendicular to the joists. Forplywood, this is usually parallel to the face grain.RKV,RKG ....Parameters for plate bending related to Poisson’s effect and torsion respectively.DX . .. .Axial (in-plane) stiffness of the cover in the direction parallel to the joists.DY ....Axial (in-plane) stiffness of the cover in the direction perpendicular to the joists.DV,DG .. . .Axial (in-plane) stiffness of the cover related to Poisson’s effect and in-planeshear respectively.In addition, the programs require the following parameters:TCOV .. ..Thickness of the cover.RHOC .. . .Mass density of the cover.Chapter 2. Material Properties 15Table 2.3: Assumptions required by the composite plate property equationsPoisson’s ratios 0.02,0.4E/G 17Epar/Eperp 20Table 2.4: Moduli values obtained from the composite plate property equations(MPa)Epar (exterior) 13442Eperp (exterior) 672Epar (interior) 10709Eperp (interior) 535G (exterior) 791G (interior) 630A number of assumptions were required in order to apply Eqs. 2.5— 2.12. Listed inTable 2.3 are the values used for Poisson’s ratios, elastic to shear modulus ratio and theE—moduli ratio parallel and perpendicular to the grain of a ply. It was also assumed thatthe interior and exterior plys may be of different species. Using Eq. 2.5 and Eq. 2.6, andthe above assumptions the E—moduli parallel to a ply’s grain were derived. Table 2.4provides a summary of the various moduli values derived for the plywood.2.3 Sheathing—joist ConnectionsGlue and nails were used for the sheathing to joist connection. The glue, an elastomericwood adhesive, caused the connection to exhibit a high strength as well as a high stiffness.Shear tests were conducted using 200mm long sections of the floors’ connections. Thedimensions of the available testing apparatus restricted the choice for specimen lengthChapter 2. Material Properties 16Table 2.5: 2x8 joist floor connection’s test results2x8 Joistglue 1 nail + gluestiffness (N/mm) peak load (N) stiffness (N/mm) peak load (N)3974 7500 5000 70005063 5500 5047 70007317 9300 7867 90004070 4000 8824 9500—— 6410 6000to 200mm. An MTS 810 testing machine was used for these tests. A frontal and sideview of a specimen clamped in the testing apparatus are shown in Fig. 2.2 and Fig. 2.3respectively. The lower table was raised thus causing the joist to shear away from theplywood.Specimens with and without a nail were tested. The tests were displacement controlled. The displacement rate was set at 0.1mm/sec. Listed in Table 2.5 and Table 2.6are the connection test results. Over the recorded strain range, the stiffness and peakloads obtained from the tests showed no evidence of whether a nail was present or not.The stiffness distributions for the two types of tests are significantly nested. The stiffness and peak loads from the tests were 3 — 4 times higher than that usually expectedfrom typical nailed connections. The connection stiffness was quite variable. An averagestiffness of 5706N/mm was calculated from all of the tests with the high and low valuesremoved. They likely depended upon the glue thickness and contact area. The threespecimens shown in Fig. 2.4 illustrate how the glue connection varied in thickness andbond width. At failure, only the residual strength of the nail remained, the glue—woodbond had been entirely destroyed.The mechanical properties of the glued connections during low amplitude vibrationsChapter 2. Material Properties 17Figure 2.2: Test setup for testing the joist/plywood connection in shear; front viewChapter 2. Material Properties 18Figure 2.3: Test setup for testing the joist/plywood connection in shear; side viewC0C)C0C)ztotCCCCCd0•-ICII0’)Cl)03c’iIChapter 2. Material Properties 20Table 2.6: I—Joist floor connection’s test resultsI—Joistglue 1 nail + gluestiffness (N/mm) peak load (N) stiffness (N/mm) peak load (N)12380 15000 3846 90003015 5000 5908 80008714 14000 1390 40006430 11500 2140 75004100 5500 9280 9500may differ from that of a nailed connection. The connector strains incurred during lowamplitude vibrations are low enough that it is believed that only the nail is contributingsignificantly to the connection stiffness. Y. H. Chui noted in his tests with glued floorsthat the fundamental frequency was virtually unchanged by the addition of glue to anailed connection [Chui and Smith, 1991]. If the high stiffness of a glued connection isutilized then the frequency distribution ought to reflect it.The characteristics of the stress—strain relationship of the 3M adhesive may explainwhy the apparent high stiffness of the glued connection is not activated during lowamplitude vibrations. The 3M adhesive is an elastomeric adhesive. Its properties arelisted in Table 2.7. Its structure is analogous to rubber, a polymeric material. A polymeris a large molecule built up by the repetition of small simple chemical units. Thesepolymers can be linked as a linear, branched or network system. By definition, the3M adhesive exhibits a rubber—like elasticity. Small scale motion is allowed by localmovement of chain segments but any large scale motion is restricted by the previouslymentioned polymer linking or network systems [Billmeyer, 1984].For a material of polymeric structure, the elastic modulus is simply a measure ofthe resistance to the uncoiling of randomly oriented chains. The application of a stressChapter 2. Material Properties 21Table 2.7: Material properties for 3M wood adhesiveBase: Synthetic elastomerSolvent: NoneColour: Rose—tanNet weight: 15.0, 0.3 N/LFlash point: NoneSolids content: 100% by weight (approximately)Consistency: Gun grade, Thixotropic pasteCaulk rate: 45 g/min (using 3 mm orifice, 350 kpa at 20C)Shear modulus: 770 kpa (20C under stress level 0—32 kpa)eventually tends to untangle the chains and align them in the direction of the stress[Cowie, 19731. The load-slip curves from the connection tests are quite linear. It hasbeen reported that for some polymer materials, up to 10% strain can be quite nonlinear[Pechhold, 1973]. Shown in Fig. 2.5 is a stress—strain curve of a drawn polyamide fiberthat exhibits an initial low modulus region. If the 3M adhesive has a similar stress—strainrelationship then one could consider the vibration strain domain to be dominated by thenail’s modulus.Listed below are the connection parameters required by the programs.ENL . . .. Connector spacing along the joists.RKFAL,RKPER ... . Connector load-slip stiffness ( modulus ) in the parallel and perpendicular directions to the joists respectively.RKROT . .. .Connector rotational stiffness between the cover and the joists.Chapter 2. Material Properties 22100 —..% 80—60—-C’2 40-U)0-C(2 20-I I I I I I I I I I I0 5 10 15 20Strain %Figure 2.5: Load deflection curve for a drawn polyamide fibre ( Pechhold, 1973)Chapter 3Experimental SetupIn order to plan the testing setup, a preliminary study of the influence of some key floorparameters was undertaken. There were a number of geometric and material parametersto be considered during the design of the lightweight wooden floors. Each of the parameters have some effect upon the floor’s dynamic response. Two important responsecharacteristics are the natural frequencies and their degree of separation.Vibration modes which are poorly spaced can interact to produce higher amplitudevibrations. This is particularly important for lightweight residential floors, which arestrongly orthotropic structures, because their lowest frequencies are generally closelyspaced.3.1 Floor Parameter StudySupporting the rim joistsOften the perimeter joists are supported in some manner. The inclusion of rim joistsupport systems in the design is not difficult or expensive. A floor with its rim joistssupported is less orthotropic, thus its dynamic response is improved. Tests performed byChui indicated a high damping value in the first mode when the rim joists were supported.The high damping in the first mode would indicate that its contribution to the floor’sresponse has been significantly reduced [Chui and Smith, 1991].Generally, the introduction of rim joist support causes an increase in all modal separations. The results from NAFFAP simulations indicate that separation between the23Chapter 3. Experimental Setup 24first and second and between the fourth and fifth frequencies are somewhat less affectedby the addition of rim joist support [Filiatrault et al, 1990].Joist end fixityEnd fixity can range from simple supports to fully fixed. Joist hangers and builtin construction are common examples. However, in practice, most construction can beconsidered as simply supported. Chui tested a floor by doubling the clamping force onthe joist ends. The results showed negligible change. It was concluded that for usualclamping loads acting on the joist ends,the natural frequencies were insensitive to changesin end fixity [Chui, 2/86].Aspect ratioThe aspect ratio is defined by the floor’s width divided by its length (joist span).The sheathing sheets are always laid with their longer length, the dimension parallel tosurface grain, spanning the joists. Studies of the effects of aspect ratio upon a floor’sresponse showed that if the aspect ratio is greater than or equal to 1 for a floor simplysupported on all four sides the floor’s frequency distribution was improved.The aspect ratio, of course, can be changed by either changing the floor’s span orwidth. These two methods do not affect the frequencies in the same manner. NAFFAPstudies showed that any increase in the number of joists past nine had no effect on thefundamental frequency and generally does not affect the modal separation. Increasingthe span caused a dramatic decrease in the fundamental frequency and an increase inmodal separation. Modal separation improvement was especially evident for the lowerfrequencies [Filiatrault et al, 1990].Joist spacingReducing the joist spacing has the