# Vibration of Timber and Hybrid Floors: A Review of Methods of Measurement, Analysis, and Design

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions of the global construction sector in 2009 contributed 23% of the total emissions produced by global economic activities [1]. The study [2] identified methods for promoting the development and use of low embodied-carbon building materials as a major opportunity to reduce the emissions in the construction sector. In Australia, the government aims to reduce the nation’s greenhouse emissions to 28% below 2005 levels by 2030 (Paris convention) and aims to achieve net zero by 2050. Since the built environment, including construction, operation, and maintenance, currently accounts for almost 25% of the country’s greenhouse gas emissions, and given that the population is projected to reach 35.9 million by 2050, these reduced carbon emission goals can only be achieved with substantial investments in renewable alternatives. The industry can be more sustainable with timber construction due to its lower greenhouse gas emissions and higher carbon storage potential than concrete and steel. Timber buildings made of large wooden components such as cross-laminated timber (CLT) and glued-laminated timber (glulam) are carbon-negative, meaning they can store more carbon than is emitted in their construction (each cubic meter of softwood glulam captures 550 kg of carbon). Furthermore, in a lightweight or mass timber wood-based hybrid solution, engineered wood products (EWPs) can be integrated with steel and concrete to combine the benefits of all materials [3,4]. The benefits of mid-rise timber and hybrid timber buildings compared with steel and concrete alternatives are notable and include a smaller carbon footprint; off-site prefabrication using advanced manufacturing methods; lighter weight, which improves construction on sites with geotechnical constraints or above underground structures; vertical extension of existing structures; and healthier indoor environments for occupants. Apart from mass timber flooring, the growing trend in lightweight timber floors for use in the multi-story market is due mainly to the surge in land prices in large cities [5].

#### 1.1. Industry Challenge

#### 1.2. Long Span Timber Floor Systems

## 2. Systems View of Vibration: Evaluation, Perceptibility, and Comfort

**Figure 2.**Different types of responses of a floor system to vibration [21] showing (

**a**) steady-state, and (

**b**) transient and impulsive.

#### 2.1. Vibration Evaluation According to ISO 2631-1 (AS 2670.1)

_{w}:

_{w}(t) is the weighted acceleration as a function of time in meters per second squared (m/s

^{2}), and T is the duration of the measurement. Equation (1) is recognized as a basic evaluation method and is known to be a good description of the steady-state response (see Figure 2a) with low crest factors, where the crest factor is defined as the modulus of the ratio of the maximum instantaneous peak value of the frequency-weighted acceleration signal to its RMS value. If the response is associated with high crest factors, i.e., greater than 9 [30] or greater than 6 [31], and in case of occasional shocks and transient vibrations (see Figure 2b), ISO 2631-1 [30] recommends alternative methods: (i) the running RMS method, and (ii) the fourth power vibration dose method. In the running RMS method, evaluation is based on using a short integration time constant, t

_{0}:

_{w}(t

_{0})].

^{1.75}) is defined as

_{k}for vertical and W

_{d}for horizontal vibration. Different frequency weightings are applied to different axes of vibration. An illustration of these factors is shown in Figure 3b, which shows larger weighting factors in the 4–8 Hz range, which correspond to the high sensitivity of vertical abdominal human body vibrations (Figure 3a) and the most onerous acceleration base limits shown in Figure 4. For evaluation of the vibration in more than one direction, individual weighted acceleration RMS values derived from orthogonal directions are combined:

_{k}, irrespective of the measurement direction. According to ISO 2631-1 [30], 50 percent of alert and fit persons can just detect a W

_{k}weighted vibration with a peak in acceleration magnitude of 0.015 m/s

^{2}.

**Figure 4.**Building vibration base curves in ISO 10137 [31] in the (

**a**) vertical z-axis (foot-to-head vibration direction), (

**b**) horizontal x- and y-axes (side-to-side and back-to-chest vibration directions), (

**c**) combined directions (x-, y-, and z-axes), and (

**d**) recommended tolerance limits in DG 11 [23] and BS 6472 [33].

**Table 1.**Frequency weighting and multiplying factors for estimation of comfort and perception [30].

Criterion | Position | Wd | Kx | Wd | Ky | Wk | Kz |
---|---|---|---|---|---|---|---|

Comfort | Seated (translational vibration) | ✓ | 1.0 | ✓ | 1.0 | ✓ | 1.0 |

Seated (rotary vibration) | ✓ | 0.4 | ✓ | 0.5 | ✓ | 0.4 | |

Standing | ✓ | 1.0 | ✓ | 1.0 | ✓ | 1.0 | |

Recumbent | ✓ | 1.0 | ✓ | 1.0 | ✓ | 1.0 | |

Perception | All | ✓ | 1.0 | ✓ | 1.0 | ✓ | 1.0 |

#### 2.2. Vibration Criteria in ISO 10137

_{e}is the number of events in a 16 h period (daytime), T is the duration of the impulse and decay signal for the event (in seconds), and d is the duration stimuli, taken as zero for T < 1 s, 0.32 for wooden floors, and 1.22 for concrete floors. Assuming a 15 s event with 50 repetitions (walking in an office scenario), Equation (7) gives F values equal to 0.1 and 0.009 for wooden and concrete floors, respectively, which is almost a tenfold difference.

Multiplying Factors to Base Curves in Figure 4 | |||
---|---|---|---|

Building Usage | Time | Continuous and Intermittent Vibration | Impulsive Vibration |

Residential | Day | 2–4 | 30–90 |

Night | 1.4 | 1.4–20 | |

Office and School | Day | 4 * | 60–128 |

Night | 4 * | 60–128 | |

Vibration Dose Values (m/s^{1.75}) in Equation (4) from BS 6472 [36] | |||

Building usage | Adverse comment unlikely | Adverse comment possible | Adverse comment probable |

Residential 16 h day | 0.2–0.4 | 0.4–0.8 | 0.8–1.6 |

Residential 8 h night | 0.13 | 0.26 | 0.51 |

## 3. Dynamic Actions Applied to the Floor

_{n}correspond to the nth harmonic, and the number of harmonics k should be adequate to model the time history of the walking load. Examples of Equation (8) are illustrated in Figure 5 [37]. Table 3 compares parameters of the periodic excitation in Equation (8) from different standards and guidelines. The harmonic force in Equation (8) assumes the walking load to be perfectly periodic.

**Table 3.**Design Fourier coefficients for periodic walking force, F, in Equation (8) outlined in standards and guidelines.

ISO 10137 [31] | CCIP-016 [22] | SCI-P354 [21] | AISC DG 11 [23] | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Harmonic Number n | Forcing Frequency fw (Hz) | Fourier Coefficient α _{n} | Harmonic Number n | Forcing Frequency fw (Hz) | Fourier Coefficient α _{n} | Harmonic Number n | Forcing Frequency fw (Hz) | Fourier Coefficient α _{n} | Harmonic Number n | Forcing Frequency fw (Hz) | Fourier Coefficient α _{n} |

1 | 1.2 to 2.4 | 0.37f − 0.37 | 1 | 1–2.8 | 0.41f − 0.3895 < 0.56 | 1 | 1.8–2.2 | 0.436f − 0.4142 | 1 | 1.6–2.2 | 0.5 |

2 | 2.4 to 4.8 | 0.1 | 2 | 2–5.6 | 0.069 + 0.0056f | 2 | 3.6–4.4 | 0.0738 + 0.012f | 2 | 3.2–4.4 | 0.2 |

3 | 3.6 to 7.2 | 0.06 | 3 | 3–8.4 | 0.033 + 0.0064f | 3 | 5.4–6.6 | 0.0364 + 0.021f | 3 | 4.8–6.6 | 0.1 |

4 | 4.8 to 9.6 | 0.06 | 4 | 4–11.2 | 0.013 + 0.0065f | 4 | 7.2–8.8 | 0.014 + 0.028f | 4 | 6.4–8.8 | 0.05 |

5 | 6.0 to 12.0 | 0.06 | >4 | >11.2 | 0 | >4 | >8.8 | 0 |

_{I}is the impulsive force in N.s, f

_{w}is the walking frequency in Hz, and fn is the frequency of the mode in consideration in Hz. A comparison between recommendations of different guidelines and standards shows significant discrepancies. Moreover, the aforementioned expressions are based on measurements of dynamic vertical forces applied to stiff ground (concrete floors) and may not be applicable to long-span timber floors [38]. The walking frequency, fw, that may occur is between 1.5 Hz and 2.5 Hz [39]; however, the probable range is between 1.8 Hz and 2.2 Hz. This pace significantly changes in staircases and can be as high as 4.5 Hz. ISO 10137 [31] recommends considering only two harmonics for the evaluation of the forces (Equation (8)) with Fourier coefficients of 1.1. The dynamic forces in Equations (8) and (9) are for a single person walking. In group activities, variations will exist in frequency, phase angle, and Fourier coefficient. ISO 10137 [31] recommends a coordination factor C(N) ≤ 1.0 be multiplied by the single person dynamic force function:

**Figure 5.**Dynamic actions F(t) on floors normalized to the walker’s weight, Q, in ISO 10137 [31] showing (

**a**) vertical force against stiff ground for one person jumping continuously, where 1 and 2 are measured and sinusoidal curve-fit, respectively; (

**b**) vertical force against stiff ground for one walking step by one person; and (

**c**) force function for one person walking across an instrumented 3 m long slab, and different measured step patterns (Wheeler [37]).

