Keywords

1 Introduction

Technology has seen an increase in floor spans and the use of lighter and higher strength materials, which has resulted in a growing concern for floor vibrations. Several design guidelines which provide methodologies for the assessment of footfall-induced floor vibrations have been developed [1,2,3]. Floor vibration design guides usually include simplified hand calculation procedures for the determination of the modal properties and vibration response of the floor. The Steel Construction Institute Publication P354 (SCI DG) [2] and the Concrete Centre Industry Publication 016 (CCIP DG) [3] are the two widely recognized design guides in the UK and other countries. In addition, a number of researchers have dedicated their work to the improvement of finite element (FE) techniques for floor vibration analysis [4,5,6,7]. Evaluation of footfall induced floor vibrations using FE modeling normally involves both modal analysis and time history analysis. The natural frequencies obtained from a modal analysis could suggest whether the floor should be treated as a low frequency or high frequency floor. The material covered in this paper relates to low frequency floors (fundamental frequency less than about 10 Hz) in which resonance may cause severe vibration amplification, rather than high frequency floors where resonance becomes less important compared with transient response [2].

The paper examines the accuracy of a semi-FE approach, in comparison with the time history analysis, in predicting the floor response to walking excitation. The semi-FE method incorporates the floor modal properties obtained from an FE modal analysis into hand calculation formulas suggested by the SCI DG and CCIP DG for response prediction. The footfall-induced vibration of an actual office floor is investigated and the results from various prediction methods along with some experimental findings are discussed.

2 Contemporary Guidelines for Floor Vibration Evaluation

2.1 Walking Force Model

The continuous steady state walking excitation is represented by a Fourier series of sine waves:

$$F\left( t \right) = P\left[ {1 + \sum\limits_{{i = 1}}^{H} {\alpha _{i} } \sin \left( {2\pi if_{p} t + \phi _{i} } \right)} \right]$$
(1)

where fp is the step frequency; P is the standard pedestrian’s weight, normally taken as 746 N (76 kg); αi and ϕi are the dynamic load factor and phase angle of the i-th harmonic component. The forcing function consists of four harmonics (H = 4) with the dynamic load factor αi corresponding to the i-th harmonic given in Table 1 as suggested by the SCI DG and CCIP DG [2, 3]. The phase angles are taken as 0, ‒π/2, π and π/2 for the first, second, third and fourth harmonics respectively.

Table 1 Dynamic load factors for walking excitation
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2.2 Response Calculation

SCI DG. The response in each vibration mode of the floor to each harmonic component of the walking force is calculated. The total acceleration response due to all harmonics in all modes is then combined using the following expression:

$$a_{RMS} = \frac{1}{\sqrt 2 }\sqrt {\mathop \sum \limits_{i = 1}^{H} \left( {\mathop \sum \limits_{n = 1}^{N} \left( {\varphi_{e,n} \varphi_{r,n} \frac{{\alpha_{i} P}}{{m_{n} }}D_{i,n} \lambda_{i} } \right)} \right)^{2} }$$
(2)

where aRMS is the root-mean-square acceleration; H, N are the number of harmonics and modes to be considered, respectively; φe,n, φr,n are the normalized mode shape values at the point of excitation and the point of response in mode n, respectively; mn is the modal mass of the floor in mode n. As human would be most sensitive to vibrations in the frequency range of 4–8 Hz, a code-defined frequency weighting factor for human perception λi is introduced, which is equal to 1.0 for frequencies within this range and less than 1.0 for those outside the range [2, 8]. Let fn and ζ be the natural frequency and damping ratio of the floor in mode n. The dynamic magnification factor Di,n in mode n to harmonic i is computed using Eq. 3 in which ri,n is the ratio of the forcing frequency ifp to the floor frequency fn. Moreover, a resonant build-up factor (≤ 1) can be included in Eq. 2 to allow for the case where a walking path is sufficient short and a steady-state condition may not be reached.

$$D_{i,n} = \frac{{r_{i,n}^{2} }}{{\sqrt {\left( {1 - r_{i,n}^{2} } \right)^{2} + \left( {2\zeta r_{i,n} } \right)^{2} } }}$$
(3)

