Vibration Performances of a Full-Scale Assembled Integral Two-Way Multi-Ribbed Composite Floor

Abstract

The static performances of an assembled integral two-way multi-ribbed composite floor system have been studied experimentally and numerically, while the dynamic characteristics and comfort analysis under a human load have not been investigated. In this article, a 9.2 m × 9.2 m floor system, composed of 16 precast panels and integrated into a whole structure through six wet joints, was designed and tested under pedestrian loads. Dynamic performances related to its natural frequencies, vibration mode shapes, and maximum acceleration were analyzed. Theoretical formulas were proposed to predict its natural frequency and maximum acceleration under a single-person load. It was found that the dynamic behavior of this innovative floor system meets the requirements of GB50010-2010 and ISO 2631. Elastic plate theory could be applied to predict the natural frequency and acceleration, with the bending stiffness obtained from the experiment. Some design and dynamic test suggestions for this floor system and similar structures are proposed based on a parametric analysis.

1. Introduction

The floor system, accounting for nearly 40% of material usage and construction time, is a critical component of modern buildings. In response to the growing demand for accelerated construction timelines and sustainable building practices, prefabricated floor systems have become increasingly essential [1,2].

Compared to cast-in-situ floors, the main challenges associated with assembled concrete floors are the reduced stiffness and increased displacement due to the joints. As a result, innovative assembled or assembled integral floor systems, such as the Vierendeel-sandwich-plate floor system [3], the PK prestressed composite slab floor system [4], the assembled monolithic hollow-ribbed floor [5], the prefabricated PC floor system [6], and other new systems [7,8,9,10,11] were proposed and investigated through experimental and theoretical methods. It was found that lightweight hollow floors with reliable wet joints are one of the best solutions for modern large-span floor systems [12,13,14,15].

To fully display the advantages of hollow floors and combine with the techniques of assembled structures, an assembled integral two-way multi-ribbed composite floor system was proposed by Zeng and Xu [16,17]. By dividing the floor system into precast 2.05 m × 2.05 m panels with multiple ribs connecting with wet joints and couplers, the hollow ratio could reach 60%, and the construction could be significantly enhanced by 40%. Static experiments and numerical simulations were conducted to verify its performance regarding stiffness and crack resistance [16,17], and satisfactory results were reached.

However, with the increase in the span, the vibration under human loads might be prominent because of its lower vertical stiffness. A dynamic performance should be conducted to check whether its natural frequency and acceleration under human loads meet the requirements of design codes, and further explore the numerical method to predict its dynamic behaviors.

Research has been conducted worldwide on dynamic behaviors for assembled integral floor systems, mainly focusing on experimental and numerical methods. Tilden [18] and Fuller [19] were among the first researchers to experimentally quantify the dynamic load effects of individuals and groups, respectively. Other researchers, such as Greimann and Klaiber [20], Tuan and Saul [21], and many others [22,23,24], extended the method to obtain other dynamic responses from other loads, including dancing, sports, and crowd harmonic loads. However, the experimental method is mainly based on field tests, not analytical or numerical methods. The finite element method (FEM) is the most commonly used method to predict human-induced vibrations. Nie et al. [25] investigated the dynamic characteristics of a long-span floor under crowd-induced rhythmic excitation by using a stochastic vibration approach combined with a FEM for modal analysis. Liu [26] studied three types of materials with different properties (orthogonally anisotropic materials, strip-shaped unidirectional materials, and isotropic materials), analyzing the vibration modes of these materials on the same hollow-core slab. They found the optimal material parameters for each mode and identified the orthogonally anisotropic material as the best choice. Wen et al. [27] carried out experimental research on the vibration comfort of a new type of precast hollow-cross rib floor system. They proposed a finite-element analysis method, and the results showed that the floor system has good vibration comfort performance. But the main problems for assembled integral floor systems are that a large number of elements and a lot of interfacial simulation treatment may slow down the calculation efficiency and increase the calculation time.

As is concluded above, it is found that the experimental methods for dynamic tests are quite mature, but there are still some problems related to the dynamic calculation under human loads with a FEM. In this paper, to enhance the calculation efficiency, a finite difference method (FDM) is used to predict its vibration modes, frequencies, and peak acceleration under a human load. To further verify the effectiveness, a 9.2 m × 9.2 m floor system, composed of 16 precast panels and integrated into a whole structure through six wet joints, was designed and tested under a pedestrian load. Dynamic performances related to its natural frequency, vibration modes, and acceleration tests under a pedestrian load were analyzed.

