Dynamic response of rooftop billboards excited by fluctuating wind loads

Abstract

As typical wind-sensitive structures, the wind-induced dynamic response of rooftop billboards and their dynamic interaction with the host building are critical issues in wind-resistant design. This study systematically investigates the dynamic response of steel billboards situated on multi-story reinforced concrete buildings by establishing a three-dimensional finite element model of the coupled billboard-building system. Fluctuating wind loads are simulated using the harmonic superposition method, and dynamic responses are analyzed through time-history analysis. The feasibility of a stiffness-proportional damping model is evaluated. A comparison with a sole billboard system reveals the significant influence of the building on the billboard’s dynamic response and its underlying mechanism. The results demonstrate a non-negligible dynamic interaction between the host building and the rooftop billboard, with the influence coefficient for the peak base bending moment exceeding ±20%. This dynamic interaction is identified as modal coupling within the coupled system. When the first and second natural frequencies of the system are close, strong modal coupling occurs, leading to a redistribution of energy that can significantly amplify or diminish the billboard’s dynamic response. The findings imply that the conventional design practice of analyzing rooftop billboards as isolated structures with fixed bases may be inaccurate, and the dynamic interaction within the coupled billboard-building system should be considered.

Keywords

rooftop billboard dynamic interaction modal coupling wind-induced response time-history analysis

1. Introduction

Outdoor billboards serve as essential advertising infrastructure in urban and suburban areas, with configurations mainly including free-standing vertical panels, single-post structures with 2-3 panels and rooftop hoardings on host buildings. Their large surface areas and elevated positions make them susceptible to wind-induced damage during typhoons, hurricanes, and severe storms. Failures, including panel detachment, column buckling, and anchor bolt fracture, pose risks to public safety and infrastructure (Salgado-Estrada et al., 2023; Wen and Xie, 2020).

Study on wind resistance for billboards began in 1994 (Letchford and Holmes, 1994). Some of these data have been incorporated into codes of practice for the design of free-standing walls. Subsequently, full-scale field tests, wind tunnel tests, and numerical simulations were conducted by researchers and the history was detailed by Giannoulis et al. (2012).

A concise overview of research on billboards in the last decade, which focuses almost exclusively on single-post and rooftop billboards, is provided here. Sponsored by the International Sign Association and the Outdoor Advertising Association of America, Smith and Zuo (Smith et al., 2014; Zuo et al., 2014), conducted full-scale field tests to provide a benchmark and then performed an extensive series of wind tunnel tests to study wind loading of sign structures with rectangular sign faces. Gu and Han (Gu et al., 2015a; Han et al., 2015) analyzed the wind load characteristics of single-post billboards based on rigid model wind tunnel tests and studied their wind-induced vibration characteristics using the time-history analysis method. Li et al. (2018); Wang et al. (2016, 2017, 2020, 2022); Wang, Li and Chen (2018) and Wang, Li and Li (2018) conducted a comprehensive study on wind loads and their characteristics for single-post billboards with double- and triple-panel configurations. So far, research about wind loads for single-post billboards has been relatively well-established. Similarly, focusing on single-post billboards, Wen and Xie (2020) carried out a field investigation and a dynamic simulation of billboards destroyed by the super-typhoon Mujigae to study the wind-induced collapse mechanism. Based on the previous studies, Salgado-Estrada et al. (2023) proposed a methodology to determine the structural safety of flexible outdoor single-post billboards.

Billboards or signboards are often installed on the rooftops of low- and middle-rise buildings for advertising. But research on rooftop billboards is extremely scarce. Based on rigid model wind tunnel tests, Gu and Han (Gu et al., 2015b; Han et al., 2017) studied, for the first time, the wind-induced responses of billboards on a high-rise building roof using the time-history analysis method. Financially supported by the Ministry of Land, Infrastructure, Transport and Tourism of Japan, Masuyama et al. (2020) conducted a series of wind tunnel experiments to collect data on wind force coefficients under various conditions, including building shapes, as well as signboard shapes and locations. To the best of the author’s knowledge, these are all the existing works about rooftop billboards. In engineering practice, rooftop billboards are usually considered to be the same as common and traditional billboards (i.e. free-standing vertical panels) and are designed according to the relevant specifications (AIJ, 2015; ASCE, 2022; MHURD, 2012, 2021). Subsequent work will demonstrate that, under certain conditions, the host building can significantly influence the rooftop billboard’s dynamic response.

