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Publisher’s version / Version de l'éditeur:

Building Acoustics, 11, September 3, pp. 1-27, 2004-09-01

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Estimating in-situ material properties of a wood joist floor: Part 1 -

Measurements of the real part of bending wavenumber and modulus of

elasticity

Nightingale, T. R. T.; Halliwell, R. E.; Pernica, G.

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Estimating in-situ material properties of a wood joist floor: Part 1 –

Measurements of the real part of bending wavenumber and

modulus of elasticity

Nightingale, T.R.T.; Halliwell, R.E.; Pernica, G.

NRCC-46882

A version of this document is published in / Une version de ce document se trouve dans :

Building Acoustics, v. 11, no. 3, Sept. 2004, pp. 1-27

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Estimating In-Situ Material Properties of a Wood Joist Floor:

Part 1 -- Measurements of the Real Part of Bending Wavenumber

T.R.T. Nightingale1, R.E. Halliwell1, and G. Pernica2

National Research Council Canada, Institute for Research in Construction,

1

Acoustics Laboratory, 2Building Envelope and Structures Laboratory

1200 Montreal Road, Ottawa, Ontario, K1A 0R6, Canada. e-mail of corresponding author: trevor.nightingale@nrc.ca

Abstract

This paper outlines and evaluates a non-destructive experimental technique used to obtain in-situ measures of the real part of the bending wavenumber in the two principal directions of a wood joist floor. In-situ measured wavenumbers are compared to those obtained from beam samples cut from the floor sheathing to identify the frequency range when the joists significantly affect vibration response in the floor sheathing.

Wavenumber measurements confirm the highly orthotropic nature. Measurements indicate that the wavenumbers in the direction parallel to the joists are a function of location between the joists. Lower wavenumbers in this direction can be expected closer to the joists. Some discussion is given regarding the rates of attenuation with distance that can be expected parallel and perpendicular to the joists given the variation in wavenumbers, however this will be the topic of a separate paper, Part 2.

1. Introduction

In wood joist floors, both with and without floor toppings, there can be a pronounced difference in the rate of vibration attenuation per unit distance parallel to the joists as compared to perpendicular [1]. Often there is considerably greater attenuation in the direction perpendicular to the joists. Previous works [2] had related the difference in attenuation to the periodic construction of the floor and additive attenuation as the wave front propagates across the series of plate-rib junctions. However, it is possible that the orthotropic nature of the floor created significantly different wavenumbers in the two directions, which would also be a contributing factor, or perhaps the dominant factor in

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the high frequencies where the floor sheathing is only effectively point connected to the joists.

In Part 1 of this paper, in-situ measures of the real part of the wavenumber for out-of-plane displacement, which is the combined effect of pure bending and shear waves, are used to gain insight into the behaviour of the floor at various frequencies and to

qualitatively explain the significantly different rates of attenuation in two orthogonal directions – parallel and perpendicular to the joists. In-situ estimates of the wavenumber are compared to those measured for the individual components (floor sheathing and joists) in isolation to identify the frequency range where the joists significantly affect vibration response in the floor sheathing. A series of assumptions are introduced to illustrate how these wavenumber estimates might be used to obtain estimates of the modulus of elasticity, although this is of secondary importance. In Part 2 of this paper the real part of the bending wavenumbers are used in conjunction with vibration mapping to obtain in-situ estimates of the loss factor for the floor.

Henceforth, we will refer to the real part of the wavenumber,

e

( )

k

B , as simply the

wavenumber recognizing that it is the component associated with the measured phase in a particular direction. The measured wavenumber is more pertinent than the

wavenumber for “pure” bending waves because it is the phase and group speeds of the measured (or effective) wave that are related to power transmission by out-of-plane displacements [3].

A review of the literature suggests that there are two methods to determine

wavenumber. The first method is implicit to structural intensity methods [4] where a finite difference approach is used to estimate the second-order spatial derivative of the transverse plate displacement, which is proportional to the square of the wavenumber. Evaluating the second order spatial derivative requires three closely spaced

measurement positions, all in a straight line. With increasing frequency, the ratio of position spacing to wavelength will become large and significant errors in the finite difference approximation arise. A correction [5] has been suggested to help counter this high-frequency effect. Conversely, when the spacing becomes very small compared to the wavelength then positioning uncertainty, which is equivalent to an induced phase mismatch, often determines the measurement accuracy [6]. The working frequency range is defined as the range where neither the high nor the low-frequency sources of

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measurement uncertainty are significant. Often this range is rather limited and multiple measurements with different point spacing are needed to cover the frequency range of interest.

The second method uses the measured phase difference between two points and relates it to the distance separating them; as such, it is simpler to apply. Implementation of this phase gradient method can take different forms for low and high frequencies. For low frequencies [3] a reference accelerometer is placed at or near the source and a field accelerometer is moved to various positions to assess the phase change with direction. The effect of the near field at the reference accelerometer is considered explicitly in the formulation derived using thin plate theory, but the field accelerometer is assumed to be free of near field effects. Near field effects are generally considered to be important when the product of the wavenumber and the distance between the source and field point is much less than unity.

