Performance of Deltabeam–CLT Composite Floors Under Human-Induced Vibration

Abstract

In this study, the human-induced vibration performance of an innovative Deltabeam–CLT composite floor system was investigated. The Deltabeam–CLT composite floors were modelled using the finite element software package ABAQUS 6.14, the dynamic performance of the composite floors was evaluated, and the simulation results were benchmarked against prior simulation data in the other literature. In order to investigate the effect of the stiffness of Deltabeam composite steel beams on the vibration performance of Deltabeam–CLT composite floors, the acceleration of the Deltabeam–CLT composite floors under the excitation of a single person walking was measured. As per the acceleration, serviceability factors, namely, response factors (RFs) and vibration dose values (VDVs), were analyzed. The results showed that an increase in the height of Deltabeam composite steel beams can decrease the RF and VDV by 53.7% and 57.7%, respectively. This study also developed an optimization scheme with steel trusses pre-embedded inside Deltabeam composite steel beams. The effects of different truss spacing and rod diameters on the vibration performance of the Deltabeam–CLT composite floors were investigated; the results showed that the RF and VDV of Deltabeam–CLT composite floors can decrease to below 4 and 0.4 m/s^−1.75, respectively, with a truss spacing smaller than 200 mm. These findings can significantly enhance the future design of the Deltabeam–CLT composite floor system, improving its serviceability and ensuring better performance in practical applications.

1. Introduction

In recent years, cross-laminated timber (CLT) (Figure 1a) has gained significant global attention as a leading engineered timber product. Its orthogonal laminated structure not only enhances overall structural stability but also simplifies and expedites the construction process [1]. Furthermore, CLT is recognized as a sustainable, low-carbon building material capable of serving as full-scale structural components that support both in-plane and out-of-plane loads [2]. Its application is particularly advantageous in the context of prefabricated floor panels [3]. Prefabricated CLT floor panels are characterized by their rapid assembly and efficiency. Therefore, they are often combined with steel support beams to create composite steel–timber floor systems [4]. The use of steel beams significantly enhances the load-bearing capacity of composite floor systems and reduces the elastic deflection of long-span floor panels [5,6,7,8]. However, traditional I-shaped steel beams have a significant sectional height, which can result in increased overall structural thickness and reduced floor-to-ceiling height. Therefore, it is necessary to reduce the building’s story height [9]. In this context, a novel shallow floor system known as the Deltabeam–CLT composite floor is gaining increasing application.

Deltabeam is a product developed by Peikko Group Corporation and is widely recognized in the industry as an effective and innovative solution for supporting CLT floor systems. This research focuses on Deltabeam with the aim of improving its performance under human loads and proposing new ideas to enhance its properties. The findings are expected to offer valuable insights for the industry and contribute to ongoing discussions for improvement in this field. Deltabeam is a lightweight composite beam for floor systems, constructed from multiple welded steel plates [10]. It derives its name from its distinctive triangular hollow cross-section (Figure 1b). The web of this steel beam features numerous circular holes, which facilitate easier passage of transverse connections between the floor and the beam, thereby providing excellent continuity for the floor system [11]. Prior to the concrete pouring on the construction site, the Deltabeam will act as an independent steel beam to support any loads that may be encountered during the construction phase. After the bottom steel plate is installed, the hollow sections within the Deltabeam can be filled with concrete through the pouring holes located at the top of the beam. Once the concrete has fully cured, the Deltabeam integrates with the concrete, forming a composite steel beam that works collaboratively with the concrete.

Assembling Deltabeam composite steel beams with CLT floor panels to create a composite floor system (Figure 2) represents a highly cost-effective and efficient solution. In this composite structure, the floor panels and beams are at the same height. This configuration not only allows for the timber floor panels to span greater distances horizontally but also reduces the number of columns required, resulting in more open space. Furthermore, due to its specific beam-to-slab positioning, this arrangement can reduce the overall building height by 10% to 30% in the vertical direction [12]. Additionally, the Deltabeam–CLT composite floor system offers adequate fire resistance. The timber floor panels, equipped with sufficient transverse connections, are capable of safely transferring loads to the Deltabeam composite beam even in the event of a fire [13].

The Peikko Group has conducted research on the structural performance of the Deltabeam–CLT composite floor system. Elena Camnasio [14] conducted load transfer experiments on ten CLT floor panels supported by Deltabeams and proposed design recommendations for the connections in the Deltabeam–CLT composite floor system. Salla-Mari West and Simo Peltonen [15] conducted experimental assessments on the load transfer capacity, degree of carbonation, and load-bearing capacity of the Deltabeam–CLT composite floor system under fire conditions.

The vibration of floors is typically induced by human activities such as walking and running. Numerous researchers have previously investigated the vibration performance of various types of floor systems under human-induced vibration. Chen et al. [16] conducted a numerical study using the finite element method to investigate the vibration behavior of long-span steel–concrete composite floors under human-induced excitations. The results indicated that long-span floors are more prone to the excitation of higher-order vibration modes during walking or rhythmic human activities. Zhang et al. [17] investigated the impact of screw size and spacing at the beam-to-slab connections on the vibrational performance of CLT floor panels under multiple load conditions through static cyclic tests and numerical simulations. The results indicate that increasing the spacing of screws at the beam-to-slab connections can significantly enhance the vibrational serviceability of CLT floor panels. Huang et al. [18] investigated human-induced vibrations in CLT floor panels under various boundary conditions. Their study found that the natural frequency and vibration acceleration of CLT floor panels are closely related to the spacing, stiffness, and support conditions of the beams. Stevanovic [19] conducted a comprehensive finite element analysis of beam-supported timber floor under human-induced loads and provided design recommendations for such systems. Huang et al. [20] investigated the human-induced vibration performance of an innovative hollow-core cross-laminated timber (HC-CLT) system. Hassanieh et al. [21] utilized numerical simulations to predict the natural frequencies of long-span steel–timber structures and conducted a parametric analysis to examine the effects of floor panel geometry, connection node mechanical performance, and superimposed load magnitude on the natural frequencies of composite steel–timber floors.

