# WOOD FLOOR VIBRATIONAL PERFORMANCE AS AFFECTED BY MSR VS. VSR LUMBER E-DISTRIBUTION.

FRANK E. WOESTE [+]

DAN J. DOLAN [+]

ABSTRACT

A simulation study was conducted to investigate the effect of the coefficient of variation of the modulus of elasticity ([[Omega].sub.E]) on the predicted vibrational performance of joist floor systems. Four floor cases were studied and two types of lumber were considered: MSR and VSR lumber where [[omega].sub.E] is 0.11 and 0.25, respectively, as defined by the National Design Specification (NDS)(1). The predicted floor vibrational performance of MSR versus VSR lumber joists with and without load sharing included in the analyses was evaluated by: 1) the probability that the fundamental frequency is less than 10Hz; and 2) the ratio of the first percentile of predicted fundamental frequency of MSR to VSR lumber. Overall, MSR lumber ([[omega].sub.E] = 0.11) floor systems exhibited improved predicted vibrational performance over VSR floor systems ([[omega].sub.E] = 0.25) based on the two measures of floor performance studied.

Traditionally, most wood floors were constructed with solid-sawn lumber joists and designed according to the L/360 live-load deflection limit. This deflection criterion was developed to prevent cracks in the ceiling plaster, and probably not to limit vibrations [7]. Vibrational performance of floor systems was not a major issue in previous decades because the floor spans were relatively short. However, recent architectural changes have created a demand for longer spans in floor systems. As a result, engineered wood products such as I-joists and parallel-chord trusses were developed to satisfy this demand for longer spans. These engineered wood products offer high strength-to-weight and high stiffness-to-weight ratios [6]. Unfortunately, the switch to longer span floors has decreased the serviceability of floors with regard to annoying vibrations. Most people are sensitive to vibrations in the 8- to 10-Hz range because some human organs have a natural frequency of about 4 to 8 Hz. Therefore, when a floor is v ibrating at the same range in frequency, it is perceived as uncomfortable. The unacceptable vibrational performance of some long-span wood floors reveals that the L/360 live-load deflection criterion published in the model building codes is not sufficient to insure that all floors are acceptable to the occupants from a vibrational standpoint.

Currently, U.S. codes do not contain design criterion for controlling floor vibrations. Much research has been conducted on this subject and several design criteria have been proposed to allow designers to check for acceptable vibrational performance of floors. However, none are widely accepted due to the complexity of some criteria and the lack of readily available information for the designer [4]. Research conducted by Johnson [5] at Virginia Tech has resulted in the development of a design criterion that will eliminate most unacceptable and marginally acceptable floor systems (floor systems in or close to the 8-to 10-Hz range) and will aid designers in designing acceptable floors with acceptable vibration performance.

Solid-sawn lumber has different grading systems. Two such grading systems are visually stress rated lumber (VSR) and machine stress rated lumber (MSR). Due to the differences in grading, MSR and VSR lumber have differences in the coefficient of variation of modulus of elasticity ([[omega].sub.E]) VSR lumber has an officially identified [[omega].sub.E] of 0.25 and MSR lumber has an officially identified [[omega].sub.E] of 0.11 [1]. Variability of the modulus of elasticity between MSR and VSR lumber is recognized in strength design and addressed in the NDS (buckling capacity of members under compression stress, truss compression chords, and beam stability). Basically, the lower the [[omega].sub.E] the higher the capacity of the member with respect to stability issues because the likelihood of a very low E-value is lessened. However, the variability of modulus of elasticity as it applies to serviceability (i.e., deflection and vibrational performance) is not addressed in the NDS; instead, the published average values of modulus of elasticity are used in serviceability calculations.

OBJECTIVE

The objective of this simulation project was to investigate the effect of the [[omega].sub.E] of lumber on the predicted vibrational performance of joist floors with and without load sharing included in the analyses. Two types of lumber were considered: MSR ([[omega].sub.E] = 0.11) and VSR ([[omega].sub.E] = 0.25). Expected floor vibrational performance of MSR lumber versus VSR lumber was evaluated by: 1) the probability that the fundamental frequency is less than 10 Hz; and 2) a comparative measure of the ratio of the first percentile of the distribution for fundamental frequency of vibration (f) of MSR lumber to the first percentile of the distribution for f of VSR lumber:

[[f.sup.MSR].sub.0.01]/[[f.sup.VSR].sub.0.01] [1]

In most cases, joists are simply supported on both ends by concrete blocks (rigid supports), or the joists are simply supported by a simple span girder (flexible support) on one end and simply supported by concrete blocks on the other end. Only the first case of rigidly supported joists are presented here; Wilson's report [11] contains the results for floor joists involving a girder support on one end. A summary of the rigid support joist cases studied is given in Table 1.

LITERATURE REVIEW

Research conducted at Virginia Tech resulted in a design criterion that predicts acceptable floor vibrational performance when the calculated frequency of the floor joists is 15 Hz or greater. The equation for calculating the fundamental frequency of a joist or a supporting girder is:

f = 1.57 [square root]386EI/[WL.sup.3] [2]

where:

f = fundamental frequency (Hz)

E = joist or girder modulus of elasticity (psi)

I = joist or girder moment of inertia ([in..sup.4])

W = total actual dead weight of the joist or girder plus the tributary-area dead weight of the floor sheathing and flooring system (hereafter referred to as the actual tributary dead weight of the joist) (lb.)

L = clear span of the joist or girder (in.)

Johnson [5] developed the 15-Hz design criterion from tests of 15 laboratory-built floors and 73 in-situ floors. Some of the in-situ floors were bare while others were part of nearly completed structures. Load-sharing tests were performed on the laboratory-built floors and deflection profiles were produced according to joist type. In general, the deflection profiles revealed that the center joist of the floor system supported about 50 percent of the load while the neighboring joists carry approximately the other 50 percent (25% each to the two adjacent joists).

The predicted fundamental frequencies of joists in the 88 Johnson [5] floors were calculated using Equation [2]. Heel drop tests were used to measure the frequency of each floor system, and the vibration performance of each floor system was subjectively rated as acceptable, marginal, or unacceptable [5]. Measured frequencies for "joists" installed in a "floor system" compared favorably with the predicted frequencies using Equation [2] and the published value of modulus of elasticity (E) for the joist grade. The test results validated Equation [2] for the purpose of calculating the frequency of an installed residential joist based on four design properties of a simple span joist: the clear span, the published B and I, and actual tributary dead weight of the joist.

It was determined from the measured data that a frequency of 15 Hz, using Equation [2], provides the distinction between unacceptable and acceptable floors for all floors evaluated. Johnson [5] concluded that the predicted frequency of a floor should be calculated based upon only the actual tributary dead weight of the joist since the proposed design criterion was based on tests of bare floors (no live loads present). If the calculated frequency by Equation [2] is greater than 15 Hz, then the vibrational performance of the floor is expected to be acceptable.

Shue's [8] study provided further validation of the Johnson [5] study. The difference between the two studies was that Shue tested occupied floors (with actual live loads present) as well as unoccupied floors (no live loading). The Shue study took into account the effect of partitions or the overall structure itself on the vibrational performance of the floor system. Shue tested 106 in-situ wood floors of which 20 were originally evaluated in the Johnson study. The results of Shue's study validated the design criterion proposed by the Johnson study for unoccupied floors. However, for occupied floors, Shue concluded that the frequency of a floor system with live loads present must be greater than 14 Hz in order for it to be considered acceptable.

