Vibrational Modal Frequencies and Shapes of Two-Span Continuous Timber Flooring Systems

. Based on classic vibrational bending theory on beams, this paper provides comprehensive analytical formulae for dynamic characteristics of two equal span continuous timber ﬂooring systems, including frequency equations, modal frequencies, and modal shapes. Four practical boundary conditions are considered for end supports, including free, sliding, pinned, and ﬁxed boundaries, and a total of sixteen combinations of ﬂooring systems are created. The deductions of analytical formulae are also expanded to two unequal span continuous ﬂooring systems with pinned end supports, and empirical equations for obtaining the fundamental frequency are proposed. The acquired analytical equations for vibrational characteristics can be applied for practical design of two-span continuous ﬂooring systems. Two practical design examples are provided as well.


Introduction
Vibrational serviceability performance of timber floors has become an important issue in the world due to their resonance frequencies and low material masses.In Europe, Eurocode 5 [1] has been widely used for design of timber floors to satisfy serviceability limit state criteria [2], in particular vibrations, because they are often governing the design of timber floors.
e fundamental frequency, unit point load deflection, and unit impulse velocity response as the key vibrational parameters need to be checked.e methods for obtaining these parameters and their limits are presented in EN 1995 Part 1-1 [2] and the National Annexes of individual European countries [3].
Human beings are considered as precarious sensors of vibrations, and their distress to timber floor vibrations concern many researchers, and human activities and machine-induced vibrations can cause distress.Human sensitivity and perception are basically related to structural vibrations.

Previous Research Studies on Timber Floor Vibration
Over past decades, extensive investigations have been conducted on evaluating the dynamic performance of timber floors and human vibrational perception in many European countries, Canada, Australia, and Japan.Ohlsson [4] investigated human-induced vibrations by testing a number of timber floors, and his work has been implemented in Eurocode 5 for timber floor design [1].Chui [5] proposed use of the root mean square (r.m.s.) acceleration for design based on his field tests on timber floors and.Hu [6] dynamically tested I-joist floors and numerically simulated the vibrational behaviour of these floors.Eriksson [7] measured the dynamic forces of low frequencies caused by walking, running, and jumping.Smith [8] explored the vibrational issues for timber floors.Bahadori-Jahromi et al. [9,10] innovated timber floors with multiwebbed joists and conducted static and dynamic tests on these floors.Homb [11] assessed human perception based on the results of impact-induced low frequency vibrations on timber floors.
Ljunggren [12] assessed the dependence of human perception on the dynamic properties of lightweight steel framed floors.Toratti and Talja [13] established body perception scales to reflect the significant dependence of disturbance on different vibrational sources.Zhang et al. [14,15] systematically assessed the effects of influencing parameters on the vibrational performance of timber floors constructed with various joists.Weckendorf [16] measured the vibrational characteristics of I-joist timber floors.Labonnote [17] categorised damping in timber components.Zhang et al. [18,19] assessed the vibrational responses of timber floors constructed with metal web joists by altering structural configurations.Jarnerö et al. [20] measured the dynamic characteristics of a prefabricated timber floor with various boundary conditions at different construction stages.Two textbooks were also written on structural timber design to Eurocode 5 [21,22].Recently, considerable attention has been paid to the dynamic responses of timber floors constructed with LVL (laminated veneer lumber) beams, glulam beams and CLT (cross-laminated timber) panels, and TCC (timberconcrete composite) floors.Basaglia et al. [23] assessed the vibrational measures on timber floors and compared the design criteria between Australia and Japan by testing a 9 m span LVL ribbed-deck cassette floor.Ebadi et al. [24] explored the vibrational response of glulam floors.Ui Chulain and Harte [25] investigated the influence of modern timber fixing systems and extra mass on the serviceability behaviour of one-way or two-way CLT floors in laboratory.Koyama et al. [26] measured the vibrational characteristics of a CLT floor subjected to walking vibrations.Bui et al. [27] experimentally investigated the natural frequencies, damping ratios, and mode shapes of multilayered timber beams.Lanata et al. [28] attempted to establish correlations between the dynamic response of in situ timber and TCC floors and human comfort perception by varying building typology.