effect of increasing the floor’s bending stiffness perunit width. The floor’s stiffness perpendicular to joist span is dependent upon sheathingstiffness, joist spacing and the type of between joist bridging. The ratio of bendingChapter 3. Experimental Setup 25stiffnesses for the floor’s width and length increases with modal separation. Generally, fora decrease in joist spacing the fundamental frequency increases and the modal separationdecreases.Nail spacingA nailed joist to sheathing connection can be considered as semi-rigid. Most construction practices fall well within the code’s specifications. The NBCC suggests a maximumspacing of 300mm at interior supports and 150mm at the sheathing’s end supports. Altering the nailing density does not affect the distribution of frequencies. NAFFAP runshave shown that an increase in nail spacing of 3 — 4 times is required before the frequency distribution is significantly changed [Filiatrault et al, 1990]. This was supportedby Chui’s test where doubling of the nail density had virtually no effect upon the fundamental frequency [Chui, 2/86]. He did find that damping had increased slightly as aresult of the stiffened connection.Nail stiffnessThe load—slip moduli of the joist to sheathing connection has a large effect uponthe stiffness and the dynamic response of the floor system. An increase in the nailhorizontal slip stiffness will increase the fundamental frequency but the modal separationis relatively unaffected. An increase in connector stiffness reduces the system’s ability todissipate energy through friction. The introduction of an elastomeric adhesive improvesthe durability of the connection. It is believed that the adhesive has a negligible effectupon the floor’s dynamic response but that it significantly reduces the “squeakiness” ofthe floor [Chui and Smith, 1991].Chapter 3. Experimental Setup 263.2 Floor DesignCurrently common residential floor construction involves either sawn lumber joists orcomposite wood I—Joists. One floor of each type was designed and tested. SPF No.22x8s and TTS Jager Industries wood I—Joists were used.Parameters that were chosen to be common between the two floors include the sheathing to joist connection nail spacing, sheathing thickness and aspect ratio. The joist spanand spacing were chosen so that the floors were designed to approximately the samepercentage of the NBCC1990 span table limits. The nail spacing for the sheathing tojoist connection was chosen to be 300mm. The floors’ sheathing was 18.5mm D—fir ply—wood. The aspect ratios were 1.13 and 1.12 for the 2x8 joist floor and the I—Joist floorrespectively.The NBCC199O has incorporated new vibration criteria for determining allowablejoist spans. The new criteria are supposed to improve the serviceability of wooden floorsby moving the floor’s fundamental frequency away from the human sensitive frequencyrange of 4 — 20Hz. The NBCC1990 suggests a span limit of 3.36m for 2x8 joists spacedat 400mm. To satisfy a reasonable aspect ratio, ten joists at a span of 3100mm werechosen.Relying upon the manufacturers span tables for the I—Joist, nine Jager joists at aspacing of 600mm and spanning 4200mm were chosen. The larger joist spacing was usedto reduce the stiffness of the floor. It was anticipated that this would cause an increasein between joist “bounciness”, but the majority of the tests were performed over joistlines.Chapter 3. Experimental Setup 273.3 The Floor’s Support SystemThe system that supports the floors had two functions. Firstly, it was to simulate typicalboundary conditions found in common residential construction. Secondly, the frame wasnot to participate with the wooden floor’s response in addition to isolating the floorsystem from the lab floor. The conceptual design for the support system followed thework of Chui, as it appeared to be successful in obtaining clean dynamic responses fromlightweight wooden floors. This permitted the identification of a significant number ofthe floors’ natural frequencies [Chui and Smith, 1991].Platform framing is the most common form of floor joist construction used over afoundation wall. The box—sill method is common in platform framing [CMHC, 1984]. Asshown in Fig. 3.1, the 2x8 header and both the 2x4 plate and sill were included in thefloor system. The 2x4 sill was bolted to the W—shape. The floors were then built uponthese sills as per the NBCC199O and CMHC recommendations. Threaded steel rods wereused as a clamping system to apply some pressure to the top of the floor. Fig. 4.3 andFig. 4.4 illustrate how the clamping system was applied. This loading could representpartial roof or wall loading. The 2x8 joist floor had each of its joist ends clamped butthe I—Joist floor had only eight of its joist ends clamped. Since the bottom flanges ofthe I—Joists were nailed directly to the 2x4 sills, it was believed that it would be unlikelythat the floor would lift from the steel frame. It is believed that the clamping system didnot significantly restrain rotation of the joist ends.The floor was supported on all four sides by the steel W—shapes as shown in Fig. 3.2.The perimeter joists were restrained from vertical and lateral movement by toe-nailingthe joists to the 2x4 sill that was bolted to the steel frame. Particularly for the 2x8 joists,this nailing may have restrained joist rotation significantly. The web of the I—Joist wasflexible enough to allow significant rotation at its top flange.Chapter 3. Experimental Setup 28Figure 3.1: Connection details at joist ends for the I—Joist joist floorChapter 3. Experimental Setup 29Figure 3.2: A view of the floor framing and the steel support systemChapter 3. Experimental Setup 30Generally, house construction allows the transmission of vibrations from one structural system to another. As a simplification, the test floors were isolated. The 2x4 sillsacted as dampers between the floor and the stiff steel W—shapes. These were laid uponthe lab’s concrete floor. Shims were used to ensure a stable and level foundation for thefloor. Floor anchors were used to eliminate any movement of the steel frame relative tothe lab floor. Shown in Fig. 3.3 is the anchoring system consisting of HSS square sectionsand channels. Vertical threaded rods embedded in the floor were the root of the system.The channels received the vertical anchor loads from the HSS square sections and thentransmitted them laterally to the lab floor and the lower flange of the W—shapes.3.4 Data AcquisitionThe raw data from the floor vibration tests were recorded as acceleration records. Two 2gIC Sensors accelerometers were used to capture the response of the wooden floors to thevarious excitations. The accelerometers were firmly screwed to the plywood sheathing.Care was taken to ensure that all cables were suspended above the floor’s surface.Vibrations of the laboratory floor could be detected by the accelerometers since thetesting apparatus was not isolated from the general lab floor. Analysis of the noise tracesidentified frequencies that may become amplified and be mistakenly identified as floorfrequencies. The real component of the complex valued spectral “Energy” density of anoise trace for the two accelerometers is shown in Fig. 3.4. Multiples of 60Hz appear tobe the only frequencies of concern.A lOOlb Lebow load cell, securely screwed to the floor’s surface, recorded the impulseimparted to the floor by the bag drop test. Full scale output for the lOOlb load cell was7.559 volts. Calibration of the load cell was set at 6.0075kg/V or 58933.575mN/V. Themass of the load cell assembly was 1.08kg.Chapter 3. Experimental Setup 31Figure 3.3: Anchoring system for the W—shape frameChapter 3. Experimental Setup 325030—a)--s_I-o 20-C) -0 40 80 120 160 200 240Frequency (Hz)Figure 3.4: Real spectral “Energy” density of a noise traceThe Labtech Notebook version 6.0 software package was used to manage the recordingof the data. The data acquisition system was limited to a sampling rate of approximately3 70Hz for four channels. Generally, the hammer and heel drop tests were recorded at0.002 seconds per sample while the bag drop tests were limited to 0.0027 seconds persample.Chapter 4TestingThe series of tests that were performed on the two floors involved simple hammer impacts, sandbag and heel drop impacts with or without a passive occupant. The fourseries of tests were performed upon each of the floors. Each test was recorded by twoaccelerometers located at the positions indicated in Fig. 4.1 and Fig. 4.2. Each of thetest series followed the common testing grid of ten impact sites shown in Fig. 4.1 andFig. Hammer Impact TestsThe hammer impact is a convenient, efficient and fast method of vibration testing thatproduces a clean response with a very clear frequency spectrum. For this reason thehammer test responses were used to identify the floor’s natural frequencies. The hammertest data were used for NAFFAP’s verification.A common hammer was used to tap the floor surface. To avoid saturating the accelerometers, given that high accelerations are associated with very sharp impacts, a12mm thick piece of porous rubber was placed on the floor at the impact site. This wassufficient to keep the maximum accelerations close to 2g. Fig. 4.3 shows the 2x8 joistfloor and the equipment in place for a hammer impact test at site number 2.33Chapter 4. Testing 344.2 Sandbag Impact TestsThe sandbag impact was chosen for its repeatability and for DYFAP’s verification. Theload time history is similar to that of a heel drop but the 151b ( 66.7N ) sandbag impartedfar less energy to the floor. Its duration falls within the range 25 — 3Oms. Fig. 4.4 andFig. 