# Analytical and Numerical Verification of Vibration Design in Timber Concrete Composite Floors

^{*}

## Abstract

**:**

_{1}< 8 Hz). The theoretical studies were accompanied by numerical analyses considering certain simplifications suitable for practical use.

## 1. Introduction

^{2}is the result (the market figures are based on Germany in 2014). The establishment of TCC for the wood-based sector makes significantly higher demands than what would normally occur with technical innovations where TCC has a considerable impact on the identity of building with wood. Such an approach to the market requires conditions that will bring the benefits of TCC closer to the user, with a sense of comfort being one of the most important factors. Therefore, the vibration conditions, or the way of the connection (between wood and concrete) principles which defined them, become the main accent in the design of floor structures.

## 2. TCC Floor System Subjected to Footfall Induced Vibration

_{rms}/0.0001) given in Table 1. For resonant vibration design situations, (when f

_{1}< 8 Hz), the minimum fundamental frequency, acceleration, and stiffness criteria of Table 1 should be fulfilled. For transient vibration design situations, (when f

_{1}≥ 8 Hz), the velocity and stiffness criteria of Table 1 should be fulfilled.

^{(f1/ς)}, in m/Ns

^{2},

_{1}may approximately be calculated as:

_{1}< π/(2l

^{2})√((EI)

_{l}/m), in Hz,

^{2}, l is the floor span in m, and (El)

_{l}is the equivalent plate bending stiffness of the floor about an axis perpendicular to the beam direction in Nm

^{2}/m. The concrete slab in timber–concrete composite systems is relatively narrow compared to pure concrete slabs. Therefore, the reinforcement is often installed near the centroid of the cross-section. The cracking of concrete will lead to a significant drop in the bending stiffness. The effective width is caused by the distribution of the normal force in the concrete cross-section as a shell and the distribution of the bending moment as a plate. Due to the decrease of the bending stiffness by cracking, the load distribution in the concrete slabs tends to be only in the shell model. However, the effective width given in [37] is comparable to the shell mode, whereas in [38], the plate mode is also considered [33,39,40]. For that reason, it is appropriate to use the effective width according to [37] instead of [38].

_{1}< π/(2l

^{2})√(((EI)

_{ef}/b

_{c})/m), in Hz,

_{c}is the spacing between the timber beams [32]. The natural frequencies of the composite beams and the spacing of the composite beams are the same as those of the composite floor [41]. For the same type of floor, the value v may, as an approximation, be taken as:

_{40})/(mbl + 200), in m/Ns

^{2},

^{2}), m is the mass in kg/m

^{2}, b is the floor width in m, l is the floor span in m, and n

_{40}is the number of first-order modes with natural frequencies up to 40 Hz which may be calculated from:

_{40}≤ (((40/f

_{1})

^{2}− 1)((b/l)

^{4})((EI)

_{l}/(EI)

_{b}))

^{0.25},

_{b}is the equivalent plate bending stiffness, in Nm

^{2}/m, of the floor about an axis parallel to the beams, where (E1)

_{b}< (EI)

_{l}. Series of studies and research, of which the results are given in [42,43,44,45,46,47,48,49], were a background for choosing values for the appropriate modal damping ratio. According to [33], for floors, unless other values are proven to be more appropriate, a modal damping ratio of ς = 0.025 (i.e., 2.5%) should be used for timber–concrete composite slabs alone and ς = 0.035 slabs with a floating screed. The instantaneous elastic bending stiffness of the composite structure should be used in vibration analysis. Alternatively, damping ratio values can be calculated based on actual loads (i.e., precomposite dead load, postcomposite dead load, and sustained live load) [50]. More accurate damping values can alternatively be obtained by samples testing, applying [51].