Design Codes in European
Countries.EN 1995-1-1 [1] has adopted three parameters proposed by Ohlsson [4] for controlling the vibrational serviceability design of timber flooring systems.For residential floors which are simply supported along all four edges, it requires that the fundamental frequency or the lowest first-order modal frequency f 1 in Hz should satisfy the following equation: where L is the floor span in m, (EI) L is the equivalent plate bending stiffness of the floor about an axis perpendicular to the beam direction in Nm 2 /m, and m is the mass per unit area in kg/m 2 .Equation ( 1) is a simplified design equation which is actually applied for two-side supported floors, and the effect of the transverse stiffness is omitted because of small errors caused.EN 1995-1-1 does not clearly indicate how the participating mass should be calculated and whether the composite effect of floor joists and deck in the floor direction should be considered.e design equations and the corresponding limits for f 1 proposed from EN 1995-1-1 and various National Annexes in European countries have been previously summarised [3].e majority of European countries have directly adopted equation ( 1) and the limit of 8 Hz specified in EN 1995-1-1 [29].However, Austria [30] only adopts equation (1) for two-side supported floors and provides a fairly accurate equation for four-side supported floors by including a quartic term about the floor span to width ratio, L/B, to reflect the effect of the transverse stiffness as Similarly, Finland [31] provides a more accurate equation for four-side supported floors by including both second-and fourth-order terms about L/B for the effect of the transverse stiffness with a raised frequency limit of 9 Hz as Both Austria and Finland specify that the floor mass m should be determined by using a quasipermanent combination of dead and imposed loads, as specified in EN 1990 [2]: where m G k is the mass due to the characteristic dead load G k and m Q k is the mass due to the characteristic imposed load Q k .ψ 2 is the factor for quasipermanent value of a variable action, and its values for different building categories can be taken from the relevant tables of EN 1990 [2] or the National Annexes to EN 1990.In general, ψ 2 can be taken as 0.3 for residential and office buildings, 0.6 for congregation areas, and 0.8 for storage areas.Spain [32] limits the values for the fundamental frequency f 1 for all construction materials including timber.More recently, Hu et al. [33] attempted to develop a baseline vibration design method for timber floors by combining all available databases in the world so as to eventually develop an ISO standard.e future work will be to harmonise the calculation methods and to rederive the baseline design criterion by using the collective floor database.

Significance of Current Research.
With the development of engineered timber products, floor joist sizes can be manufactured much larger for longer floor spans or multispans.Some research work on structural dynamic characteristics of general continuous beams has been reported [34,35].
e information for determining vibrational characteristics of continuous timber flooring systems, however, is still limited.In this paper, the characteristic equations for modal frequencies and shapes of continuous floors of two equal spans with various boundary conditions or floors of two unequal spans with pinned boundary conditions are deduced.Two design examples of two-span timber flooring systems are also presented.

2
Shock and Vibration

Equations of Motion for Two-Span Continuous Floors
For two-span continuous timber ooring systems with equal joist spacing but various end supports, they can be treated as two-span beams and analysed on a single two-span continuous timber oor joist (see Figure 1 in which 0 ≤ x 1 ≤ L 1 and 0 ≤ x 2 ≤ L 2 ).

Equations of Motion.
For a two-span continuous Bernoulli-Euler beam with a uniform cross-sectional area, mass density, and exural sti ness, the equation of motion for each beam span, i.e., transverse displacement y i (x i , t) versus time t for span i, is given as where EI is the exural sti ness, c is the viscous damping coe cient, ρ is the mass density per unit volume, A is the cross-sectional area, f i (x i , t) is the time varying external load, and i is the span number (i 1, 2) [36].Assume y i (x i , t) can be separated for x i and t as Substituting equation ( 6) in equation ( 5) and ignoring the harmonic part yields the homogeneous solutions as e natural frequencies and coe cients C i1 to C i4 of the beam can be obtained by applying appreciate boundary conditions.e natural frequencies f n are determined as where ω n is the circular modal frequency for n th mode in rad/sec, (EI) L is the equivalent plate bending sti ness of the timber oor about an axis perpendicular to the beam direction in Nm 2 /m and (EI) L is given as (EI) joist /s, (EI) joist is the exural sti ness of the oor joist in Nm 2 , s is the oor joist spacing in m, and m is the mass per unit area in kg/m 2 .