The high-frequency version of the second method has been used by several authors [7, 8] to study orthotropic plates. The method uses the same basic phase difference approach except that there are two field accelerometers, one of which becomes the reference. The wavenumber is given by,

( )

1 2 1 2 B r r k e − − = ℜ

φ

φ

Eqn. 1

where φ is the phase and r is the distance from the source, and the subscript 1 refers to the reference accelerometer and the subscript 2 refers to the field accelerometer. This method can be used when both accelerometer positions are free of the near field effects around the source (i.e., well away from the source), not in the reverberant field and positions are colinear with the source. The reverberant field has the undesirable effect of masking the phase change of the direct field and several authors [9,10] have reported that when the reverberant field is significant the phase change measured using cross-spectrum analysis increases more rapidly than if there were just a direct field. The presence of a reverberant field will tend to lead to overestimates of the wavenumber. Others have devoted considerable effort to evaluating and removing the effect of the reverberant field [9,7]. However, this technique will not be discussed here in detail since the plate under consideration has a rather high internal loss factor ( >0.03) and the plate

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is very large compared to the bending wavelengths over the frequency range of interest in building acoustics (100 to 5000 Hz) so reflections from the boundaries will have minimal effect. This was confirmed by examining the phase draw-away curves. While both the finite difference and the phase gradient approaches require very good phase matching between the measurement channels, the finite difference approach is considerably more sensitive. This coupled with the fact that finite difference

approximations lead to substantial errors for high frequencies makes the finite difference approach less desirable.

Because all methods assume a linear dependence of phase change with distance, uncertainties are introduced if the plate is not homogeneous. Despite this requirement, the method has been successfully applied to brick walls [7] and composite panels [8], both of which can be viewed as orthotropic plates that are strictly not homogeneous. A joist floor on the other hand consists of an orthotropic plate (floor sheathing) that is rib-stiffened (by the joists) so it is likely to be considerably more orthotropic and

inhomogeneous than panel or plate-like assemblies. To examine how the joists might affect the wavenumber measurements and how the wavenumber method might be applied to joist floor, equivalent plate theory [11] is used to create a framework to represent the floor (a rib-stiffened plate) as an equivalent orthotropic plate. We begin with the expression for free wave motion in a thin orthotropic plate, which is given by [11],

(

)

0

t

s

y

B

y

x

G

2

B

2

x

B

2 2 4 4 y 2 2 4 y xy 4 4 x

=

+

+

+

+

ζ

µ

ζ

ζ

ρ

ζ

Eqn. 2 where

(

)

(

)

12

h

G

G

and

1

12

h

E

B

1

12

h

E

B

3 xy yx xy 3 y y yx xy 3 x x

=

=

=

,

,

µ

µ

µ

µ

Eqn. 3

and where Bx’ and By’ are the bending stiffness in the two principal directions indicated

by the subscripts, E is Young’s modulus in the indicated direction, h is the thickness of

the plate, ρs is the surface density, Gxy is the shear modulus in the xy plane, and µ is

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(

)

ik (x ( ) y ( )) i t

e

Ae

t

y

x

B θ θ ω

ζ

cos sin

,

,

=

+ Eqn. 4

where ω is the angular frequency. Figure 1 shows the coordinates system and its relation to the joist orientation. The two principal directions of an orthotropic plate are the directions where the bending stiffness is the greatest and smallest, i.e., parallel and perpendicular to the joists in the case of framed floor.

point source and origin for draw-away lines L1 and L2 ∆r

x

y

L2 draw-away line on a joist L3 draw-away line in a bay between joists L1 draw-away line

perpendicular to joists

point source and origin for draw-away line L3 wood-I joists

Figure 1: Sketch showing the layout of the three draw-away lines to estimate the change in phase with distance. Two source locations were used. Lines L1 and L2 used a common

source location while L3 had a separate one. The spacing between points, ∆r, is 100 mm

which defines 31.4 as the maximum wavenumber that can be measured. Where possible, the source is located near the center of a draw-away line so that two sets of data are obtained and analyzed independently to assess the reproducibility of the method.

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Substitution of Eqn. 4 and Eqn. 3 into Eqn. 2 gives an expression for the angular dependant wavenumber, KB( ),

( )

( )

(

)

( )

( )

( )

4 1 4 y 2 2 y xy 4 x 2 B

B

G

2

B

2

B

s

k

+

+

+

=

θ

θ

θ

µ

θ

ρ

ω

θ

sin

sin

cos

cos

Eqn. 5

where is the angle with respect to the x-principal direction and the denominator is the angular dependant bending stiffness B( ). Because Eqn. 5 is orthogonal and the coordinate system has been chosen to align with the principal directions (see Figure 1) this allows the bending stiffness in each principal direction to be fully described by the wavenumber in the same direction.

It is likely material properties of the sheathing, joist spacing, and spacing of the screws connecting the sheathing to the joists will define three frequency regions where the vibration response of the floor will be different. These are now discussed and related to possible wavenumber measurement strategies.

Frequency Region I: Here the wavelength in the floor sheathing is considerably smaller than the spacing between the joists and the screws fastening the sheathing to the joists. This defines the region where the sheathing is effectively only point connected [12] to the joists thereby allowing vibration energy to propagate from one joist bay to another between the fastening points in the sheathing [13] with considerably less attenuation than the low frequencies where the sheathing appears to be line connected [14]. In this frequency range, it is assumed that the effect of the joists can be largely ignored and the floor response is that of a reasonably homogeneous orthotropic plate and the vibration response is determined by the properties of the floor sheathing. With this assumption the measured wavenumbers will be primarily due to the sheathing and if the phase measurements were made along a line that is parallel to a principal axis of the floor sheathing then the orthogonality of Eqn. 5 allows one to relate the measured wavenumber to the bending stiffness of the sheathing in that direction.