_{1}< 1/(2π√(g/y

_{w}), in Hz,

_{w}is the weighted average of the static deflection of the beam due to the self-weight of the floor or the dead weight of the beam–slab system, and g is the gravitational acceleration. A similar method is proposed in [53], where the fundamental frequency of simply supported beams or floor systems under uniformly distributed additional loads, as shown in the equation:

_{1}< (π/2)√((gEI)/(wl

^{4})), in Hz,

_{ef}= E

_{c}(I

_{c}+ γ

_{c}A

_{c}a

_{c}

^{2}) + E

_{t}(I

_{t}+ γ

_{t}A

_{t}a

_{t}

^{2}), in Nmm

^{2},

_{c}= 1/(1 + (π

^{2}E

_{c}A

_{c}s)/(K

_{ser}l

^{2})),

_{t}= 1,

_{ser}is the slip modulus of the connector. The authors of [54] proposed a method to calculate the effective stiffness of composite beam with an arbitrary boundary as:

_{ef}= EI

_{∞}/(1 + (EI

_{∞}/EI

_{0}− 1)/(1 + ((μ/π)

^{2})((α/L)

^{2}))), in Nmm

^{2},

_{0}the bending stiffness of the noncomposite section (K→0), and EI

_{∞}is the bending stiffness of the fully composite section. EA

_{0}and EA

_{p}are the sum and product of axial stiffness of the sub-elements, respectively. Some design methods, such as the γ-method, assume that the connectors may be smeared along the beam axis. In this case, no discrete connector is considered, but the stiffness per unit length is taken into account. For short distances, this simplification works quite well. However, with increasing distance, the discrete connector leads to local stresses. This conclusion is reported in [55,56]. The authors of [55] propose a maximum distance of 5%, whereas [56] proposes a maximum distance of 3%. For notches as the connection between timber and concrete, [57] proposes an effective distance in order to enable smearing of these connectors. Providing that the effective spacing of the connections is less than or equal to 5% of the distance between the points of contra flexure, even distribution of the connection stiffness along the beam axis (smearing) may be used. If the spacing of the connections is greater than 5% of the distance between the points of contra flexure, then, when smearing, the connections should be distributed in proportion to the shear force. Comparing the sinusoidal course and the constant course of the normal force, the different deformation can be evaluated. In the hyperstatic composite cross-section, the stiffness influences the distribution of the forces. The stiffness of the cross-section connected with discrete connectors should be reduced by the factor 2∕π to consider the same stiffness as in the smeared case. For the reason of simplification, this factor is rounded to 0.7, since the case of only one single connector at each of the beams is hardly used. However, up to now, no other values are given, since the variability of possible positions of the connectors is quite large. So, in the absence of a more precise model, only 70% of the axial stiffness of the attached cross-section should be considered for the calculation of stresses and deformation. For the calculation of the shear force in the connection, 100% of the axial stiffness of the cross-section should be considered [33]. If the spacing and/or stiffness of the connections are varied in proportion to the shear force, the effective spacing may be determined as:

_{ef}= 0.75s

_{min}(K

_{ref}/K

_{max}) + 0.25 s

_{max}(K

_{ref}/K

_{min}), in mm,

_{ef}is the effective spacing of the connections, s

_{min}is the minimum spacing of the connections, K

_{ref}is the reference stiffness of the connection used with the corresponding value of s

_{ef}, K

_{max}is the maximum stiffness of the connection, K

_{min}is the minimum stiffness of the connection, and s

_{max}is the maximum spacing of the connections or the maximum distance between the connection and the point of zero shear force.

_{ser}of connections made with dowel-type fasteners inserted perpendicular to the shear plane should be determined using:

_{ser}= 2(ρ

_{m}

^{1.5})d/23, in N/mm,

_{ser}= 2(ρ

_{m}

^{1.5})(d

^{0.9})/30, in N/mm,

_{ser}is the slip modulus for serviceability limit states, ρ

_{m}is the mean value of the timber member density in kg/m

^{3}, and d is the fastener diameter in mm. Research studies [58,59,60,61] show that this can be a good estimation when more detailed data is not available. For regular interlayers, with stiffness perpendicular to the shear plane similar to that of timber and with thickness up to 30 mm, the slip modulus of connections with dowel-type fasteners may be taken as that for a similar configuration without an interlayer, with a reduction of 30% [11]. In other cases where there is an intermediate nonstructural layer between the timber and the concrete, the slip modulus should be determined by tests or special analysis. For connections made with steel rebar glued into timber perpendicular to the shear plane using epoxy resin, the slip modulus for serviceability limit states K

_{ser}should be determined according to:

_{ser}= 0.10E

_{tim}d, in N/mm,

_{tim}is the mean modulus of elasticity of timber parallel to the grain in N/mm

^{2}, and d is the nominal diameter of the rebar in mm [33,58,60,62]. In many cases, when the manufacturer provides new, modern types of connectors, such as [63], slip modulus should be determined based on verified tests. For notched connections, the slip modulus for serviceability limit states that K

_{ser}should be determined according to:

_{ser}= 1000 N/mm/mm for h

_{n}= 20 mm,

_{ser}= 1500 N/mm/mm for h

_{n}≥ 30 mm,

_{n}is the depth of the notch. Linear interpolation may be used for h

_{n}between 20 mm and 30 mm [33,64].