Dolan and Skaggs [4] stated that one of the main goals in the Johnson study [5] in developing the design criterion was to eliminate the majority of complaints about annoying floor vibrations. Dolan and Skaggs compared the 15-Hz design criterion with the L/360 live-load deflection criterion using a parallelchord truss floor system spaced at 24 inches on center with a modulus of elasticity of 1.9 million psi. For spans between 22 and 30 feet, results from the 15-Hz design rule fall between the L/360 and L/480 live-load deflection criterion. The L/360 live-load deflection criterion is sufficient for spans up to 22 feet but after 22 feet the rule using 15 Hz as a minimum calculated frequency developed by Johnson was suggested.

Dolan et al. [3] studied the effect of imposed load on floor vibrations. Solidsawn wood joist floors were studied using two levels of [[omega].sub.E] for the joists: 9 percent and 28 percent. Three levels of live load (0 psf, 15 psf, and 40 psf) were applied and the dynamic response of the solid-sawn wood joist floors was examined in terms of three parameters: natural frequency, damping ratios, and RMS acceleration. The natural frequency of the joist floors decreased as the imposed load increased. No conclusions could be drawn for damping ratio since the results were inconsistent. The frequency weighted RMS acceleration for a joist floor decreased as the imposed load increased. The final conclusion from this study was that [[omega].sub.E] had little or no effect on the measured vibration responses of the floors tested.

Kalkert et al. [6] evaluated six design criteria for controlling floor vibrations. Deflection factors were determined for each criterion. The values for these deflection factors were two to four times larger than the deflection factor of 360 (L/360). The deflection factors ranged from 701 to 1448. The deflection factors calculated excluded the stiffening effects of sheathing. Based on the six design procedures considered, it was found that a floor system would need to be designed to a minimum L/701 live-load deflection limit in order for the predicted vibrational performance of the floor to be acceptable. The authors concluded that further research was needed to determine an appropriate deflection limit based on experimental investigations and economic analysis.

Suddarth et al. [10] performed a simulation study on the influence of [[omega].sub.E] on predicted floor performance. The simulation study involved two types of floors MSR and VSR joists. The model used in the simulation included the effect of sheathing and load sharing. The purpose of this study was to compare the deflection performance of MSR floors against the deflection performance of VSR floors using average deflection and soft-spot deflection as measures of performance. The results based on average deflection were that a 1.45 x [10.sup.6] psi MSR E-grade floor performed equivalently to a 1.6 x [10.sup.6] psi VSR E-grade floor. For soft-spot deflection, a 1.4 x [10.sup.6] psi MSR E-grade floor performed equivalently to a 1.6 x [10.sup.6] psi VSR E-grade floor.

PROCEDURE

The main focus of this study was the probabilistic relationship between independent variable E and dependent variable f in Equation [2]. The floor systems studied consisted of eight 2 by 10 joists spaced at 16 inches on center. For Cases 1 and 3, the clear span of the joists was 16 feet, 1 inch. For Cases 2 and 4, the clear span of the joists was 14 feet, 10 inches. The joists were No. 2 KD19 southern pine. An individual joist was studied for Cases 1 and 2 instead of eight joists, and thus load sharing due to floor sheathing was not taken into account. However, in Cases 3 and 4, three joists were studied collectively and thus a measure of load sharing was taken into account as described later. In order to investigate the effect of E on the expected vibrational performance of each floor system, the frequency distributions were determined for each floor case.

Once the predicted frequency distribution is available for a joist (or several joists assumed to share load), some meaningful measures of floor performance were required for the purpose of comparing VSR and MSR type joists. We selected two measures of performance: the 1st percentile of the predicted frequency of vibration distribution and the probability that the predicted frequency is less than 10 Hz. The first measure was selected based on the philosophy that joists having average or higher E will not be judged unacceptable for the vibrational response because they will have a higher frequency not detected by humans as being annoying. As stated earlier, it is the lower frequencies of vibration (8 to 10 Hz) that are most annoying to humans. Based on the design goal of having 99 percent of all joists not judged as causing annoying vibration, the 1st percentile of the predicted frequency distribution is a logical point for comparison within the two distributions derived for VSR and MSR lumber. The second meas ure, the probability that f is less than 10 Hz, was selected from the upper end of the human sensitivity range. The design goal would be to minimize the probability that any joist in a floor vibrates at a frequency less than 10 Hz.

CASE 1 (SINGLE JOIST MODEL, L/360 DESIGN)

The E was assumed to be lognormally distributed [9]. This assumption was made because it allowed us to vary the coefficient of variation easily and it is frequently accepted as the "best fit" for researchers who have fit distributions to E test data. It is also a reasonable selection for E because the values for E are positive and the lognormal distribution falls above zero along the x-axis. The parameters for the E-distribution were calculated using the following equations from Ang and Tang [2]:

[lambda] = ln [micro] - 1/2 [[xi].sup.2] = ln E - [[[xi].sub.E].sup.2] [3]

[[xi].sup.2] = ln(1 + [[omega].sup.2]) = ln(1 + [[[omega].sub.E].sup.2]) [4]

where:

[lambda] = mean of the logarithms of a variable

[[xi].sup.2] = variance of the logarithms of a variable

[micro] = mean of a variable

[omega] = coefficient of variation of a variable

E = mean of the modulus of elasticity variable, E (psi)

[[[xi].sub.E].sup.2] = variance of the logarithms of the E variable

[[omega].sub.E] = coefficient of variation of the E variable

In order to determine the distribution of f of a single joist, the natural logarithm (ln) is taken on both sides of Equation [2]:

ln f = ln 1.57 + 1/2 ln 386 + 1/2 ln I - 1/2 ln W - 3/2 in L + 1/2 ln E [5]

where for Cases 1 and 3:

I = 98.93 in. [4] (2 by 10)

W = actual tributary dead weight of the joist (joist weight plus 16-in. strip of sheathing plus carpet) = 94 lb. Total

L = 193 in.

Equations [5] can be rewritten as:

ln f = k + 1/2 ln E [6]

where:

k = a constant

E = the modulus of elasticity (psi) of an individual joist

Since E is lognormally distributed, then ln of E results in E being normally distributed. The ln f term would be normally distributed since a constant plus a normally distributed. The exponential of both sides of the equation is thus lognormally distributed. The parameters for f, using Equation [6] are:

[[lambda].sub.f] = E(ln f) = k + 1/2 E(ln E) [7]

where E denotes the expected value, and:

[[lambda].sub.f] = k + 1/2 [[lambda].sub.E] [8]

var(In f) = [[[xi].sup.2].sub.f] = var (k) +

[(1/2).sup.2] var (In E) [9]

[[[xi].sup.2].sub.f] = 0 + 1/4 [[[xi].sub.E].sup.2] = 1/4 [[[xi].sub.E].sup.2] [10]

[[xi].sub.f] = 1/2 [[xi].sub.E] [11]

The assumed E distributions for VSR and MSR lumber are plotted in Figure 1. The predicted joist frequency distributions for VSR (Fig. 2) and MSR (Fig. 3) lumber were plotted using Equations [3], [4], [8], and [11] to calculate the appropriate lognormal density function parameters. The first percentile of the frequency distribution and the probability that the frequency is less than 10 Hz were then calculated from the fitted distributions. Finally, the ratio of the values of the first percentile for each type of lumber were used to compare the vibrational performance to one another using Equation [1].

CASE 2 (SINGLE-JOIST MODEL, L/480 DESIGN)

The procedure for Case 2 was the same as for Case 1. However, the span and dead weight variables in Equation [5] were different because the joist span was determined by the L/480 live-load deflection limit. The values of the variables were:

Joist: I = 98.93 [in..sup.4] (2 X 10)

W = joist weight plus 16-inch strip of sheathing plus carpet = 87 lb.