The Deltabeam–CLT composite floor system has been applied in numerous engineering projects across Europe. However, research on its vibration serviceability under human-induced excitations remains scarce. Therefore, this study is both necessary and timely. Jaakko Yrjölä and Juuso Salonen²⁴ employed the finite element software package Robot Structural Analysis Professional to study the vibrational serviceability of Deltabeam–CLT floor systems with different component dimensions. The results indicated that a significant proportion of the response factors of Deltabeam–CLT floor systems exceeded 4, which is the limit for residential and general office floors as specified in ISO 10137 [22]. The vibration dose value, which is used to evaluate intermittent vibrations, remains unknown for the Deltabeam–CLT floor system. Overall, a comprehensive study is needed to thoroughly understand the vibration behavior of the Deltabeam–CLT composite floor system under footfall and then improve it.

To address the potential vibrational serviceability in Deltabeam–CLT composite floors that may arise from insufficient stiffness of the composite steel beams, this study aims to explore the vibrational serviceability of Deltabeam–CLT composite floors under pedestrian loads. By investigating the impact of Deltabeam composite steel beam height on the vibrational performance of the composite floor system, this study seeks to provide valuable insights for future design improvements. In this study, a finite element model of the Deltabeam–CLT composite floor system was developed using the finite element software package ABAQUS, and the model was benchmarked against simulation data in the other literature. Through numerical simulations, this study assessed various parameters, including the impact of Deltabeam composite steel beam height on the dynamic characteristics, acceleration response, and vibrational serviceability of the Deltabeam–CLT composite floor system. Based on the analysis results, an optimization was proposed to improve the vibrational performance of the composite floor system by embedding a steel truss within the Deltabeam composite steel beams. This approach retains the advantage of Deltabeam composite steel beams in reducing building height while also reducing the floor’s vibrational response under human-induced loads.

2. Numerical Simulation of the Deltabeam–CLT Composite Floor System Under Human-Induced Loads

2.1. Material Properties and Floor Panel Information

This study uses the finite element software package ABAQUS 6.14 to simulate the vibrational performance of a Deltabeam–CLT composite floor under human-induced loads. For CLT floor systems, vibration performance is typically influenced by factors such as span length, material properties, mass, and boundary conditions [18]. In this study, the simulation models were designed based on these key parameters. Figure 3 illustrates the arrangement of the Deltabeam–CLT composite floor. Since the focus of this study is the Deltabeam, which typically has a standard height of 200 mm, it is designed to be flush with the CLT floor slab. A composite floor panel with a total thickness of 200 mm was designed for this study, consisting of five-ply CLT floor panels with the following configuration: 40L-40T-40L-40T-40L. This configuration serves as the numerical model for the simulations conducted in this research. To facilitate comparison with data from the existing literature [23], the longitudinal and transverse spans of the CLT floor panels in this study are both 7 m. The floor is supported by composite steel beams made of Deltabeam and concrete on both sides. Figure 4a illustrates the transverse connection method between the Deltabeam composite steel beams and the CLT floor panels in practical engineering [14]. The connection is achieved by cutting grooves along the edges of the floor panels, placing transverse reinforcement within the grooves, and then casting concrete into the grooves to integrate the composite steel beams with the CLT floor panels. Figure 4b depicts the connection method between the Deltabeam and the columns [9]. This connection is achieved through bolts and welded joints at the ends of the column and beam. The bolts and welded joints at both end of the components interlock to form a bearing connection. In this configuration, the vertical members of the composite structure do not exhibit significant bending effects, and only the reaction forces in the vertical direction are effectively transferred.

The numerical model of the Deltabeam–CLT composite floor system in this study consists of three distinct components: the concrete sections on both sides, the Deltabeams on both sides, and the CLT floor panel in the center. Considering that all components of the composite floor are in an elastic state during normal use, only the elastic properties of each material are defined. Conventional parameters are used for both steel and concrete: the density of steel is set to 7800 kg/m³, the elastic modulus to 210 GPa, and the Poisson’s ratio to 0.3. For concrete, the density is set to 2400 kg/m³, the elastic modulus to 31.5 GPa, and the Poisson’s ratio to 0.2. The CLT floor panels are modeled using orthotropic timber materials, specifically spruce timber, with material parameters referenced from the relevant literature [21], as detailed in

Table 1. Material properties of CLT panels in the finite element model.

Density(kg/m3)	Elastic Modulus (MPa)			Poisson’s Ratio			Shear Modulus (MPa)
	E11	E22	E33	ν12	ν13	ν23	G12	G13	G23
500	11,600	730	530	0.59	0.57	0.24	610	860	120

. For the Deltabeam composite steel beams, the model was selected considering the total thickness of the CLT floor panel, which is 200 mm. The D20-400 model of the Deltabeam composite steel beam [23] was chosen; it has a height of 200 mm, which aligns with the thickness of the designed CLT floor panel. The damping ratio for the floor system is set to 2% in the software program [24].