2. Experimental Investigation

2.1. Introduction of the Innovative Floor System

The innovative floor system consists of four main components: a precast ribbed bottom slab (PRBS), lightweight infills and a cast-in-situ upper slab (CUS), as shown in Figure 1. The three different parts above and adjacent bottom slab are integrated into a reliable system through three methods (see Figure 2): (1) The connection between the PRBS and the CUS is realized by the shear strength of the rough interface, cast-in-situ joints, and stirrups. (2) The lightweight infills are fixed on the precast bottom slab through the positioning rebars and recessed cavities. (3) The rebars of the adjacent PRBSs are connected by couplers for squeezing the splicing of rebars and cast-in-situ joints where the shear keys are averagely arranged alongside the outsides of the PRBS.

2.2. Specimen Preparation

The specimen, as shown in Figure 3 and Figure 4, consists of 16 PRBs and 64 lightweight infills. The total size of the specimen is 9.6 m × 9.6 m with a joint width of 200 mm and a floor height of 300 mm. It is supported by four beams and four columns, designed by the GB50010-2010 “Code for design of concrete structures” [28]. The beam is 400 × 800 mm (width × height) reinforced with 8Φ16, while the column is 400 × 800 mm reinforced with 16Φ16. Detailed information on the prefabricated PRBs is illustrated in Figure 3g or references [16,17].

All 16 PRBSs were manufactured in factories, while the beams, columns, CUS, and joints were poured and cured at the laboratory of civil engineering of the Southeast University in China. The longitudinal rebars of adjacent PRBS panels were connected using mechanical splicing couplers (highlighted in yellow in Figure 3). The cast-in-situ concrete topping and wet joints were integrated with RC beams and columns to form a unified floor system.

2.3. Material Properties

The rebars adopted in the experiment were all HRB400, according to the Chinese standard GB50010-2010 “Code for design of concrete structures” [28]. The diameters were 6 mm, 12 mm and 20 mm, respectively. The concrete used in the cast-in-situ columns, beams, PRBC, CUS, and cast-in-situ joints was all C35. Rebars with a diameter of 20 mm were sleeved into the coupler and connected by cold extrusions.

Rebars, couplers, and concrete were tested according to Chinese standard GB 228.1-2021 “Metallic materials—Tensile tests—Part 1: methods of test under room temperature” [29], JG/T 163-2013 “Coupler for rebar mechanical splicing” [30], and GB50010-2010 [28], respectively. Mechanical properties of the rebar, coupler, and concrete were all tested in the civil engineering laboratory of Southeast University, Nanjing, China, with hydraulic machines (MTS, the US). Detailed mechanical properties are collected in

Table 1. Mechanical properties of the rebars and coupler.

Materials	Diameters of Rebar [mm]	Yield Stress [MPa]	Ultimate Stress [MPa]	Young’s Modulus [MPa]
Rebars	20	420.31	620.06	217,696
	12	451.24	616.58	209,643
	6	436.73	580.46	213,487
Coupler	20	414.57	608.02	176,867

and

Table 2. Mechanical properties of the concrete.

Positions of Concrete	Cubic Compressive Stress [MPa]	Young’s Modulus [MPa]
Cast-in-situ section	36.75	30,043
Precast panels	39.47	29,567

.

2.4. Test Method and Sensor Distribution

The dynamic responses of the assembled integral two-way multi-ribbed composite floor, including the vibration mode and acceleration time-history curve under a pedestrian load, were tested in the civil engineering laboratory of Southeast University, Nanjing, China. The vibration modes and frequencies were tested using the pulsating method and analyzed by the subspace identification (SSI) method, with all data collected at midnight to minimize disturbances from human activity. The acceleration-history curves were obtained under pedestrian loads for single-person and six-person loads, respectively. For the single-person load, the involved volunteer was 61 kg. For the six-person load, the average weight was controlled at 63 kg.

