Floor Vibrations

Floor Vibrations and Cantilevered ConstructionTHOMAS M. MURRAY AND WILLIAM E. HENDRICK

A previous paper by the senior author presented a designmethod to estimate the acceptability of a proposed floorsystem from the standpoint of occupant induced floorvibrations.1 The procedures developed therein are applicableonly to steel beam-concrete slab floor systems where thebeams can be considered simply supported at each end.However, some of the most severe vibration problems haveoccurred in construction involving free cantilevers, to whichthe proposed method of Ref. 1 is not applicable. The purposeof this paper is to present modifications to the methodsuggested in Ref. 1, so that cantilever floor systems andsystems with overhanging beams can be analyzed forannoying floor vibrations. The modifications were verified bytests on seven floor systems reported in Ref. 2. One of thetested systems is used to demonstrate the suggested analysisprocedure.

DESIGN PROCEDURE

Human sensitivity to vibration has been shown to depend onthree parameters: frequency, amplitude, and damping. Scalesrelating these parameters to human reaction have beendeveloped. The modified Reiher-Meister and the Wiss-Parmelee scales are probably the most suitable for occupantinduced floor vibration analysis (see Ref. 1 for a completediscussion). The use of either scale requires an estimate ofthe frequency and amplitude for a specified impact. Inaddition, an estimate of the critical damping is required. Thefollowing sections describe methods that can be used toestimate the three parameters for cantilever and overhangingfloor systems. For brevity, both types of systems will bereferred to as "cantilevered" floor systems in the followingdiscussion.

Damping—The damping in a cantilevered floor system canbe estimated as the sum of the damping of the separateelements in the system. From the guidelines suggested forThomas M. Murray is Associate Professor, School of Civil

Engineering and Environmental Science, University of Oklahoma,Norman, Okla.

William E. Hendrick is Design Engineer, Star ManufacturingCompany, Oklahoma City, Okla.

floor systems supported by simply supported beams: barefloor, 1%-3% (lower limit for thin slab of lightweightconcrete, upper limit for thick slab of normal weightconcrete); ceiling, 1%-3% (lower limit for hung ceiling,upper limit for sheetrock or furring attached to beams);ductwork and mechanical, 1%-10%, depending on amount;partitions, 10%-20%, if attached to the floor and spaced notmore than every five floor beams. These values wereoriginally suggested in Ref. 1 and are based on observationonly, not on the results of a systematic study.

Frequency —From test results presented in Ref. 2, thefrequency of a cantilevered system can be estimated using asingle tee-beam, if the transformed moment of inertia iscomputed assuming:

(a) Composite action, regardless of the method ofconstruction.

(b) An effective slab width, S, equal to the sum of halfthe distance to adjacent beams.

(c) An effective slab depth, de, based on an equivalentslab of rectangular cross section, is equal in weightto the actual slab including concrete in the valleys ofdecking and the weight of decking.

(a) Tee-beam model

(b) Cantilever beam (c) Overhanging beam

Fig. 1. Analytical models

85THIRD QUARTER / 1977

Figure 1(a) shows the tee-beam model for computing thetransformed moment of inertia.

The first natural frequency of a cantilevered tee-beam,Fig. 1(b), is given by:

fb = 1.875 gEIWL

t3

1 2

/

(1)

whereg = 386 in./sec2

E = modulus of elasticity, psiIt = transformed moment of inertia, in.4

W = total weight supported by the tee-beam, lbsL = length of cantilever, in.

The first natural frequency of a simply supported beamwith one overhanging end, Fig. 1(c), is given by:

fb = K gEIWL

t3

1 2

/

(2)

where g, E, It, and W are as defined previously, L = backspanlength, in., and K is a coefficient which depends on theoverhanging length to backspan ratio, H/L. The coefficient Kis determined by setting the determinant of the coefficientmatrix of the boundary condition equations equal to zero. Aclosed form solution for K is not possible and values forspecific H/L ratios were obtained numerically. The value ofthe coefficient K can be determined from Fig. 2.