2.2. Finite Element Model Development

In this study, the finite element software program ABAQUS 6.14 was used to perform numerical simulations of the Deltabeam–CLT composite floor system [25]. To account for the need to leave a certain vertical space at the contact surfaces between the composite steel beams and the CLT floor panels to simulate the transverse connections, three-dimensional solid reduced integration (C3D8R) elements [26] were employed for modeling the concrete portions, the Deltabeams, and the CLT floor panels. Reduced integration elements have one less integration point in each direction compared to fully integrated elements, which significantly reduces computational cost while having minimal impact on accuracy. The average mesh size of the finite element model is 75 mm (Figure 5a). To ensure the accuracy of modal and time history analysis results for the composite floor model, a sensitivity analysis of the mesh size was conducted. Extensive preliminary simulations were performed with mesh sizes of 150 mm, 100 mm, 75 mm, and 50 mm. The results indicated that the Deltabeam–CLT composite floor model was not highly sensitive to mesh sizes between 50 mm and 100 mm, with minimal differences in computed results. Unlike conventional modal analysis alone, this study extends to evaluating serviceability criteria such as VDV, which requires precise time history response predictions. Given that footfall-induced forces are much smaller in magnitude than seismic or impact loads and are applied at multiple, spatially discrete, and time-varying contact points, the simulation must be sufficiently detailed to capture these subtle but critical dynamic behaviors. The application of footfall loading is complex. It is based on measured gait cycle data and involves short-duration, non-uniform impulses at varying spatial intervals. A finer mesh helps ensure that these transient load effects are appropriately captured in the structural response simulation. The first natural frequency obtained was approximately 8.3 Hz. Considering computational time constraints, a mesh size of 75 mm was chosen. Additionally, to improve overall calculation accuracy, finer mesh refinement was applied at the holes in the Deltabeam web (Figure 5b).

Due to the different grain orientations of each layer in cross-laminated timber (CLT) panels, the CLT floor model was divided into five layers to ensure that the stress directions of each layer’s elements are orthogonal at 90°. The transverse connections between the CLT floor and the composite steel beams were simulated using spring connector elements in ABAQUS (Figure 5c,d). The spacing of each spring element was set to 300 mm, based on the actual spacing of the transverse reinforcement [27]. For the stiffness of the spring unit, an assumption is applied: the connection between the Deltabeam and the CLT floor slab as rigid; thus, an extremely high modulus of elasticity was assigned. This assumption is supported by a previous study [18]. In that study, the beam–CLT panel connection was modelled as rigid because the connection’s shear stiffness was significantly higher than the rotational and flexural stiffnesses of the beam and the floor. Similarly, in both that work and the current study, the assumption was suitable. Also, the focus of both studies is on the influence of the beam and CLT floor on vibration performance; therefore, assuming a rigid connection is reasonable. The influence of connection flexibility can be explored in future investigations. Currently, we lack the specific mechanical properties of the actual connections. Accurate stiffness values can only be determined through dedicated experimental tests, such as in [17]. Future experimental tests are planned to verify the connection stiffness and refine the modelling approach. Rigid connection modelling has also been adopted in previous studies on floor vibrations, where nodal degrees of freedom were coupled to simulate full interaction and deformation compatibility [16,28,29,30]. Given that human-induced vibrations remain within the elastic range, the assumption is appropriate for the current study.

Regarding boundary conditions, the CLT floor panels are assumed to be continuous along their longitudinal edges, while the ends of the Deltabeam composite steel beams are constrained in three degrees of freedom to simulate the connection between the beam and the column. The deformations at the beam–column connections due to floor vibrations are neglected in this study. The interaction between the CLT floor panels and the concrete portions of the Deltabeam composite steel beams is simulated using surface-to-surface contact settings in ABAQUS. A finite sliding model is used to represent this interaction, with hard contact defined along the normal direction of the contact surfaces. Since the impact of natural bonding and friction on the overall vibration of the composite floor is minimal, the coefficient of friction in the tangential direction is set to 0. The interaction between the Deltabeams and the concrete is modeled using embedded constraints in ABAQUS. Embedded constraints allow for specifying one or a set of elements to be constrained to the primary element, where the translational degrees of freedom of the embedded elements are constrained by the primary element, while rotational degrees of freedom of the embedded elements are not constrained by the primary element [26].

2.3. Modal Analysis

The purpose of modal analysis is to determine the structural mode shapes and eigenfrequencies. Resonance may occur when the eigenfrequencies of the structure align with the frequencies of external excitations, potentially leading to increased structural vibrations. In this study, modal analysis of the Deltabeam–CLT composite floor system was conducted using the frequency analysis step in the linear perturbation module of ABAQUS.

Based on the results of the modal analysis from the numerical simulations, the eigenfrequencies of the Deltabeam–CLT composite floor system under vertical vibrations are presented in

Table 2. Modal analysis of the composite floor from numerical simulations.

	Eigenfrequency (Hz)	Modal Mass (kg)	Damping Ratios
First-order mode	8.28	3850	2%
Second-order mode	10.44	2675	2%
Third-order mode	17.25	1245	2%

. Figure 6 illustrates the first three mode shapes of the composite floor units. The first mode is a bending mode (Figure 6a), the second mode is a torsional mode (Figure 6b), and the third mode is a transverse bending mode (Figure 6c). Here, the modal vibration diagrams from a previous experimental study on a CLT floor slab are shown in Figure 7 [31]. Figure 7a–c present the first, second, and third mode shapes, respectively. The comparison indicates that the mode shape patterns in our study closely match those observed in the prior study. This consistency provides supporting evidence for the validity of our modal analysis.

2.4. Model Validation

This study verifies the feasibility of the Deltabeam–CLT composite floor system finite element model presented in Section 2.1 by benchmarking it against data simulated by other finite element models in the literature. J. Yrjölä et al. [23] used the structural analysis software package Robot Structural Analysis Professional 2016 to construct a spatial floor model consisting of Deltabeam composite steel beams and CLT floor panels. The model included 18 CLT floor panels and 12 Deltabeam composite steel beams, with each beam spanning 7 m and the CLT floor panels having a length of 7 m, a transverse width of 3.5 m, and a thickness of 320 mm. The D50-600 model of Deltabeam was used for the steel beam elements, while the CLT floor panels were modeled as plate elements with constant thickness and orthotropic material properties. Figure 8a shows the model layout from the referenced study. We constructed an identical spatial floor model in ABAQUS, as shown in Figure 8b. All components and their interactions in the model were created using the methods described in Section 2.1. Modal analysis of the ABAQUS model resulted in a natural frequency of 9.151 Hz, which is in close agreement with the natural frequency of 9.189 Hz obtained from J. Yrjölä’s model. Additionally, the first mode shapes of both models were compared, with the comparison shown in Figure 9a,b. The first mode shapes were found to be identical, confirming the validity of the numerical simulation model used in this study. It is also noted that to reduce computational time for dynamic analysis in ABAQUS, the finite element model for the Deltabeam–CLT composite floor system proposed in Section 2.1 uses region A (7 m × 7 m), as shown in Figure 8b, for the design.