L = 178 in.

CASES 3 AND 4 (LOAD-SHARING MODEL)

Cases 3 and 4 were Cases 1 and 2 repeated using an effective joist E, [E.sub.j], based and justified by the load-sharing results from the Johnson study [5]. The model used was:

[E.sub.j] = 0.25[E.sub.L] + 0.5[E.sub.C] + 0.25[E.sub.R] [12]

where the E's represent three joists in a row in a floor. [E.sub.L] is the left joist, [E.sub.C] is the center joist impacted by a footfall, and [E.sub.R] is the right joist, and the three were considered independent random variables. Except for the procedures in obtaining the data for E and [f.sub.joist], the procedures used for Cases 3 and 4 were the same as Cases 1 and 2.

Monte Carlo simulation was used to generate 5,000 values for each variable in Equation [12]. Then, [E.sub.j] was used in Equation [2] in place of the variable E to calculate 5,000 values of [f.sub.joist]. The program Graphical Distribution Analysis (GDA) [12] was used to determine the best-fit distribution for the data for [E.sub.j] and [f.sub.joist].

RESULTS AND DISCUSSION

CASE 1 (SINGLE-JOIST MODEL, L/360 DESIGN)

The assumed E distributions for VSR and MSR lumber are shown in Figure 1. For all practical purposes, the E of MSR and VSR lumber ranged from about 700,000 psi to 3,000,000 psi and from about 1,000,000 psi to 2,250,000 psi, respectively. By visual inspection of Figure 1, the E-distribution for MSR lumber is dramatically less variable than the E-distribution for VSR lumber. The lower variability of the E-distribution for MSR is recognized by the published values of coefficient of variation [1] for each lumber type (0.11 versus 0.25).

Figure 2 represents the calculated frequency distribution of vibrational frequencies of a VSR joist with no load sharing assumed. The frequency variable is lognormally distributed because E was assumed to be lognormal. It ranges from about 10 Hz to 22 Hz. The calculated frequency distribution for MSR lumber is shown in Figure 3. It also is lognormally distributed and is noticeably less variable than [f.sub.joist] depicted in Figure 2.

By inspection of Equations [8] and [11], it is evident that the E-distribution affects the predicted fundamental frequency distribution of a joist. For example, the E for MSR lumber is less variable than VSR lumber (0.11 versus 0.25). The parameter [[xi].sup.2] in Equation [4] was calculated based on [[omega].sub.E]. For MSR lumber, a lower [[omega].sub.E] would cause the standard deviation of the logarithms of E ([[xi].sub.E]) to be lower. Since [[xi].sub.E] is the standard deviation for the lognormal density, the variability of the predicted frequency is lower (Fig. 3).

The probability that the predicted frequency will be less than 10 Hz for VSR lumber (in Fig. 3) is 0.0008. The first percentile of the distribution is 11.06 Hz. For Figure 3, the probability that the predicted frequency will be less than 10 Hz for MSR lumber was approximately zero (based on four significant figures and hereafter considered zero). The first percentile of the distribution for MSR is 13.09 Hz, which is about 18 percent more than the first percentile for VSR lumber (Fig. 3).

CASE 2 (SINGLE-JOIST MODEL, L/480 DESIGN)

The assumed E-distribution for both MSR and VSR lumber are identical to the results given for the E-distribution for Case 1 because the values used to calculate the parameters for the E-distribution in Equations [3] and [4] did not change. The two [[omega].sub.E] values (0.11 and 0.25) remained constant throughout the four cases studied.

Results for the joist frequency distribution differed from the results for the frequency distribution in Case 1 because the joists were designed to a stricter deflection limit (L/480). The predicted frequency distribution for both MSR and VSR lumber shifted to the right, producing a better expected vibrational performance than for Case 1.

For VSR lumber, the probability that the joist frequency is less than 10 Hz was zero. The first percentile of the predicted vibrational frequency distribution was 12.98 Hz. For MSR lumber, the probability that the joist frequency was less than 10 Hz was also zero. The first percentile of the predicted frequency distribution for MSR lumber was 15.36 Hz, which is 18 percent more than the first percentile for VSR lumber according to Equation [1]. In other words, the expected vibrational performance for MSR lumber in this case was 18 percent better than VSR lumber.

CASE 3 (LOAD-SHARING MODEL, L/360 DESIGN)

The two-parameter lognormal distributions shown in Figures 4 and 5 fit the effective joist E ([E.sub.j]) data well, based collectively on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smirnov test [2,12]. Again, MSR lumber effective joist E based on Equation [12] (Fig. 5) is less variable than VSR lumber effective joist E (Fig. 4).

For the predicted joist frequency ([f.sub.joist]) based on the load-sharing model (given by Equation [12]), the two-parameter lognormal distribution fit the data as well as shown in Figures 6 and 7 using the same statistical tests. Three MSR lumber joists (Fig. 7) were collectively less variable than three VSR lumber joists (Fig. 6).

The probability that [f.sub.joist] was below 10 Hz for either VSR and MSR lumber joists with load sharing considered among the joists was zero (Table 3). The first percentile frequency from Figure 7 (MSR) was 13.79 Hz compared to 12.47 Hz from Figure 6 (VSR). The ratio of the first percentile predicted frequency of MSR lumber to the first percentile of VSR lumber was 1.11. With the load-sharing model included for three joists in a row, MSR lumber in Case 3 performed 11 percent better than VSR lumber with respect to the first percentile predicted frequency as the measure of floor performance.

The effective joist E (Fig. 4 and 5) for both VSR and MSR lumber in Case 3 is slightly less variable and has a smaller range than the comparable E-distributions shown in Figure 1 for Case 1. In Figure 4, the distribution ranges from about 1.0 x [10.sup.6] psi to about 2.50 x [10.sup.6] psi. The E-distribution for VSR in Figure 1 ranges from about 700,000 psi to about 3.0 x [10.sup.6] psi. For MSR lumber in Case 3, the range in Figure 5 is from about 1.2 x [10.sup.6] psi to 2.0 x [10.sup.6] psi. For Case 1, the MSR E-distribution in Figure 1 ranges from about 1.0 x [10.sup.6] psi to 2.25 x [10.sup.6] psi. The results in Case 3 for [E.sub.j] is due to Equation [12] being a weighted average, which accounts for the distributions having a smaller range, or lower standard deviation.

The distribution for frequency of joist (Figs. 6 and 7) for both VSR and MSR lumber in Case 3 is also slightly less variable and has a smaller range than the distribution for frequency of joist in Figures 2 and 3 for Case 1. In Figure 6, the distribution ranges from about 12 to 18 Hz compared to the range of 10 to 21 Hz for VSR lumber in Figure 2. For MSR lumber (Fig. 7), the range of predicted frequency is about 13 to 16 Hz compared to 12 to 18 Hz for Case 1 (Fig. 3).

CASE 4 (LOAD-SHARING MODEL, L/480 DESIGN)

The two-parameter lognormal distribution fit the data well for effective joist E ([E.sub.j]) for both MSR and VSR lumber. The two-parameter lognormal distribution was determined to be the best fit based collectively on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smimov test [2,12].

Based on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smirnov test, the two-parameter lognormal fit the simulated [f.sub.joist] distribution with load sharing for VSR and MSR lumber. As expected, the distribution for predicted joist frequency for MSR lumber was less variable than the distribution for VSR lumber. The probability that [f.sub.joist] was less than 10 Hz for both VSR and MSR lumber was zero. The first percentile of [f.sub.joist] for MSR lumber was 16.22 Hz and the first percentile of the comparable distribution for VSR lumber was 14.66 Hz. The ratio of these two numbers using Equation [1] was 1.11, which reveals that MSR lumber has a better predicted vibrational performance than VSR lumber by 11 percent.