4.5 illustrate the test setup for a typical bagdrop test. The bag was of circularcross-section. Its diameter matched that of the impact platform that was attached tothe load cell. This ensured a balanced loading, reducing error due to a moment thatcould develop between the loading platform and the load cell. The drop height and bagmass were limited by the lOOlb ( 445N ) capacity of the load cell. A 151b ( 66.7N ) bagreleased from a height of 27mm produced peak loads of 105 — ll5lbs ( 467 — 512N ).The bag was released by sliding the looped string over a nail that was attached to thepost of the spanning frame.o Impact Site- Recording SiteFigure 4.1: Impact and recording locations used on the 2x8 joist floorChapter 4. Testing 3510 01 5 ace I24O31502 10 6 80ace 90 0 0 Impact Site- Recording SiteFigure 4.2: Impact and recording locations used on the I—Joist floorChapter 4. Testing 36Figure 4.3: The hammer impact testChapter 4. Testing 37Figure 4.4: The bagdrop impact testChapter 4. Testing 38Figure 4.5: A closeup of the bag, load cell and accelerometers (Photograph shows a dropheight approximately twice that of the 27mm test height)Chapter 4. Testing 394.3 Heel Drop TestsWalking and running are two common sources of dynamic excitation. The human footstepinduced vibration is the most common type of serviceability problem for lightweightwooden floors. Running differs from walking in that both feet will lose contact with thefloor. The force—time history for running most resembles that of an impulse. Generally,with only marginal error, the vibration can be treated as transient if only one person isactive on the floor. The type of occupancy will dictate what level of activity would betolerable. For most domestic uses, the heel drop impact has been shown to be adequate asan upper limit for evaluating floor performance. The heel drop test performs well againstthe more expensive Random, Sweep and Discrete frequency methods [Rainer, 1980].The heel drop test was performed by a 2001b ( 890N ) man. Shown in Fig. 4.6 is aperson in position to demonstrate a heel drop test. All tests were performed with shoesremoved. The person rises up on his toes, raising his heels approximately 65 — 75mmbefore suddenly shifting his weight over to his heels. He hits the floor, impacting with hisheels, while remaining as relaxed as possible. Each part of the human body vibrates atits own particular frequency. Movement of the shoulder girdle and arms can impart a significant amount of momentum to the floor. Although the load histories of the heel dropswere not recorded, a plot of the average force versus time from Lenzen and Murray shownin Fig. 4.7 indicates that a 5Oms duration could be expected [Allen and Rainer, 1976].The peak load, depending upon floor stiffness, is generally 2—4 times the heel dropper’sweight [Allen and Rainer, 1976] [Chui, 2/86].Series #4 involved a second person standing on the floor while a heel drop was performed. As shown in Fig. 4.6 the second person, the observer (passive occupant), stood500mm behind the impact site on the same joist. Each test was then repeated with theobserver now standing on the adjacent joist. The same two persons were used for theseChapter 4. Testing 40Figure 4.6: The starting position of a typical heel drop test with an observer standingon the same joist as the testerChapter 4. Testing 413000 -2500 -20001500—0 -500 —0— i I I I I I I I I I I I I I I I I I I I I I I I I0.00 0.01 0.02 0.03 0.04 0.05 0.06Time (see)Figure 4.7: Averaged plot of force versus time for heel drop impact ( Folz and Foschi,1991 )tests, as observer and heel dropper. The observer was a 2151b ( 957N ) man.The data files for all of the tests are referenced by a coded label of the form FTIR where F=type of floor ( SL, 2x8 sawn lumber floor or IJ, TTS Jager I—Joist floor),T=type of impact (H, hammer, B, bag, F, heel drop, S, heel drop with observer locatedon same joist, N, heel drop with observer located on adjacent joist), I=location of impact( see Fig. 4.1 and Fig. 4.2) and R=which accelerometer, #1 or #2, recorded the signal( see Fig. 4.1 andFig. 4.2 for location). For example, IJH6-2 denotes the test on theI—Joist floor with the hammer impact at location #6 and the signal was recorded byaccelerometer #2.Chapter 5Dynamic Response of Test Floors5.1 Evaluation of Damping RatiosWooden floors exhibit a moderate level of damping. Each connection has some capacityto dissipate energy by friction or in the case of glued connections by elastic deformation.The material itself can deform elastically under the applied loadings.The raw test data are acceleration records. An equivalent viscous damping ratio(() can be derived directly from an acceleration time history. A common method is bylogarithmic decrement, using Eq. 5.1. To gain greater accuracy the equation is expandedto consider two positive peaks separated by m positive peaks.= e2mC (5.1)An+mwhere:A trace amplituden,m counters for the trace’s positive peaks( viscous damping ratioSince the acceleration records include the full spectrum of frequencies, it is necessary tofilter the records for the floor’s natural frequencies. Applying band pass filters one canisolate particular frequencies thus creating time histories for specific frequencies. Shownin Fig. 5.1 is the decay curve for the I—Joist floor’s second frequency. The vertical scale42Chapter 5. Dynamic Response of Test Floors 43has been changed to show only the positive peaks. Not shown is the rising portion wherethe particular frequency is growing in participation from the time of the impact. Theparticipation of the lower frequencies do not peak until much of the floor motion has beencompleted. The damping of the higher frequencies has a much larger influence upon thehigher levels of motion that occur soon after the impact.The damping ratio for a frequency is dependent upon where on the decay curve themeasurements are taken. Shown in Fig. 5.1 are the damping ratios expressed as a percentof the critical damping for four regions, bounded by dashed lines, along the decay curve.The ratios generally increase with time. The low damping early in the trace likely hasa minimal influence upon the floor’s response. Approximately the second half of thedecaying portion of the free vibration trace was used to estimate the damping ratio. Thefirst peak A was taken at half the trace’s peak value. The second peak An+m was takenas near as possible to the lowest peak. Generally this meant that n was taken as 30— 50positive peaks. For the particular example of Fig. 5.1 m 51 and the damping ratio wascalculated to be 0.81%.The programs do not allow the user to specify a viscous damping ratio for any par-ticular frequency. They will only accept Rayleigh proportional damping as defined byEq. 5.2. The constants for mass and stiffness proportional damping, /3 and c respectivelyare program inputs.[C] =cr[K]+/3[M] (5.2)= + (5.3)The Rayleigh factors are related to damping ratio by the relation defined by Eq. 5.3. Thedamping ratios ( , ( ) associated with the frequencies that bound the frequency rangeof interest, first natural frequency to one close to 65Hz, were calculated. Using these twodamping ratios, a system of two Eqs. 5.3 were then used to solve for the Rayleigh factorsChapter 5. Dynamic Response of Test Floors 44a and /3.Listed in Table 5.1 are a few of the bounding damping ratios ( (, (2 ) and theircorresponding a and 3 values that were calculated for the I—Joist floor. The 2x8 joistfloor required only mass proportional damping because the higher frequencies had lowerdamping than the first frequency’s which required unacceptable negative a values. Table 5.2 lists a few of the bounding damping ratios and the /3 values that were calculatedfor the 2x8 joist floor. Only the average /3 and a values for each floor were inputted intothe programs.The calculated damping ratios were not sufficient for use in DYFAP to represent actualdamping effects. DYFAP is restricted to the Rayleigh damping equation. Variation indamping by floor location or frequency were not modelled by DYFAP. Variation by floorlocation is particularly evident in the results of Table 5.2. The averaged damping ratiosFigure 5.1: The decay curve from an acceleration trace filtered to isolate 30.4Hz, IJH6-2Chapter 5. Dynamic Response of Test Floors 45Table 5.1: A selection of proportional damping constants for the I—Joist floorRecord /3 (% (2%IJH1-2 0.000051 1.33493 0.84 1.24IJH5-1 0.000050 1.6495 0.91 1.25IJH8-1 0.000034 2.56253 1.05 1.01IJH8-2 0.000041 1.70527 0.84 1.07IJH6-1 0.000058 1.41435 0.92 1.38IJH6-2 0.000055 1.10364 0.81 1.28average 0.000047 1.7329 0.91 1.19Table 5.2: A selection of proportional damping constants for the 2x8 joist floorRecord /3 (% (2%SLH6-1 8.63898 2.01 1.35SLH6-2 22.17317 4.41 1.54SLH8-1 10.70202 2.49 1.14SLH8-2 6.06018 1.41 1.07SLII4-1 10.05732 2.34 1.70average 9.8854 2.3 1.36Chapter 5. Dynamic Response of Test Floors 46were increased in order to produce traces that reasonably matched the duration andamplitude of the experimental traces. The damping ratio (for the 2x8 joist floor was setat 4%. The I—Joist floor required its (‘ and (2 to be 2% and 2.5% respectively.5.2 Identifying the Floors’ Natural FrequenciesThe natural frequencies of the floor systems were identified from Fourier spectra derivedfrom the hammer test’s acceleration records. The hammer impact seemed to do wellin exciting the entire range of frequencies of interest. The hammer test did not involveadding any additional masses to the floor systems that could have altered the systems’frequency distributions. Spectra from all ten test locations were used to identify thepredominant frequency peaks.The complete data record for any particular test included: a 1.5 second leader ofambient noise, a 0.5 — 1 second response and about another 3.5 seconds of ambientfloor vibration following the floor’s response. The passage of time associated with thefloor’s response was only 15% of the entire record. Performing a Fast Fourier Transformon the entire record produced a