The volunteers followed a standardized walking routine from left to right with a step frequency of 2 Hz. To maintain these frequencies, an application was installed on their phones, which provided a reminder tone every 0.5 s. Upon reaching the opposite side of the floor, the volunteers waited for several seconds and then turned around and walked back, ensuring the total walking duration was more than 40 s. This procedure was followed for single-person and group walking.

To capture the acceleration-history curves of the floor, 16 acceleration meters (DASP, China Orient Institute of Noise & Vibration, Beijing, China) were evenly distributed across the specimen, as presented in Figure 5. All the data were automatically logged by a dynamic logger (DASP, China Orient Institute of Noise & Vibration, Beijing, China) at a sampling rate of 256 Hz.

The mechanism of SSI is explained by the following equations. For linear multi-DOF systems, the equation of motion could be presented as Equation (1). In the space state equation form, it could be transformed to Equations (2) and (3). Discretized to time, Equations (4) and (5) could be obtained. The main aim of the method is to obtain the A matrix based on the measurement values.

$$\mathit{M}\ddot{\mathit{X}}+\mathit{C}\dot{\mathit{X}}+\mathit{K}\mathit{X}=\mathit{u}\left(\mathit{t}\right)$$

(1)

$$\dot{\mathit{Z}}=\mathit{H}\mathit{Z}+\mathit{F}\mathit{u}$$

(2)

$$\mathit{Z}=\left\{\begin{array}{c}\mathit{X}\\ \dot{\mathit{X}}\end{array}\right\}, \mathit{H}=\left\{\begin{array}{cc}\mathbf{0}& \mathit{I}\\ \mathbf{-}{\mathit{M}}^{\mathbf{-}\mathbf{1}}\mathit{K}& -{\mathit{M}}^{\mathbf{-}\mathbf{1}}\mathit{C}\end{array}\right\}, \mathit{F}=\left\{\begin{array}{c}\mathbf{0}\\ {\mathit{M}}^{\mathbf{-}\mathbf{1}}\end{array}\right\}$$

(3)

$$\mathit{x}\left(\mathit{t}+\mathit{1}\right)=\mathit{A}\mathit{x}\left(\mathit{t}\right)+\mathit{B}\mathit{u}\left(\mathit{t}\right)$$

(4)

$$\mathit{y}\left(\mathit{t}\right)=\mathit{D}\mathit{x}\left(\mathit{t}\right)+\mathit{E}\mathit{u}\left(\mathit{t}\right)$$

(5)

where, $\mathit{M}$, $\mathit{C}$, $\mathit{K}$, and $\mathit{X}$ are the mass, damping, stiffness, and displacement matrices of a multi-DOF system. $\mathit{u}\left(\mathit{t}\right)$ is a random vibration from the earth, which could be considered as white noise. x(t), y(t) are the n-dimensional and m-dimensional state vectors at discrete sampling times t, respectively. n is the number of system model orders and m is the number of measurement meters. $\mathit{A}$ and B are the state and matrix of motion equation in a discrete form. $\mathit{D}$ is the observation matrix and $\mathit{E}$ is the feedthrough (or direct transmission) matrix while N is the number of samples. To obtain $\mathit{A}$, the following three matrices are constructed:

$${\mathit{U}}_{\mathit{p}}=\left[\begin{array}{ccc}\begin{array}{cc}u\left(0\right)& u\left(1\right)\\ u\left(1\right)& u\left(2\right)\end{array}& \cdots & \begin{array}{cc}u(N-p-1)& u(N-p)\\ u(N-p)& u(N-p+1)\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}u(p-2)& u(p-1)\\ u(p-1)& u\left(p\right)\end{array}& \cdots & \begin{array}{cc}u(N-3)& u(N-2)\\ u(N-2)& u(N-1)\end{array}\end{array}\right]$$

(6)

$${\mathit{Y}}_{\mathit{p}}=\left[\begin{array}{ccc}\begin{array}{cc}y\left(0\right)& y\left(1\right)\\ y\left(1\right)& y\left(0\right)\end{array}& \cdots & \begin{array}{cc}y(N-p-1)& y(N-p)\\ y(N-p)& y(N-p+1)\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}y(p-2)& y(p-1)\\ y(p-1)& y\left(p\right)\end{array}& \cdots & \begin{array}{cc}y(N-3)& y(N-2)\\ y(N-2)& y(N-1)\end{array}\end{array}\right]$$