Boundary Conditions. Four typical boundary conditions
for two-span continuous beams are considered with various end supports, e.g., free, sliding, pinned, and xed boundaries (Table 1), which forms sixteen combinations of ooring systems.

Vibrations for Continuous Floors with Two Equal Spans
For continuous timber ooring systems with two equal spans, L 1 L 2 L.

Frequency Equations.
To establish the frequency equations of two equal span ooring systems and to determine the modal frequencies and mode shapes, the displacement equation ( 7) should be used in conjunction with the boundary conditions listed in Table 1.To demonstrate the procedure, a typical two equal span continuous beam with left end xed and right end simply supported is used here (Figure 2).For spans 1 and 2, equation ( 7) can be rewritten as Di erentiating equations ( 9) and ( 10) for three times yields Figure 1: A typical two-span continuous timber oor.
Table 1: Boundary conditions for two-span continuous beams.

Boundary conditions Control equations
Free end Pinned middle support (16)

Shock and Vibration
Combining equations ( 15), (24), and ( 27) and letting x 2 � L yields  30) and ( 31) yields or (sin βL cosh βL − cos βL sinh βL) (cos βL − cosh βL) A nontrivial solution for equations (33) and ( 35) exists only if the determinant of the coefficient matrix vanishes, which yields 2(1 − cos βL cosh βL)sin βL (cos βL − cosh βL) us, the frequency equation can be obtained as Similarly, based on equation (7) and the boundary conditions for different two equal span flooring systems which are listed in Table 1, the frequency equations for other fifteen cases with two equal span floors and various end support conditions can be deduced, as illustrated in Table 2.

Modal Frequencies and Shapes.
From the frequency equations, the modal frequencies for two equal span timber flooring systems can be obtained numerically.Here commercial software MathCAD is used for such purpose, and Table 3 illustrates the values of β i L for the first four modal frequencies of sixteen timber flooring systems with two equal spans and various boundary conditions.e symbol * in Table 3 indicates the rigid mode for free-free boundary conditions.e vibrational mode shapes only for the first modes of two equal span continuous floor beams with various boundary conditions are illustrated in Figures 3-6. Figure 3 illustrates the first vibrational modes for two equal span continuous timber floor beams with free left ends and various boundary conditions for right ends.All the mode shapes are not in scale.e left ends sustain free drops, and the right ends have various shapes which depend on predefined boundary conditions.At the interior support, the slopes of the mode shapes for both left and right spans are continuous.In general, the fundamental frequency parameter β 1 L increases in the following order for the right end boundary conditions: free, sliding, pinned, and fixed.It should be mentioned that the first vibrational mode for the free-free boundary conditions should be rigid with a zero modal frequency.
Figure 4 illustrates the first vibrational modes for two equal span continuous timber floor beams with sliding left ends and various boundary conditions for right ends.e left ends have zero slopes, and the right spans have various shapes which depend on predefined boundary conditions.e fundamental frequency parameter β 1 L increases in the same order for the right end boundary conditions: free, sliding, pinned, and fixed.
Figure 5 illustrates the first vibrational modes for two equal span continuous timber floor beams with pinned left ends and various boundary conditions for right ends.
e left ends have zero displacements, and the right spans have various shapes which depend on predefined boundary conditions.e fundamental frequency parameter β 1 L increases in the same order for the right end boundary conditions: free, sliding, pinned, and fixed.
Finally, Figure 6 illustrates the first vibrational modes for two equal span continuous timber floor beams with fixed left ends and various boundary conditions for right ends.e left ends have zero displacement and slope, and the right spans have various shapes which depend on predefined boundary conditions.e fundamental frequency parameter β 1 L increases in the same order for the right end boundary conditions as mentioned above.
e shapes for higher modes are more complex and will not be discussed here further.

