Vibration Measurement and Numerical Simulation of the Effect of Non-Structural Elements on Dynamic Properties of Large-Span Structures

Abstract

Non-structural elements have been demonstrated to be essential for the dynamic performance of large-span structures. However, how to quantify their effect has not yet been fully understood. In this study, the contribution of non-structural elements to dynamic properties of large-span structures is systematically investigated via both field measurement and numerical simulation methods. Modal testing of an indoor stadium and an elevated highway bridge was conducted during different construction phases, and the corresponding modal characteristics were identified. Results show that the traditional capacity-based models are incapable of reflecting the actual dynamic characteristics of in-service structures since neglecting the effect of non-structural elements would result in remarkable discrepancies in modal properties. A general modeling framework incorporating the contribution of slab/deck pavement, infill walls (or crash barriers), and joints/connections for large-span structures is developed to quantitatively consider the effect of non-structural elements based on the principle of equivalence of stiffness and mass to the actual structure. The effectiveness of the method is validated by vibration measurement results.

1. Introduction

Non-structural elements (NSEs), such as partition walls, ceilings, cladding panels, facades, windows, handrails, etc., are elements that are not designed for load capacity but are used to furnish or for functional purposes [1]. The effect of NSEs has long been ignored in structural design or analysis for many reasons: (i) the unclear mechanism of interaction between main structure and NSEs; (ii) the difficulty in quantifying the mechanical properties of NSEs; (iii) a common, but not always true, belief that a model neglecting NSEs is conservative [2]. However, studies [3,4] show that NSEs significantly modify the response of main structures in terms of stiffness, strength, and ductility, and neglecting the contribution of NSEs may lead to unsafe response predictions. Great research interest has been attracted to the effect of NSCs on dynamic responses of structures in recent years to consider the potential effect of NSEs under ultimate loadings like earthquakes or strong winds [5,6,7,8,9,10]. All of these involved studies are basically bearing capacity-based since structural safety (strength) is of primary concern, followed by ductility or vulnerability.

Besides safety, vibration serviceability is another important aspect of structural performance, especially for large-span structures. Large-span structures, characterized by low natural frequencies and small damping, are increasingly favored in large public facilities like stadiums or highway bridges. For these structures, steel-concrete composite floor systems or pre-stressed reinforced concrete floor systems are usually employed to realize long spans. Due to the slenderness and low damping, these structures are susceptible to dynamic excitations, such as human activities or vehicle loading, and their performance under dynamic excitation is of great importance for serviceability-based structural design and vibration mitigation, if necessary. Dynamic properties are key parameters for structural analysis. An accurate dynamic model that can properly reflect the dynamic properties (especially the first few modes of interest) of actual in-service structure is of primary importance to structural design and analysis.

Researchers increasingly realize the contribution of NSEs to the dynamic properties of structures [11,12,13,14,15,16,17,18,19,20,21,22]. Reynolds [11] and Miskovic and Pavic [12] investigated the dynamic properties of the slender floor system and found that NSEs could have a remarkable impact on their dynamic properties. The research by Su et al. [13] revealed that compared with other parameters, NSEs were the largest contributor to structural stiffness. Petrovic and Pavic [14] reported that partition walls would greatly increase structural stiffness and damping. Studies by Devin and Fanning [15] demonstrated that infill walls and cladding panels had a key impact on the dynamic properties of the floor system. The fundamental frequency of the floor system increased by 9% after the addition of the cladding panels. To summarize, it is generally observed that NSEs would alter the natural frequencies of structures [16,17,18,19], increase structural damping [20,21,22,23,24], and sometimes even change the order of mode shapes [22].

Although there are several studies concerning the potential influence of NSEs on structural properties, most of them are qualitative evaluations, and the quantification of NSEs is still limited. There have been several attempts to properly model NSEs. The relevant work could be categorized into two types: (i) capacity-based modeling. It falls within the scope of structural safety and refers to establishing a finite element model for structural safety assessment, i.e., to consider the potential effect of NSEs under ultimate loadings like earthquakes or strong winds to ensure structural safety (strength). Telue and Mahendran [5,6] investigated the behavior of both sides of lined steel wall frames using both experimental and FE modeling methods. In the experiment work, they performed the ultimate load-capacity test and load-deformation test and analyzed the potential failure mechanism. Gaiotti [7] modeled the infill walls using the diagonal truss model to consider its contribution to structural responses to wind and seismic loading. The research by Madan et al. [8] also employed the diagonal truss model with hysteric-force deformation to consider the contribution of infill walls under seismic loading. Saifullah et al. [9] investigated the strength and stiffness of plasterboard-lined steel-framed ceiling diaphragms when subject to lateral loading like wind and earthquake loads. They first conducted experimental tests of typical ceiling diaphragms. Then, the FE model was developed using ANSYS software, and the nonlinearity of material, geometric, and element nonlinearity were included in the model. The model was capable of illustrating test results. Brandolese et al. [10] studied the behavior of suspended ceilings using the experimental fragility method. The cyclic quasi-static tests on suspended ceilings were conducted to characterize the damage progression under cyclic displacements and strength degradation. Numerical studies were also performed to validate and expand the experimental work; (ii) serviceability-based modeling. It corresponds to the field of serviceability and refers to establishing a model for vibration serviceability (or comfort) assessment, i.e., to consider the effect of NSEs on structural dynamic properties and responses under in-service or operational loadings, such as human activities or vehicles. Pavic et al. [25] measured the dynamic properties of an office floor and observed a difference of 15% between predicted natural frequencies (i.e., using the initial model developed according to construction drawings without NSEs) and measured results. Based on the initial model, they utilized Shell 63 element to represent the exterior brick walls and achieved satisfactory predictions in natural frequencies to the actual structure. Jiménez-Alonso et al. [23] studied the effect of vinyl flooring on the modal properties of a steel footbridge. They observed an increase in the damping ratio of 2.069% due to the installation of vinyl flooring through a modal test. Then, they set up the FE model of the footbridge using 3D beam elements to model the arches and the deck and 3D cable elements to model the hangers. The non-structural floor (both the glass layer and the vinyl layer) was assumed to have no effect on the structural stiffness and only affect mass. This model was capable of representing the first three modes. Different models should be employed for different calculation purposes. It is evident that the states of structures in (i) and (ii) are apparently different, and the contributions or work status of NSEs are obviously different. Thus, different modeling frameworks and modeling methods for NSEs should be employed according to the actual state (stress state and strain state) of the target structure.