SUMMARY

For analysis purposes, two measures of floor performance were selected: the 1st percentile of the predicted frequency of vibration distribution and the probability that the predicted frequency is less than 10 Hz. The first measure was selected based on the philosophy that joists having "average" or higher E will not be judged unacceptable for the vibrational response because they will have a higher frequency not detected by humans as being annoying. The lower frequencies of vibration (8 to 10 Hz) stemming from lower E's should be critical and are the analysis focus when the 1st percentile of the frequency distribution is used for comparison purposes. The second measure, the probability that f is less than 10 Hz, was selected from the upper end of the human sensitivity range.

An individual joist was studied for Cases 1 and 2 (load sharing was neglected). The joist in Case 1 was simply supported on rigid supports (typically a block wall) and designed for the L/360 live-load deflection limit. MSR lumber in Case 1 had an improved vibrational performance over VSR lumber by 18 percent based on the ratio of the first percentile of the predicted frequency in Equation [1]. The probability that [f.sub.joist] was less than 10 Hz was close to zero for both VSR and MSR lumber. However, the MSR lumber floor ([[omega].sub.E] = 0.11) had better predicted floor performance based on the measure of the probability of the joist frequency being less than 10 Hz because its probability was zero (based on four significant figures).

Case 2 differed from Case 1 only by the span limit that was designed to the L/480 live-load deflection limit. Both MSR and VSR lumber had favorable predicted vibrational performance because the probability that [f.sub.joist] is less than 10 Hz was zero for both of them. The floor performance of MSR lumber based on the ratio of the first percentile of the predicted frequency in Equation [1] was 18 percent better than VSR lumber. Case 2 provided an example of the positive effect of designing the joist to a stricter deflection limit on the vibrational performance of a floor system. The [f.sub.joist] distributions for both MSR and VSR lumber shifted to the right and increased the first percentile values. Table 2 provides a summary of the vibrational performance measures of both floor Cases 1 and 2 with respect to the [[omega].sub.E] level.

Load sharing was included in the study in Cases 3 and 4. Three joists were studied in Cases 3 and 4 instead of individual joists as in Cases 1 and 2. Case 3 involved three simply supported joists on rigid supports designed for the L/360 live-load deflection limit. The results for the predicted vibrational performance measures in Case 3 that included load sharing improved from the results for predicted vibrational performance in Case 1. The probability that [f.sub.joist] is less than 10 Hz was zero for both VSR and MSR lumber. However, MSR lumber still had better vibrational performance measures than VSR lumber. According to the ratio of first percentile predicted frequencies given by Equation [1], the predicted performance of the MSR lumber joists was 11 percent over the predicted vibrational performance of VSR lumber joists.

Case 4 involved three simply supported joists on rigid supports designed for the L/480 live-load deflection limit. The measures of floor performance improved from Case 3 since the joists were designed to a stricter deflection limit and again MSR lumber performed better than VSR lumber. The probability that is less than 10 Hz was zero for both VSR and MSR lumber. The predicted vibrational performance defined by Equation [1] for MSR lumber versus VSR lumber was 1.11, or 11 percent. The results for Cases 3 and 4 are summarized in Table 3.

CONCLUSIONS AND RECOMMENDATIONS

The variability of E, characterized by [[omega].sub.E], did have an effect on the predicted vibrational performance of floor systems. Overall, MSR lumber ([[omega].sub.E]= 0.11) floor systems had improved predicted vibrational performance over VSR floor systems ([[omega].sub.E] = 0.25) based on two measures of floor performance: fundamental frequency and probability of the fundamental frequency less than 10 Hz.

Based on the two measures of floor performance studied, the predicted vibrational performance across the floor cases improved as the design deflection limits became stricter, specifically L/360 versus L/480. A recommendation for controlling annoying residential floor vibrations is to design the floor joists to a stricter deflection limit of L/480.

The authors are, respectively, Graduate Research Assistant and Professor, Biological Systems Engineering Dept., Virginia Tech, Blacksburg, VA 24061; and Associate Professor, Wood Sci. and Forest Prod., Virginia Tech, Blacksburg, VA 24061.

(+.) Forest Products Society Member.

LITERATURE CITED

(1.) American Forest and Paper Association. 1997. National Design Specification for Wood Construction. AF&PA, Washington, D.C.

(2.) Ang, A. and W.H. Tang. 1975. Probability Concepts in Engineering Planning and Design. Vol. I. John Wiley & Sons, Inc., New York. pp.104-105.

(3.) Dolan, J.D., F.E. Woeste, and X. Li. 1995. Effect of imposed load on solid-sawn wood-joist floor vibrations. Forest Prod. J. 45(1):71-76.

(4.) __________ and T. Skaggs. 1994. Designing to reduce floor vibrations. Pap. No. 944549. American Society of Agri. Engineer, St. Joseph, Mich.

(5.) Johnson, J.R. 1994. Vibration acceptability on wood floor systems. MS. thesis. Virginia Tech, Blacksburg, Va.

(6.) Kalkert, R.E., J.D. Dolan, and F.E. Woeste. 1995. Wood-floor vibration design criteria. J.) of Structural Engineering 121(9):1294-1297.

(7.) Percival, D.H. 1979. History of L/360. Forest Prod. J. 29(8):26-27.

(8.) Shue, B.C. 1995. Some aspects of vibration serviceability in wood floor systems. M.S. thesis. Virginia Tech, Blacksburg, Va.

(9.) Suddarth, S.K., F.E. Woeste, and J.T.P. Yao. 1975. Effect of E-variability on the deflection behavior of a structure. Forest Prod. J. 25(1):17-19.

(10.) __________. W.L Galligan, and D. DeVisser. 1997. The influence of modulus of elasticity variability control in lumber grading on floor performance. Forest Prod. J. 48(6):61-65.

(11.) Wilson, A.C. 1998. Expected vibration performance of wood floors as affected by MSR vs. VSR lumber E-distribution. Master of Engineering Project Rept. Virginia Tech, Blacksburg, Va.

(12.) Worley, J.W., J.A. Bollinger, F,E. Woeste, and K.S. Kline. 1990. Graphical distribution analysis (GDA). Appl. Engineering in Agriculture 6(3):367-371.

Four floor systems constructed of either VSR or MSR lumber were studied based on a live-load deflection criteria on rigid supports, and no load sharing versus a simple load-sharing model.

The analysis of study Cases 3 and 4 included a load sharing model. The 1st percentile of the predicted joist frequency distribution was considered a measure of floor performance and the measure was impacted by MSR lumber that has a lower [[omega].sub.E].

DAN J. DOLAN [+]

ABSTRACT

A simulation study was conducted to investigate the effect of the coefficient of variation of the modulus of elasticity ([[Omega].sub.E]) on the predicted vibrational performance of joist floor systems. Four floor cases were studied and two types of lumber were considered: MSR and VSR lumber where [[omega].sub.E] is 0.11 and 0.25, respectively, as defined by the National Design Specification (NDS)(1). The predicted floor vibrational performance of MSR versus VSR lumber joists with and without load sharing included in the analyses was evaluated by: 1) the probability that the fundamental frequency is less than 10Hz; and 2) the ratio of the first percentile of predicted fundamental frequency of MSR to VSR lumber. Overall, MSR lumber ([[omega].sub.E] = 0.11) floor systems exhibited improved predicted vibrational performance over VSR floor systems ([[omega].sub.E] = 0.25) based on the two measures of floor performance studied.