In addition to rooftop billboards, studies on the secondary structures attached to buildings also involve rooftop canopies (Jiang et al., 2024, 2025), solar panels (Dai et al., 2024, 2025; Peng et al., 2023), and parapets (Huang et al., 2017; Lin et al., 2026; Qiu et al., 2019). Considering that solar panels are usually installed with a 20-30° slope while canopies are installed horizontally by pillars in the inside area of the roof, the results of these studies cannot be used for evaluating the wind loads on rooftop billboards which are installed vertically along a building edge, although they serve as a useful reference. Parapets, just like billboards, are installed vertically along the edge of a building. However, Rooftop billboards are usually installed facing the open direction of a road, and therefore, they are installed along a side of the roof or along two adjacent sides of the roof. There are few shapes of signboards that surround the entire roof like parapets. Meanwhile, billboards and parapets differ greatly in size and configuration, resulting in different wind loads. In addition, the stiffness of parapets is much larger than billboards, so the dynamic interaction of parapets between host buildings is negligible.

The present study utilizes the time history analysis method to investigate the wind-induced responses of steel billboards situated on the rooftops of multi-story reinforced concrete buildings. In the next section, the problem is stated, and the parameters and properties are defined. The calculation method and model are then briefly outlined. Structures, including billboards and buildings, are modeled using the finite element method, while fluctuating wind loads are synthesized using the harmonic superposition method. Afterward, the dynamic response of the billboard under wind loads is analyzed. To evaluate the influence of buildings on the dynamic response of billboards, the variations in the billboard’s base bending moment are taken into account, and the assessment of the calculated effects are compared with the case of sole billboard. Figure 1 shows the analysis process of this study.

Figure 1.

Analysis flowchart.

2. Problem definition

The system under investigation is composed of a common steel billboard, which is fixed on a building and three-dimensionally distributed. Considering that this study is general, the billboard and the building are derived from a representative example and then simplified. They are not a specific, built real-world project. As a numerical investigation aimed at the preliminary exploration of interaction mechanisms and parametric analysis, we chose to construct a parametric model with broad representativeness rather than replicating a particular structure. This approach allows us to systematically isolate and quantify the influence of different parameters, thereby focusing on revealing the general principles of billboard-building dynamic interaction.

Figure 2 shows a two-dimensional representation of the building’s three-dimensional model in this study. The building is 48 m long and 18 m wide, and its height ranges from 7.2 to 43.2 m. The mechanical and geometrical properties of the concrete frame structure are defined by the following parameters: story height is 3600 mm; story number (N_S) is 2/4/6/8/10/12; frame column cross-section is 600 mm × 600 mm; frame beam cross-section is 250 mm × 600 mm; floor slab thickness is 150 mm; modulus is 3 × 10⁴ MPa; density is 2700 kg/m³; Poisson’s ratio is 0.2; and damping ratio is 0.02. There are exterior walls around the frame structure and parapets on the roof. The properties of brick walls are defined by the following parameters: thickness is 240 mm; parapets height is 1200 mm; modulus is 2.4 × 10³ MPa; density is 2100 kg/m³; Poisson’s ratio is 0.15; and damping ratio is 0.02.

Figure 2.

Two-dimensional representation of the building’s three-dimensional model (unit: mm).

Figure 3 shows a three-dimensional view of the billboard for study. The billboard is 9 m high, 24 m wide, and 0.6 m thick. The mechanical and geometrical properties of the steel truss billboard are defined by the following parameters: hollow cross-section of vertical prop is 90 mm × 90 mm × 4 mm; hollow cross-section of cross beam is 60 mm × 60 mm × 3 mm; hollow cross-section of vertical prop’s bracing is 50 mm × 50 mm × 3 mm; hollow cross-section of cross beam’s bracing is 40 mm × 40 mm × 3 mm; panel thickness is 3 mm; steel modulus is 2.06 × 10⁵ MPa; steel density is 7850 kg/m³; Poisson’s ratio is 0.3; and damping ratio is 0.01.

Figure 3.

Three-dimensional view of the billboard (unit: mm).

3. Dynamic analysis

3.1. Finite element model

A commercial software product for finite element analysis, ANSYS, is employed for the finite element models. The beams, columns of the building, and the cross beams, vertical props, and bracings of the billboard are modeled by beam elements (BEAM 188), while the floor slabs and wall of the building and the panel of the billboard are modeled by shell elements (SHELL63). BEAM188 is suitable for analyzing slender to moderately stubby/thick beam structures. The element is based on Timoshenko beam theory which includes shear-deformation effects. The element is a linear, quadratic, or cubic two-node beam element in 3D. SHELL63 has both bending and membrane capabilities. Both in-plane and normal loads are permitted. The element has six degrees of freedom at each node.