2.5. Dynamic Loading Methods

In this study, the external force causing floor vibrations is the footfall force generated by a person running. Figure 10 illustrates the loading method for pedestrian moving loads in the numerical simulation [18]. The movement path, marked on the CLT floor in Figure 3, indicates that the person runs from one end of the Deltabeam–CLT composite floor to the other end, covering a total of 12 steps. The human-induced loading curve in Figure 10 is based on footfall force measurement data and characteristics proposed by Galbraith and Barton [32] and Thelandersson and Larsen [33], where each footfall is modeled with two peak values. The first peak corresponds to the impact of the heel on the floor surface, while the second peak corresponds to the contact of the toes with the floor surface. The amplitude of the second peak is approximately 2.1 kN, simulating the impact force generated at the point of footfall when a person running under a gravity force of about 1 kN contacts the floor [34].

In ABAQUS, dynamic analysis methods include explicit and implicit approaches. The explicit method is commonly used for solving short-term dynamic problems, while the implicit method is suited for static and long-term dynamic problems. Since the loading time in this study includes the entire process of moving from one end of the composite floor to the other, the implicit dynamic analysis step was chosen for simulating pedestrian loads. The pedestrian load simulation was performed by creating 12 analysis steps in ABAQUS. As shown in Figure 10, twelve points were sequentially arranged along the running path on the floor, with each point spaced 0.6 m apart to represent a step length of 0.6 m during running. The loading duration for each analysis step was set to a constant value to simulate the actual running frequency. Finally, loads were applied sequentially at these 12 points in chronological order, and the vertical acceleration at the center of the CLT floor was measured for subsequent data analysis.

3. Parametric Study of Deltabeam–CLT Composite Floor System

3.1. Theoretical Analysis

The three components of the Deltabeam–CLT composite floor system are all located in the same plane, allowing for the composite floor system to be modeled as a series of coupled springs (Figure 11). The first natural frequency of the Deltabeam–CLT composite floor is primarily influenced by the system’s first-mode modal mass and first-mode modal stiffness, with first-mode modal stiffness being positively correlated with the stiffness of the beams and the floor plate. Therefore, increasing the stiffness of the beams can enhance the overall natural frequency of the composite floor system.

3.2. Simulation Scheme

In practical engineering, increasing the height is the most direct and effective method to enhance the stiffness of Deltabeam composite steel beams. Therefore, this study conducted a parametric investigation into the impact of Deltabeam height on the vibration performance of Deltabeam–CLT composite floors. The testing schemes are summarized in

Table 3. Simulation schemes.

Test No.	DELTABEAM® Profiles	Thickness of CLT (mm)	Height of DELTABEAM® (mm)
No. 1	D20-400	200	200
No. 2	D22-400		220
No. 3	D25-400		250
No. 4	D30-400		300
No. 5	D32-400		320
No. 6	D37-400		370
No. 7	D40-400		400
No. 8	D50-600		500

. In all schemes, the thickness of the CLT floor is kept constant at 200 mm. Test No. 1 serves as a baseline reference, utilizing the D20-400 model 12 of the Deltabeam, which has a height of 200 mm, equal to the floor thickness. Figure 12 illustrates the Deltabeam composite steel beams with increased heights. In Test Nos. 2 through 8, the height of the Deltabeams is incrementally increased. Using the human-induced load application method from Section 2.5, the simulation continuously measures the vertical acceleration at the center of the CLT floor with a sampling frequency of 1000 Hz for a duration of 13 s.

3.3. Results Analysis

3.3.1. Natural Frequency Analysis

In ABAQUS, modal analysis was conducted for the eight test scenarios described.

Table 4. The first natural frequency of a Deltabeam–CLT composite floor.

Beam height (mm)	200	220	250	300	320	370	400	500
First natural frequency (Hz)	8.28	8.41	8.96	9.35	9.44	9.68	9.76	10.19

presents the first-order natural frequencies of the composite floor system corresponding to different heights of the Deltabeam composite steel beams. As the height of the Deltabeam composite steel beam increases from 200 mm to 500 mm, the first natural frequency of the Deltabeam–CLT composite floor rises from 8.28 Hz to 10.19 Hz, representing an increase of approximately 23.07%. The first natural frequency limit of the timber floor in Eurocode 5 is 8 Hz [24], and resonance may occur in timber floor with natural frequency less than 8 Hz. The first natural frequency of the Deltabeam–CLT composite floor obtained from the modal analysis increases from 8.28 Hz to 10.19 Hz, which shows that increasing the height of the beams to 500 mm reduces the risk of resonance in the timber floor. This suggests that increasing the height of the Deltabeam composite steel beam enhances the beam’s stiffness, thereby improving the overall effective flexural stiffness of the Deltabeam–CLT composite floor.

3.3.2. Time History Analysis

Figure 13 presents the time history curves of vertical acceleration recorded at the center of the Deltabeam–CLT composite floor for Test Nos. 1, 2, 4, 7, and 8, with Test No. 1 serving as the baseline for comparison with the other four test schemes. As shown in Figure 13a, the difference between Test Nos. 1 and 2 is minimal, which may be attributed to the similar beam heights in both test schemes, resulting in little variation in the composite floor’s acceleration response. However, as the height of the Deltabeam composite steel beam increases further, as illustrated in the comparisons between Test Nos. 1 and 7 and 8 in Figure 13c,d, there is a noticeable overall reduction in vertical acceleration at the center of the composite floor.

Figure 14 shows the peak acceleration for Test Nos. 1 through 8. As the height of the Deltabeam composite steel beam increases from 200 mm to 500 mm, the peak acceleration at the center of the composite floor decreases from 1.568 m/s² to 0.919 m/s², a reduction of 41.39%. This indicates that the height of the Deltabeam composite steel beam is positively correlated with the vibration performance of the composite floor. A greater beam height results in increased stiffness, which in turn reduces the vibration response of the composite floor.