_{ser}/s, where s is spacing between connectors, and K

_{ser}/s is stiffness per unit length of the element expressed in (N/mm)/mm. By applying the deflection criteria, the minimum bending stiffness fulfilling the eigenfrequency criteria can be predicted.

## 3. FEM Analysis of TCC Floor Slab

#### 3.1. Compliance of the Connection–K_{ser}

_{ser}= 825 − 250 (d

_{ZS})

^{0.2}[N/mm] per mm of expanded metal length.

_{ZS}is the thickness of the intermediate layer in mm. The calculated value of the initial displacement module of an HBV shear connector for the proof of the load-bearing capacity is 2/3 of the calculated value of the initial displacement module to accept the proof of usability. The displacement module of an HBV shear connector at the time t = ∞ may be assumed with 0.5 times the value at time t = 0.

_{k}of the HBV shear connectors with parallel loads to the expanded metal axis (longitudinal shearing) as follows:

_{k}= 160 − 8.0 (d

_{ZS})

^{0.5}in N per mm of strip length.

_{d}of the HBV shear connector may be taken as follows:

_{d}= T

_{k}/1.25

_{ser}= 825 − 250∙20

^{0.2}= 370,000 kN/m

^{2}

_{d}= (160 − 8∙20

^{0.2})/1.25 = 100 kN/m

_{cs oo}/α

_{T}

_{inst}= G + Q

_{1}+ ∑〖ψ

_{0}∙Q

_{i,k}〗

_{inst}= G + Q

_{1}

_{net,fin}= G

_{kdef}+ ∑〖ψ

_{2}∙Q

_{i,k},

_{def}− w

_{c}〗

_{net,fin}= G

_{k,def}+ ψ

_{2}∙ Q

_{1,kdef}

_{fin}= Q

_{1}∙(1 − ψ

_{2}) + ∑〖Qi∙(ψ

_{0}− ψ

_{2}) + G

_{kdef}+ ψ

_{2}Q

_{1,kdef}〗 + ∑〖ψ

_{2}∙Q

_{i,kdef}〗

_{fin}= Q

_{1}∙(1 − ψ

_{2}) + G

_{kdef}+ ψ

_{2}∙Q

_{1,kdef}

#### 3.2. Results

_{e}≈ 5/√(0.8∙w(cm)) ≈ 17.753/√3

_{e}

^{2})

#### 3.3. Stiffness Design

_{limit}= 0.5 mm. This is usually the case for floors in residential or office buildings. On the other hand, if the vibrations are quite noticeable, but not annoying, then w

_{limit}= 1.0 mm (e.g., family house). The analytical approach of this design is to create a replacement simple beam system, with an appropriate span corresponding to the largest range of the TCC floor. According to [19,67,68], the deflection may be calculated as follows:

_{l}∙b

_{w(2kN)}) ≤ w

_{limit}

_{w(2kN)}= min{(b

_{ef}@ b (width of the floor))

_{ef}= l/1.1∙∜((EI

_{b}/(EI

_{l}) = b/(1.1∙α))

#### 3.4. Connector Design

#### 3.5. Concrete Check

_{c, max}= 11.25 N/mm

^{2}, and as Figure 12 shows, maximal stress in the TCC floor slab is 2.4 N/mm

^{2}.

#### 3.6. Acceleration Design

_{dyn}/(M∙2D) = (0.4∙1∙700 N)/(m∙0.5∙l∙0.5∙b∙2∙D) = 56/(m∙l∙b∙D)

- M—the modal mass of the TCC floor
- 700 N—harmonic part of the force (see [69])
- 0.1—Fourier coefficient
- 0.4—simplification factor (person moves around)
- m—mass (kg)
- l—TCC floor span (m)
- b—the width of the floor (<1.5 l)

^{2}. Besides, the RMS value is displayed. For the floor to be deemed acceptable this value must be less than a limit based on the floor’s fundamental natural frequency. However, RMS acceleration can only be used for more resonant excitation.