From the above analysis, it is clear that the modeling of NSEs to structural dynamic properties remains limited for serviceability-based analysis and needs further exploration. Moreover, owing to the unclear boundaries between strong vibration and micro-vibration, the different contribution of NSEs in capacity-based modeling and serviceability-based modeling is not clearly understood.

The aim of this study is to properly consider the contribution of NSEs to the dynamic properties of structures within the range of serviceability-based vibration. Field measurements and modal properties identification of various full-scale structures at different construction phases were conducted. A general framework for considering the effect of NSEs on the dynamic properties of large-span structures was proposed and implemented for the structures involved.

The remainder of this paper is structured as follows. Section 2 briefly describes the test structures. Modal testing and parameter identification of test structures under different construction phases are illustrated in Section 3. Section 4 illustrates the basic principles and modeling methods of NSEs. Section 5 verifies the proposed method using full-scale structures. The main conclusions and future work are presented in Section 6.

2. Description of Test Structures

Two full-scale large-span structures, including an indoor stadium and an elevated eight-lane urban highway bridge, are introduced in this section. These structures are all located in China. They generally cover most types of flexible large-span structures favored in engineering practice. Owing to the relatively lightweight, large-span, and low-damping, these structures are susceptible to dynamic loadings like human activity or vehicle load. Thus, their dynamic properties (i.e., modal mass, natural frequency, mode shape, and damping ratio) are of great interest to both engineers and academic societies.

2.1. BU Stadium

BU Stadium, shown in Figure 1, is a large-span indoor stadium for indoor sports like basketball, badminton, and tennis. It also offers a venue for daily training. Compared with the traditional single-story stadium, BU Stadium has two floors, including a large training area of over 2000 m² on the first floor and a playing field and grandstands on the second floor. A large-span pre-stressed concrete floor with dimensions of 42 m by 45 m is adopted by the stadium on the second floor. To improve the net height of the first floor, the depth of beams supporting the 2nd floor was designed to be 1.7 m, leading to a depth-span ratio of 1/24.7 of beams. The relatively low depth-span ratio of the second floor makes its dynamic performance under human activity a great concern.

Figure 1. BU Stadium. (a) Elevated view, (b) interior of 2nd floor (before decoration), (c) interior of 2nd floor (after decoration), and (d) impacting test.

Modal tests of the large-span floor were also carried out in two different construction phases, namely, before and after decoration, as shown in Figure 1b and Figure 1c, respectively. The decoration is mainly the pavement of floors, and the construction layers for slab pavement involve a ground leveling layer (mortar), self-leveling mortar, and wooden floor on the structural concrete slab.

2.2. Dongfeng Bridge

Dongfeng Bridge, shown in Figure 2, is an elevated eight-lane urban highway bridge in Wuhan, central China, and it connects the third ring road and the fourth ring road of Wuhan City. The total length of the bridge is 5.808 km. The width of the deck is 33 m, making it one of the widest highway bridges in Wuhan. The whole bridge employs 51 continuous girders, including 16 steel box girders and 35 pre-stressed reinforced concrete box girders. The span varies from 25 m to 89 m. In this study, one three-span continuous steel-box girder (named L7), which had the longest span of 89 m, was selected for modal testing since it is expected to have low natural frequencies and closely spaced modes. Bridge L7 comprises three spans of 57.4 m, 89 m, and 57.4 m and is supported by double piers, as shown in Figure 2a,c. The steel box girder, having single-box cross sections, is divided into six rooms, as shown in Figure 2a. Eight bearings were placed on top of each pier, as illustrated in Figure 2b, including one fixed bearing (B2), four one-way bearings (i.e., B1, B3, and B4 can move freely in the longitudinal direction, and B6 in the transverse direction) and three double-way bearings (i.e., B5, B7, and B8).

Figure 2. (a) Cross section of bridge L7, (b) plane view of bridge L7 with the layout of bearings (arrows denote possible movement directions), (c) side view (in construction), and (d) deck view (deck pavement). All dimensions in m.

The construction details of the L7 Bridge are introduced as follows:

(1) Superstructure

The structural steel utilized is Q345qd. The bridge deck comprises a rectangular orthotropic plate configuration, with the top plate typically exhibiting a thickness of 16 mm, which is increased to 40 mm in proximity to both the middle and end supports. Longitudinal reinforcement of the top plate is achieved through U-shaped stiffeners spaced approximately 550 mm apart. The bottom plate generally has a thickness of 14 mm, is also thickened to 40 mm near the middle and end supports, and features T-shaped stiffeners in its longitudinal direction with spacing around 400 mm. Cross girders are positioned at intervals of 3 m, maintaining a thickness of 14 mm while being reinforced near support locations and at cantilever ends. A total volume of 437 m³ of weighted concrete is poured at both the middle and end supports. The surface treatment for the bridge deck consists sequentially, from top to bottom, of a layer comprising 9 cm SBS modified asphalt followed by an additional layer of 8 cm C50 steel fiber concrete. To ensure robust adhesion between the bridge deck surface and underlying concrete, shear pins with a diameter of 20 mm along with 10 mm welded steel mesh are strategically placed atop the steel box girder. Furthermore, the box girder incorporates a bidirectional structural slope set at an inclination rate of 2%, aligned with the centerline trajectory.