significant amount of clutter in the Fourier spectrum,particularly at the higher frequencies. The frequency content of the ambient noise leaderwas significant and not necessarily representative of the floor’s response. In order tobe able to compare the Fourier spectrum amplitude levels with those obtained fromDYFAP runs, only 4.1 seconds of recorded data were analysed. This made the DYFAPoutput and the experimental data record of the same duration. The lower time boundfor the analysis began at the beginning of the floor’s response, excluding the leadingnoise. By comparing Fig. 5.2 and Fig. 5.3 one can see how much activity was present inthe ambient noise leader and tail. The floor’s predominant natural frequencyies are nowclearly evident. The high frequency content was all but eliminated. The drop in FourierChapter 5. Dynamic Response of Test Floors 47Figure 5.2: Fast Fourier Transform of a six second acceleration recordspectrum amplitudes indicate that as expected, the ambient floor vibration included thefloor’s natural frequencies contaminated with noise frequency components. Even for arecord clipped to 4.1 seconds in duration, a lot of the record was just ambient vibrationbut the longer duration was necessary in order to achieve a reasonable 0.24Hz frequencyincrement for the Fourier spectra.Each test location produced its own unique Fourier spectrum. At any particular testlocation, the frequencies that strongly participated in the response depended heavily uponthe location of the impulse and the receiver on the floor. This was particularly evidentwhen comparing the response at two locations due to the same impulse. As an example,Fig. 5.4 and Fig. 5.5 show the Fourier spectra from two accelerometer locations producedby the impact at test location #5. These figures show that even though acceleromter#2was located near the center of the floor, it would be wrong to assume that the strong225N‘%% 180N1135D 90.—4 450Frequency (Hz)Chapter 5. Dynamic Response of Test Floors 48Figure 5.3: Fast Fourier Transform5.35 — 9.45 secondsof a bounded acceleration record, time bounds ofpeak at 40Hz was the floor’s fundamental frequency. Therefore, it is important to analyzeresponses at a number of locations to allow one to evaluate how significant a particularpeak is to the floor’s general response.For both floors, only the frequencies from 0 — 100Hz were considered since the veryhigh frequencies are not important for the free vibration of the floor. They do contributeto the extreme spikiness of the high peak accelerations during the first 2— 3 cyclesbut they dampen out very quickly. A floor’s lowest three or four frequencies typicallydominate its response.For the identification of natural frequencies, a simple approach was used. For thepurpose of this investigation, it was only necessary to be able to identify the predominantfrequencies. The many other frequencies that could be identified by more sophisticatedtechniques are of little interest since their level of participation do not significantly effectNQ)z.——Frequency (Hz)Chapter 5. Dynamic Response of Test Floors 4925-IFrequency (Hz)Figure 5.4: Fourier spectrum for hammer test, SLH5-]25 -_-:ii,iiiiiW4iiiiiFrequency (Hz)Figure 5.5: Fourier spectrum for hammer test, SLH5-2Chapter 5. Dynamic Response of Test Floors 50Table 5.3: Natural frequencies for the test floorsNatural Frequencies ( liz)Mode Number 2x8 Joist I—Joist1 34.2 26.42 40.0 30.43 44.2 34.74 50.9 37.05 55.6 47.16 64.2 50.47 66.7 55.58 83.3 67.0the floor response time traces. A peak of moderate amplitude was considered as a likelyfrequency if it occured on a number of the records. The strongest one or two peaks forany record were automatically considered to be natural frequencies. These criteria weresufficient to safely identify six frequencies and to allow a couple more to be tentative.Samples of spectral densities for the 2x8 joist floor are shown in Figs. 5.4 — 5.9. Theseplots show strong peaks for 7 of the 8 frequencies listed in Table 5.3. The peak at 55Hzoccurs in only four of the 20 records and never very strongly. It was the weakest of allthe identified frequencies and was much stronger than the next possible candidate.Samples of Fourier spectra for the I—Joist floor are shown in Figs. 5.10 — 5.14. Eightfrequencies are listed in Table 5.3 for the I—Joist floor but only four had consistently strongand sharp peaks in the spectra. The peak at 37.0Hz marked the pass fail boundary like55Hz did for the 2x8 joist floor.As previously mentioned, electrical noise at 60Hz was not to be mistaken for a floorfrequency. A narrow spike at 60Hz appears in all cases for both floors as shown in thefollowing spectral density plots. At times it may not be evident but the magnitude of thespikes are consistently between 4— 8 units. This frequency was not filtered out prior toChapter 5. Dynamic Response of Test Floors 5125-IFrequency (Hz)Figure 5.6: Fourier spectrum for hammer test, SLH6-125—_-N --20—NCl) -—‘ I ‘ I I ‘ I ‘ I I ‘ I ‘ I ‘ I ‘0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)Figure 5.7: Fourier spectrum for hammer test, SLH6-2Chapter 5. Dynamic Response of Test Floors 52—-N -20—C”-[5z.4-).— -- 0—0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)Figure 5.8: Fourier spectrum for hammer test, SLH7-125 -:1Frequency (Hz)Figure 5.9: Fourier spectrum for hammer test, SLH7-2Chapter 5. Dynamic Response of Test Floors 53100 -N-‘-. 80-060-• 40-2:i•1Frequency (Hz)Figure 5.10: Fourier spectrum for hammer test, JJH2-1the investigation of the acceleration and displacement time histories. The low amplitudehigh frequency did not significantly alter the characteristics of the traces. The I—Joistfloor had an additional extraneous frequency at 21Hz. It is likely due to some form offloor—frame interaction that was allowed as a result of some irregularity that may haveoccured during the reconstruction of the frame for the I—Joist floor. It became significantduring the heel drop tests and therefore, it was necessary to apply a narrow band passfilter to the raw acceleration records. The larger peak displacements during the heel droptests are likely the cause for the increased participation of the 21Hz component.Chapter 5. Dynamic Response of Test Floors 54100 ——S80-60-• 40—jFrequency (Hz)Figure 5.11: Fourier spectrum for hammer test, IJH2-2100 -—SQ80-60-IFrequency (Hz)Figure 5.12: Fourier spectrum for hammer test, IJH4-1Chapter 5. Dynamic Response of Test Floors 55100 —N -80—ca -060-40—j2:Frequency (Hz)Figure 5.13: Fourier spectrum for hammer test, IJH6-].100 -N80-060• 40-zFrequency (Hz)Figure 5.14: Fourier spectrum for hammer test, IJH8-i.Chapter 6NAFFAP6.1 IntroductionThe program NAFFAP, NAtural Frequency Floor Analysis Program, was developedby Andre Filiatrault and Bryan Folz at the University of British Columbia in 1989[Filiatrault and Folz, 1989]. It solves for the natural frequencies and mode shapes ofan undamped one—way stiffened floor system. The floor system is restricted to one withequidistant simply supported stilfeners attached to an orthotropic plate. Semi—rigidstiffener—to—plate connectors are allowed. The program’s solution employs a T—beamfinite strip analysis that was developed by Foschi [Foschi, 1982]. Each strip, as shown inFig. 6.1, consists of 4 nodes with 19 degrees of freedom. The DOF associated with node4 are as shown while nodes 1— 3 take u,v,w and the first derivitives of u and v. Thecurrent version can handle a floor with up to 12 stiffeners.6.2 Data FileThe boundary conditions for the perimeter joists were different for the two floors. Although their construction were essentially the same, the difference in torsional rigidityof the two types of joists warranted differences in how the degrees of freedom could bedefined. The longer spanning deeper I—Joist allowed for significant rotation about itsx—axis at midspan. By definition, the perimeter I—Joists were more closely modelled bya simple rather than a fixed support. The lower flanges were nailed to the sills but their56Chapter 6. NAFFAP 57thin webs provided little torsional stiffness. The 2x8 joists were 57.2mm shallower andhad a 60% larger torsional constant than the I—Joists. The 2x8 joist floor had a shorterspan and a smaller joist spacing than the I—Joist floor, thus joist rotation was bettercontrolled. Here the perimeter joists were best modelled as fixed supports.The floor parameters required by NAFFAP and DYFAP to model the floors were discussed in chapter 2. Values were either measured directly or were derived from equationsexcept in the case of nail stiffness. Tests were not performed for nailed connections. Thetests that were performed only provided evidence for disregarding the stiffness contribution from the elastomeric glue.It was assumed that the test floors’ connector stiffness could be modelled by a nailedconnection. A value for nail stiffness of approximately l0000lb/in (1 750N/mm ) wouldbe acceptable. To determine what value of nail stiffness would work best with NAFFAP,y,vz,wFigure 6.1: A T—beam finite stripChapter 6. NAFFAP 58a minimization procedure was followed. NAFFAP runs were done for stiffness valuesranging from 7000 —340001bs/in (1230 —5950N/mm ). The upper limit was the averagestiffness obtained from the glued connection tests.The percentage error in matching frequencies with the test results was used as aminimization parameter. Not all frequencies were used. The higher modes are moredifficult to model accurately than the first couple of modes. Besides which the lowerfrequencies generally dominated the acceleration traces. An additional acceptance criteriawas applied to the first three frequencies. They were all to be within 5% of the testfrequencies. Listed in table 6.1 are the percentage error for the 2x8 joist floor’s first threefrequencies. The table shows that a nail stiffness of 8000 — 130001b/in(1400 — 2280N/mm ) allows NAFFAP to do fairly well in matching the frequencies ofthe tests. Values in this range are all acceptable for an 8d common nail connection. Asthe nail stiffness values approach that of a glued connection one can see how NAFFAPdiverges from the first and third frequencies. Only the 10000—120001b/iri ( 1750—2100N)nail stiffnesses satisfied the sub 5% criteria. Of those, l0000lb/in ( 1750N/mm) providedthe best combined fit for the first and second frequency.The values taken for the 2x8 and I—Joist floors’ connector rotational stiffness( RKROT ) were 60001b/in ( 1.050N/mm ) and 120001b/in ( 2100N/mm ) respectively.These were minimum values to ensure that NAFFAP did not produce an overabundanceof “joist wiggling” frequencies. The low torsional rigidity of the I—Joist required the largeconnector rotational stiffness.NAFFAP had difficulty in capturing the I—Joist floor’s first frequency at 26Hz but itdid well with the strong frequencies at 30Hz, 47Hz, 55Hz and 67Hz. The frequencies above45Hz were not very sensitive to changes in the nail stiffness parameter but unwantedfrequencies between 38Hz and 45Hz were developing more pronounced mode shapes.Straying too far above the nail range of 7000 — 130001b/in (1230— 2280N/mm ) causedChapter 6. NAFFAP 59Table 6.1: Percentage difference between 2x8 joist floor tests’ and NAFFAP frequenciesNail Stiffness % Difference(lb/in) Model (34.2Hz) Mode2 (4 0.0Hz) ModeS (44.2Hz)7000 2.08 6.08 2.198000 2.63 5.63 1.819000 3.16 5.18 1.4510000 3.65 4.75 lii11000 4.15 4.33 0.7712000 4.62 3.93 0.4513000 5.09 3.53 0.1420000 7.87 1.10 1.8830000 10.91 1.65 4.1634000 11.90 2.55 4.95l000lb/in = 175N/mmNAFFAP’s first frequency to increase its error from 14% to 20%.6.3 NAFFAP’s Results for the Test FloorsNAFFAP produced a number of frequencies for both floors. Some of these are a resultof the inclusion of a rotational degree of freedom. Most often these “rotational” or “joistwiggling” modes are associated with very low vertical motions, and therefore are of littleinterest. The I—Joist floor was most susceptible to this type of mode shape. The lowertorsional rigidity of the joists coupled with the larger joist spacing were responsible.Listed in table 6.2 are the complete lists of NAFFAP frequencies and the frequenciesidentified from the tests for both floors.The mode shapes that NAFFAP suggested for the two floors are shown in Fig. 6.2and Fig. 6.3. They were drawn from the values for the vertical and rotational degreesof freedom for nodes 1, 3 and 4. The mode numbers shown in Fig. 6.2 and Fig. 6.3correspond to those listed in Table 6.2. Only the first few mode shapes for each floorChapter 6. NAFFAP 60Table 6.2: NAFFAP results compared with frequencies identified from floor testsNatural Frequencies (Hz)Mode 2x8 Joist Floor Mode I—Joist FloorNumber Test NAFFAP Number Test NAFFAP1 34.2 35.5 1 26.4 30.32 40.0 38.1 2 30.4 31.03 44.2 43.7 3 34.7 31.34 50.9 51.2 — 32.3— 52.4 — 33.15 55.6 53.3 — 33.9— 54.5 — 34.9— 55.4 — 35.0— 55.6 — 35.7— 56.7 — 36.5— 56.9 — 37.56 64.2 59.1 4 37.0 38.57 66.7 72.8 5 47.1 41.18 83.3 93.0 6 50.4 46.57 55.5 54.58 67.0 65.1Chapter 6. NAFFAP 61were drawn. The first mode shape for either floor was not symmetrical because the joistswere of different stiffnesses.Chapter 6. NAFFAP 62Mode I IMode2 11 joModeFigure 6.2: NAFFAP mode shapes for the 2x8 joist floorMode 11 2 3 4 5 6 7 8 9Mode 2Mode123456789Figure 6.3: NAFFAP mode shapes for the I—Joist floorChapter 7DYFAP7.1 IntroductionThe program DYFAP, DYnamic Floor Analysis Program, was developed at the University of British Columbia by Bryan Folz, Andre Filiatrault arid R.O. Foschi in 1990[Filiatrault et al, 1990]. It employs the same structural modelling as NAFFAP. The input data file is almost identical to that of a NAFFAP input file. The difference lies inthat DYFAP performs a time domain integration of the equations of motion in responseto a specified loading on the floor. The loading can take the form of a sinusoidal forcingfunction, step load, discretized forcing function and/or impulses from an oscillator thatis attached to the floor surface. DYFAP output consists of displacement, velocity andacceleration time histories for a designated floor location. Such an output is of courseinfluenced by the form and level of loading that is applied to the floor.7.2 Modelling of a Hammer ImpulseSince the impulses imparted to the floor by the hammer impact tests were not recorded,it was necessary to derive an estimate of a typical hammer impulse from the literature.Scaling from a trace reported in Chui’s paper, a duration of approximately l6ms and apeak load of 230N were measured [Chui, 2/86]. The peak accelerations measured fromhis floor due to the hammer impulses were roughly 0.5g. The peak accelerations for thetests reported in this thesis were on the order of 2g. To obtain the larger response one63Chapter 7. DYFAP 64Figure 7.1: Modelling of a hammer impulsecan increase the peak load and decrease the duration. The influence of changing thepeak and duration was noted and compared with the acceleration peaks of test SLH6-2.The combination of 350N and lOms was sufficient in order for DYFAP to match thetest’s peak accelerations. This loading was applied over a 2Ox2Omm area. For DYFAP,the loading was discretized as shown by Fig. 7.1. The time scale of Fig. 7.1 has beenmagnified so that the plateaued peak of the impulse model would be visible. Rather thanincreasing the peak load substantially in order to increase the impulse’s energy, the peakwas plateaued. The steeper slope of the impulse also contributed to the peak accelerationof the floor. This is partly recognized by the higher dynamic magnification factor for arectangular pulse over a half sine pulse [Clough and Penzien, 1975].1000900-‘ 800700600z5004003002001000Time (see)Chapter 7. DYFAP 657.3 The Bagdrop ImpulseThe impulses from the bagdrop impact tests were recorded by a load cell that wasmounted on the floor surface. The load was received by the floor over an area of 4780mm2.The softer impact induced lower peak accelerations. A trace of a typical bagdrop impulseis shown in Fig. 7.2. The trace does not settle back to zero over time because the bagremained atop the load cell for the duration of the test. The influence of the bag’s massupon the response of the system was small, particularly for the heavier I—Joist floor. Theentire trace was not discretized for DYFAP. The many oscillations following the initialimpulse were replaced by a flat trace as shown in Fig. 7.2. The magnitude of the bagdropimpulses were quite repeatable but each impulse’s shape was slightly different. For theexample shown in Fig. 7.2, the peak load is 1151b ( 512N ). Since the impulse traceswere available, DYFAP runs were made using the corresponding impulse for each testlocation.7.4 Comparison of Hammer and Bagdrop Tests with DYFAP ResultsThe response was unique at any particular point on the floor. Although the frequencycontent was constant, the level of participation of each frequency made each spectrumunique. A number of factors may contribute to this phenomenon. Proximity to boundary conditions such as edges and sheathing joints, testing over or between joists andnonuniform material properties may greatly affect the level of damping or the participation for any particular frequency. The level of participation chosen by the computersolution differ from those of the experimental results largely because of the program’sinability to capture all of these features in its model. Material properties such as localstiffness, connection details and mass density are considered by the computer model asmore uniform than site specific.Chapter 7. DYFAP 66Accepting the limitations of the model, there were still a number of features of theresponse and output that could be noted and compared. The peak accelerations, peakdisplacements and damping were key features.Since the test data were recorded as acceleration time histories it was necessary tointegrate twice to obtain the displacement record. The soft low pass nature of the integration process accentuates the low frequency content thus causing the integrated traceto show a low frequency drift. The majority of this disturbance was between 0 — 3Hz. Itwas necessary to apply a digital high—pass filter to the velocity and displacement tracesin order to satisfy the zeroed boundary conditions. The bag and hammer traces werefiltered up to 8Hz but this was well below any system frequencies of interest (see “Designing Digital Filters”, Charles Williams, 1986). The signal processing was performedusing the VU—POINT software package [VU—POINT, 1988].0.1250.100S 0.075z. 0.05000.0250.0000.20 0.30Time (see)Figure 7.2: Typical recorded bagdrop time history compared with that used with DYFAPChapter 7. DYFAP 67Figure 7.3: Comparison of the 2x8 floor’s test and DYFAP’s response to a hammerimpactThe fact that DYFAP was modelling the hammer impulse from an approximationresulted in a rather poor comparison with the test results. On the other hand, the loadhistory for every bagdrop test was available for discretization. Naturally each hammerimpact must have varied in intensity, particulary between floors. A fluctuation of up to0.5g from test to test is not unreasonable. The displacement time history, derived byintegrating the acceleration trace twice, was not as sensitive to variations in the loading.Therefore, DYFAP’s and the test’s results could be conveniently overplotted as shown inFig. 