#### 3.7. Constructive Design Requirements

^{2}. In the illustrative example, Table 6 shows that the sum of the loads in the Z direction is equal to 119.27 kN. If this is divided by the total area of 7.0 × 4.8 m, it obtains a satisfactory weight of 354 kg/m

^{2}, thus fulfilling this condition.

#### 3.8. Comparison Study

## 4. Discussion

## 5. Conclusions

_{2}emissions are reduced, the building process is faster, and reduced effort is needed for the props and formwork.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Decision tree. * The additional examination of acceleration is successful only in the case of timber–concrete composite systems, or other heavy systems with wide spans.

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

I | II | III | IV | V | VI | VI | |

stiffnes criteria w _{1kN} [mm] ≤ | 0.25 | 0.25 | 0.50 | 0.80 | 1.20 | 1.60 | no criteria |

response factor R ≤ | 4 | 8 | 12 | 16 | 24 | 32 | |

frequency criteria f _{1} [HZ] ≥ | 4.50 | ||||||

acceleration criteria a _{rms} [m/s^{2}] ≤ | 0.005R | ||||||

velocity criteria v _{rms} [m/s] ≤ | 0.0001R |

Limit State | Time | Timber | Concrete | Connector | LC |
---|---|---|---|---|---|

ULS | t = 0 | E_{mean} | E_{cm} | 2 K_{ser}/3 γ_{M} | LC1 |

t = ∞ | E_{mean}/(1 + k_{def}) | E_{cm}/3.5 | 0.5∙(2 K_{ser})/(3 γ_{M}) | LC2 | |

SLS | t = 0 | E_{mean} | E_{cm} | K_{ser} | LC3 |

t = ∞ | E_{mean}/(1 + k_{def}) | E_{cm}/3,5 | 0.5∙K_{ser} | LC4, RC1 |

Material | The Factor for E, G | ||

Poplar and Softwood Timber C24 | 0.625000 | ||

Glulam Timber GL32h | 0.625000 | ||

Concrete C25/30 | 0.285000 | ||

The Factor for Connector Stiffness | |||

Cux | Cuy | Cuz | Cφx |

0.5 | 1.0 | 1.0 | 1.0 |

Mode | Eigenvalue | Angular Frequency | Natural Frequency | Natural Period |
---|---|---|---|---|

No. | l [1/s^{2}] | v [rad/s] | f [Hz] | T [s] |

1 | 3921.982 | 62.626 | 9.967 | 0.100 |

2 | 15,583.944 | 124.836 | 19.868 | 0.050 |

3 | 19,639.645 | 140.142 | 22.304 | 0.045 |

4 | 21,777.285 | 147.571 | 23.487 | 0.043 |

Mode | Eigenvalue | Angular Frequency | Natural Frequency | Natural Period |
---|---|---|---|---|

No. | l [1/s^{2}] | v [rad/s] | f [Hz] | T [s] |

1 | 1561.041 | 39.510 | 6.288 | 0.159 |

2 | 1974.294 | 44.433 | 7.072 | 0.141 |

3 | 3402.945 | 58.335 | 9.284 | 0.108 |

4 | 6336.676 | 79.603 | 12.669 | 0.079 |

Description | Value | Unit |
---|---|---|

LC1—Permanent Load | ||

Sum of loads in X | 0.00 | kN |

Sum of loads in Y | 0.00 | kN |

Sum of loads in Z | 119.27 | kN |

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**MDPI and ACS Style**

Perković, N.; Rajčić, V.; Barbalić, J.
Analytical and Numerical Verification of Vibration Design in Timber Concrete Composite Floors. *Forests* **2021**, *12*, 707.
https://doi.org/10.3390/f12060707

**AMA Style**

Perković N, Rajčić V, Barbalić J.
Analytical and Numerical Verification of Vibration Design in Timber Concrete Composite Floors. *Forests*. 2021; 12(6):707.
https://doi.org/10.3390/f12060707

**Chicago/Turabian Style**

Perković, Nikola, Vlatka Rajčić, and Jure Barbalić.
2021. "Analytical and Numerical Verification of Vibration Design in Timber Concrete Composite Floors" *Forests* 12, no. 6: 707.
https://doi.org/10.3390/f12060707