Frequency Equation.
For continuous floors with two unequal spans (L 2 � αL 1 ) and pinned end supports (Figure 7), the frequency equation can be obtained based on equation ( 7) and the boundary conditions in Table 1 for pinned ends.
Shock and Vibration

Boundary conditions Control equations
Free-free

Original position Free-free Free-sliding
Free-pinned Free-fixed Figure 3: First vibrational modes for two equal span continuous timber oor beams with free left ends and various boundary conditions for right ends.Shock and Vibration (45) Solving equations ( 44) and (45) simultaneously yields us, equation ( 10) becomes (49)

Model Frequencies and Shapes.
From the frequency equations (58), (59), and (60), the modal frequencies of timber flooring systems with two unequal spans and pinned end supports for various span ratios can be obtained numerically.Again, commercial software MathCAD is used for such purpose, and Table 4 illustrates the values of β i L for the first four modal frequencies of timber flooring systems with eleven typical floor span ratios α � L 2 /L 1 between 0 and 1. Figure 8 shows the fundamental vibrational mode shapes for α � 0.00001, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.When α � L 2 /L 1 is very small, e.g., 0.00001, the mode shape for the left span approaches to the shape for a single 8 Shock and Vibration span beam with pinned left end and xed right end, and the fundamental modal frequency is the largest.e corresponding amplitude for the right span is very small.With the increase in α value, the fundamental vibrational frequency f 1 decreases and the mode shape for the right span which always follows a half sine wave becomes more extended with higher amplitude.When α 1.0, a full sine curve is obtained for the whole two equal span beam. is trend continues for α larger than 1.0.For this case, however, the right span can be treated as L 1 and the left span treated as L 2 , so the same calculations can be conducted.At the interior support, the slopes of the mode shapes for both left and right spans are continuous.

Empirical
Equations for Practical Design.Figure 9 illustrates that for α 0.0 to 1.0, a cubic relationship between the frequency parameter β 1 L and the span ratio α can be established empirically with the linear regression coe cient R 2 0.9999 as is equation can be used to determine the fundamental vibrational frequency, f 1 , of continuous timber oors with two unequal spans for practical serviceability design as soon as the span ratio α is known.
Sometimes, a linear relationship between β 1 L 1 and α L 2 /L 1 can be more convenient for practical engineers to quickly estimate the fundamental frequency of a timber oor.Here, only the values of β 1 L 1 for α 0 and 1.0 are used to form a simple linear empirical equation as follows without large errors from the accurate values (Figure 9): (62)

Design Examples
6.1.Two Equal Span Floor with Solid Timber Joists for Fixed-Pinned Ends.A two-span timber oor is designed for a domestic timber frame building.It is constructed with continuous solid timber joists (Figure 10).e oor has a width B 5.0 m and a length L 1 L 2 4.8 m for each span.It is constructed with 75 mm × 220 mm C24 solid timber joists at a spacing s 600 mm. e P5 particleboard with a thickness of 22 mm is chosen for the decking, and the Gyproc plasterboard with a thickness of 12.5 mm is chosen for the ceiling.e total self-weight of the ooring system including the timber joists is assumed to be 60 kg/m 2 , and Service Class 2 is assumed.e imposed load is Q k 1.5 kN/ m 2 from EN 1991-1-1 [37].
From equation ( 8), the fundamental frequency f 1 can be determined as where β 1 L 1 is the frequency parameter for the rst mode and β 1 L 3.3932 (quoted from Table 3), L 1 is the left span length