Traditionally, most wood floors were constructed with solid-sawn lumber joists and designed according to the L/360 live-load deflection limit. This deflection criterion was developed to prevent cracks in the ceiling plaster, and probably not to limit vibrations [7]. Vibrational performance of floor systems was not a major issue in previous decades because the floor spans were relatively short. However, recent architectural changes have created a demand for longer spans in floor systems. As a result, engineered wood products such as I-joists and parallel-chord trusses were developed to satisfy this demand for longer spans. These engineered wood products offer high strength-to-weight and high stiffness-to-weight ratios [6]. Unfortunately, the switch to longer span floors has decreased the serviceability of floors with regard to annoying vibrations. Most people are sensitive to vibrations in the 8- to 10-Hz range because some human organs have a natural frequency of about 4 to 8 Hz. Therefore, when a floor is v ibrating at the same range in frequency, it is perceived as uncomfortable. The unacceptable vibrational performance of some long-span wood floors reveals that the L/360 live-load deflection criterion published in the model building codes is not sufficient to insure that all floors are acceptable to the occupants from a vibrational standpoint.

Currently, U.S. codes do not contain design criterion for controlling floor vibrations. Much research has been conducted on this subject and several design criteria have been proposed to allow designers to check for acceptable vibrational performance of floors. However, none are widely accepted due to the complexity of some criteria and the lack of readily available information for the designer [4]. Research conducted by Johnson [5] at Virginia Tech has resulted in the development of a design criterion that will eliminate most unacceptable and marginally acceptable floor systems (floor systems in or close to the 8-to 10-Hz range) and will aid designers in designing acceptable floors with acceptable vibration performance.

Solid-sawn lumber has different grading systems. Two such grading systems are visually stress rated lumber (VSR) and machine stress rated lumber (MSR). Due to the differences in grading, MSR and VSR lumber have differences in the coefficient of variation of modulus of elasticity ([[omega].sub.E]) VSR lumber has an officially identified [[omega].sub.E] of 0.25 and MSR lumber has an officially identified [[omega].sub.E] of 0.11 [1]. Variability of the modulus of elasticity between MSR and VSR lumber is recognized in strength design and addressed in the NDS (buckling capacity of members under compression stress, truss compression chords, and beam stability). Basically, the lower the [[omega].sub.E] the higher the capacity of the member with respect to stability issues because the likelihood of a very low E-value is lessened. However, the variability of modulus of elasticity as it applies to serviceability (i.e., deflection and vibrational performance) is not addressed in the NDS; instead, the published average values of modulus of elasticity are used in serviceability calculations.

OBJECTIVE

The objective of this simulation project was to investigate the effect of the [[omega].sub.E] of lumber on the predicted vibrational performance of joist floors with and without load sharing included in the analyses. Two types of lumber were considered: MSR ([[omega].sub.E] = 0.11) and VSR ([[omega].sub.E] = 0.25). Expected floor vibrational performance of MSR lumber versus VSR lumber was evaluated by: 1) the probability that the fundamental frequency is less than 10 Hz; and 2) a comparative measure of the ratio of the first percentile of the distribution for fundamental frequency of vibration (f) of MSR lumber to the first percentile of the distribution for f of VSR lumber:

[[f.sup.MSR].sub.0.01]/[[f.sup.VSR].sub.0.01] [1]

In most cases, joists are simply supported on both ends by concrete blocks (rigid supports), or the joists are simply supported by a simple span girder (flexible support) on one end and simply supported by concrete blocks on the other end. Only the first case of rigidly supported joists are presented here; Wilson's report [11] contains the results for floor joists involving a girder support on one end. A summary of the rigid support joist cases studied is given in Table 1.

LITERATURE REVIEW

Research conducted at Virginia Tech resulted in a design criterion that predicts acceptable floor vibrational performance when the calculated frequency of the floor joists is 15 Hz or greater. The equation for calculating the fundamental frequency of a joist or a supporting girder is:

f = 1.57 [square root]386EI/[WL.sup.3] [2]

where:

f = fundamental frequency (Hz)

E = joist or girder modulus of elasticity (psi)

I = joist or girder moment of inertia ([in..sup.4])

W = total actual dead weight of the joist or girder plus the tributary-area dead weight of the floor sheathing and flooring system (hereafter referred to as the actual tributary dead weight of the joist) (lb.)

L = clear span of the joist or girder (in.)

Johnson [5] developed the 15-Hz design criterion from tests of 15 laboratory-built floors and 73 in-situ floors. Some of the in-situ floors were bare while others were part of nearly completed structures. Load-sharing tests were performed on the laboratory-built floors and deflection profiles were produced according to joist type. In general, the deflection profiles revealed that the center joist of the floor system supported about 50 percent of the load while the neighboring joists carry approximately the other 50 percent (25% each to the two adjacent joists).

The predicted fundamental frequencies of joists in the 88 Johnson [5] floors were calculated using Equation [2]. Heel drop tests were used to measure the frequency of each floor system, and the vibration performance of each floor system was subjectively rated as acceptable, marginal, or unacceptable [5]. Measured frequencies for "joists" installed in a "floor system" compared favorably with the predicted frequencies using Equation [2] and the published value of modulus of elasticity (E) for the joist grade. The test results validated Equation [2] for the purpose of calculating the frequency of an installed residential joist based on four design properties of a simple span joist: the clear span, the published B and I, and actual tributary dead weight of the joist.

It was determined from the measured data that a frequency of 15 Hz, using Equation [2], provides the distinction between unacceptable and acceptable floors for all floors evaluated. Johnson [5] concluded that the predicted frequency of a floor should be calculated based upon only the actual tributary dead weight of the joist since the proposed design criterion was based on tests of bare floors (no live loads present). If the calculated frequency by Equation [2] is greater than 15 Hz, then the vibrational performance of the floor is expected to be acceptable.

Shue's [8] study provided further validation of the Johnson [5] study. The difference between the two studies was that Shue tested occupied floors (with actual live loads present) as well as unoccupied floors (no live loading). The Shue study took into account the effect of partitions or the overall structure itself on the vibrational performance of the floor system. Shue tested 106 in-situ wood floors of which 20 were originally evaluated in the Johnson study. The results of Shue's study validated the design criterion proposed by the Johnson study for unoccupied floors. However, for occupied floors, Shue concluded that the frequency of a floor system with live loads present must be greater than 14 Hz in order for it to be considered acceptable.

Dolan and Skaggs [4] stated that one of the main goals in the Johnson study [5] in developing the design criterion was to eliminate the majority of complaints about annoying floor vibrations. Dolan and Skaggs compared the 15-Hz design criterion with the L/360 live-load deflection criterion using a parallelchord truss floor system spaced at 24 inches on center with a modulus of elasticity of 1.9 million psi. For spans between 22 and 30 feet, results from the 15-Hz design rule fall between the L/360 and L/480 live-load deflection criterion. The L/360 live-load deflection criterion is sufficient for spans up to 22 feet but after 22 feet the rule using 15 Hz as a minimum calculated frequency developed by Johnson was suggested.

Dolan et al. [3] studied the effect of imposed load on floor vibrations. Solidsawn wood joist floors were studied using two levels of [[omega].sub.E] for the joists: 9 percent and 28 percent. Three levels of live load (0 psf, 15 psf, and 40 psf) were applied and the dynamic response of the solid-sawn wood joist floors was examined in terms of three parameters: natural frequency, damping ratios, and RMS acceleration. The natural frequency of the joist floors decreased as the imposed load increased. No conclusions could be drawn for damping ratio since the results were inconsistent. The frequency weighted RMS acceleration for a joist floor decreased as the imposed load increased. The final conclusion from this study was that [[omega].sub.E] had little or no effect on the measured vibration responses of the floors tested.