Equations (1) and (2) were derived for free lateralvibration of prismatic, straight, elastic beams consideringbending deformations only.

In practice, overhanging beams are usually supported byflexural members rather than rigid supports. The flexibilityof these members can significantly affect the frequency of thefloor system. In Ref. 2, it is shown that the system frequencyfor such cases can be approximated by:

1 1 12 2 2f f fs b g

= + (3)

wherefs = the system frequency, Hzfb = overhanging beam frequency, Hz

fg = 1.57 gEI

W Lg

g g3

1 2

/

(4)

whereWg = total supported weight, lbsIg = girder moment of inertia, in.4

Lg = girder span, in.

In the computation of Ig, composite action should not beassumed unless the slab or deck rests directly on the girderflange. The effective slab width should be estimated as fornormal composite construction, even if shear connections arenot used.

Table 1. Dynamic Load Factors for Heel-Drop Impact

f, Hz DLF f, Hz DLF f, Hz DLF

1.00 0.1541 5.50 0.7819 10.00 1.17701.10 0.1695 5.60 0.7937 10.10 1.18311.20 0.1847 5.70 0.8053 10.20 1.18911.30 0.2000 5.80 0.8168 10.30 1.19491.40 0.2152 5.90 0.8282 10.40 1.20071.50 0.2304 6.00 0.8394 10.50 1.20651.60 0.2456 6.10 0.8505 10.60 1.21211.70 0.2607 6.20 0.8615 10.70 1.21771.80 0.2758 6.30 0.8723 10.80 1.22311.90 0.2908 6.40 0.8830 10.90 1.22852.00 0.3058 6.50 0.8936 11.00 1.23392.10 0.3207 6.60 0.9040 11.10 1.23912.20 0.3356 6.70 0.9143 11.20 1.24432.30 0.3504 6.80 0.9244 11.30 1.24942.40 0.3651 6.90 0.9344 11.40 1.25452.50 0.3798 7.00 0.9443 11.50 1.25942.60 0.3945 7.10 0.9540 11.60 1.26432.70 0.4091 7.20 0.9635 11.70 1.26922.80 0.4236 7.30 0.9729 11.80 1.27402.90 0.4380 7.40 0.9821 11.90 1.27873.00 0.4524 7.50 0.9912 12.00 1.28343.10 0.4667 7.60 1.0002 12.10 1.28793.20 0.4809 7.70 1.0090 12.20 1.29253.30 0.4950 7.80 1.0176 12.30 1.29703.40 0.5091 7.90 1.0261 12.40 1.30143.50 0.5231 8.00 1.0345 12.50 1.30583.60 0.5369 8.10 1.0428 12.60 1.31013.70 0.5507 8.20 1.0509 12.70 1.31433.80 0.5645 8.30 1.0588 12.80 1.31853.90 0.5781 8.40 1.0667 12.90 1.32274.00 0.5916 8.50 1.0744 13.00 1.32684.10 0.6050 8.60 1.0820 13.10 1.33084.20 0.6184 8.70 1.0895 13.20 1.33484.30 0.6316 8.80 1.0969 13.30 1.33884.40 0.6448 8.90 1.1041 13.40 1.34274.50 0.6578 9.00 1.1113 13.50 1.34664.60 0.6707 9.10 1.1183 13.60 1.35044.70 0.6835 9.20 1.1252 13.70 1.35414.80 0.6962 9.30 1.1321 13.80 1.35794.90 0.7088 9.40 1.1388 13.90 1.36155.00 0.7213 9.50 1.1454 14.00 1.36525.10 0.7337 9.60 1.1519 14.10 1.36885.20 0.7459 9.70 1.1583 14.20 1.37235.30 0.7580 9.80 1.1647 14.30 1.37585.40 0.7700 9.90 1.1709 14.40 1.3793

Amplitude —The "heel-drop" impact has been used todevelop acceptability criteria when the modified Reiher-Meister scale is used. The amplitude of a single tee-beamsubjected to a heel-drop impact can be computed from:

Aot = (DLF)max ∆s (5)

whereAot = amplitude(DLF)max = maximum dynamic load factor∆s = static deflection caused by a 600-lb force

86

ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

Fig. 2. Frequency coefficients for overhanging beams

Therefore, for a cantilever tee-beam,

Aot = (DLF)max × 6003

3LEIt

(6)

and for an overhanging tee-beam,

Aot = (DLF)max × 600

3

2H L HEI t

( )+(7)

Equations for (DLF)max are given in Ref. 1 and values of(DLF)max are given in Table 1.