## 4. Floor Dynamics

_{1}and E

_{2}, corresponding Poisson ratios of ν

_{1}and ν

_{2}(E

_{1}ν

_{2}= E

_{2}ν

_{1}), and a shear modulus of G, under the action of a transient dynamic force function F(x,y,t), which is (kN/m

^{2}), takes the following form [41]:

_{1}to D

_{3}are flexural stiffness, and D

_{t}is the torsional stiffness defined as

_{x}and L

_{y}:

**Table 4.**Definition of general floors in buildings based on the fundamental natural frequency f

_{n}of the floor system.

Standard/Guideline | Low-Frequency Floor | High-Frequency Floor |
---|---|---|

ISO 10137 [31] | 8 Hz < f_{n} < 10 Hz or smaller | 10 Hz < f_{n} |

SCI-P354 [21] | f_{n} < 10 Hz | 10 Hz < f_{n} |

CCIP-016 [22] | f_{n} < 10.5 Hz | 10.5 Hz < f_{n} |

AISC DG11 [23] | f_{n} < 9 Hz | 9 Hz < f_{n} |

Toratti and Talja [40] | f_{n} < 10 Hz | 10 Hz < f_{n} |

BS 6472-1:2008 [34] | 7 Hz < f_{n} < 10 Hz or smaller | 10 Hz < f_{n} |

Allen and Murray [43] | f_{n} < 9 Hz | 9 Hz < f_{n} |

Wyatt and Dier [44] | f_{n} < 7 Hz | 7 Hz < f_{n} |

Ohlsson [45] | f_{n} < 8 Hz | 8 Hz < f_{n} |

#### 4.1. Frequency, Mode Shape, and Modal Mass in One-Way and Two-Way Spanning Floor Systems

_{n}is the boundary condition factor, equal to π

^{2}, 22.4, and 3.52 for the first mode of idealized conditions: simply supported, fixed and cantilevered beams, respectively. If the end-fixities cannot be idealized, the supports can be modelled with elastic springs, and some k

_{n}values are recommended in [46]. Of interest in floor vibration analysis is the fundamental natural frequency, f

_{1}, the lowest frequency normally associated with a bending mode shape. The fundamental frequency of the floor (beam analogy with appropriate kn) is related to its mid-span deflection δ (mm), under self-weight (uniformly distributed), through

_{s}, supported by secondary beams and primary beams of frequencies f

_{b}and f

_{p}, respectively, a more accurate expression normally known as Dunkerly’s modal decomposition [47] is used:

_{n}(a value normalized to unity for the first mode), and are shown in a sinusoidal form at different positions denoted by x, to calculate the shape function μ

_{n}(x), which can be multiplied by a time function, ψ

_{n}(t), to yield w

_{n}(t) from a modal superposition:

_{n}and Ø

_{n}are related to the dynamic walking force, calculated from methods explained in Section 3. Using a beam analogy, the mode shapes of the floor system will only account for the flexural modes, where the torsional and transverse modes are ignored. However, a recent report for Forest and Wood Products Australia [48] shows experimental results of a simply supported CLT slab with a torsional second mode that has a significant modal contribution in the vibration response. Adopting a two-way spanning floor system analogy, frequency and mode shapes can be approximated by mathematical equations, though modal parameters are normally calculated from a finite element (FE) modal analysis. The fundamental frequency of a CLT floor (strong direction-x) from the unforced governing equation (Equation (11)) and parameters defined in Equation (12), and corresponding mode shapes are

_{3}should be used to consider the torsional rigidities of the supporting beams. In both one-way and two-way spanning systems, dynamic values of the Young’s modulus of the materials should be used. In hybrid systems with concrete sections, allowance should be made for the cracked stiffness under service loading [22]. The modal mass of the floor is related to the kinetic energy in the system and is an indication of the amount of mass involved in each mode shape. In each mode shape, the mass contribution, Mn, can be calculated by finding the maximum kinetic energy in the mode:

_{i}(ω) is the Fourier transform of the response x

_{i}(t), and F

_{j}(ω) is the Fourier transform of the excitation f

_{i}(t). In the SIMO method, using a modal hammer, excitation at each point on the floor gives information for a single row, whereas using a shaker yields data for a unique column of H. By using a grid of accelerometers and/or moving the hammer and shaker, the complete frequency response matrix can be generated. Defining cross-spectral density, GXF, load signal spectral density, GFF, and response signal spectral density, GXX, the validity of the frequency response function is evaluated by the coherence function γ

^{2}:

_{r}represents residues, ω

_{r}is the un-damped natural frequency, and ζ

_{r}is the equivalent viscous modal damping ratio. SDOF curve fitting is based on the analysis of a single mode at a time, and MDOF curve fitting is applied to the several frequencies or entire set of frequencies, simultaneously, and is more suitable for heavily coupled modes. The modal parameters can be calculated from numerical modal analysis (NMA), typically carried out using FEA. The consistency between NMA and EMA is assessed by measuring the linear consistency between numerical and experimental mode shapes using a statistical indicator, modal assurance criterion (MAC) analysis. The MAC is a normalized quantity, calculated as the scalar product:

_{E}} and {ψ

_{N}} are the experimental and numerical modal vectors, respectively. A MAC presentation of a CLT floor is shown in the case study (I) in Part 2 of this study.

#### 4.2. Modal Damping

_{1}and ω

_{2}are half-power points from the resonant peak frequency ω

_{0}and correspond to −3 dB down from ω

_{0}on a dB scale. Previous studies [58,59] have shown the sensitivity of the hysteretic damping η on the accuracy of the peak location, which in turn is highly dependent on the sampling rate. The relationship between the damping loss factor η for the hysteretic case and the damping ratio ζ

_{r}for the viscous case is easy-to-understand via η = 2 ζ

_{r}[60].

## 5. Vibration Design Methods

#### 5.1. Rules of Thumb

_{1}kN, is recommended. A study by Woeste and Dolan [65] on timber joist floors at Virginia Tech University showed that a “code compliant” floor can be problematic if the floor system has components each with natural frequencies in the 7–10 Hz range. They [65] recommend using wider spacing between joists (600 mm instead of 300 mm), since it requires deeper members with larger flexural stiffness (EI), which in turn increases the natural frequency of the system. Based on a comprehensive study, Woeste and Dolan [65] recommend (1) using a 40 psf live load for design of residential floors, (2) increasing the joist depth (larger EI) or using a deflection limit of L/600 (or L/480 for solid joists or in trusses with Strongbacks), and (3) gluing floor sheathing and using screws instead of nails to improve long-term vibration performance.

#### 5.2. Empirical and Simplified Analytical Methods

_{1}kN is the measured 1 kN point load deflection in mm, and f is the measured floor fundamental natural frequency in Hz.

**Figure 6.**Empirical expression for acceptability of floor vibration in ISO/TR21136 [66] showing: (

**a**) logistic regression on the database of field light frame timber floors in the across Canada occupant survey [16] and testing and validated Equation (25) (solid dark line), and (

**b**) EN 1995-1-1:2004 [15] showing criteria in Equation (29). (Better and poorer performances are shown with arrows 1 and 2).

#### 5.2.1. Empirical Method in ISO/TR21136

_{app}is

_{eff}(Nm

^{2}) and GA

_{eff}(N) are the flexural and shear stiffness values of a 1 m wide CLT panel with a density of ρ (kg/m

^{3}), cross-section of A (m

^{2}), and length L (m). This criterion is limited to (1) CLT slab floors without ribs; (2) bare CLT floors and ignoring the stiffening effects from the finish, partition, continuity of the multi-span, and ceiling; and (3) simply supported conditions.

#### 5.2.2. Hamm et al.

^{2}), length of L (m), and width of b (m), the method suggests a cut-off frequency, f (in Equation (27)), of 8 Hz with d

_{2}kN ≤ 0.5 mm for higher performance floors and 6 Hz with d

_{2}kN ≤ 1.0 mm for lower performance floors, where the deflection under a 2 kN static load is derived from

_{L}and EI

_{b}are the effective stiffness along the span and in the transverse direction in Nm

^{2}/m, respectively. Hamm et al. [63] do not specify formula for finding the stiffness. For low-frequency floors, with f in the range between 4.5 Hz and 8 Hz, maximum acceleration values are a

_{max}≤ 0.05 m/s

^{2}and a

_{max}≤ 0.10 m/s

^{2}, for higher performance floors and lower performance floors, respectively, where the maximum acceleration is defined as

#### 5.2.3. Combined Frequency, Deflection, and Impulse Velocity Method in EN 5:2004

^{2}), L and b are the span and width (m), respectively, and v is the unit impulse velocity response, calculated as the maximum initial vertical velocity (m/s) caused by an ideal unit impulse (1 Ns) applied at a location on the floor that gives the maximum response. The equation for calculating v in a square floor simply supported on all four edges is given in Equation (29). Similar to Hamm et al. [63], EC 1995-1-1:2004 [15] does not propose equations for finding EI

_{L}and EI

_{b}. EC 1995-1-1:2004 [15] accounts for the modal input in Equation (29) by introducing n

_{40}, which is the number of first-order modes with natural frequencies up to 40 Hz. Fs is the static vertical force applied at any point on the floor, and d is the corresponding maximum instantaneous deflection under the load. The deflection limit, a, is shown in Figure 6b. EC 1995-1-1:2004 [15] specifies that the method is applicable for an unloaded floor (only self-weight is included) and does not clearly give a method for finding deflection under the static load.