7.3 and Fig. 7.4. All of the following comparisons used the output from the same floorlocation. The impact location was #6 and the response was taken from accelerometer#2.The Fourier spectrum, acceleration and displacement time histories for the bagdropwere comparable. The Fourier spectra overplots of Fig. 7.5 and Fig. 7.6 do not match0.30.20.1a)E 0.0a)C)0-0.2—0.3Time (sec)Chapter 7. DYFAP 68Figure 7.4: Comparison of the I—Joist floor’s test and DYFAP’s response to a hammerimpactclosely over the entire range of frequencies but the DYFAP output does recognize themore important, dominant, low test frequencies. This is particularly evident in the caseof the I—Joist floor. The strong DYFAP peak at about 30Hz and the smaller ones at 38Hzand 47Hz correspond well with the trends of the test data. Upon comparing the spectrafor hammer and bag tests, the DYFAP results were able to show the shift towards thelower frequencies. The softer, longer duration of the bag drop impact allowed the lowerfrequencies to receive relatively more energy than the higher frequencies. The bag impactdid not induce such high amplitude peaks as did the hammer impact. As a result of thelower y—axis bounds, the spectra appear to be not as clear as those from the hammerimpacts.The acceleration records from the bagdrop tests indicate that the assumed floor damping for the two floors were appropriate. The overplots of the DYFAP outputs and test0.4$0.3• 0.2o.ia)C.)c1-0.1—0.20.50Time (see)Chapter 7. DYFAP 6910— I- J Peak Amplitude of 16 mrn/s2/HzN --- —TestoB DYFAP-1:11 ‘2Frequency (Hz)Figure 7.5: Fourier spectrum of the 2x8 floor’s acceleration response to a bagdrop impact50 —.—% Peak Amplitude of 128 rnm/s/Hz40’—Test°30-• 20-)Frequency (Hz)Figure 7.6: Fourier spectrum of the I—Joist floor’s acceleration response to a bagdropimpactChapter 7. DYFAP 7020000 -i—.. 15000 -— Testi. —15000-—20000— i i i i i i i i0.00 0.25 0.50 0.75Time (sec)Figure 7.7: The 2x8 floor’s acceleration response to a bagdrop impactdata for the 2x8 and 1—Joist floors are shown in Fig. 7.7 and Fig. 7.8 respectively. Particularly for the 2x8 floor, the DYFAP trace matched the test’s amplitude levels, dampingto zero at 0.6 seconds.The displacement time histories for the bagdrop tests are shown together with theDYFAP results in Fig. 7.9 and Fig. 7.10. The static displacement listed in the figureswas caused by the 151b ( 66.7N ) sandbag. Once it was dropped it remained upon thefloor for the duration of the test.7.5 Excitation by an OscillatorThe human heel drop involves a very complicated system, namely the human. The massof the human was a significant percentage of the entire system’s mass. Particularly forthe 2x8 joist floor, a 2001b ( 890N ) man represented 29% of the system’s total mass.Chapter 7. DYFAP 7110000 —B Test5000 DYFAP•. •: •.. •. •f-5000—10000—15000— i i i0.00 0.25 0.50 0.75 1.00Time (sec)Figure 7.8: The I—Joist floor’s acceleration response to a bagdrop impact0.5 —‘ 0.4 — Test Peak Dispi. = 0.428mmDYFAP Peak Dispi. 0.404mm0.3-f- 0.2—H.,01ilIIfk\f0.0 -! !I1iIYRrr’<’ ./JJr.. Static Dispi. 0.037mm—0.3— i i i i i i i i0.00 0.25 0.50 0.75Time (see)Figure 7.9: The 2x8 floor’s displacement response to a bagdrop impactChapter 7. DYFAP 720.4 -Test. Peak Dispi. = 0.309mm0.3 DYFAP Peak Dispi. = 0.353mmit0.2-I1L P0.10.0— fliij—0.1-Static Displ. 0.033mm—0.2- i i i i i i i i i I I I0.00 0.25 0.50 0.75 1.00Time (see)Figure 7.10: The I—Joist floor’s displacement response to a bagdrop impactIt can be assumed that as this percentage rises, the influence of the occupant upon thefloor’s response also increases. An oscillator consisting of viscous dashpots, elastic springsand lumped masses was used with DYFAP to represent the occupant. To model the heeldrop action an initial velocity equal to was assigned to the oscillator’s masses tosimulate their fall to the floor. DYFAP doesn’t allow the oscillator to lose contact withthe floor surface.7.5.1 DYFAP Results using the ISO ModelThe ISO, International Standards Organization, have adopted a lumped parameter vibratory model for deriving the driving point impedance of the human body in a standingposition [ISO 5982]. This is a simple two DOF model that was proposed by Coermannand is illustrated in Fig. 7.11 [Folz and Foschi, 19911. Listed in Table 7.1 are the valuesChapter 7. DYFAP 73for the oscillator’s lumped masses, linear springs and viscous dashpots.DYFAP runs were done using the ISO model to assess how the output comparedwith the heel drop data. Although the ISO model has less mass than the test subject,displacements and damping ought to still give an indication of how well DYFAP and theISO model work together.The experimental results showed a significant response in the lower frequency range.Table 7.1: Parameter values for ISO human modelElement Mass Stiffness Damping(kg) (N/mm) (Ns/mm)1 62.0 62.0 1.462 13.0 80.0 0.93Figure 7.11: ISO, idealized lumped parameter vibratory human modelChapter 7. DYFAPFigure 7.12: Fourier spectrum of the 2x8 floor’s test response to a heel drop74The Fourier spectra of Fig. 7.12 and Fig. 7.13 show the magnitude of the human’s influence when compared to the corresponding bagdrop spectra of Fig. 7.5 and Fig. 7.6.It is difficult to identify individual frequencies that belong to the human but peaks, orsignificant trends at 4.75Hz, 8 — 9Hz, 11 — 12Hz, and 16 — 19Hz may be attributed to thehuman for they appear in both floor’s responses. The large peak at 19Hz may indicatea slightly stiffer human response on the I—Joist floor when compared to the 16— 17Hzactivity for the 2x8 joist floor. The ISO will contribute frequencies of 5.03Hz and 12.49Hzto the system’s response. As mentioned on page 66, filtering of the test’s velocity anddisplacement traces was required to satisfy the zeroed boundary conditions. The heeldrop data required filtering up to 2 — 3Hz but this was still well below any expectedfrequencies to be associated with the human or the floor.DYFAP displacement time histories using the ISO model were produced for each10N 8S‘— 5.——3S0Frequency (Hz)Chapter 7. DYFAP 7550-N40-Cl)30-°Frequency (Hz)Figure 7.13: Fourier spectrum of the I—Joist floor’s test response to a heel dropfloor. As shown in Fig 7.14 and Fig 7.15 the ISO results compared well with the testdata. The damping appears to be at about the proper level and the trace is appropriatelydominated by the low frequencies of the oscillator. The peak displacements are somewhathigh but this is due to the high stiffness of the model. They are still in better agreementwith the tests than the forcing function of Fig. 4.7. The forcing function failed to induceappropriate peak displacements. The forcing function exceeded the 2x8 and the I—Joistfloors’ displacements by 86% and 62% respectively. The high frequency oscillations of theforcing function traces also illustrate how poorly a forcing function does in representinga human impulse upon a lightweight floor. Previous research by Foschi and Folz foundthat the forcing function was adequate for heavy floors but that it was likely that as theoccupants’ mass became significant it would be necessary to replace the forcing functionwith an oscillator [Folz and Foschi, 1991]. Fig. 7.16 illustrates how the ISO oscillator andChapter 7. DYFAP 76Figure 7.14: Comparison of the I—Joist floor’s response to a heel drop with DYFAP’sresults using the ISO model or a forcing function, IJF6-2the forcing function produced equivalent responses in the case of a heavy floor. Eventhough Fig. 7.16 does not include a comparison with a test trace, the fact that theoscillator and forcing function traces are very different for the lightweight floors indicatethat the method of excitation does become important as the occupants’ mass percentageincreases.The ISO model was shown to be appropriate but it required six input parameters. Itwas not clear whether the two DOF model was necessary in order for DYFAP to produceaccurate results. The ISO model’s second frequency was not close to the frequency rangeof interest, therefore a single degree of freedom model may be sufficient. A single DOFmodel would be preferable since tuning the ISO model would involve adjusting up to sixparameters whereas a single DOF model would require only two since the mass would be3.0Ia)1.0a)C)-—1.0Test Peak Dispi. = 1.42mmDYFAP with ISO model, Peak Dispi. = 1.75mm- - -- Forcing Function, Peak Dispi. = 2.36mmInput: Joist #5 (x y)=(2140.0)Output: Joist 05 (x,y)=(3150.O)0.20 0.30Time (see)well known.Chapter 7. DYFAP 773.0 —-— Test Peak Dispi. = 1.41mm. (I DYFAP with ISO model. Peak Dispi. = 1.91mm- ‘I - - -- Forcing Function. Peak Dispi. = 2.62mm20— •1- 01,- Input: Joist #5 (x y)=(16000)- Output: Joist #5 (x.y)=(230.0)Q)1.0a)C)—-. 0.0I,•1.0 — 1111111 I 11111111111111111111111111 11111 I liii—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (sec)Figure 7.15: Comparison of the 2x8 floor’s response to a heel drop with DYFAP’s resultsusing the ISO model or a forcing function, SLF6-20.30-— I’0.20 - ‘ ‘ Heel Drop Forcing Function0 Human Modela)0.10Cl).— _ an—u.uu—0.10—0.20 - 1111111111111 I lilIllIllIlIllIll I liii I I I 11111111110.