(2) Substructure

The bridge features an H-shaped double-column solid pier, with heights measuring 9.894 m, 8.491 m, 9.081 m, and 11.106 m (corresponding to bridge piers 19–22), and a center-to-center spacing of 8.5 m between the columns. The concrete used for the piers is classified as C40. A low-profile abutment is implemented, with dimensions of 10 × 8.5 × 3 m constructed from C30 concrete. The pile foundation consists of drilled and cast-in-place concrete piles designed for end bearing, approximately 50 m in length, and also utilizes C30 concrete.

(3) Other Structures

The expansion joint on both sides of the beam is of type SF320, and the support is a JQGZ universal ball-type support. The adjacent L6 beam (19th pier) is a four-span continuous concrete box girder (4 × 30 m), while the L8 beam is a variable-width-and-height four-span continuous steel box girder (34 m + 36 m + 40 m + 44 m).

Three construction phases of bridge L7 were considered: (i) Phase 1, the structural construction of the bridge has been completed, but without deck pavement; (ii) Phase 2, 8 cm thick steel fiber reinforced concrete was paved on the steel deck; (iii) Phase 3, another 9 cm thick modified asphalt concrete was paved on Phase 2 structure, and the bridge is ready for in-service use. The construction method of the bridge deck pavement is shown in Figure 3.

Figure 3. Construction method for bridge deck pavement.

3. Modal Testing and Parameter Identification of Test Structures Under Different Construction Phases

Table 1 summarizes the dynamic testing and analysis details of all involved structures. The ambient vibration (AV) testing method [26,27,28,29,30] was employed in modal testing of all involved structures owing to their huge masses or flexibility. The heel impact (HI) excitation method [31,32] was also introduced in the modal testing of BU Stadium to check the performance of AV testing. During the experiment, the test subject (Gender: male; Age: 24 years; Body mass: 71.3 kg) was instructed to jump off the chair and impact the floor with heels, as shown in Figure 1d.

Table 1. Modal testing and analysis methods for the tested structures.

Test Structure	Testing Method	Accelerometer	Sampling Frequency (Hz)	Identification Method
BU Stadium	AV + HI	VSE-15-D1	200	SSI + PP
Dongfeng Bridge	AV	B&K 4507B	256	SSI + PP

HI: heel impact testing; AV: ambient vibration testing; ID: initial displacement method; SSI: stochastic subspace identification; PP: peak-picking.

Two types of accelerometers were used for data acquisition, including Bruel & Kjaer 4507B—piezoelectric accelerometers (nominal sensitivity of 1 mV/g and frequency range from 0.3 Hz to 6000 Hz) and VSE-15-D1 accelerometers (nominal sensitivity of 5 mV/gal and frequency range from 0.1 Hz to 200 Hz). Data were acquired using PULSE data acquisition software. The data acquisition system and test instruments are depicted in Figure 4.

Figure 4. Data acquisition system and test instruments. (a) Bruel & Kjaer DAQ system, (b) 4507B accelerometers, and (c) VSE-15-D1 accelerometer.

3.1. Modal Testing of BU Stadium

Both ambient excitation and heel impact methods were employed for modal testing of BU stadium. For ambient vibration testing, modal properties of the floor were obtained using the data-driven SSI method [33,34]. The identified first three bending natural frequencies of the floor at Phase 1 stage are 5.00 Hz, 9.21 Hz, and 9.42 Hz, respectively, and they decreased to 4.61 Hz, 8.71 Hz, and 8.92 Hz accordingly after decoration. It should be noted that the mode shapes of the floor did not change after decoration. The 2nd and 3rd natural frequencies are close, owing to the similar length and width of the floor (45 m by 42 m).

In the following, results of the impacting method are presented to check the reliability of identified modal properties using the data-driven SSI method. The testing program was conducted in BU Stadium at different construction phases. TPs were installed at quarter points and midpoints in a grid, as shown in Figure 5. Figure 6 shows an example of the measured vertical acceleration response of the floor under a single-person (71.3 kg) excitation. As shown in the figure, the floor acceleration significantly increased once the heels touched the floor, then it gradually attenuated in a free decay manner to zero due to the presence of structural damping.

Figure 5. Plan view of slab B4 and sensor arrangements. All dimensions in mm.

Figure 6. Measured floor responses with single-person impacting.

Figure 7a,b illustrate the Fourier spectrum of vertical accelerations at TP 3 (see Figure 5) under the same person’s excitation before and after decoration, respectively. The first vertical bending frequency decreased from 5.00 Hz to 4.61 Hz, and the second bending frequency decreased from 9.21 Hz to 8.71 Hz after structural decoration. The structural damping increased since the spectrum became wider and flatter after structural decoration. It should be noted that the presence of a person’s excitation on the floor would affect measured dynamic properties, known as human–structure interaction [35]; however, it should be noted that this effect could be ignored since the mass ratio of the single person to the test floor is relatively small (0.00018) [35]. Since TP 3 was placed at a quarter point along the length of the floor (see Figure 5), only two peaks (corresponding to the first symmetric bending mode and the second anti-symmetric bending mode along the slab length) were observed in Figure 7. After a careful examination of spectra at all TPs, we found that the natural frequencies identified using both methods agreed well with each other.

Figure 7. Measured floor responses with single-person impacting. (a) Time history; (b) frequency sepectrum.