3.3.3. RF and VDV Analysis

This study uses two indicators, vibration dose value (VDV) and response factor (RF), to evaluate the serviceability of the Deltabeam–CLT composite floor. The RF has long been used as a design standard for concrete and steel structures and is included in design codes such as BS 6472 [35] and ISO 10137 [22]. The calculation method for RF is provided in Equation (1).

$$R={\displaystyle \frac{a_{w,RMS}}{a_{R-1}}}$$

(1)

In the equation, a_w,RMS represents the frequency-weighted root mean square acceleration, which can be determined using the frequency weighting curves proposed in BS 6472 (Figure 15a). a_R−1 represents the reference impact value, which can be determined from Figure 15b.

RF primarily evaluates the vibration response under continuous vibration, with specific limits, called multiplying factors, shown in

Table 5. Multiplying factors specified in ISO 10137 for low probability of adverse comment [22].

Place	Time	Multiplying Factor for Exposure to Continuous Vibration 16 h Day 8 h Night	Impulsive Vibration Excitation with up to 3 Occurrences
Critical working areas (e.g., hospital operating theatres)	Day	1	1
	Night	1	1
Residential	Day	2 to 4	60 to 90
	Night	1.4	20
Office	Day	4	128
	Night	4	128
Workshops	Day	8	128
	Night	8	128

.

VDV primarily evaluates the vibration response under intermittent vibration. It is incorporated into the assessment standards of BS 6472 and ISO 10137. The VDV can be calculated from the floor’s acceleration time history response after applying frequency-weighted filtering:

$$VDV={[{\int }_0^T a_w^4\left(t\right)dt]}^{{\displaystyle \frac{1}{4}}}$$

(2)

In the equation, a_w(t) represents the weighted acceleration, which can be frequency-weighted using the curves proposed in BS 6472 (Figure 15a). The weighted time history data for Test No. 1 is illustrated in Figure 16. t denotes the total duration of the excitation applied to the floor. The evaluation criteria for VDV are provided in

Table 6. VDV range associated with the likelihood of negative feedback [35].

Location Time	Unlikely to Have Negative Reviews (m/s−1.75)	There May Be Negative Reviews (m/s−1.75)	Likely to Have Negative Reviews (m/s−1.75)
16 h of daylight	0.2–0.4	0.4–0.8	0.8–1.6
8 h at night	0.1–0.2	0.2–0.4	0.4–0.8

.

Figure 17a,b show the VDV and RF for Test Nos. 1 through 8 as presented in Table 4. The results indicate that both the VDV and the RF decrease by approximately 50% with the increase in the height of the Deltabeam composite steel beams, suggesting that increasing the height of the Deltabeam composite steel beams can effectively enhance the vibration serviceability of the Deltabeam–CLT composite floor. Additionally, a comparison reveals a similar overall downward trend for both VDV and RF. This similarity is due to both metrics being calculated based on frequency-weighted acceleration.

As shown in Figure 17a, when the height of the Deltabeam composite steel beam is set at 200 mm, the RF of the Deltabeam–CLT composite floor reaches a peak value of 7.533, which only meets the RF limit of 8 for workshop RF as delineated in Table 5. Upon increasing the beam height to 400 mm and 500 mm, the RF decreases to 4.040 and 3.491, respectively, reflecting a maximum reduction of 53.7%. This reduction aligns with the RF limit of 4 for residential areas specified in Table 5. Consequently, the vibration response of the Deltabeam–CLT composite floor under continuous vibration adheres to the comfort criteria established for typical environments. For scenarios requiring more stringent response factor thresholds, augmenting the height of the Deltabeam composite steel beam can effectively enhance the vibration performance of the composite floor to satisfy the requisite standards.

Regarding VDV, as shown in Figure 17b, when the height of the Deltabeam composite steel beam is within the range of 200 to 300 mm, the VDV decreases from 0.629 m/s^−1.75 to 0.424 m/s^−1.75. When the beam height increases to between 320 mm and 500 mm, the VDV falls within the range of 0.394 m/s^−1.75 to 0.266 m/s^−1.75, indicating that the serviceability level of the composite floor is in a state where negative feedback is less likely, according to Table 6. This demonstrates that the vibration serviceability of the Deltabeam–CLT composite floor improves with increasing height of the Deltabeam composite steel beams. Additionally, as the beam height increases from 200 mm to 500 mm, the VDV decreases by 57.7%, indicating that increasing the beam height significantly reduces the intermittent vibration of the Deltabeam–CLT composite floor.

3.4. Discussion

Section 3.3 analysis indicates that the vibration performance of the Deltabeam–CLT composite floor is positively correlated with the stiffness of the Deltabeam composite steel beams. Therefore, increasing the height of the Deltabeam composite steel beams can enhance the effective bending stiffness of the composite floor, thereby improving its vibration performance and serviceability.

However, this optimization approach would cause the Deltabeam composite steel beams to resemble traditional reinforced concrete T-beams, thus losing the advantage of reducing the overall building height. It would also compromise the aesthetic appeal and make it difficult for ducts to pass beneath the floor, contrary to the architect’s original intent for selecting Deltabeam composite steel beams. Therefore, while maintaining the original height of the Deltabeam, we need to explore a vibration serviceability optimization solution that aligns better with the unique characteristics of the Deltabeam.