CCIP DG. The real and imaginary acceleration (areal,i,n and aimag,i,n) due to each harmonic, i, in each mode, n, are calculated as:

$$a_{real,i,n} = \varphi_{e,n} \varphi_{r,n} \frac{{\alpha_{i} P}}{{m_{n} }}\left[ {\frac{{r_{i,n}^{2} \left( {1 - r_{i,n}^{2} } \right)}}{{\left( {1 - r_{i,n}^{2} } \right)^{2} + \left( {2\zeta r_{i,n} } \right)^{2} }}} \right]$$
(4)
$$a_{imag,i,n} = \varphi_{e,n} \varphi_{r,n} \frac{{\alpha_{i} P}}{{m_{n} }}\left[ {\frac{{r_{i,n}^{2} \left( {2\zeta r_{i,n} } \right)}}{{\left( {1 - r_{i,n}^{2} } \right)^{2} + \left( {2\zeta r_{i,n} } \right)^{2} }}} \right]$$
(5)

The total real and imaginary acceleration (areal,i and aimag,i) due to each harmonic i in all modes are then computed using Eq. 6 from which the magnitude ai of the response to this harmonic can be obtained as Eq. 7.

$$a_{real,i} = \mathop \sum \limits_{n = 1}^{N} a_{real,i,n} \ldots \ldots \ldots a_{imag,i} = \mathop \sum \limits_{n = 1}^{N} a_{imag,i,n}$$
(6)
$$a_{i} = \sqrt {a_{real,i}^{2} + a_{imag,i}^{2} }$$
(7)

A multiplying factor of 0.71 (=1/√2) is needed to obtain the RMS acceleration aRMS,i from the peak acceleration ai. Also, a frequency weighting factor λi may also be multiplied by aRMS,i as described in the SCI DG. Finally, the total response aRMS to all harmonics is estimated by combining the aRMS,i values using the Square Root of the Sum of the Squares (SRSS) method.

Acceptance Criteria. The SCI DG and CCIP DG use the baseline for perception of vibration given in the standard ISO 10137 [9]. A response factor RF is defined as the ratio of the frequency weighted acceleration aRMS to the “base value” of 0.005 m/s2. From human comfort perspective, the limit RF = 8 is suggested for general offices.

3 Vibration Analysis of a Case Study Floor

Figure 1 shows the framing layout of a typical floor in a real office building. The floor was fully fitted with demountable partitions, ducting, ceiling, services and a raise floor above the concrete slab. The present study focused on a floor bay, shown shaded in Fig. 1, with beam spans of up to 12.7 m and with long corridors crossing the midspan. The walking path was long enough for the vibration energy to build up and a walking induced steady state vibration to occur. The floor is of composite construction, consisting of a 120 mm lightweight concrete slab supported on 610UB101 (approximately 610 mm deep I-section with a weight of 101 kg/m) secondary steel beams B2 which are in turn supported on 610UB113 primary steel beams B1 spanning 8.1 m between columns or concrete walls.

Fig. 1
figure 1

Plan view of floor framing layout

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3.1 Modal Analysis

An FE model of the floor was created using SAP2000 software [10]. Figure 2 depicts the selected four mode shapes with significant participation of the investigated bay. The corresponding natural frequencies fn, modal mass mn, and modal displacement at the bay center φn are given in Table 2. The 6.22 Hz mode was found to be the resonance mode which showed antinodes with maximum modal displacements being located around the center of the bay of interest. The fourth mode (8.37 Hz) was still worth being considered as it could match the fourth harmonic of a common step frequency of around 2.1 Hz. However, natural modes higher than four were found to have either too high frequencies to be excited by normal walking or negligible modal displacements at the bay of interest (Fig. 2).

Table 2 Modal properties of case study floor
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Fig. 2
figure 2

Mode shapes critical to investigated bay

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3.2 Semi-FE Versus Time History Analysis in Prediction of Footfall-Induced Floor Vibration

Semi-FE Analysis using SCI DG and CCIP DG. The modal frequencies, masses and displacements of the four natural modes determined from the FE modal analysis were utilized in the determination of floor response using Eqs. 2 and 3 from the SCI DG and Eqs. 4–7 from the CCIP DG. Both φe,n and φr,n were taken as φn given by Table 2 for a simple and conservative design. The damping ratio was assumed to be 3% as recommended by the guidelines for composite floors with typical fitout [1,2,3].

Time History Analysis. For a walking event at a step frequency fp, an equivalent concentrated time-dependent force was applied at the floor bay center in the FE model. Let vp and L be the walking speed and length of the floor span. The relationship between the step frequency and speed is given by [2]:

$$v_{p} = 1.67f_{p}^{2} - 4.83f_{p} + 4.5$$
(8)

The excitation time for a walking event was taken as L/vp, the duration required for a person to walk from one end of the span to the other end. A half-sine function, which was a simple representation of the fundamental mode shape configuration, was incorporated to the dynamic part of the one-spot walking force of Eq. 1 to allow for the movement of the walker along the walking path. An example of the simulated walking force is shown in Fig. 3a, which can be applied to the midpoint of the span to represent a walking event. This force was for step frequency of 2.07 Hz and used the dynamic load factors suggested by the SCI DG.