3.2. Modal Testing of Dongfeng Bridge

The AV testing method was utilized for the L7 bridge owing to its flexibility and large size. A sufficiently dense grid of TPs was employed for L7 to identify the first few modes of interest. The sensor deployment of the L7 bridge is shown in Figure 8. Sensors installed on the bridge deck were arranged in three lines along the longitudinal direction of the bridge considering the large width of the deck, i.e., two lines on both sides of the bridge and the middle line was placed 0.6 m away from the central line, which was occupied by the central separation barrier. The testing campaign was divided into several setups due to limited accelerometers and coverage of wire length. Three reference TPs (marked as triangles in Figure 8) were placed on the bridge deck, i.e., one on each span, and the position of the reference TPs was carefully checked, such that the targeted vibration modes were observable. Since modal testing is not the main focus of this study, the details of test setups will not be shown here. Interested readers are referred to [36]. A time series of 60 min was recorded during individual trials in data acquisition.

Figure 8. Test grid and sensors deployment of L7 bridge. Note: 1. All dimensions are in m; TP: test point; 2. △ Fixed reference TP, in three orthogonal directions; mobile TP, in vertical and transverse directions; mobile TP, in vertical direction.

Modal parameters of the L7 bridge were identified based on the reference-based data-driven SSI method as well as the peak-picking (PP) method [23]. The averaged normalized power spectrum density (ANPSD) [37] was employed in PP to increase the accuracy of identified modal frequencies. Figure 9 gives an example of a stabilization diagram at the reference point based on a data-driven SSI method combined with ANPSD. The red star (☆) represents stable points for frequency, the green quincunx (*) denotes stable points for both frequency and damping ratio, and the blue circle (○) denotes stable points for frequency, damping ratio, and mode shape. Obviously, the mode frequencies of the L7 bridge are closely spaced from 1 to 5 Hz. Table 2 compares the natural frequencies and damping ratios identified using the PP method and SSI method for the Phase 3 bridge, and the corresponding mode shapes are illustrated in Figure 10. It is observed that both methods can effectively identify the first few modes of interest. The identified frequencies by both methods agree well with each other, and the modal assurance criterion (MAC) values of the first 12 modes (except for mode 2, see Table 2) are close to the unit, offering promising results for the dynamic properties of the L7 bridge. The identified modes for transverse bending and torsion modes (see Figure 10b,d,f,g,j) are not as smooth as bending modes, partly because the excitation energy in longitudinal and transverse directions is much smaller than in vertical components (the amplitudes in longitudinal and transverse directions are about 25% of vertical one) under ambient vibration.

Figure 9. Stabilization diagram at reference point: (a) vertical direction; (b) transverse direction. Note: ☆ represents stable points for frequency, * denotes stable points for frequency and damping ratio, ○ denotes stable points for frequency, damping ratio, and mode shape.

Table 2. Comparison of identified mode results using the PP method and SSI method for the Phase 3 bridge.

Mode No.	PP	SSI-DATA		Frequency Difference/%	MAC Value	Mode Description
	Frequency/Hz	Frequency/Hz	Damping Ratio/%
1	1.318	1.320	0.60	−0.160	0.999	1st VB
2	NA	1.705	2.02	NA	NA	1st longitudinal
3	2.053	2.064	1.78	−0.568	0.985	1st TB+ torsion
4	2.402	2.409	1.08	−0.299	0.994	2nd VB
5	2.563	2.582	2.90	−0.726	0.943	2nd TB+ torsion
6	2.789	2.794	1.42	−0.175	0.971	3rd VB
7	3.136	3.151	1.99	−0.465	0.993	1st torsion
8	3.656	3.683	1.20	−0.744	0.941	3rd TB+ torsion
9	3.775	3.787	1.53	−0.323	0.978	4th TB+ torsion
10	4.403	4.415	2.66	−0.281	0.969	4th VB
11	4.879	4.901	1.55	−0.463	0.942	5th TB+ torsion
12	5.657	5.706	1.41	−0.843	0.939	2nd torsion

Note: MAC-modal assurance criterion; NA-not available; VB-vertical bending; TB-transverse bending.

Figure 10. The first few mode shapes of the L7 bridge based on the SSI method.

Based on these two methods, Table 3 summarizes the first 12 modes of the L7 bridge under different construction phases. As shown in Table 3, the bridge has low natural frequencies and small damping ratios, and the first few modes are closely spaced. The natural frequencies of the L7 bridge decreased after each stage of decoration. Specifically, they decreased by 15% to 21% in Phase 3 compared with Phase 1, whereas the damping ratios dramatically increased as non-structural components were included. It should be noted that introducing the pavements at Phase 2 and Phase 3 did not alter the mode shapes of the L7 bridge.

Table 3. Identified dynamic properties of the L7 bridge under different construction phases.

Mode No.	Phase 1		Phase 2		Phase 3		Mode Description
	Frequency /Hz	Damping ratio/%	Frequency /Hz	Damping Ratio/%	Frequency /Hz	Damping Ratio/%
1	1.57	0.21	1.42	0.35	1.32	0.60	1st VB
2	2.07	0.68	1.90	1.91	1.71	2.02	1st longitudinal
3	2.38	0.62	2.24	1.29	2.06	1.78	1st TB+ torsion
4	2.76	0.48	2.58	0.69	2.41	1.08	2nd VB
5	3.16	1.12	2.84	2.20	2.58	2.90	2nd TB+ torsion
6	3.31	0.35	2.99	0.89	2.79	1.42	3rd VB
7	3.80	0.78	3.33	1.51	3.15	1.99	1st torsion
8	4.18	0.64	3.65	1.71	3.68	1.20	3rd TB+ torsion
9	4.52	0.52	3.96	0.80	3.79	1.53	4th TB+ torsion
10	5.49	0.67	4.85	1.47	4.42	2.66	4th VB
11	6.22	0.73	5.57	1.07	4.90	1.55	5th TB+ torsion
12	7.23	0.85	6.44	1.03	5.71	1.41	2nd torsion

Note: VB-vertical bending; TB-transverse bending.