Labonnote et al. [36] conducted experimental studies on the dynamic performance of timber floor specimens connected with screws using impact testing. Their findings revealed that timber floors with higher connection stiffness exhibited higher natural frequencies. Wang et al. [37] investigated the influence of floor geometry on the vibration comfort of simply supported CLT floors using OPENSEES simulations. The results demonstrated that as the aspect ratio of the floor increased, the VDV increased significantly, indicating that floor dimensions have a substantial impact on the vibration serviceability of CLT floors. Therefore, in addition to enhancing the stiffness of the Deltabeam, improving the intrinsic vibration performance of the CLT floor such as by reducing the aspect ratio or increasing the connection stiffness between CLT panels can further improve the overall vibration serviceability of the composite floor system. In addition to passive design strategies, vibration control represents another important approach to improving floor comfort. Huang et al. [38] conducted a semi-active vibration control study on CLT floors using a tuned mass damper (TMD) system based on shape memory alloys (SMA). The results showed that by adjusting the stiffness of the SMA through temperature regulation, acceleration amplitude could be reduced by up to 26% within a specific frequency range. In future work, this type of damper could be experimentally applied to Deltabeam–CLT composite floors to evaluate and compare its effectiveness, aiming to develop the optimal vibration control solution for such systems.

4. An Optimization Approach for Vibration Serviceability of Deltabeam–CLT Composite Floors Based on Embedded Steel Trusses

4.1. Optimization Scheme

This study aims to improve the stiffness of composite beams and, consequently, the overall vibration serviceability of Deltabeam–CLT composite floors by embedding steel trusses within the Deltabeam. Therefore, an optimization scheme for vibration serviceability was proposed by incorporating steel trusses within the Deltabeam composite steel beams. This modified Deltabeam with steel trusses is similar to composite steel truss–concrete (CSTC) beams [39,40], where CSTC beams consist of prefabricated steel trusses embedded within cast-in-place concrete. Zhang et al. [41] found through experimental and theoretical studies that, compared to conventional reinforced concrete beams, the inclusion of steel trusses improved the elastic deflection stiffness and elastic–plastic deflection stiffness of the concrete beams by 93.28% and 495.721%, respectively. X Zhang et al. [42] conducted low-cycle repeated loading tests on eight steel–concrete composite beams with various types of steel trusses. The test results indicated that the steel truss concrete composite beams exhibited favorable load–displacement hysteresis curves and higher bending stiffness. Figure 18a,b show traditional steel truss concrete composite beams and the composite floor structures supported by these beams. Additionally, the construction process of steel truss concrete composite beams is very similar to that of Deltabeam composite steel beams. Initially, the steel trusses independently support the structure, and later, they work in conjunction with the hardened concrete. Based on this, this study proposes an optimization scheme for the vibration serviceability of composite floors by embedding steel trusses within Deltabeam composite steel beams.

The optimization scheme proposed in this study involves combining the Deltabeam with two rows of Pratt trusses (N-shaped trusses) [43], resulting in a Deltabeam composite steel beam with embedded steel trusses. As shown in Figure 19a,b, two types of truss support configurations were designed. These are named RE–Deltabeam (Deltabeam with rectangular cross-section trusses) and TR–Deltabeam (Deltabeam with triangular cross-section trusses), based on the different cross-sectional shapes of the trusses.

4.2. Numerical Simulation of Deltabeam–CLT Composite Floors with Embedded Trusses

To investigate the vibration serviceability of Deltabeam–CLT composite floors with embedded trusses under human-induced loads, this section establishes two finite element models of steel trusses in ABAQUS, as shown in Figure 19, based on the finite element model proposed in Section 2.2. The material properties of the truss members are the same as those of the Deltabeam, with steel defined by an elastic modulus of 210 GPa and a density of 7800 kg/m³. Three-dimensional truss elements (T3D2) from the software are used to simulate the truss members. Rigid connections are applied between the truss members and between the truss members and the Deltabeam. Similar to the contact settings defined for the Deltabeam and concrete in Section 2.3, embedded constraints are used to define the interaction between the truss elements and the concrete.

A parametric analysis of the finite element models of Deltabeam–CLT composite floors with embedded trusses was conducted. Considering that the stress in trusses is often influenced by the spacing between internal members and the diameter of the members, this study explores the impact of these two factors on the vibration serviceability of the composite floors.

4.3. Impact of Truss Spacing on the Vibration Serviceability of Composite Floors

4.3.1. Test Schemes

This section investigates the impact of truss spacing on the vibration serviceability of Deltabeam–CLT composite floors, while keeping the truss member diameter fixed at 10 mm. In this study, the truss spacing is determined based on the span of the composite steel beam. A total of eight different spacings were defined, and both the RE–Deltabeam and TR–Deltabeam proposed in Section 4.1 were tested.

Table 7. Numerical simulation test scheme for truss spacing.

Test No.	DELTABEAM® Profiles	B (mm)	Ø (mm)	Truss Section	Truss Spacing (b)
RE100	D20-400	100	10	RE-Deltabeam
RE140		140
RE200		200
RE250		250
RE280		280
RE350		350
RE500		500
RE700		700
TR100		100		TR-Deltabeam
TR140		140
TR200		200
TR250		250
TR280		280
TR350		350
TR500		500
TR700		700

provides the specific test schemes. For example, in the first test scheme, RC100 refers to placing two rows of steel trusses with a rectangular cross-section and a spacing of 100 mm within the Deltabeam. During the numerical simulation, the same human-induced load application method as described in Section 2.5 was used.

4.3.2. Analysis of Results

After conducting modal and time history analyses on the test schemes listed in Table 7, the first natural frequency of the RE— and TR—Deltabeam–CLT composite floors, as well as the vertical acceleration responses at the center of the floors under human-induced loads, was obtained. The first natural frequency corresponding to different truss spacings is shown in

Table 8. First natural frequency of composite floor slabs with different truss spacings.

Spacing of Truss	100	140	200	250	280	350	500	700	No Truss
Rectangular truss (Hz)	8.85	8.66	8.63	8.57	8.49	8.44	8.43	8.41	8.28
Triangular truss (Hz)	8.67	8.63	8.62	8.35	8.31	8.32	8.31	8.31

. Compared to the Deltabeam–CLT composite floor without trusses, the first natural frequency of the composite floor increased by up to 6.88% and 4.71% with the addition of rectangular and triangular trusses, respectively. Furthermore, as the truss spacing decreases, the first natural frequency of the composite floor shows an increasing trend. This suggests that incorporating denser steel trusses within the Deltabeam composite steel beam can enhance the beam’s bending stiffness to some extent, thereby improving the dynamic performance of the composite floor.