Kalkert et al. [6] evaluated six design criteria for controlling floor vibrations. Deflection factors were determined for each criterion. The values for these deflection factors were two to four times larger than the deflection factor of 360 (L/360). The deflection factors ranged from 701 to 1448. The deflection factors calculated excluded the stiffening effects of sheathing. Based on the six design procedures considered, it was found that a floor system would need to be designed to a minimum L/701 live-load deflection limit in order for the predicted vibrational performance of the floor to be acceptable. The authors concluded that further research was needed to determine an appropriate deflection limit based on experimental investigations and economic analysis.

Suddarth et al. [10] performed a simulation study on the influence of [[omega].sub.E] on predicted floor performance. The simulation study involved two types of floors MSR and VSR joists. The model used in the simulation included the effect of sheathing and load sharing. The purpose of this study was to compare the deflection performance of MSR floors against the deflection performance of VSR floors using average deflection and soft-spot deflection as measures of performance. The results based on average deflection were that a 1.45 x [10.sup.6] psi MSR E-grade floor performed equivalently to a 1.6 x [10.sup.6] psi VSR E-grade floor. For soft-spot deflection, a 1.4 x [10.sup.6] psi MSR E-grade floor performed equivalently to a 1.6 x [10.sup.6] psi VSR E-grade floor.

PROCEDURE

The main focus of this study was the probabilistic relationship between independent variable E and dependent variable f in Equation [2]. The floor systems studied consisted of eight 2 by 10 joists spaced at 16 inches on center. For Cases 1 and 3, the clear span of the joists was 16 feet, 1 inch. For Cases 2 and 4, the clear span of the joists was 14 feet, 10 inches. The joists were No. 2 KD19 southern pine. An individual joist was studied for Cases 1 and 2 instead of eight joists, and thus load sharing due to floor sheathing was not taken into account. However, in Cases 3 and 4, three joists were studied collectively and thus a measure of load sharing was taken into account as described later. In order to investigate the effect of E on the expected vibrational performance of each floor system, the frequency distributions were determined for each floor case.

Once the predicted frequency distribution is available for a joist (or several joists assumed to share load), some meaningful measures of floor performance were required for the purpose of comparing VSR and MSR type joists. We selected two measures of performance: the 1st percentile of the predicted frequency of vibration distribution and the probability that the predicted frequency is less than 10 Hz. The first measure was selected based on the philosophy that joists having average or higher E will not be judged unacceptable for the vibrational response because they will have a higher frequency not detected by humans as being annoying. As stated earlier, it is the lower frequencies of vibration (8 to 10 Hz) that are most annoying to humans. Based on the design goal of having 99 percent of all joists not judged as causing annoying vibration, the 1st percentile of the predicted frequency distribution is a logical point for comparison within the two distributions derived for VSR and MSR lumber. The second meas ure, the probability that f is less than 10 Hz, was selected from the upper end of the human sensitivity range. The design goal would be to minimize the probability that any joist in a floor vibrates at a frequency less than 10 Hz.

CASE 1 (SINGLE JOIST MODEL, L/360 DESIGN)

The E was assumed to be lognormally distributed [9]. This assumption was made because it allowed us to vary the coefficient of variation easily and it is frequently accepted as the "best fit" for researchers who have fit distributions to E test data. It is also a reasonable selection for E because the values for E are positive and the lognormal distribution falls above zero along the x-axis. The parameters for the E-distribution were calculated using the following equations from Ang and Tang [2]:

[lambda] = ln [micro] - 1/2 [[xi].sup.2] = ln E - [[[xi].sub.E].sup.2] [3]

[[xi].sup.2] = ln(1 + [[omega].sup.2]) = ln(1 + [[[omega].sub.E].sup.2]) [4]

where:

[lambda] = mean of the logarithms of a variable

[[xi].sup.2] = variance of the logarithms of a variable

[micro] = mean of a variable

[omega] = coefficient of variation of a variable

E = mean of the modulus of elasticity variable, E (psi)

[[[xi].sub.E].sup.2] = variance of the logarithms of the E variable

[[omega].sub.E] = coefficient of variation of the E variable

In order to determine the distribution of f of a single joist, the natural logarithm (ln) is taken on both sides of Equation [2]:

ln f = ln 1.57 + 1/2 ln 386 + 1/2 ln I - 1/2 ln W - 3/2 in L + 1/2 ln E [5]

where for Cases 1 and 3:

I = 98.93 in. [4] (2 by 10)

W = actual tributary dead weight of the joist (joist weight plus 16-in. strip of sheathing plus carpet) = 94 lb. Total

L = 193 in.

Equations [5] can be rewritten as:

ln f = k + 1/2 ln E [6]

where:

k = a constant

E = the modulus of elasticity (psi) of an individual joist

Since E is lognormally distributed, then ln of E results in E being normally distributed. The ln f term would be normally distributed since a constant plus a normally distributed. The exponential of both sides of the equation is thus lognormally distributed. The parameters for f, using Equation [6] are:

[[lambda].sub.f] = E(ln f) = k + 1/2 E(ln E) [7]

where E denotes the expected value, and:

[[lambda].sub.f] = k + 1/2 [[lambda].sub.E] [8]

var(In f) = [[[xi].sup.2].sub.f] = var (k) +

[(1/2).sup.2] var (In E) [9]

[[[xi].sup.2].sub.f] = 0 + 1/4 [[[xi].sub.E].sup.2] = 1/4 [[[xi].sub.E].sup.2] [10]

[[xi].sub.f] = 1/2 [[xi].sub.E] [11]

The assumed E distributions for VSR and MSR lumber are plotted in Figure 1. The predicted joist frequency distributions for VSR (Fig. 2) and MSR (Fig. 3) lumber were plotted using Equations [3], [4], [8], and [11] to calculate the appropriate lognormal density function parameters. The first percentile of the frequency distribution and the probability that the frequency is less than 10 Hz were then calculated from the fitted distributions. Finally, the ratio of the values of the first percentile for each type of lumber were used to compare the vibrational performance to one another using Equation [1].

CASE 2 (SINGLE-JOIST MODEL, L/480 DESIGN)

The procedure for Case 2 was the same as for Case 1. However, the span and dead weight variables in Equation [5] were different because the joist span was determined by the L/480 live-load deflection limit. The values of the variables were:

Joist: I = 98.93 [in..sup.4] (2 X 10)

W = joist weight plus 16-inch strip of sheathing plus carpet = 87 lb.

L = 178 in.

CASES 3 AND 4 (LOAD-SHARING MODEL)

Cases 3 and 4 were Cases 1 and 2 repeated using an effective joist E, [E.sub.j], based and justified by the load-sharing results from the Johnson study [5]. The model used was:

[E.sub.j] = 0.25[E.sub.L] + 0.5[E.sub.C] + 0.25[E.sub.R] [12]

where the E's represent three joists in a row in a floor. [E.sub.L] is the left joist, [E.sub.C] is the center joist impacted by a footfall, and [E.sub.R] is the right joist, and the three were considered independent random variables. Except for the procedures in obtaining the data for E and [f.sub.joist], the procedures used for Cases 3 and 4 were the same as Cases 1 and 2.

Monte Carlo simulation was used to generate 5,000 values for each variable in Equation [12]. Then, [E.sub.j] was used in Equation [2] in place of the variable E to calculate 5,000 values of [f.sub.joist]. The program Graphical Distribution Analysis (GDA) [12] was used to determine the best-fit distribution for the data for [E.sub.j] and [f.sub.joist].