4. Modeling of Non-Structural Elements

The aim of considering the contribution of NSEs is to let the dynamic properties of the numerical model approach the actual in-service structure, especially for the first few modes of interest. This is of great importance for serviceability-based performance evaluation and vibration mitigation, if necessary, of large-span flexible structures. The dynamic characteristics of a structure mainly depend on the distribution of its mass and stiffness, and connections and/or joints linking individual components would affect the damping of the system. In Section 3, experiments demonstrate that NSEs change the natural frequencies of a structure and increase structural damping. Note that the damping properties and mechanisms in structures are complex and would be affected by many factors [32,37], and the most effective and convincing method to obtain structural damping is through measurement or experimental tests. Hence, in the present study, we only focus on the contribution of NSEs on structural natural frequencies and mode shapes.

4.1. Main Types of NSEs in Large-Span Structures

For large-span structures investigated in this study, NSEs are mainly categorized into the following three types:

(1) slab/deck pavement;

(2) infill walls or crash barriers;

(3) joints/connections, such as temperature/settlement joints of large-span floors or bridge joints connecting two girders. Joints/connections, generally treated as free boundaries in structural design, are, in fact, filled with flexible materials (i.e., asphalt) or set as flexible connections to facilitate the normal use of structures. Since structures are usually in a linear elastic stage under in-service conditions, these joints/connections would behave much closer to rigid or semi-rigid connections rather than free boundary conditions (BCs).

The first two types of NSEs would alter the distribution of mass and stiffness of the structure, and the third one mainly affects structural stiffness through BCs, leading to the alteration in structural dynamic properties.

It should be noted that the contribution of additional mass, such as the presence of workers, construction machines, or preloading (construction materials stacked on floors), will not be included here since it will disappear soon after completion of construction.

4.2. Basic Principle and Modeling Methods of NSEs

The basic principle of appropriate modeling of NSEs is the equivalence of stiffness and mass as the actual structure, especially for the first few modes of interest.

For type (1), two methods could be employed here based on the equivalence of mass and stiffness as the actual structure. The first method is that the composite section of pavement and structural slab could be represented using a new section only if it has similar mass and moment of inertia as the original composite section such that one can update the parameters of structural slabs (i.e., thickness and material properties like density and elastic modulus) to consider the contribution of pavement without introducing new shell elements. The other method is using multi-layer shell elements with different thicknesses and material properties in the numerical model to consider the contribution of individual pavement. Obviously, the first method is an indirect method, and the latter is a direct method. It should be mentioned that these two methods are equivalent. These two methods will be introduced in the following section.

For type (2), the stiffness of infill walls or crash barriers should also be considered rather than additional mass since infill walls/crash barriers would not detach from the primary structure and would behave as a whole with the main structure. Both the in-plane and out-of-plane stiffness of the infill wall or crash barrier should be considered. For infill walls (i.e., masonry walls or lightweight partition walls), the equivalent pin-jointed strut model [2] (see Figure 11) is always employed to consider the stiffness of the structure. The in-plane and out-of-plane stiffness could be determined using the linear segment of the experimental results of a load–displacement curve under horizontal load, which are of increasing interest in the area of earthquake engineering, namely, to evaluate the effect of infill walls on seismic performance of main structures [5,6,7,8,9,10,38,39,40]. For a crash barrier that is made of reinforced concrete, one can treat it as a shear wall and use the elastic modulus and shear modulus to directly determine the in-plane and out-of-plane stiffness.

Figure 11. Infill walls under horizontal load: (a) actual system, (b) simplified model.

For type (3), the stiffness provided by NSEs in connections should be appropriately considered in the prediction model. The contribution of stiffness by NSEs could be modeled using additional linear spring elements, as shown in Figure 12. However, it remains challenging to accurately determine the stiffness of springs owing to the complex characteristics of joints and connections and the difficulty in quantifying the strains in BCs. In the current study, the trial-and-error method is suggested in determining the stiffness of springs.

Figure 12. Schematic of boundary conditions to be processed.

It should be mentioned that although it is challenging to accurately determine the stiffness of the springs in joints/connections, one can roughly determine it by comparing the correlation of the first few local modes between predicted results against measured ones using the Aver_mac indicator by the trial-and-error method. Aver_mac is defined in the following form as:

$$Ave{r}_{mac}=\min (MAC(1),\dots ,MAC(n))$$

(1)

in which n is the number of modes of interest and MAC (modal assurance criterion) is the correlation between measured the predicated mode shapes, defined as follows [41,42,43]:

$$MAC\left(i\right)={\displaystyle \frac{{\left[{\left({\varphi }_i^A\right)}^T{\varphi }_i^B\right]}^2}{\left[{\left({\varphi }_i^A\right)}^T{\varphi }_i^A\right]\left[{\left({\varphi }_i^B\right)}^T{\varphi }_i^B\right]}}$$

(2)

where ${\varphi }_i^A$ is the simulated ith modal shape vector by finite element method, and ${\varphi }_i^B$ is the ith identified modal shape from measurement.

$MAC$ value represents the correlation between two modal vectors, and a perfect correlation is achieved for $MAC$ = 1 and no correlation when $MAC$ = 0. In practice, it is generally accepted that the two vectors have acceptable correlations when $MAC$ > 0.80. Thus, the $Ave{r}_{mac}$ considers the overall correlation between the first few modes of interest. The larger the $Ave{r}_{mac}$ is, the closer the predicted mode shapes to the measured ones. By setting the proper range of stiffness of springs and using $Ave{r}_{mac}$ as an indicator, the stiffness of springs could be determined.

5. Validation of the Proposed Method

The performance of numerical models developed considering the effect of NSEs using the above-mentioned method will be presented in this section.