Two Unequal Span Floor with I-Joists for Pinned-Pinned
Ends.A two-span timber oor is designed for an o ce timber frame building.has a width B 6.0 m, together with lengths L 1 7.3 m and L 2 2.92 m for left and right spans, which gives a span ratio α L 2 /L 1 2.92/7.30.4.It is constructed with the engineered I-joists (JJI joists) produced by James Jones & Sons Ltd in the UK [39] (Figure 11).e top and bottom anges are manufactured from C24 solid timber with the width b ranging from 47 mm to 97 mm (A to D) and a constant height of h f 45 mm. e web is manufactured from 9 mm OSB3 which is embedded into the anges by 12 mm.e 22 mm P5 particleboard is chosen for the decking, and the Gyproc plasterboard with a thickness of 12.5 mm is chosen for the ceiling.
e total self-weight of the ooring system including the engineered I-joists is assumed to be 75 kg/m 2 , and also Service Class 2 is assumed.e imposed load is taken as Q k 2.5 kN/m 2 [37].e adopted JJI joists for this study are JJI 400D joists at a spacing s 300 mm and with an overall joist depth h 400 mm.
From equation ( 8), the fundamental frequency f 1 can be determined as where β 1 L 1 is the frequency parameter for the rst mode and β 1 L 3.6070 (quoted from Table 4), L 1 is the left span length of the oor and L 1 7.5 m, L 2 is the right span length of the oor and L 2 3.0 m, α is the span ratio and a L 2 /L 1 3.0/ 7.5 0.4, b is the width of the solid timber anges of JJI 400D joists and b 97 mm, h f is the depth of the solid timber anges of JJI 400D joists and h 45 mm, t w is the thickness of the OSB3 web of JJI 400D joists and t w 9 mm, h is the overall depth of JJI 400D joists and h 400 mm, E is the elastic modulus for C24 timber and E 11 GPa [38], E P5 is the mean elastic modulus for 9 mm OSB3 web and E P5 3000 MPa, I is the equivalent second moment of area with respect to C24 solid timber for a JJI 400D joist and I 2.8345 × 10 −4 m 4 , (EI) L is the same as above and (EI) L EI/s 11 × 10 9 × 2.8345 × 10 −4 /0.3 10.3932 × 10 6 Nm 2 /m, s is the oor joist spacing and s 0.3 m, and m is the oor mass per unit area and m 75 kg/m 2 .

Conclusions
Based on classic vibrational bending theory on beams, comprehensive analytical formulae for dynamic characteristics of two-span continuous timber ooring systems have been established, including frequency equations, modal frequencies, and mode shapes.Four practical boundary conditions are considered for end supports, including free, sliding, pinned, and fixed supports, and a total of sixteen combinations of flooring systems are created.e characteristic equations for modal frequencies of two equal span continuous beams with various boundary conditions have been deduced, and four sets of mode shapes have been illustrated.A rigid mode exists for free-free end boundary conditions.A full example for fixed-pinned boundary conditions has been presented to show the deduction procedure.
e characteristic equations for modal frequencies of two unequal span continuous beams with pinned-pinned end conditions have been deduced, and one set of the corresponding mode shapes for various span ratios has been illustrated.A full example for pinned-pinned boundary conditions with a general span ratio has been presented to show the deduction procedure.Also, two empirical equations, cubic and linear, for determining the modal frequency parameters with respect to varying span ratios have been proposed and can be used directly for practical design.
Finally, two practical design examples for determining fundamental modal frequencies have been presented, one for a two equal span continuous floor constructed with solid timber joists and fixed-pinned ends and the other for a two unequal span continuous floor constructed with JJI 400D Ijoists pinned at both ends.

2C 11 β 2 ( 2 LFigure 2 :
Figure 2: A two equal span continuous timber oor beam with left end xed and right end simply supported.

Figure 4 :Figure 5 :Figure 6 :
Figure 4: First vibrational modes for two equal span continuous timber oor beams with sliding left ends and various boundary conditions for right ends.

Figure 7 :
Figure 7: A continuous timber oor beam with two unequal spans and pinned-pinned end supports.

Figure 10 :Figure 11 :
Figure 10: A two-span oor constructed from solid timber joists with xed left end and pinned right end.(a) A two-span oor constructed with solid timber joists.(b) A two-span oor with xed left end and pinned right end.

Table 3 :
Frequency parameters for two-span beams with various end support conditions.

Table 4 :
Frequency parameters for continuous beams with two unequal spans and pinned-pinned end supports.