#### 5.2.4. Vibration-Controlled Span Method in CSA 086:19

_{v}(m) is

_{eff}(Nm

^{2}) is the effective flexural stiffness of the floor system in the span direction, k

_{tss}is a factor that accounts for the flexural stiffness in the transverse direction (for k

_{1}see A.5.4.5.1.3 from CSA 086:19 [16]), and m

_{L}is the mass per unit length (kg/m) of the composite floor system. A composite floor system includes joists, the subfloor, and the topping (if any). The stiffness in longitudinal and cross-span directions are calculated using equations in CSA 086:19 [16], which account for composite actions from bending and axial contributions of each module in the composite floor system. For vibration control of prefabricated wood I-joist floors, other conditions are imposed in CSA 086:19: (1) subfloor thickness should be less than 28.5 mm (typical subfloors used in Canada are OSB, while in Australia particle board is more common), (2) in multi-span floors the effective composite joist bending stiffness factor should be 1.2, and (3) the stiffness contribution of a concrete topping should not be applied.

#### 5.2.5. One Step Root Mean Square Method in HIVOSS:2007

#### 5.2.6. Floor Performance vs. Floor Usage in prEN 1995-1-1: 2025 (Final Draft)

^{2}), including weight of partitions and a 10% additional imposed load; and k

_{e}

_{1}and k

_{e}

_{2}are multi-span and transverse stiffness factors, respectively, and both can be taken as one for a single span and one way spanning floors. Floor stiffness in the span direction and transverse to it are represented with EI

_{L}and EI

_{T}, respectively. No clear guide is given for the calculation of these stiffness values. The simpler formula for deriving the fundamental frequency in Equation (32), which uses w

_{sys}, is the deflection of a single bay of the floor system under floor mass, m, where k

_{e}

_{1}and k

_{e}

_{2}can both be presumed to be equal to one in floors with flexible supports if transverse bending is considered. For a two-span floor system, values of k

_{e}

_{1}are represented in prEN 1995-1-1:2025 (Final Draft) [70] and vary from 1 to 1.31 for floor span ratios of 1.0 and 0.2, respectively. The deflection criterion in prEN 1995-1-1:2025 (Final Draft) [70] is based on a simply supported single span assumption of length L and an effective width of b

_{ef}:

_{0}of 700 N, is recommended, and a simple expression for calculation of the Fourier force coefficient, α, is given. In the absence of on-site testing applying EN 16929 [71], the following modal damping ratios, ζ, are proposed: 2% for joisted floors; 2.5% for timber–concrete, rib type, and slab type floors; 3% for joisted floors with a floating floor; and 4% for timber–concrete, rib type, and slab type with a floating floor. The reduction factor of 1/7 in Equation (34) comes from 0.4 × 1/√2, where the factor of 0.4 assumes that the resonance may not be due to an excitation in the center of the floor. This non-conservative assumption may be altered by the design engineer, for example, in long corridors. The derivation of velocity vibration response is given in Equation (35). The mean modal impulse, I

_{m}(Ns), is calculated based on a walking frequency, fw, of 1.5 Hz in residential floors and 2.0 Hz in other floors. The peak fundamental velocity, v

_{1,peak}, is amplified using a multiplier factor, k

_{imp}, that accounts for higher modes in the transient response.

_{1kN}and acceleration criteria need to be checked. The RMS acceleration in m/s

^{2}(Equation (34)) should be greater than 0.005R, where R is the response factor given in Table 6 for each performance level. In floors with a fundamental frequency greater than 8 Hz, d

_{1}kN and velocity criteria must be checked. The RMS velocity (Equation (35)) should be greater than 0.0001R. prEN 1995-1-1:2025 (Final Draft) [70] suggests that for the different building use categories, the assignment of floor performance levels to be applied can be stated in the national annex for use in a country.

**Table 6.**Floor performance levels and recommended selection for different use categories in prEN 1995-1-1:2025 (Final Draft) [70].

Floor Performance Levels | ||||||
---|---|---|---|---|---|---|

Criteria | Level I | Level II | Level III | Level IV | Level V | Level VI |

d_{1kN} (mm) ≤ | 0.25 | 0.25 | 0.5 | 0.8 | 1.2 | 1.6 |

Response factor (R) | 4 | 8 | 12 | 20 | 30 | 40 |

Floor Usage | Quality Choice | Base Choice | Economy Choice | |||

Multi-family residential | Levels I, II, III | Level IV | Level V | |||

Single-family residential | Levels I, II, III, IV | Level V | Level VI | |||

Office | Levels I, II, III | Level IV | Level V |

#### 5.3. Modal Superposition Methods

CCIP-016 [22] | ||||
---|---|---|---|---|

Low-Frequency Floor | High-Frequency Floor | |||

Floor Use | Peak Acceleration | Response Factor, R | V_{RMS}(m/s) | Response Factor, R |

Commercial (offices, retail, restaurants, and airports) | 0.57% g | 8 | 8 × 10^{−4} | 8 |

Residential (day) | 0.28–0.57% g | 4 to 8 | 4–8 × 10^{−4} | 4 to 8 |

Premium quality office, open office with busy corridors near midspan, heavily trafficked public areas with seating | 0.28% g | 4 | 4 × 10^{−4} | 4 |

Residential (night) | 0.2% g | 2.8 | - | - |

Hospitals and critical work areas | 0.071% g | 1 | - | - |

AISC DG 11 [23] | ||||

Low-Frequency Floor | High-Frequency Floor | |||

Floor Use | Peak Acceleration | Response Factor, R | V_{RMS}(m/s) | Response Factor, R |

Outdoor pedestrian bridges | 5% g | 70 | - | - |

Indoor pedestrian bridges, shopping malls | 1.5% g | 21 | - | - |

Offices, residences, quiet areas | 0.5% g | 7 | - | - |

Ordinary workshops | - | - | 8 × 10^{−4} | 8 |

Offices | - | - | 4 × 10^{−4} | 4 |

Residences | - | - | 2 × 10^{−4} | 2 |

Hospital patient rooms | - | - | 1.5 × 10^{−4} | 1.5 |

#### 5.3.1. CCIP-016

- Find the frequency, f
_{n}, modal mass, m*_{n}, and modal damping, ζ_{m}, of each mode. The mode shape values at the excitation, μ_{en}, and response, μ_{rn}, locations in each mode on the floor are also needed. While this is normally conducted through FEA, it can also be calculated using simple equations such as those provided in Equations (17) and (18). - Calculate the harmonic forcing frequency, f
_{h}= h f_{w}, for harmonic numbers, h, from 1 to 4, and the harmonic force, F_{h}, from coefficients in Table 3 and using Equation (8) at each harmonic. - Find the real and imaginary parts of the acceleration from Equation (36).
- Sum the real and imaginary responses in all modes and find the magnitude of the acceleration at the harmonic, ah, by summing the square root of the real and imaginary accelerations at this harmonic.
- Convert ah to a response factor, R
_{h}, for this harmonic by dividing ah by the base acceleration, a_{R}_{= 1,h}(m/s^{2}), given in Equation (37) for the corresponding f_{h}. - Find the total response factor, R, which is the “square root sum of the squares” combination of the response factor for each of the four harmonics.

_{s}.

- The impulsive footfall force, F
_{I}, in Ns is calculated from Equation (9) [22] for all modes with frequencies up to twice the fundamental frequency of the floor. - Find the peak velocity of each mode, v
_{n}, and the time history of the velocity response, v_{n}(t), over the period of one footfall, Tw, from Equation (38). - Add the velocity response in each mode, v
_{n}(t), in the time domain to find the total response, v(t) (m/s). - Calculate the RMS velocity (VRMS) and divide it by the baseline RMS velocity, V
_{R}= 1 (m/s), at the fundamental frequency, f_{1}, to determine the response factor, R, and compare it against the target values in Table 7 for the desired floor use (see Equation (38)).

#### 5.3.2. SCI-P354

_{wrms}, of each force harmonic, h, at every mode, n, of the response at a location, r, from excitation at a point, e, on the floor is calculated. Then, the total acceleration response function, a

_{w}(t), is found by summing a

_{wrms}of each mode of vibration at each harmonic of the forcing function:

_{nh}is the phase of the response of the nth mode relative to the hth harmonic. In high-frequency floors, SCI-P354 [21] suggests that all modes with frequencies up to twice the fundamental frequency be accounted for. The upper limit of twice the fundamental frequency is due to the significance of the frequency weighting factors (see Figure 3b) in reducing the response at high frequencies. The acceleration due to an impulse force, F

_{I}, (Equation (9)) is calculated by summing the acceleration responses of each mode using the following superposition:

_{w}. The RMS acceleration can be found from Equation (2), with T equal to 1/f

_{h}. In order to determine the sum of accelerations from individual acceleration value calculated for each mode in each harmonic, SCI-P354 [21] recommends the following methods: (i) sum of peaks (SoP), and (ii) square root sum of squares (SRSS). SoP is conservative and assumes that all of the components of response will always peak at the same time. This will not be a true representation of an accurate summation in LSTFs, where multiple mode shapes of different frequencies actively contribute to the vibration response [38]. A better indicator, which is very closely related to a full-time history analysis, is the SRSS given in Equation (41). A comparison between SoP and SRSS for a typical response in the time domain is shown in Figure 8.