5.1. BU Stadium

An original FE model of BU stadium at Phase 1 stage (named M1) was set up using ANSYS software. The details of this model can be found in Ref. [44]. Figure 13 shows the first three bending modes of the large-span floor obtained from the modal analysis of the FE model. The natural frequencies for the second and third modes are close to each other owing to the similar dimensions (45 m by 42 m).

Figure 13. First three bending modes of large-span floors in BU stadium (Phase 2) by the FE method.

For BU Stadium, NSEs are mainly the pavement (ground leveling layer, self-leveling mortar, and wooden floor) on the structural concrete slab. The contribution of the pavement was achieved by modifying the parameters of the original concrete slabs according to the principle of equivalent stiffness and mass compared with the actual structure. The new model developed here was named M2. It is worth noting that only the mass of the wooden floor was included in M2, and the stiffness was ignored. As previously observed in a field measurement [45], the effective stiffness provided by the wooden floor to the overall structure is small due to the sliding between the wooden floor and underneath slabs. Modal properties, specifically the natural frequencies and modal shapes of M2, were obtained by modal analysis using ANSYS software. The mode shapes for M2 are similar to M1; namely, the mode shapes did not change after structural decoration.

Table 4 evaluates the performance of the M1 and M2 models on the prediction of the first three bending modes of the floor. As shown in Table 4, both M1 and M2 have satisfactory predictions for the first three modes. The natural frequencies of the slab slightly decreased after decoration owing to the more pronounced increase in structural mass compared with stiffness. The predicted natural frequencies of the M2 model are slightly smaller than the measured ones, partly because of neglecting the stiffness of wooden floors.

Table 4. Comparison of measured and calculated first three bending frequencies of large-span floors in BU stadium.

Mode No.	Calculated [Hz]		Measured [Hz]		Variance/%
	M1	M2	T1	T2	(M1–T1)/T1	(M2–T2)/T2
1	4.82	4.39	5.00	4.61	−3.55	−4.68
2	8.70	8.41	9.21	8.71	−5.58	−3.49
3	8.80	8.59	9.42	8.92	−6.59	−3.72

Note: T1: test results for structure before decoration; T2: test results for structure after decoration.

5.2. Dongfeng Bridge

The numerical model of the L7 bridge under different construction phases was developed using ANSYS software according to the construction drawings. The steel-box girder was modeled using shell-181 elements to represent the top, bottom, and web plate. Beam-188 elements with self-defined sections were employed to model U-shape, T-shape, and rectangle stiffeners in the girder. The piers were also modeled using beam-188 elements with self-defined sections, and the effect of bearing platforms and piles was not included in the model. Deck pavements were considered using multi-layer shell elements (shell 181) by attributing each layer with different thicknesses and material properties according to the actual construction method. The separating and crash barriers were also considered in the model. The spherical bearing was modeled using linear springs, and stiffness was determined based on the experimental results of the mechanical properties test (provided by the bearing supplier). The contribution of stiffness provided by bridge joints on both edges of the L7 bridge was considered using one-dimensional (longitudinal) spring elements, i.e., only the longitudinal constraint effect was considered. The stiffness of the spring was determined by comparing the first few modes of interest between simulated results against measured ones using the Aver_MAC value [see Equation (1)] by the trial-and-error method. In the current study, only the first four modes were considered. A threshold value of Aver_mac ≥ 0.9 is suggested after careful examination of all involved modes. The involved calculation parameters are listed as follows: elastic modulus of steel and concrete are 2.06 × 10¹¹ and 3.25 × 10¹⁰ N/m², respectively; density for steel, concrete, and asphalt are 7850, 2500, and 2300 kg/m³, respectively; the stiffness of springs at temperature joints along the longitudinal direction of the bridge is 2.01 × 10⁶ N/m. Figure 14a,b illustrate the cross section of the bridge deck and the overall FE model of the L7 Bridge (Phase 3), respectively.

Figure 14. (a) cross section and (b) FE model of the L7 bridge.

Modal properties of the bridge were obtained using modal analysis using the ANSYS software.

Taking Phase 3 as an example, modal analysis was performed on the initially established finite element model of the L7 bridge, and the simulated and measured frequencies are presented in Table 5. Figure 15 depicts the first 12 mode shapes of the L7 bridge (Phase 3) obtained from the initial FE model.

Table 5. Comparison of measured and simulated first 12 modes of L7 bridge under Phase 3.

Mode No.	fm	fs	Variance (%)	Mode Description
1	1.32	1.36	3.58	1st VB
2	1.71	1.72	1.04	1st longitudinal
3	2.06	1.97	−4.50	1st TB+ torsion
4	2.41	2.44	1.18	2nd VB
5	2.58	2.65	2.63	2nd TB+ torsion
6	2.79	2.88	3.12	3rd VB
7	3.15	3.24	2.96	1st torsion
8	3.68	3.75	1.80	3rd TB+ torsion
9	3.79	3.88	2.64	4th TB+ torsion
10	4.42	4.60	4.21	4th VB
11	4.9	4.99	1.72	5th TB+ torsion
12	5.71	5.99	4.98	2nd torsion

Note: f_m—measured natural frequencies; f_s—simulated natural frequencies.

Figure 15. First 12 mode shapes of the L7 bridge (Phase 3) based on the initial FE model.

It is evident the predicted mode shapes are close to the ones identified in modal testing (shown in Figure 10). The relative errors between the frequencies obtained from finite element analysis and those measured ones are all within 5%, indicating a strong correlation between the simulated modal characteristics and the measurements.