Frequency Region II: Here the bending wavelength is long enough that floor sheathing appears effectively line connected to the joists, and the joist spacing is at least one half wavelength. It is often assumed that the transition between line and point connected occurs when the wavelength is less than twice the fastener spacing [12]. Orrenius and

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Finnveden when studying line connected rib-stiffened plates in ships showed [15] that , stiffening ribs provide simply supported boundary conditions which effectively divide the structure into a series of subpanels each of which can be thought of as being a

waveguide for propagation parallel to the joists. The plate wavenumber corresponds to that of a thin beam with a stiffness equal to that of the orthotropic plate in the direction of the ribs. They also showed that joists play prominently below the cut-on of the first cross mode between the joists given by,

2 B 2 1 y c

k

y

s

h

B

=

ρ

ω

' Eqn. 6

where kBy is the plate wavenumber in the y-direction, normal to the ribs. For Region II the

measured wavenumbers are expected to be meaningful for the direction parallel to the joists, but in the direction perpendicular to the joists the floor will appear less

homogenous and the results may be affected.

Frequency Region III: Below the cut-on of the first cross mode between the joists, the wavelength will be greater than twice the joist spacing so it is likely that the sheathing and joists move together. This region is considerably more complex than Regions III or II because the stresses in sheathing decay with distance from the rib and the concept of an effective joist width needs to be introduced [15]. (The joists may also affect the distribution of stresses in region II). The properties of Region III are such that the

twisting moment relation (Mxy=Myx) may not hold [16] so the orthogonality experienced in

Regions II and III no longer holds and wavenumber in the principal direction is no longer strictly determined by the bending stiffness in that direction alone.

In this paper wavenumbers are measured using a phase gradient approach, which is a variant of the low and high-frequency measurement methods. It does not require phase matched measurement channels to obtain estimates of the in-situ wavenumbers. The method is used to measure the wavenumbers in two orthogonal directions on joist floors. Data are then compared to wavenumbers measured from beam specimens cut from the floor sheathing to assess the effect of the joists on the vibration propagation parallel and perpendicular to the joists.

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2. Implemented Method

The implemented method is similar to the low-frequency method in that a reference accelerometer is placed at the source. However, rather than a single field position, the field accelerometer is systematically moved away from the source along a line, as shown in Figure 2. For each field position, the phase difference relative to the reference point (at the source) is computed and no effort to correct for the near field at the source is made. The slope of the best-fit line of measured phase change with field point distance approximates,

( )

dr d k e B =

φ

ℜ Eqn. 7

and gives a direct estimate of the wavenumber.

The draw-away approach has several benefits. First, there is no need to correct for the phase mismatch between the measurement channels and the near field at the reference point. The phase match is constant for all measurements, and does not affect the slope of the best-fit line (estimate of the wavenumber), only the intercept. Second, the draw-away approach enables the user to assess the suitability of the data i.e., the presence of abrupt phase shifts due to discontinuities and possibly the presence of other near fields. This enables the user to compute the slope using data where these effects are not present.

The draw-away approach also allows the absolute phase difference between the measurement points to be determined from cross spectra data obtained from an FFT analyzer. The phase of the cross-spectrum is always in the principal branch of the arctangent function, which for the B&K 2144 analyzer used in this study was –π to π radians. The device can not determine the absolute, or “unwrapped”, phase relative to the source because the motion of field points located at nλ+∆r will have the same relative phase but will be wrapped by 2nπ radians where is n is an integer, where λ is the wavelength. However, by selecting a series of closely spaced measurement positions that extend away from the source, as shown in Figure 2, it is possible to track the incremental increase in phase between consecutive points. By observing when the phase has changed sign from positive to negative, it is possible to unwrap the phase by

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adding nπ radians to the reported value, where n is an integer equal to the number of times the phase has changed from positive to negative.

In this paper, we have chosen the x and y axes as being parallel and perpendicular to the joists, respectively, which allows us to make use of the orthogonality of thin orthotropic plates (i.e., Eqn. 5).

point source

∆r

x y

Figure 2: Sketch showing the layout of measurement points along a draw-away line from the source for an orthotropic plate. Each draw-away line is chosen parallel to a principal direction of the plate, which simplifies resolving the stiffness in an arbitrary direction. The spacing between points, ∆r, determines the highest frequency that can be measured because it must be much smaller than the bending wavelength in order to accurately unwrap the phase.

2.1 Measured Wavenumbers and Pure Bending

It is instructive to examine the effect of rotary inertia and shear deformation for the floor sheathing because in-situ wavenumbers measured on the floor will be compared to those obtained from small beam samples. For an arbitrary plate or beam, the transverse or out-of-plane displacement will be the result of a contribution from bending and shear waves. The importance of shear waves in determining the effective phase speed (and hence measured wavenumber) is determined by the relative speeds of the pure bending wave and the shear wave. Determining the pure bending wave speed or wavenumber has received considerable attention.

Rindel [3] provided a modified dispersion relationship that related the measured wave speed to the pure bending wave speed using the shear wave speed, which can be written in terms of wavenumbers

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3 shear B 3 B k k k pure measured       + =

α

Eqn. 8

where α is the constant which Rindel took to be unity. However, Mindlin [17] showed that for high frequencies the measured wavenumber tended to approach values that were higher than the shear wavenumber so the factor α was introduced which is a function of the Poisson’s ratio of the material, (α=0.841 for a Poisson’s ratio of 0.3 which is typical for wood materials). Estimates of the shear wavenumber can be obtained from

2 1 2 m shear

G

k





=

ρ

ω

Eqn. 9

where

ρ

m is the mass density, ω is the angular frequency, and G is the shear modulus.

Thus, the measured wavenumber will tend to overestimate the pure bending wavenumber.