Figure 20 shows the comparison of acceleration time history responses for Test Nos. RE700, RE100, TR700, and TR100 with the no-truss composite floor (Test No. 1 in Table 3). The amplitude of the curves indicates that the vertical acceleration response at the center of the composite floor significantly decreases with the addition of steel trusses with a spacing of 100 mm.

The RF and VDV corresponding to each truss spacing are calculated using Equations (1) and (2), and the results are plotted in Figure 21. Compared to the composite floor in Test No. 1, which has no trusses, the composite floors in Test No. RE100 with rectangular trusses and Test No. TR100 with triangular trusses exhibit a significant reduction in both RF and VDV. The RF for the two types of trusses were reduced by approximately 53.05%, reaching the RF limit values for residential areas in Table 5. Meanwhile, the VDV was reduced by about 56.92%, also meeting the limit range of 0.2–0.4 m/s^−1.75 as specified in Table 6. This indicates that the incorporation of trusses into the Deltabeam composite steel beam enhances the vibrational serviceability of the composite floor. When the truss spacing is reduced from 700 mm to 100 mm, the RF and VDV for the RE–Deltabeam–CLT composite floor are reduced by 37.56% and 49.34%, respectively, while for the TR–Deltabeam–CLT composite floor, the RF and VDV are reduced by 45.54% and 54.21%, respectively. This suggests that decreasing truss spacing is an effective way to enhance the vibration performance of trussed composite floors. Overall, when the truss spacing exceeds 200 mm, the RF and VDV of the rectangular truss are lower than those of the triangular truss. However, when the spacing is less than 200 mm, the RF and VDV of the triangular truss are slightly lower than those of the rectangular truss. Therefore, the rectangular truss generally provides better improvement in the vibration serviceability of the Deltabeam–CLT composite floor compared to the triangular truss.

It is noteworthy that when the truss spacing is within the range of 200 mm to 350 mm, both RF and VDV exhibit an exponential decrease. Once the spacing reaches 200 mm, further reduction in truss spacing does not significantly lower RF and VDV. This indicates that, below a spacing of 200 mm, the spacing of the trusses has a limited impact on the floor’s vibration performance. Therefore, in practical design, regardless of the type of truss cross-section used, a truss spacing of 200 mm is a preferable choice for optimizing the vibration serviceability of the Deltabeam–CLT composite floor.

4.4. Impact of Truss Diameter on the Vibration Serviceability of Composite Floors

4.4.1. Test Schemes

Section 4.3 of the simulation results indicates that reducing the truss spacing to 200 mm significantly improves the vibration serviceability of the Deltabeam–CLT composite floor. However, in practical engineering, excessively close truss spacing may lead to difficulties in installing transverse connections. Therefore, it is necessary to consider other factors related to truss loading to explore more suitable design options. This subsection investigates the impact of truss diameter on the vibration serviceability of the Deltabeam–CLT composite floor while keeping the truss spacing constant. Six different truss rod diameters—6 mm, 10 mm, 14 mm, 20 mm, 30 mm, and 40 mm—were tested for both types of truss cross-sections.

4.4.2. Results Analysis

Modal analysis was performed on the composite floor with various truss diameters using the frequency analysis step in ABAQUS to obtain their first-order natural frequencies. The results for the two types of truss cross-sections are listed in

Table 9. First natural frequencies f₁(Hz) of composite floors with rectangular cross-section trusses of different diameters.

Diameter(mm)		6	10	14	20	30	40
Truss spacing	100 mm	8.66	8.85	8.89	8.76	8.71	8.83
	280 mm	8.49	8.49	8.51	8.50	8.47	8.40
	350 mm	8.42	8.44	8.49	8.52	8.53	8.53
	700 mm	8.40	8.41	8.45	8.46	8.45	8.45

and

Table 10. First natural frequencies f₁(Hz) of composite floors with triangle cross-section trusses of different diameters.

Diameter(mm)		6	10	14	20	30	40
Truss spacing	100 mm	8.65	8.67	8.74	8.75	8.83	8.71
	250 mm	8.30	8.35	8.39	8.39	8.41	8.41
	350 mm	8.32	8.32	8.32	8.34	8.33	8.33
	700 mm	8.30	8.31	8.35	8.31	8.32	8.31

. Overall, the frequency variations for the two types of trusses are similar. As the truss rod diameter increases, the first natural frequency of the composite floor shows a slight increase, but the change does not exceed 2%. This indicates that the effect of truss diameter on the dynamic characteristics of the composite floor is minimal and can be considered negligible. Additionally, for trusses with smaller spacings of 100 mm and 280 mm, the relationship between the first-order natural frequency and truss rod diameter is nonlinear. This is likely due to the fact that increasing the truss diameter not only enhances the stiffness of the composite steel beam but also significantly increases the overall mass of the composite floor, leading to a reduction in its natural frequency.

The RF and VDV for different truss diameters were calculated using Equations (1) and (2), with the results illustrated in Figure 22 and Figure 23. The data show that as the truss diameter increases from 6 mm to 40 mm, there is generally a noticeable decrease in both the RF and VDV for Deltabeam–CLT composite floors with both types of trusses. For instance, the RF and VDV for triangular trusses with a 700 mm spacing decreased by 27.44% and 32.39% by increasing truss diameter form 6 mm to 40 mm, respectively.