To improve the results, the model updating procedures are further employed. Due to various assumptions, idealization, discretization, and parameterizations that are introduced in numerical modelling, the obtained numerical model may not always reflect the actual structural behavior. Currently, various methods for bridge finite element model calibration have been proposed, including sensitivity-based, maximum likelihood, non-probabilistic, probabilistic, response surface, and regularization methods. The sensitivity-based method treats the error between the theoretical and measured data as the objective function and modifies the sensitive parameter values to make the objective function optimal. This method requires iterative solutions during the process and needs to solve the sensitivity matrix of the system, which is of high cost for large structures. The response surface calibration method obtains the sample values by experimental design and obtains the explicit expression of the objective function and calibration parameter through regression analysis of the sample values. It approximates the complex implicit function relationship between the objective function and the calibration parameter using the explicit expression, which greatly improves the calibration efficiency. Suzana et al. [46] comprehensively reviewed the finite element model updating methods for structural applications. The whole process of model updating is described step by step: selection of updating parameters (design variables), definition of the model updating problem, and solution using different FEMU methods. Davide Raviolo et al. [47] systematically compared the optimization algorithms for finite element models, updating them on numerical and experimental benchmarks. Recently, they proposed a Bayesian sampling optimisation strategy for finite element model updating [48].

This paper uses the response surface method to calibrate the finite element model based on the modal test results of the bridge corresponding to each construction phase.

Taking Phase 3 as an example, five parameters are identified as correction factors based on sensitivity analysis and practical engineering insights: the elastic modulus of the steel plate (E_s), the mass density of the steel plate (ρ_s), the mass density of asphalt pavement (ρ_a), the elastic modulus of bridge piers (E_c), the longitudinal spring stiffness of expansion joints (k). Initially, value ranges for each parameter are established based on empirical data, followed by normalization of these parameters. The normalized levels for these parameters are presented in Table 6.

Table 6. Calibration items and level values.

Calibration Items	Level −1	Level 0	Level 1	Δi
Elastic modulus of steel plate Es (1011 N/m2)	1.85	2.06	2.26	0.21
Mass density of the steel plate ρs (kg/m3)	7065	7850	8635	785
Mass density of asphalt pavement ρa (kg/m3)	1840	2300	2760	460
Elastic modulus of bridge piers Ec (1010 N/m2)	2.93	3.25	3.58	0.33
Longitudinal stiffness of expansion joints k (106 N/m)	1.8	2.0	2.2	0.2

As shown in Table 6, the density of asphalt concrete exhibits significant variation, which may be attributed to the influence of the mixture proportions on its density.

Firstly, the frequency residual is enlarged by 100 times as the objective function to avoid the rounding error. Subsequently, sensitivity analyses are conducted using the central difference method with regard to the five correction parameters. A center composite design method is used to design 27 orthogonal experiments for response surface equation fitting, and the F test is used to detect the significance of each coefficient. Insignificant correction coefficients and their combinations are eliminated, and after elimination, all the response surface equations have only one significant coefficient. The precision of the equations is verified using the R² test and the relative root mean square error (RMSE) value, and finally, the genetic algorithm in the MATLAB optimization toolbox is used to optimize the objective function and obtain the corrected values of the five parameters, as shown in Table 7.

Table 7. Selected parameters before and after updating.

Calibration Items	Initial Value	After Updating	Variance
ρs (kg/m3)	7850	8493.8	8.2%
Es (Pa)	2.06 × 1011	1.90 × 1011	−7.7%
Ec (Pa)	3.25 × 1010	3.24 × 1010	−0.03%
ρa (kg/m3)	2300	1840.1	−19.9%
k (N/m)	2 × 106	1.91 × 106	−4.3%

The modal frequencies of the finite element model were recalculated using the updated parameters, and the results were compared with the measured values, as presented in Table 8.

Table 8. The relative error before and after updating.

Mode No.	fm	fs′	Initial Error (%)	Error After Updating (%)
1	1.32	1.31	3.58	−0.11
2	1.71	1.70	1.04	0.10
3	2.06	1.96	−4.50	−5.05
4	2.41	2.36	1.18	−2.06
5	2.58	2.61	2.63	1.16
6	2.79	2.79	3.12	−0.27
7	3.15	3.14	2.96	−0.06
8	3.68	3.67	1.80	−0.27
9	3.79	3.80	2.64	0.27
10	4.42	4.44	4.21	0.53
11	4.9	4.86	1.72	−0.90
12	5.71	5.84	4.98	2.40

Note: f_m—measured natural frequencies; f_s′—simulated natural frequencies after model updating.

From Table 8, it can be seen that, except for the 3rd and 4th-order frequencies, the errors of the bridge’s higher-order frequencies after correction are significantly reduced and are all within 3%. The reasons for the increased errors of the 3rd and 4th-order frequencies are as follows: (1) The optimal solution of the multi-objective function optimization is the overall optimal solution, which may result in some objectives achieving the optimal solution while others deviate further from the optimal solution; (2) Sensitivity analysis can only guarantee effectiveness within the initial value neighborhood, and cannot objectively evaluate the global sensitivity of parameters. The correction parameters obtained through sensitivity analysis may not be the parameters with the largest errors, which may result in suboptimal correction effects for some objectives. The 3rd-order frequency error slightly increased, while the other higher-order frequency errors are extremely small (within 3% for the 3rd-order, within 1% for the 8th-order), indicating that the precision of the finite element model of the bridge has been significantly improved after correction.

Using similar methods, Table 9 shows the comparison of all simulated and measured dynamic properties of the L7 bridge at three different construction phases. The measured and simulated natural frequencies, as well as the mode shape correlation index, namely MAC values, are presented in the table. It is noted that the modal shapes have not changed for Phase 2 and Phase 3 compared with the Phase 1 structure.

Table 9. Comparison of measured and simulated first 12 modes of L7 bridge under different construction phases.