(7)

$${\mathit{Y}}_{\mathit{f}}=\left[\begin{array}{ccc}\begin{array}{cc}y\left(p\right)& y(p+1)\\ y\left(1\right)& y\left(0\right)\end{array}& \cdots & \begin{array}{cc}y(N-2)& y(N-1)\\ y(p-1)& y(p-2)\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}y(p-2)& y(p-1)\\ y(p-1)& y(p-2)\end{array}& \cdots & \begin{array}{cc}y\left(0\right)& y\left(1\right)\\ y\left(1\right)& y\left(0\right)\end{array}\end{array}\right]$$

(8)

Based on reference [31,32], we know ${\mathit{U}}_p\approx {\mathit{O}}_{\mathit{f}}{\mathit{X}}_{\mathit{f}}+input terms$. To eliminate the influence of input terms, the following step in Equation (10) is conducted, and $\overline{{\mathit{Y}}_{\mathit{f}}}$ is obtained. Then, we perform a singular value decomposition on $\overline{{\mathit{Y}}_{\mathit{f}}}$. Solving Equations (11) and (12), $\mathit{A}$ could be obtained. ${\mathit{\lambda }}_{\mathit{i}}$ stands for the eigenvalue of $\mathit{A}$, and the corresponding eigenvectors of A are the vibration shaking modes. And from Equation (5), the natural frequency ${\omega }_i$ and damping ratio ${\xi }_i$ could be calculated as presented in Equation (13).

$${\mathit{Z}}_p=[{\mathit{U}}_p;{\mathit{Y}}_{\mathit{p}}]$$

(9)

$$\overline{{\mathit{Y}}_{\mathit{f}}}={\mathit{Y}}_{\mathit{f}}\ast \left(\mathit{I}-{{\mathit{Z}}_{\mathit{p}}}^{\mathit{T}}{\left({{{\mathit{Z}}_{\mathit{p}}\mathit{Z}}_{\mathit{p}}}^{\mathit{T}}\right)}^{\mathbf{-}\mathbf{1}}{\mathit{Z}}_{\mathit{p}}\right)=\mathit{U}\sum {\mathit{V}}^{\mathit{T}}.$$

(10)

$${\mathit{O}}_{\mathit{p}}=\mathit{U}(:,\mathbf{1},\mathit{n})\sum {(1:\mathit{n},\mathbf{1}:\mathit{n})}^{\mathbf{1}/\mathbf{2}}$$

(11)

$${\mathit{O}}_{\mathit{p}}=\left[\begin{array}{ccc}\mathit{D}& \mathit{D}\mathit{A}& \cdots \mathit{D}{\mathit{A}}^{\mathit{p}\mathbf{-}\mathbf{1}}\end{array}\right]$$

(12)

$$\frac{ln\left({\lambda }_i\right)}{∆t}={\alpha }_i+{\beta }_i\mathit{i}$$

(13)

$${\omega }_i=\sqrt{{{\alpha }_i}^2+{{\beta }_i}^2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\xi }_i=\frac{{\alpha }_i}{{\omega }_i}$$

(14)

3. Experimental Results

3.1. Vibration Modes, Frequencies, and Damping Ratios

The first three vibration modes of the full-scale specimen are shown in Figure 6. Additionally, the damping ratios and frequencies of each vibration mode are collected in

Table 3. Summary of the vibration mode shapes, frequencies, and damping ratios.

NO of Vibration Modes	Frequencies [Hz]	Damping Ratio [%]	Vibration Mode Shapes
1	11.718	3.304
2	27.314	0.855
3	43.510	0.793

. It is observed that the first frequency of this assembled integral floor system is 11.718 Hz with a damping ratio of 3.304%. The first vibration mode is symmetric, and the deflection at the middle point is greater than in the other parts. The second frequency is 27.314 Hz, with a relatively small damping ratio of 0.855%. The second vibration mode shape is antisymmetric. For the third mode, the frequency is 43.510 Hz, and the damping ratio is 0.793%.

3.2. Acceleration Response of the Single-Person Tests

The acceleration-time history curves of the specimen under single-person walking are presented in Figure 7. Even though we want to control the walking frequency at 2 Hz, the average frequency in the experiment is about 1.83 Hz. This would not influence the comfort analysis as a larger frequency will generate greater acceleration. Also, for the numerical simulation, we can directly use the actual interval of each step.