The maximum number of elements is 8,178, and the maximum number of degrees of freedom is 30,084. One of the three-dimensional finite element models for the billboard with building couple system is shown in Figure 4. Meanwhile, finite element models of the sole billboard system are established for comparison. These models consist only of the billboard itself, not the host building, and are fully constrained at the bottom. In the real world, the billboard is secured to the roof of the host building using high-strength friction bolts. In the finite element model, this is simulated by constraining the displacement of the connection nodes between the billboard and the host building. In addition, the dynamic interaction between the soil and the structure is neglected and the assumption that the stiffness of the foundation is infinite is adopted. Therefore, the nodes at the base of the host building are fully constrained.

Figure 4.

The three-dimensional finite element model for the billboard with building couple system.

The fundamental frequencies of the 2/4/6/8/10/12-story frame structure are 6.6/3.6/2.4/1.8/1.4/1.1 Hz along the lateral axis, along which the wind load acts on the dynamic system. The fundamental frequency of the billboard is 4.2 Hz along the direction of the wind load force on the dynamic system.

3.2. Dynamic equation

The dynamic response of the aforementioned system can be obtained through time-history analysis, with the dynamic equation expressed as follows (Clough and Penzien, 2003):

[M] {\ddot{x} (t)} + [C] {\dot{x} (t)} + [K] {x (t)} = {P (t)}

where

[M]

[C]

, and

[K]

is the mass, damping, and stiffness matric of the dynamic system, respectively;

{x (t)}

{\dot{x} (t)}

, and

{\ddot{x} (t)}

is the displacement, velocity, and acceleration column vector of the dynamic system, respectively; and

{P (t)}

is the wind load applied on the dynamic system.

In linear elastic conditions, the system’s inputs and outputs can be treated according to the principle of linear superposition. The system’s overall response can be regarded as the sum of the static response and the dynamic response. This paper focuses on the system’s dynamic response; therefore, static wind loads are not discussed herein, and only the dynamic response of the system under fluctuating wind loads is analyzed. In this paper, the dynamic response of the billboard is analyzed in terms of its dimensionless base moment $M (t)$ , which is defined as $M_{b} (t) / m_{0} {\bar{v}}_{10}^{2}$ , where $M_{b} (t)$ is the bending moment at the base of the billboard, $m_{0}$ is the mass of the billboard, and ${\bar{v}}_{10}$ is the 10-min average reference wind speed at a height of 10 m.

3.3. Damping model

Stiffness-proportional damping is employed and expressed as:

[C] = \frac{2 ξ}{ω_{1}} [K]

where

ξ

is the damping ratio and

ω_{1}

is the first-order natural circular frequency.

Hysteretic damping is more appropriate for structural dynamic analysis (Clough and Penzien, 2003), and thus, is generally considered a more accurate method for identifying calculation errors caused by stiffness-proportional damping. It is evident that for the two damping models, the time histories of billboard’s dimensionless base moment are almost identical with only minimal discrepancies; the peak errors are also approximately 5%. The utilization of a stiffness-proportional damping model for the dynamic system in this paper is demonstrably feasible, with a concomitant very limited error.

3.4. Wind load

The harmonic superposition method (Wang, 2018) is used for the simulation of fluctuating wind speeds. The fluctuating wind speed time history ( $V (t) = {v_{1} (t), v_{2} (t), \dots, v_{m} (t)}$ ) for the selected nodes in the spatial domain can be simulated using Equation (3) to (5).

v_{j} (q Δ t) = 2 \sqrt{Δ ω} \cdot Re (\sum_{k = 1}^{j} G_{j k} (q Δ t) \cdot \exp (i \frac{k}{m} Δ ω q Δ t)) (j = 1, 2, 3, \dots, m)

G_{j k} (q Δ t) = \sum_{l = 0}^{M - 1} B_{j k} (l Δ ω) \cdot \exp (i l Δ ω q Δ t) (q = 0, 1, 2, \dots, N - 1)

B_{j k} (l Δ ω) = H_{j k} (l Δ ω) \cdot \exp (i ϕ_{k l}) (l = 0, 1, 2, \dots, M - 1)

where

Δ t

is the step size for discrete time;

m

is the number of focused nodes for simulated time history;

M

is the number of discrete frequency point;

ϕ_{k l}

is a random phase uniformly distributed over the interval

[0, 2 π)

;

Δ ω = 2 π \cdot Δ n

;

Δ n = f_{u} / M

is the step size for discrete frequency;

f_{u}

is the upper cutoff frequency;

H_{j k} (ω_{k l})

is the element in the Cholesky decomposition matrix for the power spectrum matrix (

S (ω)

) of the fluctuating wind;

ω_{k l}

is the double-indexed circular frequency, which is expressed by Equation;

θ_{j k} (ω_{k l})

is the phase for

H_{j k} (ω_{k l})

, which is expressed by Equation;

N

is the number of discrete time point and determined by Equation; and

μ

is a positive integer.