RESULTS AND DISCUSSION

CASE 1 (SINGLE-JOIST MODEL, L/360 DESIGN)

The assumed E distributions for VSR and MSR lumber are shown in Figure 1. For all practical purposes, the E of MSR and VSR lumber ranged from about 700,000 psi to 3,000,000 psi and from about 1,000,000 psi to 2,250,000 psi, respectively. By visual inspection of Figure 1, the E-distribution for MSR lumber is dramatically less variable than the E-distribution for VSR lumber. The lower variability of the E-distribution for MSR is recognized by the published values of coefficient of variation [1] for each lumber type (0.11 versus 0.25).

Figure 2 represents the calculated frequency distribution of vibrational frequencies of a VSR joist with no load sharing assumed. The frequency variable is lognormally distributed because E was assumed to be lognormal. It ranges from about 10 Hz to 22 Hz. The calculated frequency distribution for MSR lumber is shown in Figure 3. It also is lognormally distributed and is noticeably less variable than [f.sub.joist] depicted in Figure 2.

By inspection of Equations [8] and [11], it is evident that the E-distribution affects the predicted fundamental frequency distribution of a joist. For example, the E for MSR lumber is less variable than VSR lumber (0.11 versus 0.25). The parameter [[xi].sup.2] in Equation [4] was calculated based on [[omega].sub.E]. For MSR lumber, a lower [[omega].sub.E] would cause the standard deviation of the logarithms of E ([[xi].sub.E]) to be lower. Since [[xi].sub.E] is the standard deviation for the lognormal density, the variability of the predicted frequency is lower (Fig. 3).

The probability that the predicted frequency will be less than 10 Hz for VSR lumber (in Fig. 3) is 0.0008. The first percentile of the distribution is 11.06 Hz. For Figure 3, the probability that the predicted frequency will be less than 10 Hz for MSR lumber was approximately zero (based on four significant figures and hereafter considered zero). The first percentile of the distribution for MSR is 13.09 Hz, which is about 18 percent more than the first percentile for VSR lumber (Fig. 3).

CASE 2 (SINGLE-JOIST MODEL, L/480 DESIGN)

The assumed E-distribution for both MSR and VSR lumber are identical to the results given for the E-distribution for Case 1 because the values used to calculate the parameters for the E-distribution in Equations [3] and [4] did not change. The two [[omega].sub.E] values (0.11 and 0.25) remained constant throughout the four cases studied.

Results for the joist frequency distribution differed from the results for the frequency distribution in Case 1 because the joists were designed to a stricter deflection limit (L/480). The predicted frequency distribution for both MSR and VSR lumber shifted to the right, producing a better expected vibrational performance than for Case 1.

For VSR lumber, the probability that the joist frequency is less than 10 Hz was zero. The first percentile of the predicted vibrational frequency distribution was 12.98 Hz. For MSR lumber, the probability that the joist frequency was less than 10 Hz was also zero. The first percentile of the predicted frequency distribution for MSR lumber was 15.36 Hz, which is 18 percent more than the first percentile for VSR lumber according to Equation [1]. In other words, the expected vibrational performance for MSR lumber in this case was 18 percent better than VSR lumber.

CASE 3 (LOAD-SHARING MODEL, L/360 DESIGN)

The two-parameter lognormal distributions shown in Figures 4 and 5 fit the effective joist E ([E.sub.j]) data well, based collectively on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smirnov test [2,12]. Again, MSR lumber effective joist E based on Equation [12] (Fig. 5) is less variable than VSR lumber effective joist E (Fig. 4).

For the predicted joist frequency ([f.sub.joist]) based on the load-sharing model (given by Equation [12]), the two-parameter lognormal distribution fit the data as well as shown in Figures 6 and 7 using the same statistical tests. Three MSR lumber joists (Fig. 7) were collectively less variable than three VSR lumber joists (Fig. 6).

The probability that [f.sub.joist] was below 10 Hz for either VSR and MSR lumber joists with load sharing considered among the joists was zero (Table 3). The first percentile frequency from Figure 7 (MSR) was 13.79 Hz compared to 12.47 Hz from Figure 6 (VSR). The ratio of the first percentile predicted frequency of MSR lumber to the first percentile of VSR lumber was 1.11. With the load-sharing model included for three joists in a row, MSR lumber in Case 3 performed 11 percent better than VSR lumber with respect to the first percentile predicted frequency as the measure of floor performance.

The effective joist E (Fig. 4 and 5) for both VSR and MSR lumber in Case 3 is slightly less variable and has a smaller range than the comparable E-distributions shown in Figure 1 for Case 1. In Figure 4, the distribution ranges from about 1.0 x [10.sup.6] psi to about 2.50 x [10.sup.6] psi. The E-distribution for VSR in Figure 1 ranges from about 700,000 psi to about 3.0 x [10.sup.6] psi. For MSR lumber in Case 3, the range in Figure 5 is from about 1.2 x [10.sup.6] psi to 2.0 x [10.sup.6] psi. For Case 1, the MSR E-distribution in Figure 1 ranges from about 1.0 x [10.sup.6] psi to 2.25 x [10.sup.6] psi. The results in Case 3 for [E.sub.j] is due to Equation [12] being a weighted average, which accounts for the distributions having a smaller range, or lower standard deviation.

The distribution for frequency of joist (Figs. 6 and 7) for both VSR and MSR lumber in Case 3 is also slightly less variable and has a smaller range than the distribution for frequency of joist in Figures 2 and 3 for Case 1. In Figure 6, the distribution ranges from about 12 to 18 Hz compared to the range of 10 to 21 Hz for VSR lumber in Figure 2. For MSR lumber (Fig. 7), the range of predicted frequency is about 13 to 16 Hz compared to 12 to 18 Hz for Case 1 (Fig. 3).

CASE 4 (LOAD-SHARING MODEL, L/480 DESIGN)

The two-parameter lognormal distribution fit the data well for effective joist E ([E.sub.j]) for both MSR and VSR lumber. The two-parameter lognormal distribution was determined to be the best fit based collectively on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smimov test [2,12].

Based on the visual test, maximum log-likelihood, Chi-Squared test, and Kolmogorov-Smirnov test, the two-parameter lognormal fit the simulated [f.sub.joist] distribution with load sharing for VSR and MSR lumber. As expected, the distribution for predicted joist frequency for MSR lumber was less variable than the distribution for VSR lumber. The probability that [f.sub.joist] was less than 10 Hz for both VSR and MSR lumber was zero. The first percentile of [f.sub.joist] for MSR lumber was 16.22 Hz and the first percentile of the comparable distribution for VSR lumber was 14.66 Hz. The ratio of these two numbers using Equation [1] was 1.11, which reveals that MSR lumber has a better predicted vibrational performance than VSR lumber by 11 percent.

SUMMARY

For analysis purposes, two measures of floor performance were selected: the 1st percentile of the predicted frequency of vibration distribution and the probability that the predicted frequency is less than 10 Hz. The first measure was selected based on the philosophy that joists having "average" or higher E will not be judged unacceptable for the vibrational response because they will have a higher frequency not detected by humans as being annoying. The lower frequencies of vibration (8 to 10 Hz) stemming from lower E's should be critical and are the analysis focus when the 1st percentile of the frequency distribution is used for comparison purposes. The second measure, the probability that f is less than 10 Hz, was selected from the upper end of the human sensitivity range.

An individual joist was studied for Cases 1 and 2 (load sharing was neglected). The joist in Case 1 was simply supported on rigid supports (typically a block wall) and designed for the L/360 live-load deflection limit. MSR lumber in Case 1 had an improved vibrational performance over VSR lumber by 18 percent based on the ratio of the first percentile of the predicted frequency in Equation [1]. The probability that [f.sub.joist] was less than 10 Hz was close to zero for both VSR and MSR lumber. However, the MSR lumber floor ([[omega].sub.E] = 0.11) had better predicted floor performance based on the measure of the probability of the joist frequency being less than 10 Hz because its probability was zero (based on four significant figures).