Figure 8 presents the acceleration time-history curve of V11 under single-person walking. The wave is similar to a free vibration due to the 3.304% damping ratio, and there is no obvious wave superposition. In the two back-and-forth processes, the acceleration waves are of similar shape, and the three acceleration waves are also similar. The maximum acceleration of 15 m was all less than 0.01 m/s², meeting the vibration serviceability requirements of GB 50010-2010 and ISO 2631 [33]. Under the same load, the accelerations of V2, V14, and V3 were greater than the others. The first frequency is 11.718 Hz, five times larger than the load frequency. Thus, the conclusion that no resonance exists under one-person loads could be roughly reached.

3.3. Acceleration Response of the Multi-Person Tests

The acceleration-history curves of the specimen under multi-person loads are presented in Figure 9. Compared with the time-history curves of a one-person load, the maximum acceleration of 15 m was much greater. It reached 0.15 m/s² for multi-person walking and 0.45 m/s² for multi-person walking, larger than the specification requirements of vibration serviceability of GB 50010-2010 and ISO 2631. Figure 10 presents the acceleration time-history curve of V11 under six-person walking. The waves are no longer like a free vibration, which means the six accelerations may add up in the walking process.

4. Theoretical Derivation

4.1. Simplification of Beam–Plate Connection and Boundary Conditions

To derive the analytical solution for this floor system, some simplifications or assumptions must be made. The main assumption contains two aspects, i.e., the properties of the floor, beam-plate connection, and boundary conditions:

The floor system is considered as an orthogonally anisotropic elastic shell, where the differences between the hollow part and ribbed beam part are neglected. This assumption is inherited from some previous studies on assembled floor systems.

The influence of beams is neglected. As mentioned, the beam height is 800 mm, about 1/11.5 of the span, and the distance of the stirrups is 100 mm along the beam, which is considered to be stiff enough to ignore its bending deflections under human loads. Furthermore, even though some simplifications were made, satisfactory results were still reached, demonstrating the efficiency of this simplification.

The boundary condition is idealized as fixed ends for the four sides. From Figure 3, it could be observed that there are only tensile rebars at the bottom of the beam–plate connections. According to GB 50010-2010, this design could be considered as fixed ends.

In this way, the innovative floor system could be considered as a simply supported shell on four sides. The boundary conditions are x = 0, y = 0, M = 0 at the four sides. In mathematical form, boundary conditions are presented as follows:

$$\mathrm{x}=0 \mathrm{or} \mathrm{x}=9.2 \mathrm{m} w=0, \frac{{\partial }^{2}w}{\partial x^2}=0$$

(15)

$$\mathrm{x}=0 \mathrm{or} \mathrm{y}=9.2 \mathrm{m} w=0, \frac{{\partial }^{2}w}{\partial x^2}=0$$

(16)

4.2. Natural Frequency and Vibration Modes

In reference [16,17], the static performance of this assembled integral two-way multi-ribbed composite floor system could be divided into elastic, elastic–plastic, and damaged stages. Compared to the live loads of the elastic limit, the pedestrian loads were minimal. Therefore, the elastic mechanics method could be used in this section to calculate the vibration modes and frequency. The equations controlling its vibration are shown as follows:

$$D\left[\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}\right]+m\frac{{\partial }^{2}w}{\partial t^2}=0$$

(17)

where E is the modulus of the concrete, w is the out-of-plane deflection of the floor system, y is the in-plane vertical coordination of the floor system, x is the in-plane horizontal coordination of the floor system, D is the bending stiffness of the floor system, and m is the average mass of the floor system. Equation (14) could be transformed to a fourth-order partial difference equation. The actual stiffness of this full-scale 9.2 m × 9.2 m integral two-way multi-ribbed composite floor system can be determined from static experiments.