Beginning with the wave equation, Timoshenko [20] provided expressions to correct for the effect of rotary inertia and shear deformation, which when rewritten in terms of the wavenumbers, are 1 2 2 2 B inertia B h 12 1 k k pure measured −       − =

λ

π

Eqn. 10 and 1 2 2 2 B shear B G E 1 h 12 1 k k pure measured −             ℵ + − =

λ

π

Eqn. 11

respectively, where h is the thickness of the beam, E is the modulus of elasticity, λ is the

measured wavelength, and ℵis taken to be 1.2. Inspection indicates that shear

deformation will be considerably more important than inertia because the ratio

G E ℵ

will typically be much greater than unity. Roelens et al. [7] conducted a similar analysis to provide modified dispersion relationships for thick isotropic plates.

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0.4 0.5 0.6 0.7 0.8 0.9 1 100 1000 10000 Frequency, [Hz] Correcti on, [-] Shear Deformation (Timoshenko) Rotary Inertia (Timoshenko) Shear Deformation (Modified Rindel)

OSB beam sample

Material properties

E = 6.8x109 pa

G = 2.8x108 pa

thickness = 0.0184 m poisson's ratio = 0.3

For the special case of an isotropic beam having Poisson’s ratio of 0.3, a ratio of E to G

of 2.6 and ℵ taken as 1.2 Eqn. 11 reduces to the commonly used relationship [18] that

the wavelength must be greater than six times the thickness error if the underestimation of the pure bending wave speed is to be less than 10 percent.

Figure 3 shows the correction that the measured wavenumber should be multiplied by to remove the effect of the indicated factor – shear deformation or rotary inertia. The corrections are for the 18mm thick Oriented Strand Board (OSB) commonly used as the sheathing on wood joist floors. Inspection indicates that for this material there should be no need to correct for rotary inertia. However, shear deformation correction factors are significant, with the results of Eqn. 8 and Eqn. 11 in close agreement despite the very different approaches used. At about 1.5 kHz the measured wavenumber will be about 10% greater than would be expected if the motion were pure bending.

Thus, wavenumber measurements on the OSB can be considered to be reasonable estimates of the pure bending wavenumber for frequencies below 1.5 kHz, if a 10% overestimation can be tolerated. For frequencies above about 1.5 kHz, the measured wavenumber will be a poor estimate of the pure bending wavenumber and material properties derived from wavenumbers that have not been corrected for shear deformation will be significantly underestimated.

Figure 3: Correction that should be applied to the measured wavenumber to obtain an estimate that is corrected for deformation and rotary inertia. The shear deformation correction denoted as “Timoshenko” was obtained from Eqn. 11, while the other denoted by “modified Rindel” was obtained from Eqn. 8.

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3. Measurements and Results

In this section, wavenumber data for a wood joist floor in the two principal directions are presented and used to assess the effect of joists in the vibration response of the floor sheathing. In the high frequencies where the floor sheathing appears as an orthotropic but reasonably homogeneous plate, estimates of Young’s modulus are calculated for the two principal directions.

The floor system consisted of 1.21 x 2.42 x 0.018 m OSB sheets applied with the long axis at right angles to 305 mm deep wood I-beam joists spaced 406 mm on center. The gypsum board ceiling was attached to the bottom of the joists via resilient channels spaced 610 mm on center. Figure 4 shows a section through the floor with complete construction details given in Appendix A.

Figure 4: Sketch showing the wood joist floor assembly. Construction details are provided in Appendix A.

The floor was excited at a single point using a Wilcoxon shaker (Model F3/F9) and the phase difference between a reference accelerometer placed near the source and a series of field positions was measured. Both the reference accelerometer and the field accelerometers were PCB 352C33 devices (mass 4 grams) having low impedance compared to the point impedance of the plate. The phase difference between the two points was obtained in fractional octave bands from the phase angle of the cross-spectrum using a B&K 2144 dual channel cross-spectrum analyzer. One-third octave bands were chosen because they are consistent with the intended application of the

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Narrow band data could have been averaged manually [9] in the frequency domain to obtain results for an arbitrary fractional octave bandwidth.

The floor is assumed to be orthotropic with the axis of greatest stiffness (lowest wavenumber) parallel to the direction of the joists and is taken as the x-principal direction. The direction perpendicular to the joists is taken as the y-principal direction. Parallel to the joists one might expect the effective stiffness to increase as the line defined by the field points and source is moved toward a joist, with the maximum

occurring immediately above a joist. The minimum should be obtained midway between two joists. To test this hypothesis two draw-away lines were measured parallel to the joists – one immediately on top of a joist and the second midway between the joists, as shown in Figure 1. In addition to these, a third draw-away line was oriented

perpendicular to the joists. Table 1 provides a summary of the orientation of the draw-away lines with respect to floor elements as well as the x and y-principal directions.

Orientation with respect to Draw-away line of

Figure 1

Floor joists Principal direction of floor

Long axis of OSB sheets

Line 1 perpendicular Y parallel

Line 2 parallel

and on a joist

X perpendicular

Line 3 parallel

and between joists

X perpendicular

Table 1: Orientation of the three draw-away lines of Figure 1 with respect to the elements forming the floor and the principal directions of the floor.

The spacing between the field points must be chosen such that the phase can be

accurately unwrapped. This requires that there be at least one measurement point each half wavelength otherwise the change in sign of the phase will go undetected. In this

study the spacing between points, ∆r, was 100 mm which effectively defines 31.4 as the

largest wavenumber that can be accurately measured. Consequently, wavenumber data and all calculated quantities will not be given for frequencies where the wavenumber is greater than 30. Wavenumber estimates for higher frequencies can be obtained by appropriately reducing the spacing between measurement points, although this quickly becomes impractical using accelerometers, and requires the use of a laser vibrometer.