Case 2 differed from Case 1 only by the span limit that was designed to the L/480 live-load deflection limit. Both MSR and VSR lumber had favorable predicted vibrational performance because the probability that [f.sub.joist] is less than 10 Hz was zero for both of them. The floor performance of MSR lumber based on the ratio of the first percentile of the predicted frequency in Equation [1] was 18 percent better than VSR lumber. Case 2 provided an example of the positive effect of designing the joist to a stricter deflection limit on the vibrational performance of a floor system. The [f.sub.joist] distributions for both MSR and VSR lumber shifted to the right and increased the first percentile values. Table 2 provides a summary of the vibrational performance measures of both floor Cases 1 and 2 with respect to the [[omega].sub.E] level.

Load sharing was included in the study in Cases 3 and 4. Three joists were studied in Cases 3 and 4 instead of individual joists as in Cases 1 and 2. Case 3 involved three simply supported joists on rigid supports designed for the L/360 live-load deflection limit. The results for the predicted vibrational performance measures in Case 3 that included load sharing improved from the results for predicted vibrational performance in Case 1. The probability that [f.sub.joist] is less than 10 Hz was zero for both VSR and MSR lumber. However, MSR lumber still had better vibrational performance measures than VSR lumber. According to the ratio of first percentile predicted frequencies given by Equation [1], the predicted performance of the MSR lumber joists was 11 percent over the predicted vibrational performance of VSR lumber joists.

Case 4 involved three simply supported joists on rigid supports designed for the L/480 live-load deflection limit. The measures of floor performance improved from Case 3 since the joists were designed to a stricter deflection limit and again MSR lumber performed better than VSR lumber. The probability that is less than 10 Hz was zero for both VSR and MSR lumber. The predicted vibrational performance defined by Equation [1] for MSR lumber versus VSR lumber was 1.11, or 11 percent. The results for Cases 3 and 4 are summarized in Table 3.

CONCLUSIONS AND RECOMMENDATIONS

The variability of E, characterized by [[omega].sub.E], did have an effect on the predicted vibrational performance of floor systems. Overall, MSR lumber ([[omega].sub.E]= 0.11) floor systems had improved predicted vibrational performance over VSR floor systems ([[omega].sub.E] = 0.25) based on two measures of floor performance: fundamental frequency and probability of the fundamental frequency less than 10 Hz.

Based on the two measures of floor performance studied, the predicted vibrational performance across the floor cases improved as the design deflection limits became stricter, specifically L/360 versus L/480. A recommendation for controlling annoying residential floor vibrations is to design the floor joists to a stricter deflection limit of L/480.

The authors are, respectively, Graduate Research Assistant and Professor, Biological Systems Engineering Dept., Virginia Tech, Blacksburg, VA 24061; and Associate Professor, Wood Sci. and Forest Prod., Virginia Tech, Blacksburg, VA 24061.

(+.) Forest Products Society Member.

LITERATURE CITED

(1.) American Forest and Paper Association. 1997. National Design Specification for Wood Construction. AF&PA, Washington, D.C.

(2.) Ang, A. and W.H. Tang. 1975. Probability Concepts in Engineering Planning and Design. Vol. I. John Wiley & Sons, Inc., New York. pp.104-105.

(3.) Dolan, J.D., F.E. Woeste, and X. Li. 1995. Effect of imposed load on solid-sawn wood-joist floor vibrations. Forest Prod. J. 45(1):71-76.

(4.) __________ and T. Skaggs. 1994. Designing to reduce floor vibrations. Pap. No. 944549. American Society of Agri. Engineer, St. Joseph, Mich.

(5.) Johnson, J.R. 1994. Vibration acceptability on wood floor systems. MS. thesis. Virginia Tech, Blacksburg, Va.

(6.) Kalkert, R.E., J.D. Dolan, and F.E. Woeste. 1995. Wood-floor vibration design criteria. J.) of Structural Engineering 121(9):1294-1297.

(7.) Percival, D.H. 1979. History of L/360. Forest Prod. J. 29(8):26-27.

(8.) Shue, B.C. 1995. Some aspects of vibration serviceability in wood floor systems. M.S. thesis. Virginia Tech, Blacksburg, Va.

(9.) Suddarth, S.K., F.E. Woeste, and J.T.P. Yao. 1975. Effect of E-variability on the deflection behavior of a structure. Forest Prod. J. 25(1):17-19.

(10.) __________. W.L Galligan, and D. DeVisser. 1997. The influence of modulus of elasticity variability control in lumber grading on floor performance. Forest Prod. J. 48(6):61-65.

(11.) Wilson, A.C. 1998. Expected vibration performance of wood floors as affected by MSR vs. VSR lumber E-distribution. Master of Engineering Project Rept. Virginia Tech, Blacksburg, Va.

(12.) Worley, J.W., J.A. Bollinger, F,E. Woeste, and K.S. Kline. 1990. Graphical distribution analysis (GDA). Appl. Engineering in Agriculture 6(3):367-371.

Four floor systems constructed of either VSR or MSR lumber were studied based on a live-load deflection criteria on rigid supports, and no load sharing versus a simple load-sharing model.

Joist Load sharing Case design included 1 L/360 No 2 L/480 No 3 L/360 Yes 4 L/480 Yes Predicted vibrational performance of floor Cases 1 and 2 was improved by using lumber having a lower [[omega].sub.E]. Case 1: Joist (L/360) with rigid supports Lumber 1st percentile Probability E-grade [[omega].sub.E] of [f.sub.joist] [f.sub.joist] [less than] 10 Hz MSR 0.11 13.09 [approximate] 0.0000 VSR 0.25 11.06 0.0008 Case 2: Joist (L/480) with rigid supports Lumber [[f.sup.MSR].sub.0.01]/ 1st percentile E-grade [[f.sup.VSR].sub.0.01] of [f.sub.joist] MSR 1.18 15.36 VSR 12.98 Lumber Probability [[f.sup.MSR].sub.0.01]/ E-grade [f.sub.joist] [less than] 10 Hz [[f.sup.VSR].sub.0.01] MSR 0.0000 1.18 VSR 0.0000

The analysis of study Cases 3 and 4 included a load sharing model. The 1st percentile of the predicted joist frequency distribution was considered a measure of floor performance and the measure was impacted by MSR lumber that has a lower [[omega].sub.E].

Case 3: Joist (L/360) with rigid supports; load sharing included Lumber 1st percentile E-grade [[omega].sub.E] of [f.sub.joist] MSR 0.11 13.79 VSR 0.25 12.47 Lumber Probability [[f.sup.MSR].sub.0.01]/ E-grade [f.sub.joist] [less than] 10 Hz [[f.sup.VAR].sub.0.01] MSR 0.0000 1.11 VSR 0.0008 Case 4: Joist (L/480) with rigid supports; load sharing included Lumber 1st percentile Probability E-grade of [f.sub.joist] [f.sub.joist] [less than] 10 Hz MSR 16.22 0.0000 VSR 14.66 0.0000 Lumber [[f.sup.MSR].sub.0.01]/ E-grade [[f.sup.VSR].sub.0.01] MSR 1.11 VSR

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Author: | WILSON, ANN C.; WOESTE, FRANK E.; DOLAN, DAN J. |
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Publication: | Forest Products Journal |

Date: | Apr 1, 2000 |

Words: | 5775 |

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