Usually more than one tee-beam is effective in resistingan impact. The first maximum amplitude of a floor systemcan be estimated from:

Ao = Aot/Neff (8)

where Neff = number of effective tee-beams. For a series oftee-beams with equal effective flange width and simplesupports, it was shown in Ref. 3 that:

Neff = 2.967 – 0.05776(S/de) + 2.556 × 10–8(L4/It)+ 0.00010(L/S)3 (9)

Unless L/S is very large, i.e., greater than 10, Neff can beapproximated from:

Neff = 2.97 – S

dLEIe t17 3 135

4

. .+ (10)

where E = 29 × 106 psi and S, de, L, It are in inch units. As adesign approximation, it is assumed here that the number ofeffective tee-beams for cantilevered construction is the sameas for simple supports and no overhang.

Equation (9) was developed assuming at least fiveidentical tee-beams exist and the impact location is at thecenter of the five beams. Frequently, the framing forcantilevered balconies is irregular and Eq. (9) cannot beused. For such cases, it is suggested that a static finiteelement analysis be used to determine Ao. A computerprogram such as STRUDL4 can be used by dividing the slabinto a mesh and treating the beams as line elements with amoment of inertia equal to the transformed moment of inertia,and determining the maximum static deflection caused by a600-lb concentrated load. In lieu of a finite element analysis,the designer may conservatively take Neff = 1.


Fig. 3. Modified Reiher-Meister scale

Proposed Design Method1. Estimate the damping in the finished floor system; if

greater than 8%-10% there is no need for a vibrationanalysis.

2. Compute the transformed moment of inertia of a singletee-beam, It, using the guidelines presented.

3. Compute the frequency from Eqs. (1) or (2) and (3), asapplicable.

4. Compute the heel-drop amplitude of a single tee-beam,Aot, using Eq. (6) or (7) and Table 1.

5. If the effective slab widths are equal, estimate thenumber of effective tee-beams, Neff, using Eq. (10);otherwise, perform a static finite element analysis or

conservatively use Neff = 1.6. Compute the amplitude of the floor system, using Ao =

Aot/Neff.7. Estimate perceptibility, using the modified Reiher-

Meister scale, Fig. 3.8. If the system plots below the lower half of the distinctly

perceptible range, the system is satisfactory if thedamping is less than 3%-4%. If the system plots in theupper half of the distinctly perceptible range and thedamping is relatively low, less than 6%-8%, complaintsfrom occupants may occur. If the system plots above thedistinctly perceptible range, the system will beunacceptable if the damping is less than 10%-12%.

88


(a) Initial amplitude (b) Frequency

Fig. 4. Comparison of theoretical and experimental results

EXPERIMENTAL VERIFICATION

Tests were conducted in seven buildings at a total of 15locations to verify the proposed design method.2 Nine of thelocations were on church balconies, three were on the upperlevel of a shopping mall, two were on the second floorexterior walkway of a motel, and one location was on thesecond floor exterior balcony of an office building. At eachlocation the floor was impacted by an approximately 190-lbman executing a heel-drop. The resulting floor motion wasrecorded, together with timing lines, on light sensitive paperusing an engineering seismograph. From the record, it waspossible to determine the initial amplitude, frequency, anddamping of the floor system.

Comparisons of predicted and measured frequencies andamplitudes are shown in Figs. 4(a) and 4(b), respectively.Considering there was no laboratory-type control of the floorsystem construction or tolerances, the results are consideredto be excellent. Furthermore, some of the beams supportingthe church balconies were not prismatic, varying in bothsection and slope to meet pew location and walkwayrequirements, and engineering judgment was used to obtainequivalent stiffness for use in the proposed design formulas.