Using separation of variables in the form $w\left(x,t\right)=\sum _{i=1}^N{\varphi }_i(x,y){\theta }_i\left(t\right)$, where ${\theta }_i$ is only related to t, ${\varphi }_i$ is the shape function only related x and y. Equation (17) could be transformed to Equation (18). In Equation (18), only when $D\left[\frac{{\partial }^4}{\partial x^4}+2\frac{{\partial }^4}{\partial x^2\partial y^2}+\frac{{\partial }^4}{\partial y^4}\right]{\varphi }_i$ and $m\frac{{\partial }^2}{\partial t^2}{\theta }_i$ equal a same constant, this equation could be solved. Introducing this constant as ${{\omega }^2}_i$, we then get Equation (19). Solving Equation (19), the shape function and natural frequencies could be obtained as Equations (20) and (21).

$$D\left[\frac{{\partial }^4}{\partial x^4}+2\frac{{\partial }^4}{\partial x^2\partial y^2}+\frac{{\partial }^4}{\partial y^4}\right]{\varphi }_i(x,y)+m\frac{{\partial }^2}{\partial t^2}{\theta }_i=0$$

(18)

$$D\left[\frac{{\partial }^4}{\partial x^4}+2\frac{{\partial }^4}{\partial x^2\partial y^2}+\frac{{\partial }^4}{\partial y^4}\right]{\varphi }_i\left(x,y\right)=-m\frac{{\partial }^2}{\partial t^2}{\theta }_i={{\omega }_i}^2$$

(19)

$$W_{ij}\left(x,y\right)=A_{ij}sin\frac{i\pi x}{a}sin\frac{i\pi y}{b}$$

(20)

$$f_{ij}=\frac{\pi }{2a^2}\left(i^2+j^2\right)\sqrt{\frac{D}{m}}$$

(21)

where, $W_{ij}\left(x,y\right)$ is the vibration mode and $f_{ij}$ is the frequency, i and j indicate the number of half-waves in the x and y direction.

It can be found from the figures above that the vibration mode shapes of the analytical and experimental results are almost identical (see Figure 11, Figure 12 and Figure 13). The absolute errors of the first three modes are 4%, 11%, and 122%, respectively, as collected in

Table 4. Comparison of experiment, simulation, and GB50010 results.

Frequencies [Hz]	First Order	Errors [%]	Second Order	Errors [%]	Third Order	Errors [%]
Test	11.718	-	27.314	-	43.510	-
Analytical method	12.20	+4.11%	30.49	+11.63	48.78	+12.11%

. It can be concluded that Equations (17) and (18) could be used to predict the frequencies and vibration mode shapes of this assembled integral two-way multi-ribbed composite floor.

4.3. Calculation of Acceleration Under Human Loads

Since the equations in Section 4.1 prove that the vibration mode shapes and frequencies could be well predicted by elastic theory, the vibrations under human loads could also be calculated. Taking into consideration the human loads, the vibration equations could be revised as Equation (8). The human load is modeled by using a single-footfall force loading on the footprints [34,35]. Assuming the foot force is exerted on the spot (x_i, y_i), then the single-footfall force $F\left(t\right)$ could be expressed as:

$$D\left[\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}\right]+m\frac{{\partial }^{2}w}{\partial t^2}+C\frac{\partial w}{\partial t}=F(t)\delta (x-x_i)\delta (y-y_i)$$

(22)

$$F\left(t\right)=G{\sum }_{k=1}^3{B}_k\sin \left(\frac{k\pi }{t_e}t\right)$$

(23)

$$B_1=-\frac{0.0698}{t_e}+1.211; B_2=\frac{0.10528}{t_e}-0.1284; B_3=\frac{0.3002}{t_e}-0.1534$$

(24)

where $F\left(t\right)$ is the force from a single footfall, $\delta$ is the Dirac function, G is the weight of a human body, B_k is the Fourier coefficient, f_s is the walking step rate, and T_e is the duration of a single footstep, while C is the damping matrix.

When the step frequency equals 2 Hz, and T_e is 0.5 s, the single-footfall force could be obtained, as illustrated in Figure 14. The curve of the single-footfall force is characterized by two peaks, which represent the moments that the first-foot contacts the floor with “heel strike” and the other leaves with “toe off”. There is an overlap between the two contacts of “heel strike” and “toe off” during the walking process. Thus, the duration of the single-footfall T_e will be greater than the period of human walking t_s = 1/f_s. The loading scheme considering the overlap periods is shown in Figure 15 [36,37].