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3.1 Measured wavenumbers

Figure 5 shows the measured wavenumbers obtained from draw-away data collected

along L1 draw-away line, which is perpendicular to the joists. Two sets of data are given

– one for points to the left of the source (y<0), and one for points to the right of the source (y>0). There is reasonable agreement over most of the frequency range suggesting the measurement method is reasonably reproducible and the physical properties of the plate are relatively uniform for a particular direction.

Figure 5: Estimated wavenumbers obtained from draw-away phase data collected along the L1

draw-away line in the y<0 and y>0 directions, both of which are perpendicular to the joists. The best estimate is the mean of the two data sets.

Figure 6 compares three sets of measured wavenumber data. The first set is obtained using the best estimate (average of the data in Figure 5) measured in the direction of the

L1 draw-away line, which is perpendicular to the joists and the second set using a

procedure [19] for computing low-order modal frequencies in composite floor systems.

In the computation procedure the effective surface density is taken to be 20.0 kg/m2

which includes the mass of the joists but not the resiliently suspended ceiling. The third set are obtained from measurements of OSB beam samples cut from the long axis of the

sheet, which is in the direction L1 draw-away line (see Figure 1 and Table 1). The

wavenumbers for the OSB beam samples were obtained using a free-free beam [20] and the relationship between beam length and mode shape at resonance frequencies. To obtain a wide frequency range it was necessary to use samples of varying length, typically between 0.7 and 1.2 m.

0 5 10 15 20 25 30 35 100 1000 10000 Frequency, [Hz] Wavenumber , [1/m]

Draw-away line L1, y<0

Draw-away line L1, y>0

Best Estimate perpendicular to joists

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0 5 10 15 20 25 30 35 40 45 100 1000 10000 Frequency, [Hz] W a v e num be r, [ 1 /m ] In-Situ Measurement (draw-away line, L1)

Composite Floor System Computed using Forintek method Limit imposed by 100 mm point spacing

OSB beam sample

Figure 6 shows that over the entire frequency range, the wavenumber from the computational procedure is lower than that measured using a beam sample. This means that the predicted stiffness of the complete floor system in the y-direction is greater than the stiffness of the OSB sheathing alone. This would be consistent for a floor with the resilient channels attached perpendicular to the bottom of the joists as the channels would increase the floor stiffness in the y-principal direction.

The in-situ estimates of the wavenumbers lie between the estimates for the OSB beam samples and those calculated for a composite floor. The uncertainty in the in-situ measurement is such that estimates for the OSB beam samples fall within the 95% confidence interval for all but three bands over the 100-2000 Hz frequency range. Above about 1000 Hz the in-situ measured wavenumber increases rapidly with increasing frequency and approaches the curve for wavenumbers measured on OSB beam samples. The implication is that over the measured frequency range of this study the strapping at the bottom of the joists in the form of resilient channels may not be particularly effective at stiffening the floor, especially in the high frequencies. Thus, as a first order approximation, the stiffness of the floor in the y-direction could be assumed to be the same as the OSB floor sheathing in the same direction.

Figure 6: Three estimates of the wavenumber for the direction perpendicular to the joists, ky.

The error bars on the in-situ data are the 95% confidence limits.

In the direction parallel to the joists, draw-away lines L2 and L3 of Figure 1 were used to

estimate the wavenumbers which are shown in Figure 7. From Figure 7 it is evident that above about 315 Hz the wavenumber is considerably lower when measured on the OSB

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0 5 10 15 20 25 30 35 100 1000 10000 Frequency, [Hz] Wavenumber , [1/m] Parallel On a Joist (draw-away line L2) Parallel Between Joists (draw-away line L3)

Limit imposed by 100 mm point spacing

floor sheathing immediately on top of a joist as compared to the OSB floor sheathing mid-way between the joists. This has the important implication that in addition to being orthotropic by having different wavenumbers (and bending stiffness) in the two principal

directions, kx≠ky, the wavenumber (and bending stiffness) in the x-direction is a function

of location with respect to the joists, so kx(y). It is important to recognize that when

measuring on the joist the effective thickness will be greater than that of the OSB sheathing. It is thus possible that shear deformation is important and the measured wavenumber will be an overestimate of the pure bending wavenumber. Considering this

effect, would suggest that the measured variation of kx(y) along the joist shown in Figure

7 is a conservative estimate.

Figure 7: Estimates of the in-situ wavenumber for the direction parallel to the joists, kx,

measured using draw-away lines L2 and L3. The error bars indicate the 95% confidence limits

for the in-situ data.

Below about 315 Hz the wavenumbers tend to become more similar as the frequency decreases suggesting that below this frequency the joists are significant in determining the effective stiffness of the OSB between the joists. Based on the discussion in the introduction, this should mark the transition between frequency Region II (sub panel waveguides) to Region III (global response of the floor where the joists and sheathing more together). Eqn. 6 predicts the transition frequency to be about 270 Hz when measured input data are used.

The rate of structure borne attenuation per unit distance traveled is proportional to the product of the real part of the bending wavenumber and internal loss factor. Thus, if the

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0 5 10 15 20 25 30 35 40 100 1000 10000 Frequency, [Hz] Wavenumber , [1/m]

In-Situ Best Estimate On a Joist (draw-away line L2)

In-Situ Best Estimate on OSB Between Joists (draw-away line L3)

Estimated using Low-Order Modes OSB beam sample

Limit imposed by 100 mm point spacing

joists and the OSB have approximately the same internal loss factor then, there will be considerably greater attenuation of structure borne energy in the OSB than in the joists because the bending wavenumber is considerably greater in the OSB than the joist. This has a very important implication for semi-empirical models of structure borne power and flanking transmission in the direction parallel to the joists. It suggests that near a point source it may be sufficient to consider only the power flow in the sheathing, but at greater distances the power flow in the joists may become relatively more important leading to a situation where power from the joists is fed back into the sheathing.