**Figure 8.**Different methods for estimation of the peak and root mean square (RMS) accelerations [21].

#### 5.3.3. American Institute of Steel Construction AISC/CISC DG 11

_{p}, is calculated from Equation (44) and is compared against recommended acceleration tolerances in Figure 4d.

_{ESPA}is calculated as follows:

#### 5.3.4. Harmonized Peak Acceleration and VDV Approach

_{e}is the estimated number of events and is used to relate a single event VDV to a total VDV for the entire exposure period (day or night). The frequency weighting factor, W

_{b}, is equal to 16/f

_{n}for modal frequencies, f

_{n}> 16 Hz and is equal to 1 for f

_{n}between 8 Hz and 16 Hz. The method is simple to use and is a better predictor of vibration performance of joist floors. The proposed floor modification factor, K, is equal to 2.8 for CLT floors and 4.6 for joist floors. From Equation (46) and in conjunction with criteria in SCI-P354 [21] (see Table 2), the number of allowed events can be estimated.

#### 5.4. Time History Analysis Method

**Figure 9.**Loading protocol in a time history analysis showing walking and running (

**a**,

**b**) footfall excitations on a hybrid CLT/steel floor system [76].

## 6. Hybrid Floor Systems

_{t}, modulus of elasticity of E

_{t}, concrete topping with a thickness of h

_{c}, and a modulus of elasticity of E

_{c}and simply supported at two ends, the vibration-controlled span L

_{v}is:

_{t}and (EA)

_{t}are, respectively, bending and axial stiffness of a 1 m wide mass timber panel (in (Nm

^{2}and N, respectively), obtained from the producer’s specification or calculated according to CSA 086-14 [86]; (E)

_{c}and (EA)

_{c}, respectively, are bending and axial stiffness of a 1 m wide concrete panel (in Nm

^{2}and N, respectively); K is the load-slip modulus in the major direction (N/m/m), which can be calculated from the shear stiffness of the timber–concrete connection [84]; t is the thickness of insulation, acoustic or construction layer (m); and a

_{c}and a

_{t}are the distance (m) between the centroid of the concrete section and the timber section to the neutral axis of the composite section, respectively. The design method outlined in the U.S. Mass Timber Floor Vibration Design Guide [9] is very similar to the Canadian CLT Handbook [50] and is extended to nail- or dowel-laminated timber (NLT or DLT) and tongue-and-groove (T&G) decking as well. For NLT, more details are given in the NLT U.S. Design & Construction Guide [87]. The American Institute of Steel Construction (AISC) recently published Hybrid Steel Frames with Wood Floors (DG 37) [88]. DG 37 [88] follows the method described in CCIP-016 [22] and recommends a minimum fundamental frequency over 8 to 9 Hz for hybrid steel–timber floors. The suggested mass timber-to-steel connection fasteners are bolts, screws (partially threaded and fully threaded), nails, and pins, which provide the shear connection between the steel beam and mass-timber panel. DG 37 [88] allows for the use of simplified methods such as vibration-controlled spans for preliminary designs but recommends more rigorous methods of design at later stages.

**Figure 10.**Hybrid floor systems: (

**a**) lightweight sandwich panel made from plywood and bamboo [79]; (

**b**) Kielsteg LVL/CLT and CLT box floors [82]; (

**c**) STC floor made of CLT on steel girders [88], photo courtesy of Odeh Engineers, Inc. (North Providence, RI, USA), also see Figure 1c; and (

**d**) timber–concrete composite (TCC) floor [84] showing AISC-SOM composite timber deck and steel beam detail [83].

## 7. Case Study I: EMA and NMA of a CLT Floor

_{1}

^{EMA}= 11.7 Hz and f

_{2}

^{EMA}= 16.5 Hz, respectively. The (i,j) labelling of mode shapes corresponded to the number of half waves in the major and minor directions, respectively. The NMA frequencies of the aforementioned modes were f

_{1}

^{NMA}10.0 Hz and f

_{2}

^{NMA}15.4 Hz, respectively, which differed by 15% and 7% from the EMA frequencies, respectively. Using a more computationally costly NMA, and modelling each single board with solid elements, a 1.5 mm gap between the boards and a friction coefficient (µ) of 0.1, in a fine mesh setting, the difference between NMA and EMA frequencies became as low as 5%. However, the computational effort required to model a full-scale CLT floor system in a real building makes it an inefficient exercise. Other EMA and NMA frequencies and differences are represented in Table 8. The NMA frequencies were lower than the EMA frequencies, which may be related to the adopted static modulus of elasticity (MOE) for defining mechanical properties of timber in FEA. Previous research on concrete floors suggests using the dynamic MOE of concrete. Results from vibration testing of softwood timber boards found the dynamic MOE to be 9–10% higher than the static moduli [92]. The other source of discrepancy between EMA and NMA frequencies may be attributed to the adopted boundary conditions. In the FEA, the CLT panel was assumed to be free (on all edges and top/bottom surfaces), whereas in the physical tests, the CLT panel was supported on four inflated air bladders.

**Table 8.**EMA and NMA frequencies of the CLT panel in Case study I, taken from AS 1720.1 [93].

Rank | Mode Shape (i,j) | EMA Frequency f ^{EMA} (Hz) | NMA Frequency f ^{NMA} (Hz) | Difference |
---|---|---|---|---|

1 | 1,1 | 11.7 | 10.0 | 15% |

2 | 0,2 | 16.5 | 15.4 | 7% |

3 | 1,2 | 27.9 | 25.1 | 10% |

4 | 2,0 | 34.2 | 28.8 | 16% |

5 | 2,1 | 40.6 | 34.7 | 15% |

6 | 0,3 | 44.7 | 42.3 | 5% |

7 | 2,2 | 55.3 | 50.7 | 8% |

8 | 1,3 | 58.4 | 51.0 | 13% |

9 | 3,0 | 85.6 | 75.1 | 12% |

10 | 2,3 | 88.3 | 76.2 | 14% |

11 | 3,1 | 94.0 | 79.9 | 15% |

12 | 0,4 | 112.8 | 82.2 | 27% |

## 8. Case Study II: EMA and NMA of a 6 m × 6 m Cassette Floor

#### 8.1. The Floor System and FEA Model

^{3}MPa for MOE and 80 × 10

^{3}MPa for shear modulus, with a Poisson’s ratio of 0.25. In the FEA (see Figure 2), MGP boards, particleboard, PFCs, and the supporting short columns were modelled using 8-noded ANSYS shell-181 elements with 5 integration points, and the metal webs were modelled with link elements [91]. The connection between the bearer beam and the PFC, and the connection between the PFC and the short column was Mohr–Coulomb frictional contact with a coefficient of friction calculated at 0.45 (calculated from a simple sliding test, not discussed here). From a mesh sensitivity analysis (not shown here for the sake of brevity) a mesh with a total of 182,550 nodes was selected, which had mid-span deflection and a fundamental frequency of less than 5% different from an FEA model with 303,015 nodes.

Property | MGP 12 | MGP 10 | Particle Board |
---|---|---|---|

Density (kg/m^{3}) | 594 | 550 | 748 |

E_{L} (MPa) | 12,700 | 10,000 | 3000 |

E_{R} (MPa) | 1435 | 1130 | 3000 |

E_{T} (MPa) | 991 | 780 | 3000 |

G_{LR} (MPa) | 1029 | 810 | 1360 |

G_{LT} (MPa) | 165 | 130 | 1360 |

G_{RT} (MPa) | 1041 | 820 | 1360 |

μ_{LR} | 0.292 | 0.292 | 0.103 |

μ_{LT} | 0.382 | 0.382 | 0.103 |

μ_{RT} | 0.328 | 0.328 | 0.103 |

#### 8.2. Dynamic Properties

**Table 10.**Dynamic properties of the floor system of Case study II from the digital hammer excitations.

Accelerometer A | Accelerometer B | Accelerometer C | Damping | |||||
---|---|---|---|---|---|---|---|---|

favg (Hz) | CoV | favg (Hz) | CoV | favg (Hz) | CoV | ζ | CoV | |

1 | 9.08 | 0.59% | 9.08 | 0.01% | 9.08 | 0.01% | 0.90% | 0.14% |

2 | 17.05 | 3.77% | 16.68 | 2.81% | 16.77 | 0.50% | 1.08% | 0.66% |

3 | 17.84 | 0.01% | 17.45 | 2.65% | 17.98 | 0.31% | 1.03% | 0.17% |

4 | 19.41 | 0.01% | 19.49 | 0.19% | 19.40 | 0.34% | 0.84% | 0.55% |

5 | 20.98 | 0.01% | 21.03 | 0.20% | 21.04 | 0.24% | 0.91% | 0.14% |

6 | 22.01 | 0.26% | 22.31 | 0.33% | 22.23 | 0.37% | 0.89% | 0.14% |

7 | 24.86 | 0.01% | 23.69 | 2.41% | 23.55 | 0.40% | 0.95% | 0.21% |

#### 8.3. Walking Tests and Evaluation to ISO 2631.2 and BS 6472-1

_{w}of 1.80 Hz (brisk walking) and 2.25 Hz (fast walking), were selected such that the harmonics of the fundamental frequency (9.08 Hz), the 4th harmonic with a walking frequency of 2.25 Hz, and the 5th harmonic at a walking frequency of 1.80 Hz were excited. Walker 1 (800 N) and walker 2 (750 N) followed a constant stride length of 700 mm at both walking speeds. The walking configurations are outlined in Table 11.