In the multi-person load calculation, the same single-footfall model in Figure 14 is assumed for each volunteer. In the experiment, six people are walking at the same speed. Thus, the multi-person load could be expressed as the six single-footfall forces. However, it is necessary to note that the simulation using the multi-person load above is not so satisfactory. It should belong to the simulation method of the crowd load, which will be studied in the following article.

By using the single-footfall curve, the human load could be simplified to 14 point loads with an interval of 0.54 s. Experimental results show that the damping ratio is 3%, and the first frequency of the floor system is 11.72 Hz, five times larger than the human-load frequency. Thus, it can be assumed that the energy of each step could be partly dissipated by the free vibration of the floor system at the interval of two steps. In other words, each step could be seen as a single impact force on the floor. This assumption could also be proven through the time–acceleration history curves from the vibration tests.

Equation (22) is solved numerically by the finite difference method (FDM); the discretized forms are presented as Equation (25)–(29) and solved by the Newton–Raphson method.

$$\frac{{\mathit{\partial }}^{4}w}{\partial x^4}=\frac{w_{i+2,j}-4w_{i+1,j}+6w_{i,j}-4w_{i-1,j}+w_{i-2,j}}{{∆x}^4}$$

(25)

$$\frac{{\partial }^{4}w}{\partial y^4}=\frac{w_{i,j+2}-4w_{i,j+1}+6w_{i,j}-4w_{i,j-1}+w_{i,j-2}}{{∆x}^4}$$

(26)

$$\frac{{\partial }^{4}w}{\partial x^2\partial y^2}=\frac{w_{i+1,j+1}-2w_{i+1,j}+w_{i+1,j}-2w_{i,j+1}+4w_{i,j}-2w_{i,j-1}+w_{i-1,j+1}-2w_{i-1,j}+w_{i-1,j-1}}{{∆x}^2{∆y}^2}$$

(27)

$$\frac{{\partial }^{2}w}{\partial t^2}=\frac{w_{t+1}-2w_t+w_{t-1}}{{∆t}^2}$$

(28)

$$\frac{\partial w}{\partial t}=\frac{w_{t+1}-w_t}{∆t}$$

(29)

The acceleration–time history curves of single-person loads are presented in Figure 16 and compared with the experimental results. Since the layout of all acceleration meters is symmetric, only the acceleration time–history curves of V11, V9 and V13 are illustrated in Figure 16. It was found that they are of similar shape under single-person walking and running, and the peak acceleration calculated by the method above is about 0.0081 m/s². The single-person peak acceleration obtained from the vibration test is 0.075 m/s², where the absolute error is less than 8%.

According to JGJ 3-2010 “Technical specification for concrete structures of tall buildings” [38], the maximum acceleration of floor systems can be calculated by Equation (30), where g is the gravity acceleration and $F_p$ is the contact force calculated by Equation (22). The maximum acceleration predicted by JGJ 3-2010 is 0.0295 m/s² with an absolute error of 60%. This specification is used for cast-in-situ concrete floor systems, which are stiffer than this innovative floor system. Furthermore, the damping ratio for cast-in-situ concrete floor systems is assumed to be 0.05, while the test indicates the first-order damping ratio for this floor is 0.033, which is also a main reason for the larger accelerations.

$${\mathrm{a}}_{max}=\frac{F_p}{{\xi }_{i}m}g$$

(30)

4.4. Parametric Analysis

When calculating the natural frequency, it is evident that the bending stiffness D is very important. In our experimental study, it is straightforward to obtain D through a static experiment. When applied to other floors with different sizes, the finite element method should be employed to determine the bending stiffness of the innovative floor system.

Based on the framework above, influence factors, such as damping ratio, single-person weight, step frequency and bending stiffness, are studied, and details of these parameters are illustrated in

Table 5. Parametric analysis of single-person loads.

Stepping Frequencies [Hz]	Weight of a Single Person [N]	Damping Ratio [%]	Bending Stiffness [D]
1.25	600	3	0.9
1.5	700	4	1.0
1.75	800	5	1.1
2	900	6	1.2

. The relationships between these parameters and the maximum accelerations are illustrated in Figure 17.

It is observed that the weight and stepping frequency on the maximum acceleration is linear, while the influence of the damping ratio and bending stiffness is nonlinear. Increasing the bending stiffness and the damping ratio can both decrease the vibration of this innovative floor system. But increasing the bending stiffness always means a thicker plate or higher beam, which would increase the costs of construction. Thus, it is recommended that some construction strategies be adopted to enhance the damping ratio and control the vibration of this innovative floor system.