Mode No.	Phase 1				Phase 2				Phase 3				Mode Description
	fm /Hz	fs /Hz	Variance /%	MAC	fm /Hz	fs /Hz	Variance /%	MAC	fm /Hz	fs /Hz	Variance /%	MAC
1	1.57	1.55	−1.05	0.999	1.42	1.44	1.59	0.999	1.32	1.31	−0.11	0.999	1st VB
2	2.07	1.99	−3.90	-	1.90	1.86	−2.22	-	1.71	1.70	0.10	-	1st longitudinal
3	2.38	2.45	3.10	0.964	2.24	2.29	2.01	0.965	2.06	1.96	−5.05	0.958	1st TB+ torsion
4	2.76	2.69	−2.45	0.991	2.58	2.53	−2.07	0.989	2.41	2.36	−2.06	0.983	2nd VB
5	3.16	3.34	5.60	0.877	2.84	2.91	2.46	0.867	2.58	2.61	1.16	0.848	2nd TB+ torsion
6	3.31	3.23	−2.35	0.995	2.99	2.91	−2.67	0.992	2.79	2.79	−0.27	0.987	3rd VB
7	3.8	3.95	3.90	0.931	3.33	3.18	−4.53	0.922	3.15	3.14	−0.06	0.921	1st torsion
8	4.18	4.07	−2.70	0.956	3.65	3.84	5.13	0.956	3.68	3.67	−0.27	0.943	3rd TB+ torsion
9	4.52	4.64	2.60	0.947	3.96	4.06	2.40	0.942	3.79	3.80	0.27	0.931	4th TB+ torsion
10	5.49	5.33	−2.85	0.979	4.85	5.06	4.41	0.975	4.42	4.44	0.53	0.965	4th VB
11	6.22	6.45	3.65	0.866	5.57	5.75	3.21	0.865	4.9	4.86	−0.90	0.849	5th TB+ torsion
12	7.23	7.53	4.10	0.943	6.44	6.64	3.09	0.943	5.71	5.84	2.40	0.939	2nd torsion

Note: f_m—measured natural frequencies; f_s—simulated natural frequencies. The 1st longitudinal mode shape is not available, and MAC for this mode is not listed in the table.

As shown in Table 9, the proposed method can effectively predict the first 12 modes of the L7 bridge at different construction phases. The largest discrepancies in natural frequencies for Phase 1–3 are 5.6%, 5.13%, and −5.05%, respectively. The MAC values for the first four vertical bending modes are over 0.96, indicating the predicted modes are well correlated with the measured ones. The predicted modes for transverse bending and torsion modes are not as perfect as vertical bending modes, partly because the excitation energy in longitudinal and transverse directions is much smaller than the vertical component (the amplitude is about 25% of the vertical one) under ambient vibration environment such that the identified modes are not as smooth (see Figure 10b,d,f,g,j), but they are generally acceptable since the minimum MAC value exceeds 0.84.

The natural frequencies of the L7 Bridge at Phase 3 decreased by 15–35% compared with Phase 1. In particular, the first vertical bending natural frequency decreased by 18.56% after deck pavement, implying the remarkable impact of NSEs on structural dynamic properties. Although the modes of the L7 bridge are closely spaced, the proposed method shows promising capability of predicting the first few modes of interest.

In a word, using the proposed method, one can accurately model the contribution of NSEs on structural dynamic properties.

6. Conclusions

The contribution of NSEs to the dynamic properties of large-span structures is systematically investigated in this study through field measurement, numerical simulation, and experimental techniques. Full-scale structures, including an indoor stadium and an elevated highway bridge, were measured at different construction phases to obtain the dynamic properties of interest. Results show that NSEs have a remarkable impact on serviceability-based dynamic properties of the target structure. In particular, it would alter natural frequencies, increase the structural damping, and sometimes even change the order of mode shapes of the structure. The evident difference in dynamic properties of structures under different construction phases reveals that the traditional load capacity models are incapable of reflecting the actual dynamic characteristics of in-service structures since they ignore the contribution of NSEs.

Another important conclusion is that different models should be employed for different calculation purposes. The main difference between the capacity-based model (i.e., for seismic performance assessment) and the serviceability-based model lies in the treatment of NSEs. Both models should consider the contribution of NSEs; however, since NSEs are in different states under these two apparently different conditions, different modeling methods should be employed to appropriately take into account the effect of NSEs.

A general modeling framework incorporating the contribution of slab/deck pavement, infill walls (or crash barriers), and joints/connections for large-span structures is developed to quantitatively consider the effect of NSEs based on the principle of equivalence of stiffness and mass on the actual structure. The effectiveness of the method is validated by field measurement results. It should be noted that the proposed method is somewhat empirical, especially in processing the boundaries (i.e., determining the stiffness of springs on BCs), since one needs to check the first few modes of interest by the trial-and-error method based on measured modal properties of the as-built structure. For structures to be built, it may show weakness since the actual dynamic properties (especially the mode shapes) are not available in advance. However, compared with the traditional load-bearing capacity model, the model developed here, taking into account the contribution of NSEs, would be much closer to the actual dynamic properties of the target structure. For structures having similarities with the test structures, the parameters obtained in this study could be utilized as useful references, which may further improve the accuracy of the predictions.

The contribution of this study is as follows: first of all, it is demonstrated that the traditional capacity-based models are incapable of reflecting the actual dynamic characteristics of in-service structures since they ignore the effect of NSEs. Secondly, this study provides a general framework for modeling the contribution of NSEs for serviceability-based modeling, which could serve as the first step toward a better understanding of the characteristics of NSEs for structures in operational stages. Thirdly, this study provides a detailed dataset of dynamic properties of various full-scale structures, giving light to further investigate the mechanism of interaction between main structure and NSEs. Still, much work needs to be conducted in the future, such as theoretical modeling of NSEs and quantitatively determining the potential influencing parameters.