**Table 11.**Measured vibration responses of the floor system of Case study II from different walking configurations.

Walking Configuration | ||||||
---|---|---|---|---|---|---|

f_{w}(Hz) | Walker 1 (80 kg) | Walker 2 (75 kg) | Single Walker W1 @ St. 1 | Double Walkers W1 Start @ St.1 W2 Start @ St. 2 | Double Walkers W1 Start @ St.1 W2 Start @ St. 3 | |

T1 | 1.80 | ✓ | - | ✓ | - | - |

T2 | 2.25 | ✓ | - | ✓ | - | - |

T3 | 1.80 | ✓ | ✓ | - | ✓ | - |

T4 | 2.25 | ✓ | ✓ | - | ✓ | - |

T5 | 1.80 | ✓ | ✓ | - | - | ✓ |

T6 | 2.25 | ✓ | ✓ | - | - | ✓ |

Vibration Response Parameters | ||||||

a_{w}(m/s ^{2}) Equation (2) | a_{w,max}(m/s ^{2}) | MTVV (m/s ^{2}) Equation (3) | VDV (m/s ^{1.75}) Equation (4) | $\frac{VDV}{{a}_{w}{T}^{1/4}}>1.75$ $\frac{MTVV}{{a}_{w}}>1.5$ Equation (5) | Max number of events, n_{e} Equation (43) | |

T1 | 0.75 | 1.75 | 0.75 | 0.92 | No | 11 |

T2 | 0.91 | 2.05 | 0.91 | 1.10 | No | 10 |

T3 | 1.17 | 2.66 | 1.17 | 1.39 | No | 10 |

T4 | 1.54 | 3.72 | 1.54 | 1.89 | No | 11 |

T5 | 0.78 | 1.79 | 0.78 | 0.92 | No | 9 |

T6 | 1.29 | 2.66 | 1.29 | 1.52 | No | 9 |

_{k}, (Figure 3b) of ISO 10137 [31]. The experimental and ISO-weighted acceleration time histories of T1 to T6 walking configuration tests are plotted in Figure 14. The acceleration time histories in Figure 14 showed a steady-state (refer to Figure 2 for the definition) response much more significant than a transient response in all walking configurations. A clear resonant excitation response was seen in all configurations. It is worth noting that the measured fundamental frequency of 9.08 Hz in comparison with the recommended frequency cut-off rates in Table 4 suggests that the floor can be assumed to either have a steady state or transient response.

_{w}, was calculated using an integration time constant of 1 s. The maximum acceleration, a

_{w,max}, corresponded to the sum of peaks (SoPs) in the time history (see Figure 8, for an illustration of SoP). The maximum transient vibration value (MTVV) and vibration dose value (VDV) were calculated from Equations (3) and (4), respectively. In all walking configurations, the conditions of Equation (5), ISO 2631-1 [30] were not satisfied. Therefore, in the evaluation of comfort, only a

_{w}, needed to be checked [30]. However, for the sake of comparison, the equivalent maximum number of events (n

_{e}) that may produce a “low probability of adverse comment” according to BS 6472-1 [34] was also calculated from Equation (43). The maximum acceleration and RMS accelerations in Table 11 were in T4, where the walkers followed a diagonal walking pattern. As shown in the frequency spectrum of T4 in Figure 14, many higher frequency modes were incorporated in the vibration response.

_{w}, (in Equation (2)) by the base acceleration of 0.05 m/s

^{2}(Figure 4a), the multiplying factors of 15 and 18.2 were calculated, respectively. Both factors were outside the recommended multiplying factors (<4) for continuous and intermittent vibrations for residential and office buildings in Table 2. Assuming a transient vibration, the limit multiplying factors for residential and office were 90 and 128, respectively. Using Equation (7), as specified in ISO 10137 [31] to account for the number of events, assuming a 15 s event with 11 repetitions (n

_{e}of T1 in Table 4), Equation (7) gave an F value equal to 0.21. Multiplying F by the limit value of 90, it was reduced to 18.9, which means the floor was acceptable in residential and office settings with 11 events during the day. Comparing single person walking VDVs with those in Table 2, the floor was acceptable for residential usage during a 16 h day period.

#### 8.4. Vibration Design to AS 1170.0 and IRC

_{1kN}, from FEA was equal to 1.99 mm and was very close to the 2 mm limit recommended in the AS 1170.0 [19]. Under a combination of self-weight and a 40% live load, with distributed live loads of 1.5 kPa (residential) and 3 kPa (commercial), the floor center deflections were 13.9 mm and 23.1 mm, respectively, compared to the 20 mm limit (L/300) serviceability criterion recommended in AS 1170.0 [19]. The predicted deflection under live loads of 1.44 kPa (30 psf) and 1.91 kPa (40 psf) for residential and mixed-use dwellings were 22.3 mm and 29.5 mm, respectively, greater than the 16.7 mm (L/360) limit in the International Residential Code (IRC) [61].

#### 8.5. Vibration Design Based on Empirical and Simplified Analytical Methods

#### 8.5.1. Simplified Method in ISO/TR 21136

_{1}value equal to 9.08 Hz, which gave a deflection limit of 0.6 mm and was well below the measured and FEA deflections. With regards to the test results from the Canadian survey as depicted in Figure 6a of the companion paper 1, it is clear that the floor system did not meet the acceptance criteria. The measured frequency–deflection is presented in blue on Figure 6a and indicates that the floor response was far from the acceptable curve [66].

#### 8.5.2. Empirical Method of Hamm et al.

^{2}) including PFC beams and short columns and considering a measured damping of 0.9% (from Table 10), an a

_{max}value of 1.09 m/s

^{2}was calculated, which was 38% smaller than the measured a

_{max}of 1.75 m/s

^{2}at a f

_{w}of 1.80 Hz. Regardless, the maximum acceleration was larger than the 0.05 m/s

^{2}and 0.10 m/s

^{2}recommended for higher performance floors and lower performance floors, respectively.

#### 8.5.3. Combined Frequency, Deflection, and Impulse Velocity in EN 1995-1-1:2004

_{L}= 2.35 × 10

^{6}Nm

^{2}/m and (EI)

_{b}= 1.60 × 10

^{3}Nm

^{2}/m, respectively. Using Equation (29), the fundamental frequency was 10.6 Hz, n

_{40}was equal to 12, and the unit impulse velocity response was v equal to 18.6 mm/Ns

^{2}. With respect to the graph in Figure 6b, the floor system had an x-coordinate of 1.54 mm and a y-coordinate of 97 mm and fell into the better performance region. However, the third condition in Equation (29) suggests that v should be less than 16.2 mm/Ns

^{2}. If the FEA results, f

_{1}= 9.90 Hz and a deflection of 1.99 mm with n

_{40}= 15 (calculated from FEA) were used, the unit impulse velocity response, v, of 23.0 mm/Ns

^{2}would still have been larger than the 19.5 mm/Ns

^{2}criterion. Therefore, the floor system did not satisfy the velocity requirement of EN 1995-1-1 [15].

#### 8.5.4. Vibration-Controlled Span in CSA 086:19

_{1}and K

_{tss}were equal to 0.15 and 0.49, respectively, and EI

_{eff}was equal to 2.35 × 10

^{6}Nm

^{2}(from the FEA). A maximum recommended span l

_{v}of 5310 mm was calculated, which suggests that the floor system was not acceptable.

#### 8.5.5. One Step Root Mean Square Method (HIVOSS)

_{1}= 9.90 Hz and modal mass of 359 kg (m* = m/4), the OS-RMS

_{90}was almost 13 (see contours in Figure 7). The floor was classified in the border of E and F. Assuming Class E, the floor is not recommended in critical workspace or health and education buildings, and it is critical for residential, office, retail, hotels, and meeting rooms. If used in industrial and sport facilities the floor will have acceptable vibration performance.

#### 8.5.6. Floor Performance vs. Floor Usage in prEN 1995-1-1: 2025 (Final Draft)

_{1}, of the floor systems were calculated as 6.45 Hz (residential) and 5.90 Hz (office). The corresponding w

_{sys}values for residential and office floors from the FEA were 7.78 mm and 9.31 mm, respectively. The frequency multiplier factors k

_{e1}and k

_{e2}were equal to one. The calculated frequencies were larger than the minimum allowable fundamental floor frequency of 4.5 Hz prEN 1995-1-1 2025 (Final Draft) [70]. Since the fundamental frequency was below 8 Hz, the d

_{1kN}and acceleration criteria had to be checked. The a

_{rms}values of the residential and office floors from Equation (34) were 1.05 m/s

^{2}and 1.31 m/s

^{2}, respectively (assuming 2% modal damping ratios ζ recommended for joisted floors). In comparison with the performance levels in Table 6, d

_{1kN}> 1.6 mm, and from the calculated response factors, R (R = a

_{rms}/0.005) values of 210 and 262 for residential and office floors, respectively, the floor did not fall within any recommended class according to prEN 1995-1-1 2025 (Final Draft) [70].