Figure 8: Measured and calculated estimates of the in-situ wavenumber for the direction parallel to the joists, kx, measured using draw-away lines L2 and L3. The error bars indicate the

95% confidence limits.

Figure 8 compares the measured in-situ wavenumbers in the direction parallel to the joists to those obtained from calculation. The wavenumbers measured in-situ on the floor immediately on top of a joist are similar to those calculated for the floor in the same direction. The calculation procedure was intended for low-order floor modes (i.e., Region III) where it is assumed that the vibration response of the floor is determined by the complex interaction of all the joists and the OSB, which might explain the better agreement for low than for high frequencies.

For high frequencies, the in-situ wavenumbers on the joist are higher than those predicted by the calculation method. This might be caused by two factors. First, an overestimation of the wavenumber is consistent with the additional effect of shear deformation, as the portion of the floor immediately above a joist does not behave

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0 5 10 15 20 25 30 35 100 1000 10000 Frequency, [Hz] Wavenumber , [1/m] Best Estimate Parallel On a Joist (draw-away line L2) Best Estimate Parallel

on OSB Between Joists (draw-away line L3)

Best Estimate Perpendicular (draw-away line L1)

Limit imposed by 100 mm point spacing

principally like a thin plate in the frequency range of interest. Second, the OSB floor sheathing upon which the measurements are made is only point-connected to the joist at each screw location. Full composite action between the OSB and joists may not be achieved resulting in a lower effective stiffness (higher effective wavenumber) than if there were no slippage. Point connected plates tend to behave independently between fastening points when the spacing between the screws is greater than a half wavelength [12] so that the OSB is free to move independently with respect to the joist away from fastening locations.

The wavenumbers measured between the joists in the X-direction (L2 and L3 draw-away

lines) for frequencies greater than about 500 Hz are in good agreement with those obtained from OSB beam samples cut parallel to the X-direction of floor . (See Figure 1 and Table 1, which indicate that this direction is perpendicular to the long axis, the direction of least stiffness for an OSB sheet in isolation.) This similarity in wavenumbers suggests that for this frequency range and direction the stiffness of the floor between the joists is determined predominantly by the OSB floor sheathing with little effect from the joists.

Figure 9: Measured estimates of the in-situ wavenumber for the direction perpendicular to the joists, ky, using draw-away line L1 and parallel to the joists, kx, using draw-away lines L2 and

L3.

Figure 9 allows comparison of the measured in-situ wavenumbers along the three draw-away lines. Perhaps it is most interesting to compare the wavenumbers for the direction

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the joists (draw-away line L3) on the OSB floor sheathing. Of particular interest is the

fact that the two wavenumber curves cross indicating that in the frequency range (315 to 500 Hz) around the crossing point the floor might be effectively isotropic because of the similarity in the OSB wavenumbers in the x and y-principal directions.

Above about 500 Hz in the direction parallel to the joists, (draw-away line L2), the OSB

wavenumber between the joists is higher than in the direction perpendicular to the joists indicating that the bending stiffness of the OSB is lower in the direction parallel to the joists than perpendicular. This is fully consistent with OSB panels having a Young’s modulus (and stiffness) that is considerably greater in the long axis of the panel which is oriented perpendicular to the joists when installed on a framed floor. Below about 315 Hz the wavenumber is greater in the direction perpendicular to the joists than in the direction parallel to the joists. This indicates that below about 315 Hz the stiffness of the OSB is now greater in the direction parallel to the joists than perpendicular. Presumably, this is due to the effect of the joists, as hypothesized earlier.

3.2 Young’s Modulus of Elasticity

Measured wavenumbers in each principal direction of an orthotropic plate can be used with Eqn. 3 and Eqn. 5 to determine estimates of Young’s modulus in each direction if the plate can be considered homogeneous and acoustically thin. As discussed in the previous section the wood joist floor only approximates a homogeneous orthotropic plate in the frequencies where the sheathing is effectively point connected to the joist.

Assuming that the transition between line and point connection [12] occurs when the

wavelength is less than twice the point spacing (300 mm), then KB>10 defines the

frequency range where the floor might be approximated by an orthotropic plate whose properties are those of the floor sheathing. Below this wavenumber (or the

corresponding frequency), the floor appears inhomogeneous and requires special treatment, which is often done using the concept of an equivalent plate.

This section uses draw-away data collected along lines L1, L2 and L3 to explore the

possibility of estimating Young’s modulus for pure bending from measured

wavenumbers (KB>10) and thin orthotropic plate expressions (Eqn. 3 and Eqn. 5).

In-situ estimates are compared to those obtained from small beam samples cut from the structural floor elements. The first estimate for the beam sample is frequency dependent

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OSB Floor Sheathing 1E+09 1E+10 1E+11 1E+12 100 1000 10000 Frequency, [Hz] Y oung's Modul us, [ P a] In-situ estimate

OSB beam sample estimate OSB beam sample corrected for shear deformation

and is obtained by finding the modulus that when used with thin beam theory gives the same wavenumber as measured for a free-free beam sample, without correction for shear deformation. The second is an estimate for pure bending which is obtained by a least squares fit of estimated wavenumbers (using thin beam theory) to the measured wavenumbers corrected for shear deformation (using Eqn. 11). It is assumed that the product of the two Poisson’s ratios appearing in Eqn. 3 can be approximated by 0.12

and that the surface density of the OSB sheathing is 11.4 kg/m2.