ω_{k l} = (l - 1) Δ ω + \frac{k}{m} Δ ω

θ_{j k} (ω_{k l}) = \arctan (\frac{Im (H_{j k} (ω_{k l}))}{Re (H_{j k} (ω_{k l}))})

N = 2^{μ}

Davenport spectrum (Davenport, 1961) of fluctuating wind speed is employed and expressed as:

\frac{n S (ω)}{v_{*}^{2}} = \frac{4 x^{2}}{{(1 + x^{2})}^{4 / 3}}

x = 1200 n / {\bar{v}}_{10}

where

{\bar{v}}_{10}

is the 10-min average reference wind speed at a height of 10 m;

n = ω / 2 π

is the frequency;

v_{*} = \sqrt{k} {\bar{v}}_{10}

is the friction speed;

k = {(\frac{κ}{\ln (10 / z_{0})})}^{2}

is the surface drag coefficient;

z_{0}

is the roughness length; and

κ

is the Karman constant, taken as 0.4.

Wind load acting vertically on the billboard panel represents the most critical condition for its structural response. Along the vertical direction, the primary wind-bearing zone of the billboard (the panel) is divided into 5 zones as illustrated in Figure 3, while the wind-loaded zones of the building are divided by story. Thus, the number of selected nodes for simulated time history ( $m$ ) is the sum of the story number ( $N_{S}$ ) and 5. For each case, the above-mentioned harmonic superposition method is employed to generate 10 sets of time history samples of fluctuating wind speed. Subsequently, the corresponding time histories of the fluctuating wind pressure (denoted as WL1 to WL10), converted from the wind speeds, are applied to the dynamic system to conduct dynamic response analysis.

The key parameters of wind are as the follows: the 10-min average reference wind speed at a height of 10 m ( ${\bar{v}}_{10}$ ) and the exponent of the average wind speed profile ( $α$ ) are relatively selected as $36 m / s$ and 0.15 in accordance with Chinese code (MHRUD, 2012); the roughness length ( $z_{0}$ ) is 0.05 m; the pressure coefficients on the windward ( $C_{F}^{1}$ ) and leeward sides ( $C_{F}^{2}$ ) are 0.8 and −0.5; the total duration of the simulated time history ( $T$ ) is $600 s$ ; the time step ( $Δ t$ ) is $0.01 s$ ; the number of discrete time points ( $N$ ) is 60,000; the number of discrete frequency points ( $M$ ) is 30,000.

The frequency-independent coherence function proposed by Shiotani (1967) is employed and expressed as:

C o h (ω) = \exp (- | \frac{z_{a} - z_{b}}{60} |)

where

z_{a}

and

z_{b}

are, respectively, the height of Point a and b.

4. Numerical results and analysis

The dynamic response in time domain of the billboard is addressed in this section.

Figure 5 shows the peak values of billboard’s upper oscillation amplitude for different story numbers of the building. Figure 6 shows the peak values of billboard’s dimensionless base moment for different story numbers of the building. Figure 7 shows the time histories of billboard’s dimensionless base moment for different story numbers of the building. To conserve space, only the time-history responses for the WL1 wind load sample are presented here.

Figure 5.

The peak values of billboard’s swing amplitude atop for different story numbers of the building.

Figure 6.

The peak values of billboard’s dimensionless base moment for different story numbers of the building.

Figure 7.

The time histories of billboard’s dimensionless base moment for different story numbers of the building (WL1).

The time-history responses of the two dynamic systems show significant differences. The presence of buildings influences the time-history response of the base bending moment of the billboard, leading to marked variations. Under identical wind loads, the peak values vary without a discernible pattern. Generally, when buildings are excluded from consideration, the peak values predominantly fall within the range of 0.10 to 0.15. However, when buildings are included, the primary distribution range expands slightly.

All the dimensionless base moment ( $M (t)$ ) samples of billboards, totaling 600,000, are analyzed for probability statistics. Figure 8 shows the probability distributions of billboard’s dimensionless base moment, and Table 1 shows the standard deviations of billboard’s dimensionless base moment for different story number of the building.