_{1}= 9.08 Hz, and a d

_{1kN}value of 1.54 mm, the vibration velocity requirements of Equation (35) need to be checked. The calculated RMS velocities, v

_{rms}, of the residential and office floors (with f

_{w}of 1.5 Hz in residential floors and 2.0 Hz in other floors) were 0.0039 m/s and 0.0083 m/s, respectively. The corresponding response factors, R (R = v

_{rms}/0.0001) values of 40 and 83, were calculated, which placed the floor in a level VI, and “economic choice” for residential and “non-categorized” for office usage.

#### 8.6. Vibration Design Based on the Modal Superposition Methods

#### 8.6.1. AISC/CISC DG 11

^{2}, which returned a response factor, R, of 5.7. If Equation (45) was used, the equivalent sinusoidal peak acceleration, a

_{ESPA}, ranged between 0.065 m/s

^{2}and 0.112 m/s

^{2}at walking frequencies between 1.5 and 2.2 Hz, respectively. The corresponding response factors, R, ranged from 12 to 21 at the mentioned walking frequencies. Comparing the response factors to the performance targets in Table 7, at low frequency, the floor would be acceptable for offices and residential buildings. At a high frequency, the floor would not be acceptable for any building usage.

#### 8.6.2. CCIP-016

**Figure 15.**Case study II, showing the response factors from modal superposition methods: (

**a**) CCIP-016 [22] for low-frequency floors; (

**b**) CCIP-016 [22], SCI-P354 [21], and DG 11 [23] for high-frequency floors; and (

**c**) the total response, v(t), using CCIP-016 [22] Equation (38) at experimental walking frequencies.

#### 8.6.3. SCI-P354

_{w}

_{(t)}, was calculated by summing the a

_{wrms}values of each mode of vibration at each harmonic (up to four harmonics) using Equation (39) with a critical damping of 1.1%, as suggested in the guideline. That yielded an RMS acceleration of 0.4 m/s

^{2}(response factor of 80), which is shown in Figure 15a and compared to other modal predictions for low-frequency floors. Compared with the recommended tolerance limits of BS 6472-1 [34] shown in Figure 4d, the calculated acceleration (4% g) was outside the acceptable region for offices and residencies.

#### 8.6.4. Harmonized Peak Acceleration and VDV Approach

_{e}equal to 1, the total VDVs were higher than the tolerances in Table 2, and thus the floor would not be acceptable for residential buildings. It should be noted that the peak acceleration calculated from Equation (46) was not very sensitive to the adopted damping ratio. A comparison between VDVs from physical tests in Table 11 and in Figure 16 [18] suggested that the harmonized method provided a higher bound of the experimental observations.

**Figure 16.**Total VDVs from Equation (46) proposed by Chang et al. [18] at different walking frequencies and for different numbers of events (ne).