Once the acceleration time trace was obtained from the time history analysis, a rolling RMS acceleration trace can be produced using Eq. 9 in which each aRMS(t) value was calculated from a set of acceleration values with the integration time T being taken as 1 s [2, 3, 9]. Figure 3b shows the time history and rolling RMS acceleration for walking at the resonance step frequency of 2.07 Hz.

$$a_{RMS} = \sqrt {\frac{1}{T}\mathop \int \limits_{0}^{T} a(t)^{2} dt}$$
(9)
Fig. 3
figure 3

Walking force and floor response time histories (fp = 2.07 Hz)

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Comparison of Predicted Response. A step frequency range of 1.5–2.4 Hz was used in the vibration analysis to cover pacing rates from slow to normal and fast walking. The variation of the maximum RMS acceleration as a function of the step frequency was found as shown in Fig. 4 where each point on the plot was obtained from an appropriate semi-FE analysis or time history FE analysis. The response spectrum shows two peaks at the 1.55 and 2.07 Hz step frequencies. The third harmonic of the normal walk 2.07 Hz step frequency and the fourth harmonic of the slow-walk 1.55 Hz pacing rate matched the fundamental frequency of the floor bay, thus producing a resonance condition. The resonance at the third harmonic was more critical than the resonance at the fourth harmonic. The worst-case RMS acceleration (fp = 2.07 Hz) was estimated at 0.0348 m/s2 (RF = 6.96) and 0.0382 m/s2 (RF = 7.64) using the time history analysis with dynamic load factors based on the CCIP DG and SCI DG respectively. The response levels acquired from the semi-FE approach was 5–10% higher, being 0.0364 m/s2 (RF = 7.28) and 0.0422 m/s2 (RF = 8.44) as per the CCIP DG and SCI DG formulas respectively.

It would now be useful to discuss the contribution of multi modes and multi harmonic forcing components to the total response when using the semi-FE approach. In the event that a simplified prediction considered only the fundamental vibration mode (6.22 Hz) and the harmonic component in resonance with that frequency (the third harmonic), the calculated RMS acceleration would be 0.0312 m/s2 for the CCIP DG (α3 = 0.073) and 0.0341 m/s2 for the SCI DG (α3 = 0.080). These response values were 14–19% lower than those obtained using multi modes and multi harmonics (Fig. 4).

Fig. 4
figure 4

Predicted floor response spectrum a CCIP DG, b SCI DG

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3.3 Measured Response

Several walking tests were carried out in which the test person managed to maintain a normal pacing rate of around 2.0–2.2 Hz when walking along the bay’s corridor. The recorded acceleration response was filtered to remove high frequency content above 15 Hz to which humans are insensitive [1]. The response was also scaled to take account of the difference between the weight of the test person and the standard pedestrian weight of 746 N. Figure 5 depicts the accelerometers installed on the test floor and a typical acceleration time trace due to walking excitation. A rolling acceleration time history was generated using Eq. 9 from which the maximum RMS acceleration was found to be 0.032 m/s2. Higher response levels would, however, be expected because it was unlikely that the exact resonance step frequency was continuously performed on the test floor. A good agreement between the predicted response and the measured one was hence achieved (Figs. 4 and 5).

Fig. 5
figure 5

Measured response to walking

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4 Conclusions

The semi-FE analysis which does not require a time history analysis and modeling of walking force may be an efficient approach in predicting the floor response to walking excitation. The presented semi-FE method involves utilizing the SCI DG and CCIP DG equations with modal properties obtained from an FE modal analysis, considering multi mode responses to multi harmonic components of the footstep frequency. The dynamic load factors in accordance with the CCIP DG are a bit lower than those given by the SCI DG, hence the corresponding predicted response levels. Comparisons between the semi-FE method and the full time history analysis were made for both resonance and off-resonance walking excitations. It was found that the discrepancy in response calculation between the semi-FE method and the time history analysis was insignificant for the case study floor with a straight walking path. Furthermore, response prediction using either the semi-FE approach or time history analysis was reasonably conservative when compared with the vibration level measured on the real floor. Of the two UK guidelines discussed, the CCIP DG method appeared to exhibit a more accurate response prediction for the floor examined.