5. Discussion and Further Application

5.1. Error Discussion

Even though satisfactory results were reached, there are still some errors in the numerical simulation. The errors mainly come from two aspects, which are the incorrect prediction of single-person loads and the calculation error of the bending stiffness from the theoretical derivation.

For the first error, the single foot force–time curve is presented in Figure 14, where the point loads from each step are considered as fixed values. In real cases, the foot force depends on the height of each leg lifted while walking, which makes it unable to be captured by a fixed function.

For the latter one, the error mainly comes from two aspects: (1) over-simplification of boundary conditions; the beam around the four sides should be taken into consideration for more accurate prediction; (2) the whole floor system is simulated with a uniform shell, which cannot fully capture the different vibrations when stepping on the ribbed beam or the hollow lightweight infills.

5.2. Further Application

The method proposed in this article could be applied not only to predict the natural frequency and maximum acceleration for similar assembled integral floor systems, but also to walls and decks with ribbed beams. But the key problem is to check if these components could be simulated by a uniform plate with fixed ends, especially the interface behavior between the precast panels and cast-in-situ parts.

Design parameters, such as the thickness of the PRB, height and distance of the ribbed beam, are the key factors that directly change the bending stiffness of this innovative assembled integral two-way multi-ribbed composite floor system. With the increase in thickness of the PRB and the height of the ribbed beam, the frequency would increase correspondingly due to a larger bending stiffness, and the maximum acceleration would decrease. However, in this paper, we simulate the floor system as a uniform shell with fixed ends, where the bending stiffness could be obtained from static tests. When applied to other spans or different designs, the relationship between the bending stiffness and design parameters is not clear. A 3d finite element method should be the potential method, but there might be too many solid elements slowing down the calculation efficiency and precision. Thus, a simplified model is needed, and this study will be the key part of our next article.

The damping ratio is the key to controlling the vibration of the floor system with the same bending stiffness. For this type of assembled integral two-way multi-ribbed composite floor system, the ribbed beam distance and thickness of the whole floor system would significantly enhance the stiffness, through which the maximum acceleration could also be controlled. Considering the costs, it is recommended that some construction strategies be adopted to enhance the damping ratio and control the vibration of this innovative floor system.

For the same vibration tests, it is recommended to put acceleration meters at both the top and bottom of the test specimen. Furthermore, it is recommended that more volunteers than the design code requires are needed, since the relationship between the maximum acceleration and multi-person load is nonlinear. Single-person, multi-person walking and running should also be a necessary choice for further study of the dynamic behaviors of floor systems.

6. Conclusions

This paper presented the dynamic performances of a 9.2 m × 9.2 m assembled integral two-way multi-ribbed composite floor system, both experimentally and theoretically. The natural frequency, vibration mode shapes, and acceleration–time history curves were analyzed, and simplified formulas were derived to predict their dynamic performances. The main conclusions are presented as follows:

• The dynamic behavior of this innovative floor system meets the requirements of GB50010-2010 and ISO 2631. The first-order vibration frequency of the 9.2 m × 9.2 m floor is 11 Hz, larger than 3 Hz. The maximum acceleration for a single person is 0.07 m/s², smaller than the limited value of 0.35 m/s². For multiple persons, the maximum acceleration is 0.45 m/s².
• Elastic plate theory could be applied to predict the natural frequency and acceleration, with the bending stiffness obtained from the experiment. The errors of elastic vibration theory in calculating the first three vibration mode shapes and natural frequencies are approximately 10%. And the absolute error of the acceleration time–history curve is less than 10%, demonstrating its efficiency in predicting the vibration behaviors of this innovative assembled integrated two-way multi-ribbed composite floor system.
• The bending stiffness and damping ratio are the key factors influencing the vibration acceleration of this innovative floor system. Considering the costs, it is recommended that some construction strategies be adopted to enhance the damping ratio and control the vibration of this innovative floor system.
• For similar vibration tests, it is recommended to put acceleration meters at both the top and bottom of the test specimen. Furthermore, it is recommended that more volunteers than the design code requires are needed, since the relationship between the maximum acceleration and multi-person load is nonlinear.