Figure 10 indicates that for most of the frequency range of interest where KB>10 in-situ

estimates of Young’s modulus of the OSB floor sheathing in the direction perpendicular to the joists is greater than that of the OSB beam samples in the same direction.

Figure 10: Estimates of Young’s modulus of the OSB floor sheathing in the direction

perpendicular to the joists obtained from in-situ measurement (draw away line L1) and beam

samples. Estimates for Young’s modulus are not shown for frequencies where KB<10, i.e.,

where the sheathing is effectively line connected to the joists.

If one assumes that the phase response measured on the OSB floor sheathing

immediately above the joist is due only to bending waves propagating in the joist then it should be possible to obtain an estimate of Young’s modulus for the joist using,

( )

[

]

4 B l 2

k

e

I

I

B

E

=

=

ω

ρ

Eqn. 12

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Estimate of Young's Modulus for floor joist 1E+09 1E+10 1E+11 1E+12 1E+13 100 1000 10000 Frequency, [Hz] M odul us, [ P a] Estimate obtaind

from manufacturer's literature

In-situ estimate

where

ρ

l is the mass per unit length, and I is the moment of inertia for a beam. The

moment of inertia for the “I-joist” is obtained by subtracting from the estimate the appropriate moments for a beam of solid rectangular cross section [21]. Figure 11 compares the estimate of Young’s modulus obtained from the manufacturer’s product data sheet and the measured in-situ from wavenumbers. There is reasonable

agreement for frequencies greater than about 800 Hz.

Figure 11: Estimates of Young’s modulus for the floor joist using in-situ wavenumbers (draw-away line L2) and from manufacturer’s literature. Estimates for Young’s modulus are not

shown for frequencies where KB<10, i.e., where the sheathing is effectively line connected to

the joists.

Similarly, if one assumes that the phase response measured on the OSB floor sheathing between the joists is not significantly influenced by waves travelling in the joists then it should be possible to estimate Young’s modulus for the OSB from draw-away line data

measured between the joists along line L3. This is likely to be the case when the

spacing between the joists is at least one-half wavelength, which is satisfied for KB>10,

(λB < 800 mm or frequencies about 315 Hz). Figure 12 compares the Young’s modulus

estimates obtained from OSB beam samples and the measured in-situ estimate for the OSB sheathing. The in-situ estimates are higher than those for the OSB beam samples suggesting that the joists might be affecting the wavenumbers. The trend of the curve suggests that the in-situ estimate of Young’s modulus at frequencies above those shown in Figure 12 may asymptotically approach those for the OSB measured in isolation. However, this cannot be verified due to the upper frequency limit imposed by the 100-mm spacing of the measurement points.

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Estimate of Young's Modulus for OSB floor sheathing Parallel to the Joist

1E+08 1E+09 1E+10 1E+11 1E+12 100 1000 10000 Frequency, [Hz] Modul us, [ P a] In-situ estimate

OSB beam sample corrected for shear deformation

OSB beam sample estimate

Figure 12: Estimates of Young’s modulus of the OSB floor sheathing in the direction parallel to the joists obtained from in-situ measurement (draw away line L3) and beam samples.

Estimates for Young’s modulus are not shown for frequencies where KB<10, i.e., where the

sheathing is effectively line connected to the joists.

4. Conclusions

Measuring the phase change along a draw-away line and conducting a regression analysis to determine the real part of the wavenumber is a relatively simple task. Wavenumber measurements along draw-away lines parallel and perpendicular to the joists confirmed the highly orthotropic nature of a wood joist floor. Measurements suggest that the degree of orthotropy in the direction parallel to the joists could be partially determined by the complex interaction of the OSB floor sheathing between the joists. It is suggested that below the cut-on frequency of the first cross mode, the joists significantly affect the stiffness of the OSB floor sheathing parallel to the joists. Above the frequency of the first cross mode, the effect of the joists is significantly reduced and in the limit, as the bending wavelength becomes much smaller than the joist spacing, the wavenumber (and bending stiffness) approaches that of the OSB floor sheathing in isolation. Wavenumbers in the direction parallel to the joists also approached those measured on OSB floor sheathing in isolation.

The significantly different wavenumbers in the joists and OSB floor sheathing provide the potential for the joists to transmit vibration energy with less attenuation than the OSB floor sheathing. This means that at some distance from the source the joists may transfer energy back to the OSB floor sheathing.

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For the tested wood joist floor, the wavenumbers measured perpendicular to the joists is primarily determined by the stiffness of the OSB floor sheathing. The measured

wavenumber was slightly lower than one derived solely from the OSB floor sheathing indicating that the joists are only weakly “coupled” to adjacent ones thereby creating a marginally stiffer floor in this direction.

These preliminary results suggest that measurement of in-situ wavenumbers can be a very useful method to assess the relative magnitude of wavenumbers in the two principal directions and also to determine the direction in which the rate of vibration energy will be greatest.

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Appendix A – Floor Construction Details

Total floor dimension: Lx: 4.6 m

Ly: 4.5 m

Floor Sheathing: Oriented Strand Board Thickness: 0.018 m Lx: 1.21 m Ly: 2.43 m Ex: 2.5 x 10 9 pa Ey: 6.5 x 10 9 pa ρm: 590 kg/m 3

Attachments: Fastened with 51 mm or longer #10 straight-shank wood screws placed 150 mm o.c. at edges and 305 mm o.c. in the field.