## 9. Conclusions and Suggestions for Future Work

- Measure and formulate load functions of continuous and impulsive excitations as well as rhythmic activities, tailored for long-span timber floors. The existing dynamic vertical forces are based on measurement of footfall forces using force plates and on stiff grounds. Furthermore, these force models do not account for the human-induced response between the walker and the floor, such as the feed-back phenomenon observed between the user and the structure with the Millennium Bridge in [98]. These models can be improved by using more accurate measurement equipment such as digital pressure mats. In laboratory testing, the walker can be placed in a virtual reality (VR) environment using VR goggles to achieve more realistic walking dynamics.
- Characterize the dynamic properties (frequencies, modal mass, mode shape, and damping) and response to vibration of long-span floor panels (i) in the laboratory environment, and (ii) floor systems in selected constructed or completed buildings. This will help in understanding the difference between a slab design analogy and the actual performance of the floor within the mass timber or light-frame structural system.
- Develop experimentally validated analytical models that can reliably predict the dynamic properties and vibration response of the floor systems.
- Assess occupant comfort with different floor usages and identify acceptance criteria for the investigated floor systems. Selected completed buildings with long-span timber floors, an office floor for instance, can be instrumented and monitored during working hours, and the occupants’ experiences can be collected from surveys. There will be vast benefit in gathering the data from field tests and occupant surveys to establish an international database for researchers and practitioners worldwide for vibration design of long-span timber floor systems.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Yu, M.; Wiedmann, T.; Crawford, R.; Tait, C. The carbon footprint of Australia’s construction sector. Procedia Eng.
**2017**, 180, 211–220. [Google Scholar] [CrossRef] - Australian Industry Skills Committee. Construction; Australian Industry Skills Committee: Canberra, Australia, 2022. [Google Scholar]
- Mai, K.Q.; Park, A.; Nguyen, K.T.; Lee, K. Full-scale static and dynamic experiments of hybrid CLT–concrete composite floor. Constr. Build. Mater.
**2018**, 170, 55–65. [Google Scholar] - Hassanieh, A.; Valipour, H.; Bradford, M. Experimental and numerical investigation of short-term behaviour of CLT-steel composite beams. Eng. Struct.
**2017**, 144, 43–57. [Google Scholar] [CrossRef] - Carradine, D. Multi-Storey Light Timber-Framed Buildings in New Zealand—Engineering Design; BRANZ: Judgeford, New Zealand, 2019. [Google Scholar]
- Masaeli, M.; Gilbert, B.P.; Karampour, H.; Underhill, I.D.; Lyu, C.; Gunalan, S. Experimental assessment of the timber beam-to-column connections: The scaling effect. In Proceedings of the World Conference on Timber Engineering 2021, WCTE 2021, Santiago, Chile, 9–12 August 2021. [Google Scholar]
- TTW. Atlassian Sydney Headquarters. Available online: http://www.ttw.com.au/projects/atlassian-sydney-headquarters (accessed on 9 February 2023).
- United Nations. Department of Economic and Social Affairs, The 2030 Agenda for Sustainable Development. Available online: https://www.un.org/en/development/desa/publications/global-sustainable-development-report-2015-edition.html (accessed on 9 February 2023).
- WoodWorks. U.S. Mass Timber Construction Manual. Available online: https://www.woodworks.org/resources/u-s-mass-timber-construction-manual/ (accessed on 9 February 2023).
- Bazli, M.; Heitzmann, M.; Ashrafi, H. Long-span timber flooring systems: A systematic review from structural performance and design considerations to constructability and sustainability aspects. J. Build. Eng.
**2022**, 48, 103981. [Google Scholar] [CrossRef] - Hosseini, S.M.; Peer, A. Wood Products Manufacturing Optimization: A Survey. IEEE Access
**2022**, 10, 121653–121683. [Google Scholar] [CrossRef] - Hu, L.J.; Chui, Y.H.; Onysko, D.M. Vibration serviceability of timber floors in residential construction. Prog. Struct. Eng. Mater.
**2001**, 3, 228–237. [Google Scholar] [CrossRef] - Pavic, A. Results of IStructE 2015 survey of practitioners on vibration serviceability. In Proceedings of the SECED 2019 Conference: Earthquake Risk and Engineering towards a Resilient Word, London, UK, 9–10 September 2019. [Google Scholar]
- Dolan, J.; Murray, T.; Johnson, J.; Runte, D.; Shue, B. Preventing annoying wood floor vibrations. J. Struct. Eng.
**1999**, 125, 19–24. [Google Scholar] [CrossRef] - EN 1995-1-1; Eurocode 5: Design of Timber Structures—Part 1-1: General—Common Rules and Rules for Buildings. European Committee for Standardization: Brussels, Belgium, 2004.
- CSA 086:19; Engineering Design in Wood. National Standard of Canada: Ottawa, ON, Canada, 2019.
- Hu, L. Serviceability design criteria for commercial and multi-family floors. In Canadian Forest Service Report; Forintek Canada Corporation: Sainte-Foy, QC, Canada, 2000. [Google Scholar]
- Chang, W.; Goldsmith, T.; Harris, R. A new design method for timber floors–peak acceleration approach. In Proceedings of the International Network on Timber Engineering Research Meeting 2018, Tallinn, Estonia, 13–16 August 2018. [Google Scholar]
- AS/NZS 1170.0; Structural Design Actions, Part 0: General Principles. Australian Standard: Sydney, Australia, 2002.
- CEAS. Consulting Engineering Advancement Society. Available online: https://ceas.co.nz/ (accessed on 9 February 2023).
- Smith, A.L.; Hicks, S.J.; Devine, P.J. Design of Floors for Vibration: A New Approach; Steel Construction Institute Ascot: Berkshire, UK, 2007. [Google Scholar]
- Concrete Centre. A Design Guide for Footfall Induced Vibration of Structures; Concrete Centre: London, UK, 2006. [Google Scholar]
- Murray, T.; Allen, D.; Ungar, E.; Davis, D.B. Steel Design Guide Series 11: Vibrations of Steel-Framed Structural Systems Due to Human Activity; American Institute of Steel Construction: Chicago, IL, USA, 2016; Available online: https://www.aisc.org/Design-Guide-11-Vibrations-of-Steel-Framed-Structural-Systems-Due-to-Human-Activity-Second-Edition (accessed on 1 June 2023).
- Feldmann, M.; Heinemeyer, C.; Butz, C.; Caetano, E.; Cunha, A.; Galanti, F.; Goldack, A.; Hechler, O.; Hicks, S.; Keil, A. Design of Floor Structures for Human Induced Vibrations. JRC–ECCS Joint Report 2009. Available online: https://publications.jrc.ec.europa.eu/repository/bitstream/JRC55118/jrc_design_of_floor100208.pdf (accessed on 1 June 2023).
- Ebrahimpour, A.; Sack, R.L. A review of vibration serviceability criteria for floor structures. Comput. Struct.
**2005**, 83, 2488–2494. [Google Scholar] [CrossRef] - Sitharam, T.; Sebastian, R.; Fazil, F. Vibration isolation of buildings housed with sensitive equipment using open trenches–case study and numerical simulations. Soil Dyn. Earthq. Eng.
**2018**, 115, 344–351. [Google Scholar] [CrossRef] - Sanayei, M.; Zhao, N.; Maurya, P.; Moore, J.A.; Zapfe, J.A.; Hines, E.M. Prediction and mitigation of building floor vibrations using a blocking floor. J. Struct. Eng.
**2012**, 138, 1181–1192. [Google Scholar] [CrossRef] - Wiss, J.F.; Parmelee, R.A. Human perception of transient vibrations. J. Struct. Div.
**1974**, 100, 773–787. [Google Scholar] [CrossRef] - Chopra, A.K. Dynamics of Structures; Pearson Education India: Chennai, India, 2007. [Google Scholar]
- ISO 2631-1–AS 2670.1; Evaluation of Human Exposure to Whole-Body Vibration—Part 1: General Requirements. Australian Standard: Sydney, Australia, 2001.
- ISO 10137; Basis for Design of Structures-Serviceability of Buildings and Walkways against Vibrations. International Standards Organisation: Geneva, Switzerland, 2007.
- AS ISO 2631.2; Mechanical Vibration and Shock—Evaluation of Human Exposure to Wholebody Vibration, Part 2: Vibration in Buildings (1 Hz to 80 Hz). Australian Standard: Sydney, Australia, 2014.
- BS 6472; Guide to the Evaluation of Human Exposure to Vibration in Buildings (1Hz to 80 Hz). British Standards Institution: London, UK, 1992.
- BS 6472-1; Guide to Evaluation of Human Exposure to Vibration in Buildings. British Standards Institution: London, UK, 2008.
- Sedlacek, G.; Heinemeyer, C.; Butz, C.; Veiling, B.; Waarts, P.; Van Duin, F.; Hicks, S.; Devine, P.; Demarco, T. Generalisation of criteria for floor vibrations for industrial, office, residential and public building and gymnastic halls. EUR
**2006**, 1–343. Available online: https://op.europa.eu/en/publication-detail/-/publication/a2fc45db-6b9a-49e6-9f36-72fc25a8eded/language-en (accessed on 1 June 2023). - Kerr, S.C. Human Induced Loading on Staircases; University of London: London, UK; University College London: London, UK, 1999; Available online: https://discovery.ucl.ac.uk/id/eprint/1318004/1/312827.pdf (accessed on 1 June 2023).
- Wheeler, J.E. Prediction and control of pedestrian-induced vibration in footbridges. J. Struct. Div.
**1982**, 108, 2045–2065. [Google Scholar] [CrossRef] - Basaglia, B.M.; Li, J.; Shrestha, R.; Crews, K. Response prediction to walking-induced vibrations of a long-span timber floor. J. Struct. Eng.
**2021**, 147, 04020326. [Google Scholar] [CrossRef] - Ji, T.; Pachi, A. Frequency and velocity of people walking. Struct. Eng.
**2005**, 84, 36–40. [Google Scholar] - Toratti, T.; Talja, A. Classification of human induced floor vibrations. Build. Acoust.
**2006**, 13, 211–221. [Google Scholar] [CrossRef] - Rahbar Ranji, A.; Rostami Hoseynabadi, H. A semi-analytical solution for forced vibrations response of rectangular orthotropic plates with various boundary conditions. J. Mech. Sci. Technol.
**2010**, 24, 357–364. [Google Scholar] [CrossRef] - El-Dardiry, E.; Ji, T. Modelling of the dynamic behaviour of profiled composite floors. Eng. Struct.
**2006**, 28, 567–579. [Google Scholar] [CrossRef] - Allen, D.; Murray, T. Design criterion for vibrations due to walking. Eng. J.
**1993**, 30, 117–129. [Google Scholar] - Wyatt, T.; Dier, A. Floor serviceability under dynamic loading. In Proceedings of the International Symposium “Building in Steel-The Way Ahead”; ECCS Publication: Stratford-upon-Avon, UK, 1989; pp. 19–20. Available online: https://www.google.com.au/books/edition/Building_in_Steel/o95RAAAAMAAJ?hl=en (accessed on 1 June 2023).
- Ohlsson, S. Ten years of floor vibration research—A review of aspects and some results. In Proceedings of the Symposium/Workshop on Serviceability of Buildings (Movements, Deformations, Vibrations), Ottawa, ON, Canada, 16 May 1988; pp. 419–434. [Google Scholar]
- Karnovsky, I.A.; Lebed, O.I. Formulas for Structural Dynamics: Tables, Graphs and Solutions; McGraw-Hill Education: New York, NY, USA, 2001. [Google Scholar]
- Middleton, C.; Brownjohn, J. Response of high frequency floors: A literature review. Eng. Struct.
**2010**, 32, 337–352. [Google Scholar] [CrossRef] - Basaglia, B.M. Dynamic Behaviour of Long-Span Timber Ribbed-Deck Floors. Ph.D. Thesis, University of Technology Sydney, Ultimo, Australia, 2019. [Google Scholar]
- Stürzenbecher, R.; Hofstetter, K.; Eberhardsteiner, J. Structural design of Cross Laminated Timber (CLT) by advanced plate theories. Compos. Sci. Technol.
**2010**, 70, 1368–1379. [Google Scholar] [CrossRef] - Karacabeyli, E.; Gagnon, S. Canadian CLT Handbook; FPInnovations: Pointe-Claire, QC, Canada, 2019; Volume 1, Available online: https://web.fpinnovations.ca/wp-content/uploads/clt-handbook-complete-version-en-low.pdf (accessed on 1 June 2023).
- Fink, G.; Honfi, D.; Köhler, J.; Dietsch, P. Basis of design principles for timber structures. In A State-of-the-Art Report by COST Action FP1402/WG1; Shaker: Aachen, Germany, 2018; Available online: https://www.cost.eu/uploads/2018/11/Basis-of-Design-Principles-for-Timber-Structures.pdf (accessed on 1 June 2023).
- Jarnerö, K.; Brandt, A.; Olsson, A. Vibration properties of a timber floor assessed in laboratory and during construction. Eng. Struct.
**2015**, 82, 44–54. [Google Scholar] [CrossRef] - Ewins, D.J. Modal Testing: Theory, Practice and Application; John Wiley & Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- ISO 18324; Timber Structure—Test Methods—Floor Vibration Performance. International Standards Organisation: Geneva, Switzerland, 2016.
- Labonnote, N. Damping in Timber Structures; NTNU-trykk: Trondheim, Norway, 2012. [Google Scholar]
- Rijal, R.; Samali, B.; Shrestha, R.; Crews, K. Experimental and analytical study on dynamic performance of timber floor modules (timber beams). Constr. Build. Mater.
**2016**, 122, 391–399. [Google Scholar] [CrossRef] - Conta, S.; Homb, A. Sound radiation of hollow box timber floors under impact excitation: An experimental parameter study. Appl. Acoust.
**2020**, 161, 107190. [Google Scholar] [CrossRef] - Ouis, D. On the frequency dependence of the modulus of elasticity of wood. Wood Sci. Technol.
**2002**, 36, 335–346. [Google Scholar] [CrossRef] - Craik, R.J.; Barry, P.J. The internal damping of building materials. Appl. Acoust.
**1992**, 35, 139–148. [Google Scholar] [CrossRef] - Lin, R.; Zhu, J. On the relationship between viscous and hysteretic damping models and the importance of correct interpretation for system identification. J. Sound Vib.
**2009**, 325, 14–33. [Google Scholar] [CrossRef] - Code, B. International Residential Code; International Energy: Paris, France, 2018. [Google Scholar]
- RFS2-CT-2007-00033; Design of Footbridges Guideline. Human Induced Vibrations of Steel Structures. HIVOSS: Luxembourg, 2009.
- Hamm, P.; Richter, A.; Winter, S. Floor vibrations–new results. In Proceedings of the 11th World Conference on Timber Engineerig (WCTE2010), Riva del Garda, Italy, 20–24 June 2010. [Google Scholar]
- AS/NZS 1170.1; Structural Design Actions, Part 1: Permanent, Imposed and Other Actions. Australian Standard: Sydney, Australia, 2002.
- Woeste, F.; Dolan, J. Design to minimize annoying wood-floor vibrations. Struct. Eng.
**2007**, 8, 24–27. [Google Scholar] - ISO/TR 21136; Timber Structures—Vibration Performance Criteria for Timber Floors. International Standards Organisation: Geneva, Switzerland, 2017.
- Onysko, D. Serviceability Criteria for Residential Floors Based on a Field Study of Consumer Response; FPInnovations: Ottawa, ON, Canada, 1986; Available online: https://library.fpinnovations.ca/en/permalink/fpipub38186 (accessed on 1 June 2023).
- Smith, I.; Chui, Y.H. Design of lightweight wooden floors to avoid human discomfort. Can. J. Civ. Eng.
**1988**, 15, 254–262. [Google Scholar] [CrossRef] - Hu, L. Serviceability of Next Generation Wood Buildings: Laboratory Study of Vibration Performance of Cross-Laminated-Timber (CLT) Floors; FPinnovations: Québec, QC, Canada, 2013; Available online: https://library.fpinnovations.ca/en/permalink/fpipub39703 (accessed on 1 June 2023).
- prEN 1995-1-1:2025; Vibrations. European Committee for Standardization: Brussels, Belgium, 2025.
- EN 16929; Test Methods—Timber Floors—Determination of Vibration Properties. European Committee for Standardization: Brussels, Belgium, 2018.
- ASHRAE. Noise and Vibration Control; Handbook A49; ASHRAE: Peachtree Corners, GA, USA, 2019. [Google Scholar]
- Ellis, B. Serviceability Evaluation of Floor Vibration Induced by Walking Loads; Structural Engineer: London, UK, 2001. [Google Scholar]
- Rainer, J.; Pernica, G.; Allen, D.E. Dynamic loading and response of footbridges. Can. J. Civ. Eng.
**1988**, 15, 66–71. [Google Scholar] [CrossRef] [Green Version] - Huang, H.; Gao, Y.; Chang, W.-S. Human-induced vibration of cross-laminated timber (CLT) floor under different boundary conditions. Eng. Struct.
**2020**, 204, 110016. [Google Scholar] [CrossRef] - Wang, C.; Chang, W.-S.; Yan, W.; Huang, H. Predicting the human-induced vibration of cross laminated timber floor under multi-person loadings. Structures
**2020**, 29, 65–78. [Google Scholar] [CrossRef] - Chen, F.; Li, Z.; He, M.; Wang, Y.; Shu, Z.; He, G. Seismic performance of self-centering steel-timber hybrid shear wall structures. J. Build. Eng.
**2021**, 43, 102530. [Google Scholar] [CrossRef] - Schneider, J.; Tannert, T.; Tesfamariam, S.; Stiemer, S. Experimental assessment of a novel steel tube connector in cross-laminated timber. Eng. Struct.
**2018**, 177, 283–290. [Google Scholar] [CrossRef] - Karampour, H.; Bourges, M.; Gilbert, B.P.; Bismire, A.; Bailleres, H.; Guan, H. Compressive behaviour of novel timber-filled steel tubular (TFST) columns. Constr. Build. Mater.
**2020**, 238, 117734. [Google Scholar] [CrossRef] - Darzi, S.; Karampour, H.; Gilbert, B.P.; Bailleres, H. Numerical study on the flexural capacity of ultra-light composite timber sandwich panels. Compos. Part B Eng.
**2018**, 155, 212–224. [Google Scholar] [CrossRef] - Darzi, S.; Karampour, H.; Bailleres, H.; Gilbert, B.P.; McGavin, R.L. Experimental study on bending and shear behaviours of composite timber sandwich panels. Constr. Build. Mater.
**2020**, 259, 119723. [Google Scholar] [CrossRef] - Trummer, A.; Krestel, S.; Aicher, S. KIELSTEG-Defining the design parameters for a lightweight wooden product. In Proceedings of the World Conference on Timber Engineering: WCTE 2016, Vienna, Austria, 22–25 August 2016. [Google Scholar]
- Skidmore, O.M.S. Timber Tower Research Project. Available online: https://www.som.com/research/timber-tower-research/ (accessed on 25 January 2023).
- Auclair, S.C.; Hu, L.; Gagnon, S.; Mohammad, M. Effect of type of lateral load resisting system on the natural frequencies of mid-to high-rise wood buildings. In Proceedings of the WCTE 2018, World Conference on Timber Engineering, Seoul, Republic of Korea, 20–23 August 2018. [Google Scholar]
- Jiang, Y.; Crocetti, R. CLT-concrete composite floors with notched shear connectors. Constr. Build. Mater.
**2019**, 195, 127–139. [Google Scholar] [CrossRef] - CSA 086-14; Supplement: Engineering Design in Wood. Canadian Standards Association: Toronto, ON, Canada, 2016.
- Council Binational Softwood Lumber. Nail-Laminated Timber: US Design & Construction Guide; Council Binational Softwood Lumber: Surrey, BC, Canada, 2017; Available online: https://info.thinkwood.com/nlt-design-and-construction-guide-u.s.-version-think-wood-0 (accessed on 25 January 2023).
- Barber, D.; Blount, D.; Hand, J.J.; Roelofs, M.; Wingo, L.; Woodson, J.; Yang, F. Design Guide 37: Hybrid Steel Frames with Wood Floors; The American Institute of Steel Construction: Chicago, IL, USA, 2022. [Google Scholar]
- Bitter, R.; Mohiuddin, T.; Nawrocki, M. LabVIEW: Advanced Programming Techniques; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- MathWorks. MATLAB: The Language of Technical Computing: Computation, Visualization, Programming: Installation Guide for UNIX Version 5; MathWorks: Natwick, MA, USA, 2022. [Google Scholar]
- DeSalvo, G.J.; Swanson, J.A. ANSYS Engineering Analysis System: User’s Manual; Swanson Analysis Systems: Houston, TX, USA, 2022. [Google Scholar]
- Brancheriau, L.; Baillères, H. Natural vibration analysis of clear wooden beams: A theoretical review. Wood Sci. Technol.
**2002**, 36, 347–365. [Google Scholar] [CrossRef] - Faircloth, A.; Brancheriau, L.; Karampour, H.; So, S.; Bailleres, H.; Kumar, C. Experimental modal analysis of appropriate boundary conditions for the evaluation of cross-laminated timber panels for an in-line approach. For. Prod. J.
**2021**, 71, 161–170. [Google Scholar] [CrossRef] - Nemli, G.; Aydın, A. Evaluation of the physical and mechanical properties of particleboard made from the needle litter of Pinus pinaster Ait. Ind. Crops Prod.
**2007**, 26, 252–258. [Google Scholar] [CrossRef] - AS 1720.1; Timber Structures—Part 1: Design Methods. Australian Standard: Sydney, Australia, 2010.
- Ross, R. Wood Handbook: Wood as an Engineering Material; Forest Products Laboratory: Madison, WI, USA, 1987. Available online: https://www.fpl.fs.usda.gov/documnts/fplgtr/fplgtr282/fpl_gtr282.pdf (accessed on 25 January 2023).
- AS 4100; Steel Structures. Australian Standard: Sydney, Australia, 2020.
- Dallard, P.; Fitzpatrick, A.; Flint, A.; Le Bourva, S.; Low, A.; Ridsdill Smith, R.; Willford, M. The London millennium footbridge. Struct. Eng.
**2001**, 79, 17–21. [Google Scholar]