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Time (sec)Figure 7.16: Displacement time histories for composite floor (2% damping)Chapter 7. DYFAP 78miHcFigure 7.17: Single degree of freedom model‘7.5.2 Modelling a Heel Drop with a Single Mass OscillatorSince the floor parameters were set, only the oscillator parameters, mass, stiffness anddamping were available for any adjustments in an attempt to match the DYFAP outputwith the experimental heel drop data. The single DOF oscillator is shown in Fig. 7.17.The mass need not be changed since the mass of the heel dropper was a well knownparameter. The stiffness and damping were unknowns.DYFAP runs were done for a number of stiffness and damping combinations. Acalibration of the model was attempted using the 2x8 joist floor. The I—Joist floor wouldlater be used to verify the applicability of the model. The acceleration time histories werenot sensitive enough to changes in the oscillator parameters. DYFAP’s difficulty withhigh accelerations when using the oscillator as an exciter overshadowed any influencethat the oscillator parameters may have had. Damping has a larger influence than usualChapter 7. DYFAP 79Table 7.2: Grid of model parameters for oscillator calibrationStiffness Viscous Damping(N/mm) (Ns/mm).75 1.25 1.5 1.7520 x40 x x60 x x x80 x100 xupon the response in the case of the DYFAP model because of how it models the heeldrop action. The contribution to acceleration from the damping term in the equations ofmotion are quite substantial. The initial velocity that is given to the oscillator results ina high acceleration since the floor’s response to this initial velocity is quite rigid. Ratherthan allowing some local deformation, the model forces a large portion of the floor tomove immediately with an initial velocity.The displacement time histories were much better behaved. The peak displacement,damping and the dominant period were used as keys for the calibration. Table 7.2 showsthe grid of model parameters that were investigated. The results of these runs are shownin Fig. 7.19 and Fig. 7.18.Upon comparing the plots of Fig. 7.18 one can note that as the model’s stiffnessincreased so did the amplitude of the floor’s oscillations. A stiffer model induced higheraccelerations thus larger deflections. Similarly the plots of varying damping in Fig. 7.19show increased oscillatory motion for higher damping. A highly damped oscillator willinduce high accelerations just as the stiff model did. Damping and stiffness had the sameeffect upon the floor’s peak displacements. This was quite evident when comparing theirFourier spectra. The amplitude of the oscillator’s frequency became extremely large asChapter 7. DYFAP 80the viscous damping or stiffness was increased. The combination of k = 40N/mm andC = 1.25Ns/mm did the best at controlling the peak displacement without inducingtoo much oscillatory motion. For the I—Joist floor, in order to achieve better agreementwith the test’s peak displacement, a slightly stiffer model, k = 60N/inm, was required.It is known that the human body can adjust its stiffness by changing the degree ofmuscle tension. As previously proposed on page. 74 the “bouncier” I—Joist floor mayhave solicited a stiffer response from the human tester.7.5.3 DYFAP Results using the Calibrated Single DOF ModelAs with the bagdrop tests the following comparisons for the heel drop tests were madefrom the midfloor impact location #6. The Fourier spectra and the displacement tracesfor the cases in which the accelerometer was located on the same joist and two joistsaway from the impact site were used to compare with the DYFAP results.The displacement traces for both floors were compared with DYFAP results using either the oscillator or a forcing function. The three traces are plotted together in Fig. 7.20and Fig. 7.21. The time scale is labelled such that time= 0.0 marks the time of the heel’simpact. The leading time of 0.16 — 0.17 seconds corresponds to the time required for theheel dropper to shift his weight over to his heels. Starting from time= 0.0 the traces areplotted for 0.70 seconds.The oscillator’s response does a very good job with peak displacements but also inmatching the general low frequency nature of the trace.The displacement record derived from the floor’s response to the heel drop impacttwo joists away are shown in Fig. 7.22 and Fig. 7.23. As expected the displacement amplitudes have decreased substantially. The test traces now show more influence from thefloor’s natural frequencies. DYFAP did well in predicting an appropriately low peak displacement. DYFAP introduced too much response participation of the floor’s frequencies,I-.CD -1 CD CD 0I 04 .—I 0. —I2.0I.00.0—1.0Time(sec)—1.0—1.0Time(sec)—1.0I a.Time(sec)Time(sec)00.——1.0‘1I 0 I 0Q1:4 CD CD 0 CD 0 oq CD Cl) 0 0 Cl) oq 0 CD 0 Cl) 0—1.0—1.0Time(sec)Time(sec)I 0 QTime(see)Note:ReducedSWiness.40N/mmTime(sec)Chapter 7. DYFAP 833.0 -— Test Peak Dispi. = 1.4 1mmDYFAP Peak Dispi. = 1.20mm2.0—- - -- Forcing Function Peak Dispi. 2.62mmInput: Joist #5 (x y)=(1600.O)Output: Joist 05 (x.y)=(23001,‘ ,.‘ ,. - -- - -——1.0— i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (see)Figure 7.20: Comparison of the 2x8 floor’s response to a heel drop with DYFAP’s resultsusing the 1 DOF model or a forcing function, SLF6-13.0 -— Test Peak Dispi. = 1.46mmDYFAP Peak Dispi. = 1.42mmS 2 0 — ii - - - - Forcing Function Peak Dispi. = 2.36mm- IIInput: Joist #5 (x y)=(2 140.0)—1.0— I I I I I I I I”I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (see)Figure 7.21: Comparison of the 1—Joist floor’s response to a heel drop with DYFAP’sresults using the 1 DOF model or a forcing function, IJF6-2Chapter 7. DYFAP 84Figure 7.22: The 2x8 floor’s response two joists away from the heel drop, SLF6-lwhich is particularly evident for the 2x8 joist floor in Fig. Comparison of Results for the Case of Two OccupantsUp to this point all comparisons had been made for the case of only one occupant.Generally, a vibration is reported to be annoying for a passive occupant rather than theactive occupant. The goal of any design criteria or the tolerance charts of Chapter 1 isto satisfy the passive occupant.The test series #4 dealt with the case of an observer standing on the same joist oron the joist adjacent to the heel dropper. As shown by Figs. 7.24—7.27, the locationof the observer relative to the heel dropper affected the output. DYFAP did well inmatching the tests’ peak displacements regardless of where the observer was located.The oscillators used for each floor were the same ones that were derived previously0.40.3..? 0.2S 0.1C)0-0.1—0.2Time (see)Chapter 7. DYFAP 85Figure 7.23: The I—Joist floor’s response two joists away from the heel drop, IJF6-1(see pg. 78).Lastly, to relate this data with the subjective rating schemes of Wiss and Parmalee andChui, the displacement—frequency and the RMS accelerations were calculated. Fig. 7.28shows the test results plotted on the Wiss and Parmalee scale. The four data pointscorrespond to a passive occupant standing on the same or adjacent joist relative tothe heel dropper for each floor. Both floors received appropriate ratings of “stronglyperceptible”. The RMS accelerations for the plot of Fig. 7.29 were calculated fromfrequency weighted acceleration traces as recommended by the ISO [ISO 2631]. Thefrequency weighting that was followed is listed in Table 7.3. The ISO guidelines donot specify a weighting factor for frequencies greater than 80Hz due to lack of test data.Three of the four tests produced RMS values that were greater than Chui’s recommendedvalue of 0.375m/s2 for a satisfactory floor. The I—Joist floor was noticeably “bouncier”0.20.1-I-)0.0a)C).— ——0.20.10 0.20 0.30Time (see)Chapter 7. DYFAP 862.0 -Test Peak Dispi. = 1.30mm—1.0— i i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (sec)Figure 7.24: Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on the same joist as the heel dropper, 2x8 floor2.0 -Test Peak Dispi. = 1.13mmDYFAP Peak Dispi. = 0.96mm1.0— i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (see)Figure 7.25: Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on a joist adjacent to the heel dropper, 2x8 floorChapter 7. DYFAP 872.0 -I I, 1\ Test Peak Dispi. = 1.62mmDYFAP Peak Dispi. 1.76mm. 1.0— Input: Joist #5 (x,y)=(2140,O)Output: Joist #5 (x,y)(2640.O)EC.)/ .— •. . ./ --U) /-/— I I I I I I I I I I I I I f I I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (sec)Figure 7.26: Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on the same joist as the heel dropper, I—Joist floor2.0 -Test Peak Dlspl. = 0.98mmDYFAP Peak Dispi. = 0.79mmi::__________—1.0— , I I j I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I—0.20 —0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70Time (sec)Figure 7.27: Comparison of DYFAP with the 1 DOF model and the test data for thecase of a passive observer located on a joist adjacent to the heel dropper, I—Joist floorChapter 7. DYFAP 88Table 7.3: Frequency weighting factors used for calculation of RMS accelerationFreq. Range Weighting FactorOHz< fo <4Hz fo/34Hz< fo <8Hz 18Hz< fo <80Hz 8/fo80Hz< fo <500Hz 0.1than the 2x8 joist floor. The RMS plot does illustrate this distinction but the I—Joistfloor’s values were expected to be somewhat higher.Based on survey results for living room floors, Onysko has proposed static criteria toensure satisfactory performance [Onysko, 1986]. The maximum floor system deflection,under a static concentrated load of lkN, at the joist’s midspan, should be limited by:(5mL2 = (6.7/L’.22)mmforL > 3.Om (7.1)As a final check, since the floors were designed as per the NBC 1990, which incorporateda form of Onysko’s criteria, the program FAP was used to determine the floors’ staticdeflection under a concentrated load of lkN. The criteria limits the 2x8 and I—Joistfloors’ deflection to 1.69mm and 1.16mm respectively. The floors were satisfactory whenbased on FAP’s results of 0.90mm and 0.83mm for the 2x8 and I—Joist floor respectively.Chapter 7. DYFAP 891—— R=4Cl)________aR=3C) 0.1U) R=2.