EXAMPLE

Figure 5 shows structural details of a test location on theupper level of the shopping mall.2 The floor is 1½-in. claytile laid over a 2½-in. concrete slab on a steel deck. The slabis supported by W8 × 15 beams at 30 in. on center, whichrest on the top flange of a W27 × 94. The beams are assumedto act compositely with the concrete slab and the clay tile fortransformed moment of inertia calculations. Composite actionis not assumed for the girder.

Damping: Slab and beam 2%Soffit 3

5% < 8%

∴ Need to investigate floor system.

Beam Transformed Section Properties (see Fig. 5):

de = 3.5 in.; n = 7.6Acn

=30 35

7 6( . ).

= 13.82 in.2

W8 × 15: A = 4.43 in.2; I = 48.1 in.4;d = 8.12 in.


(a) Framing plan

(b) Section

(c) Tee-beam model

Fig. 5. Framing system for example

Yb =+ + +

+=

4 43 812 2 1382 812 05 1754 43 1382

884

. ( . ) . ( . . . ). .

. in.

It =1382 35

12

2. ( . ) + 13.82(1.53)2 + 48.1

+ 4.43 884812

2

2

..

−

= 195.8 in.4

Beam Frequency:

Floor Weight:

Clay tile = 20 psfConcrete = 150 pcf

W =

+

150

212

20 (23.34)(2.5) + 23.34(15)

= 2976 lbs

90


HL

= =9 20

1414065

..

.

From Fig. 2: K = 0.72

fb =×

=

0 72386 29 10 1958

2976 1414 12884

6

3

1 2

.( )( . )

( )[ . ( )].

/

Hz

Girder Frequency:W27 × 94: I = 3270 in.4

W = + +

+

=

( . )( . )

( ).

.

,

23342 1414

2 15012

201525

94 3883

41 796

2

lbs

f g =×

=

157386 29 10 327041796 83 12

4 62

6

3

1 2

.( )( )

( )[38. ( )].

/

Hz

System Frequency:

1 14 62

1884

0 05962 2 2fs

= + =( . ) ( . )

.

fs = 4.09 Hz

Amplitude:Neglect influence of girder.

∆ s =+

×=

600 9 2 12 9 2 1414 123 29 10 1958

01202

6[ . ( )] ( . . )( )

( )( . ). in.

From Table 1, with fs = 4.09 Hz:(DLF)max = 0.604Aot = 0.604(0.12) = 0.072 in.

Neff = − +×

=

2 9730

17 3 351414 12

135 29 10 1958258

4

6.( . )( . )

[ . ( )]. ( )( . )

.

Ao = 0.072/2.58 = 0.028 in.

Perceptibility:With a frequency of 4.09 Hz and an initial amplitude of0.028 in., the system plots in the upper third of the"distinctly perceptible" range on the modified Reiher-Meister scale, Fig. 3.

Field Measurements:The system was measured before the soffit wascompleted:

Damping = 1.39 %f = 4.72 Hz

Ao = 0.029 in.

REFERENCES

1. Murray, Thomas M. Design to Prevent Floor VibrationsEngineering Journal, American Institute of Steel Construction,Vol. 12, No. 3, Third Quarter, 1975.

2. Hendrick, William E. Floor Vibrations in CantileveredConstruction A thesis submitted to the graduate faculty inpartial fulfillment of the requirements for the degree of Masterof Science, University of Oklahoma, Norman, Okla., 1976.

3. Saksena, S. K., and T. M. Murray Investigation of a FloorVibration Parameter School of Civil Engineering andEnvironmental Science, University of Oklahoma, Norman,Okla., Feb. 1972.

4. ICES STRUDL-II, The Structural Design Language CivilEngineering Systems Laboratory, Massachusetts Institute ofTechnology, Cambridge, Mass., June 1972.


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