**Figure 1.**Examples of long-span timber floors (LSTFs): (

**a**) lightweight LSTFs with and without Strongbacks; (

**b**) mass timber LSTFs with (

**left**) ribbed-deck CLT floor, and the (

**right**) CLT band-beam system concept; and (

**c**) hybrid floors: (

**left**) delta beam–CLT–concrete, and (

**right**) CLT floor on steel frame; courtesy: David Barber.

**Figure 11.**Case study I: EMA and NMA of a CLT slab showing (

**a**) support points, impact location, and accelerometer positions; (

**b**) signal processing and FFT from the time domain signal; and (

**c**) mode shapes and the modal assurance criteria (MAC) results.

**Figure 12.**Case study II: the lightweight 6 m × 6 m floor system; plan layout; walking path (in green) showing dot points of step footprints; accelerometers A, B, and C locations (circled in red); and the FEA geometry and mesh.

**Figure 13.**Case study II showing (

**a**) time history and frequency domain responses of a digital hammer excitation measured at accelerometer C in Figure 12, and (

**b**) mode shapes and frequencies from the FEA.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Karampour, H.; Piran, F.; Faircloth, A.; Talebian, N.; Miller, D.
Vibration of Timber and Hybrid Floors: A Review of Methods of Measurement, Analysis, and Design. *Buildings* **2023**, *13*, 1756.
https://doi.org/10.3390/buildings13071756

**AMA Style**

Karampour H, Piran F, Faircloth A, Talebian N, Miller D.
Vibration of Timber and Hybrid Floors: A Review of Methods of Measurement, Analysis, and Design. *Buildings*. 2023; 13(7):1756.
https://doi.org/10.3390/buildings13071756

**Chicago/Turabian Style**

Karampour, Hassan, Farid Piran, Adam Faircloth, Nima Talebian, and Dane Miller.
2023. "Vibration of Timber and Hybrid Floors: A Review of Methods of Measurement, Analysis, and Design" *Buildings* 13, no. 7: 1756.
https://doi.org/10.3390/buildings13071756