Modulii were measured at NRC using free-free beam samples and applying the correction for shear deformation (Eqn. 8) Joists: Proprietary having an “I” profile

Joist depth (total): 0.302 m Web width: 0.013 m Flange width: 0.038 m Flange depth: 0.038 m

Bending stiffness (EI): 7.91x105 pa m4 Spacing between joist centres: 406 mm Density: 3.4 kg/m

Bending stiffness obtained from manufacturer. Resilient channels: Proprietary steel having a “z” profile

Steel thickness: 0.5 mm Depth: 0.013 m

Spacing between channels centres: 406 mm

Orientation: Perpendicular to joists fastened at each joist by 38 mm screw Ceiling: Gypsum board resiliently suspended by metal channels

Number of layers: 2

Thickness each layer: 15.9 mm Density each layer: 10.5 kg/m2

Attachment to resilient channels: Base layer with 42 mm, or longer, screws placed 305 mm o.c. at the edges and 610 mm o.c. in the field, and exposed layer with 52 mm, or longer, screws placed 305 mm o.c. at the edges and in the field.

38 mm

38 mm

226 mm 302 mm 13 mm

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References

1 Nightingale, T.R.T., Halliwell, R.E., Quirt, J.D., (2002) “Vibration response of floors and the effectiveness of toppings to control flanking transmission,” Proceedings of the INTERNOISE 2002, 19-21 August.

2 Nightingale, T.R.T., Bosmans, I., “Vibration response of lightweight wood frame building elements,” Building Acoustics, Vol. 6, No.3/4, pp. 269-288.

3 Rindel, J.H., (1994) “Dispersion and absorption of structure-borne sound in acoustically thin plates,” Applied Acoustics, Vol. 41, pp. 97-111.

4 Pavic, G., (1976) “Measurement of structure borne wave intensity, Part 1: Formulation of the methods,” Journal of Sound and Vibration, Vol. 49, No. 2, pp. 221-230.

5 Carniel, X., Pascal, J.C., (1985), “Characteristiques de propagation et measure du flux d’energie vibrotoire and les barres,” Proceedings of the 2nd. International Congress on Acoustic Intensity,” Senlis France, 23-25 September.

6 Buman, P.D., (1994) “Measurements of structural intensity: Analytic and experimental evaluation of various techniques for the case of flexural waves in one-dimensional structures,”

Journal of Sound and Vibration, Vol. 174, No. 5, pp. 677-694.

7 Roelens, I., Nuytten, F., Bosmans, I., Vermeir, G., (1997) “In-situ measurement of the stiffness properties of building components,” Applied Acoustics, Vol. 52, pp. 289-309.

8 Thwaites, S., Clark, H., (1995) “Non-destructive testing of honeycomb sandwich structures using elastic waves,” Journal of Sound and Vibration, Vol. 187, pp. 253-269.

9 Clark, N.H., Thwaites, S., (1995) “Local phase velocity measurements in plates,” Journal of

Sound and Vibration, Vol. 187, pp. 241-252.

10 Lyon, R.H., (1983) “Progressive phase trends in multi-degree of freedom systems,” Journal of

the Acoustical Society of America, Vol. 73. No. 5, pp. 1223-1228.

11 Troitsky, M.S., (1976) “Stiffened plates bending, stability, and vibrations,” Elsevier Scientific Publishing Co., Amsterdam.

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12 Craik, R.J.M., Smith, R.S., (2000), “Sound transmission through lightweight parallel plates. Part II: Structure borne sound,” Applied Acoustics Vol. 61, pp. 247-269.

13 Schoenwald, Stefan, Nightingale, T.R.T, (2001) “Measurement of structural intensity on plate structures,” Canadian Acoustics, Vol. 29, No. 3, pp. 102-103.

14 Bosmans, Ivan, Nightinagle, T.R.T., (2001) “Modeling vibrational energy transmission at bolted junctions between plate and a stiffening rib,” Journal of the Acoustical Society of America, Vol. 109, No. 3, pp. 999-1010.

15 Orrenius, U., Finnveden, S., (1996) “Calculation of wave propagation in rib-stiffened plate structures,” Journal of Sound and Vibration, Vol. 198, No. 2, pp. 203-224.

16 Fung, Y.C., (1962), “On corrugated-stiffened panels,” GALCIT SM 62-63 (CFSTI No. AD 429 770) Calif. Inst. Tech., June 1962.

17 Mindlin, R.D., (1951), “Influence of rotatory inertia and shear on flexural motion of isotropic elastic plates,” Journal of Applied Mechanics, Vol. 18, pp. 31-38.

18 Cremer, L., Heckl, M., “Structure borne sound,” Springer Verlag, Berlin, Second edition 1998.

19 Hu, L. J., Chui, Y. H., (2004), “Development of a design method to control vibrations induced by normal walking action in wood-based floors”, Proceedings of the 8th World Conference on Timber Engineering, June 14-17, Lahti, Finland.

20 Timoshenko, S., Young, D.H., Weaver, W., “”Vibration problems in engineering,” John Wiley & Sons, New York, fourth ed. 1974

21 Timoshenko, S.P., Gere, James, M., (1972) Mechanics of materials, Van Norstrand Reinhold Company, New York.

Figure

Figure 1: Sketch showing the layout of the three draw-away lines to estimate the change in  phase with distance
Figure 2: Sketch showing the layout of measurement points along a draw-away line from the  source for an orthotropic plate
Figure 3 shows the correction that the measured wavenumber should be multiplied by to  remove the effect of the indicated factor – shear deformation or rotary inertia
Figure 4: Sketch showing the wood joist floor assembly.  Construction details are provided in  Appendix A
+7

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