It is noteworthy that for trusses with a spacing of 100 mm, increasing the truss diameter does not lead to a significant reduction in the corresponding RF and VDV. This phenomenon is primarily attributed to the force distribution and stiffness characteristics of the truss. Figure 24a,b illustrate the stress variations in the truss members located at the center of the Deltabeam during loading for two different cross-sectional truss configurations. It can be observed that, compared to a truss spacing of 700 mm, when the spacing is reduced to 100 mm, the peak stress in the truss members of the triangular section truss decreases from 0.192 MPa to 0.021 MPa, and the average stress decreases from 0.082 MPa to 0.001 MPa. Similarly, for the rectangular section truss, the peak stress drops from 0.189 MPa to 0.018 MPa, and the average stress drops from 0.069 MPa to 0.001 MPa.

4.5. Discussion

Based on the aforementioned analysis, it can be inferred that embedding trusses with adequate spacing and diameter within the Deltabeam composite steel beam can significantly enhance the beam’s stiffness while preserving its original height. This modification leads to an improved vibration comfort for the Deltabeam–CLT composite flooring system.

Compared to the tests that involves increasing the height of the Deltabeam composite beam, the incorporation of triangular or rectangular trusses with a spacing of 100 to 200 mm results in a reduction in VDV and RF for the Deltabeam–CLT composite floor to 2.614 m/s^−1.75 and 3.511, respectively. These values are comparable to the VDV (2.66 m/s^−1.75) and RF (3.491) recorded in Test No. 8, which involved raising the beam height to 500 mm, and both measurements fall within the minimum limits established in Table 5 and Table 6. This observation indicates that reducing the spacing of the rectangular or triangular trusses to 100 mm can yield a vibration attenuation effect analogous to that achieved by increasing the beam height to 500 mm. Additionally, for trusses with spacings ranging from 250 to 700 mm, increasing the truss diameter to over 30 mm can reduce the VDV and RF of the Deltabeam–CLT composite floor system to the minimum allowable limits. Therefore, when economically feasible, it is preferable to use trusses with spacings of less than 200 mm or with member diameters greater than 30 mm.

When steel material is sufficient, the effect of increasing the truss spacing is superior to that of increasing the member diameter. At the same time, it is important to note that further reduction in the VDV and RF of the composite floor system becomes challenging once the truss spacing is reduced to 200 mm. In practice, the vibration performance of the Deltabeam–CLT composite floor system is determined by both the Deltabeam composite steel beam and the CLT floor. Thus, solely increasing the stiffness of the Deltabeam composite steel beam has limited impact on enhancing the vibration comfort of the floor system. Future design considerations should ensure satisfactory vibration performance of the Deltabeam–CLT composite floor by further investigating the effects of the stiffness series mechanism shown in Figure 11.

5. Conclusions

This study investigated the impact of the stiffness of Deltabeam composite steel beams on the vibration serviceability of Deltabeam–CLT composite floors using the finite element software program ABAQUS to examine the vibration response of these floors under human-induced loading. The accuracy of the simulation results was validated by comparison with data from the existing literature. The modeling process primarily considered the impact of the height of the Deltabeam composite steel beam. A comprehensive assessment of the vibration performance of the Deltabeam–CLT composite floor was conducted through modal, time history, VDV, and RF analysis, and a comparison was made with the vibrational serviceability criteria outlined in the standards. The following conclusions can be drawn:

(1)

According to the VDV and RF analysis of the CLT floor in this study, when the height of the Deltabeam composite steel beam is 200 mm, the RF of the Deltabeam–CLT composite floor is 7.533, and the VDV is 0.629 m/s^−1.75.

(2)

Increasing the height of the Deltabeam effectively enhances the stiffness of the Deltabeam–CLT composite floor, thereby reducing its vibration response under human-induced vibration. When the height of the composite steel Deltabeam is increased from 200 mm to 500 mm, the first natural frequency of the Deltabeam–CLT composite floor increases by 23.07%. At the same time, its VDV and RF decrease by 57.7% and 53.7%, respectively, meeting the maximum limit requirements outlined in BS 6472 and ISO 10137.

(3)

An optimization scheme for vibration serviceability was proposed by embedding a steel truss within the Deltabeam. Two types of truss cross-sections, rectangular and triangular, were designed. The results of the numerical simulation indicate that, compared to the standard Deltabeam composite beam with a height of 200 mm, the incorporation of a steel truss with a spacing of 100 mm and a rod diameter of 10 mm leads to a maximum reduction in the RF and VDV of the Deltabeam–CLT composite floor by 45.54% and 54.21%, respectively, achieving the same vibration reduction effect as increasing the beam height to 500 mm.

(4)

Reducing the spacing of the trusses can effectively enhance the serviceability of the Deltabeam–CLT composite floor. By decreasing the truss spacing from 700 mm to 100 mm, the RF and VDV of the Deltabeam–CLT composite floor with rectangular section trusses decreased by 37.56% and 49.34%, respectively, while for the floor with triangular section trusses, the RF and VDV decreased by 45.54% and 54.21%, respectively.

(5)

Changing the diameter of the truss members shows a better vibration reduction effect when the truss spacing is under 100 mm. Although the natural frequency of the Deltabeam–CLT composite floor did not change significantly as the truss member diameter increased from 6 mm to 40 mm, its dynamic performance improved considerably, with RF and VDV decreasing by up to 27.44% and 32.39%, respectively.

(6)

According to the RF and VDV analyses, increasing the rod diameter of the steel trusses along with increasing the height of the Deltabeam can increase the stiffness of the Deltabeam and thus improve the overall vibration serviceability of the combined Deltabeam–CLT composite floor.

The vibration performance of the Deltabeam–CLT composite floor is determined by both the Deltabeam composite steel beam and the CLT floor. Merely increasing the stiffness of the Deltabeam has its limits, as it is impractical to infinitely raise the height of the Deltabeam or make the truss spacing excessively dense, which would hinder lateral connections. Therefore, future designs should also consider altering the mass and stiffness of the CLT floor itself to ensure sufficient vibration resistance in the Deltabeam–CLT composite floor.

In addition to CLT floor slabs, concrete floor slabs are often used as structural floors, and these concrete floor slabs can also be combined with Deltabeams to work together. The research outcomes also contribute to circumstances where avoiding excessive vibration is the focus.