—sification< 0.011 Imperceptible2 Barely Derceptible& R=5.08(FA/D°217)°65 3 Distinctly perceptible4 Strongly perceptible5 Severe0.001— I I I I I I I II4 I I0.01 0.1Damping, Re:Critical DampingFigure 7.28: Series #4 test data, a:SLN6, b:SLS6, c:IJN6, d:IJS6, plotted on Wiss andParmalee’s mathematical model0.800CE)SDISTURBING0.500 Unacceptable to all occupants0* dPERCEPTIBLE0.375 a Unacceptable to most occupants* b SUGHTLY PERCEPTIBLE—0.200 Acceptable to many occupantsC)NOT PERCEPTIBLE0.100 Acceptable to nearly all occupantsS.Figure 7.29: Series #4 test data, a:SLN6, b:SLS6, c:IJN6, d:IJS6, plotted on Chui’s RMSscaleChapter 8Summary and Conclusion8.1 SummaryLightweight wooden floors have become prone to poor vibrational performance. Theintroduction of lighter, longer spanning structural members caused the floor’s naturalfrequencies to draw closer to those that have been shown to disturb occupants. Previouswork had been done in evaluating floors by a subjective procedure. People were asked torate a floor’s level of vibration on scales of perception (acceptability).For this thesis, tests measuring the dynamic response of lightweight wooden floorswere conducted for the purpose of determining the applicability of the programs NAFFAPand DYFAP for lightweight floor systems. The floor vibration problem was addressed byanalytical modelling of a floor’s dynamic response. The floors were impacted by meansof a hammer, dropping a sandfilled bag and a human heel drop. These three types ofexcitation produced data that was useful in verifying and establishing the applicabilityof the programs NAFFAP and DYFAP.8.2 Concluding RemarksNAFFAP’s output includes all possible frequencies for the defined floor system. Anoccupant is generally only aware of vertical motion, so many of the frequencies associatedwith degrees of freedom such as joist rotation are insignificant. The I—Joist floor, whichhad a relatively low joist torsional stiffness, produced many such frequencies. The output90Chapter 8. Summary and Conclusion 91for the 2x8 joist floor was relatively clean of these torsional frequencies.NAFFAP’s first four frequencies matched well with those from the tests. The modeshapes were useful for identifying the first two or three important frequencies but thehigher modes became difficult to distinguish from insignificant modes. It was unclearfrom NAFFAP’s output which were the predominant frequencies. The spectral analysisoffered by NAFFAP is not sufficient by itself because its output of frequencies and modeshapes do not allow one to fully realize a particular frequency’s level of participation.DYFAP’s time histories allows one to establish which are the more important frequencies, thus making it an appropriate program to follow a NAFFAP study. A Fourierspectrum of a DYFAP acceleration trace shows oniy those frequencies that are strongand are associated with vertical motion. Since DYFAP and NAFFAP employ the samefloor parameters, their calculated natural frequencies are also the same. Upon comparingthe two programs’ frequency output, one can identify which are the predominant naturalfrequencies.Reliability of DYFAP’s peak accelerations depends upon how well the excitationsource was modelled and what type of floor excitation was used. DYFAP’s success withthe bagdrop test showed that the bagdrop impulse was easily modelled as a discretizedforcing function. Difficulties with the heel drop simulation using oscillators led to problems of extremely high accelerations. The oscillator results were only comparable usingthe displacement time histories since they were less susceptible to gross errors resultingfrom a poor exciter model.The ISO have adopted a two—mass oscillator to represent a human. A DYFAP runwith the ISO model was performed to show DYFAP’s performance with this standardizedoscillator model. DYFAP managed to provide reasonable agreement with the test’s peakdisplacement. The dominant low frequency nature of the test traces were successfullyreproduced by DYFAP.Chapter 8. Summary and Conclusion 92A simpler human model, a single mass oscillator, was then used to gage how wellDYFAP could match the experimental heel drop data. Runs were made for a range ofoscillator damping and stiffnesses. The significant influence upon the floor’s response ofaltering the oscillator’s damping or stiffness parameters made convergence possible toa “best fit” model. It was shown that the simpler single degree of freedom model wassufficient for DYFAP to approximately reproduce the heel drop test displacement traces.Previous research by Folz and Foschi [Folz and Foschi, 1991] demonstrated that aforcing function was sufficient to represent a heel drop impulse for the case of a heavyfloor system. It was easily established that for lighter floors, where the occupants’ massis up to 30% of the system’s mass, a forcing function was insufficient in modelling a heeldrop. The mass and low natural frequency of the oscillator were important elements forthe successful matching of the test’s displacement time histories.8.3 Further Areas of StudyThe preliminary testing with multiple occupants showed that their relative position onthe floor was an important factor for determining the response. DYFAP’s success withthe displacement time histories show promise for multiple occupants. The next logicalavenue of research would be to assess DYFAP’s ability to reproduce test data of two ormore occupants, both passive and active, distributed randomly on the floor. Modellingof a heel drop impact may be improved by developing a time dependent damping and/orstiffness capacity for the oscillators. Modelling the impact as more of a plastic ratherthan a rigid impact may prove beneficial.Bibliography[Allen and Rainer, 1976] Allen, D.E. and Rainer, J.H., “Vibration Criteria for LongSpan Floors”, Canadian Journal of Civil Engineering, Vol.3, No. 2, June, 1976, pp. 165—173.[Billmeyer, 1984] Billmeyer, Fred W., “Textbook of Polymer Science— ThirdEdition”, John Wiley & Sons, USA, 1984.[Chui, 1986] Chui, Y.H., “Vibrational Performance of Timber Floorsand the Related Human Discomfort Criteria”, Journal ofthe Institute of Wood Science, Vol. 10, No. 5, 1986, pp.183—188.[Chui, 2/86] Chui,Y.H., “Evaluation of Vibrational Performance ofLight-Weight Wooden Floors: Determination of Effects ofChanges in Construction Variables on Vibrational Characteristics”, TRADA Research Report 2/86, HughendenValley, TRADA, U.K.[Chui, 15/86] Chui,Y.H., “Evaluation of Vibrational Performance ofLight-Weight Wooden Floors: Design To Avoid AnnoyingVibrations”, TRADA Research Report 15/86, HughendenValley, TRADA, U.K.[Chui, 1988] Chui,Y.H., “Evaluation of Vibrational Performance ofLight-Weight Wooden Floors”, Proceedings of the 198893Bibliography 94International Conference on Timber Engineering, Vol. 1,September 19—22, 1988, pp. 707—715.[Chui and Smith, 1991] Chui, Y.H. and Smith, I., “Method of Wooden FloorConstruction for Minimizing Levels of Vibration”, CMHC1989 External Research Program Project, January, 1991.[Clough and Penzien, 1975] Clough, R., and Penzien, J., “Dynamics of Structures”,McGraw—Hill, USA, 1975, pg. 94.[Cowie, 1973] Cowie, J.M.G., “Polymers: Chemistry and Physics ofModern Materials”, Intext Educational Publishers, USA,1973.[Filiatrault and Folz, 1989] Filiatrault, A. and Folz, B., “NAFFAP: Natural FrequencyFloor Analysis Program USER’S MANUAL”, version 2.0,Dept. of Civil Engineering, U.B.C., September 1989.[Filiatrault et al, 1990] Filiatrault, A., Folz, B. and Foschi, R.O., “Finite—StripFree—Vibration Analysis of Wood Floors”, Journal ofStructural Engineering, Vol. 116, No. 8, August, 1990,pp. 2127—2142.[Folz and Foschi, 19911 Folz, B. and Foschi, R.O., “Coupled Vibrational Responseof Floor Systems with Occupants”, Journal of EngineeringMechanics, Vol. 117, No. 4, April, 1991, pp. 872—892.[Foschi, 1969] Foschi, R.O., “Buckling of the Compressed Skin of a Plywood Stressed-Skin Panel with Longitudinal Stiffeners”,Bibliography 95Department of Fisheries and Forestry Canadian ForestryService Publication no. 1265, 1969, pp. 49—50.[Foschi, 1982] Foschi, R.O., “Structural Analysis of Wood Floor Systems”, Journal Structural Division, ASCE, Vol. 108, No.7, July, 1982, pp. 1557—1574.[Grether, 1971] Grether, Walter F., “Vibration and Human Performance”,Human Factors, Vol. 13, No. 3, June, 1971, pp. 203—216.[Malik and Nigam, 1987] Malik, M. and Nigam, S.P., “A Study on a VibratoryModel of a Human Body”, Journal of Biomechanical Engineering. Trans. ASME, Vol. 109, May, 1987, pp. 148—153.[Ohisson, 1982] Ohisson, S., “Floor Vibrations and Human Discomfort”,Ph.D. Thesis, Chalmers University of Technology, Div. ofSteel and Timber structures, Gothenburg, Sweden, 1982.[Onysko, 1986] Onysko, D., “Serviceability Criteria for Residential FloorsBased on a Field Study of Consumer Response”, Formtek Canada Corp., Ottawa Laboratory, Ottawa, Ontario,1986.[Pechhold, 1973] Pechhold, W., “Deformation of Polymers as Explained bythe Meander Model”, Battelle Institute Materials ScienceColloquia, September 11—16, 1972, pp. 301—313.[Pernica and Rainer, 1986] Pernica, C. and Rainer, J.H., “Vertical Dynamic Forcesfrom Footsteps”, Canadian Acoustics, Vol. 14, No. 2,April, 1986, pp. 12—21.Bibliography 96[Polensek, 1970] Polensek, Anton, “Human Response to Vibration ofWood-Joist Floor Systems”, Wood Science, Vol. 3, No.2, October, 1971, pp. 111—119.[Rainer, 19801 Rainer, J.H., “Dynamic Tests on a Steel-Joist Concrete-Slab Floor”, Canadian Journal of Civil Engineering, Vol.7, No. 2, June, 1980, pp. 213—224.[Sazinski and Vanderbilt, 1979] Sazinski, R.J. and Vanderbilt, M.D., “Behaviour and Design of Wood Joist Floors”, Wood Science, Vol. 11, No.4, April, 1979, pp. 209—220.[Wiss and Parmelee, 1974] Wiss, J.F. and Parmelee, R.A., “Human Perception ofTransient Vibrations”, ASCE Journal of the StructuralDivision, Vol. 100, No. ST4, April, 1974, pp. 773—787.[CMHC, 1984] CMHC, “Canadian Wood—frame House Construction”,Publication NHA 5031 M 08/84, Canada Mortgage andHousing Corporation, Ottawa.[TRUS JOIST] “Residential Products Reference Guide”,TRUS JOIST CANADA LTD., January, 1991.[VU—POINT, 19881 “VU—POINT: A Digital Data Processing System for IBMPC/XT/AT and Compatible Personal Computers”, version 1.21, S—CUBED, a division of Maxwell Laboratoriesinc., March 1988.Bibliography 97[ISO 2631] “Guide for the evaluation of human exposure to whole-body vibration”, (1987) Standard No. 2631, Tnt. Organization for Standardization, Geneva, Switzerland.[ISO 5982] “Vibration and Shock-Mechanical driving pointimpedance of the human body”, (1987) Standard No.5982, Tnt. Organization for Standardization, Geneva,Switzerland.


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