Figure 8.

The probability distributions of billboard’s dimensionless base moment.

Table 1.

The standard deviations of billboard’s dimensionless base moment for different story numbers of the building.

Story number ( $N_{S}$ )	Standard deviation ( $σ$ )		Influence coefficient ( $(σ^{B B} - σ^{B}) / σ^{B B}$ )
Story number ( $N_{S}$ )	The sole billboard system ( $σ^{B}$ )	The billboard with building system ( $σ^{B B}$ )	Influence coefficient ( $(σ^{B B} - σ^{B}) / σ^{B B}$ )
2	0.0267	0.0275	3.0%
4	0.0282	0.0249	−11.8%
6	0.0300	0.0305	1.4%
8	0.0305	0.0319	4.4%
10	0.0314	0.0312	−0.5%
12	0.0320	0.0318	−0.8%

For the sole billboard system, the standard deviation of the dimensionless base moment increases progressively with the story number. Specifically, the standard deviation for $N_{S} = 12$ is 1.2 times greater than that for $N_{S} = 2$ . This observation is both reasonable and expected. As the story number rises, the elevation of the billboard also increases, leading to higher wind speeds and enhanced dynamic responses. As the story number escalates from 2 to 12, the elevation of the billboard’s vertex increases from 16.2 m to 52.2 m, while the average wind speed rises from 39 m/s to 46 m/s.

The inclusion of the building results in deviations from the aforementioned trend. The probability distributions of the dimensionless base moment for both the sole billboard system and the billboard with the building system adhere to a normal distribution. The presence of the building does not alter the form of the probability distribution for the dimensionless base moment, but it does affect its standard deviation. Specifically, after incorporating buildings into the dynamic system (i.e., taking into account the dynamic interaction between billboards and buildings), the standard deviation for $N_{S} = 4$ decreases by 11.8%—a significant change—and while that for $N_{S} = 8$ increases by 4.4%.

At this stage, it is unclear how the standard deviation for the coupled system, and thus the dynamic interaction, varies systematically with the number of stories. This relationship is explored in the following sections.

To explicitly show the dynamic interaction between billboards and buildings, the influence of building on billboard is illustrated with the influence coefficient ( $e$ ). The influence coefficient of peak value ( $e_{M}$ ) is defined as $(M_{\max}^{B B} - M_{\max}^{B}) / M_{\max}^{B}$ , where $M_{\max}^{B}$ and $M_{\max}^{B B}$ are the peak values of billboard’s dimensionless base moment for the sole billboard system and the billboard with building system, respectively. And the influence coefficient of standard deviation ( $e_{σ}$ ) is defined as $(σ^{B B} - σ^{B}) / σ^{B B}$ , where $σ^{B}$ and $σ^{B B}$ are the standard deviation of billboard’s dimensionless base moment for the sole billboard system and the billboard with building system, respectively.

Figure 9 shows the influence coefficients of peak value for different story numbers of the building. The influence coefficient fluctuates around zero, with the minimum influence coefficient being −26% and the maximum influence coefficient being 23%. This suggests that the interaction between billboards and buildings, when subjected to wind loads, may contribute to over 20% of the billboard’s base bending moment. This influence must not be overlooked in engineering practice.

Figure 9.

The influence coefficients of peak value for different story numbers of the building.

It’s worth noting that when the story number is 4 ( $N_{S} = 4$ ), the influence coefficients range between −26% and −1%, all of which are negative values. In order to eliminate randomness, 10 additional time histories of wind load, which are formed by the same method, are generated and applied on the dynamic systems. The influence coefficients are found to be 2%, −6%, −23%, −1%, −14%, −1%, −22%, −13%, −7%, −19%, −10%, and −23%, which are also predominantly negative. Therefore, for a 4-story building, the influence coefficients are predominantly negative. This finding indicates that the building-billboard interaction can reduce the billboard’s base bending moment. It is necessary to employ a perspective of frequency domain to comprehend this phenomenon.

Figure 10 shows the Fourier spectra of billboard’s dimensionless base moment for different story numbers of the building. Along the wind load direction, the first and second modes exhibit strong coupling because the first order frequency ( $f_{1} = 3.5 H z$ ) and second order frequency ( $f_{2} = 3.8 H z$ ) of the billboard with building system are relatively close. This results in a slight reduction in the contribution of the second mode to the billboard’s base bending moment response, while the contribution from the first mode increases. For the billboard with building system, the second mode exerts a dominant influence on the billboard’s base bending moment, while the first mode contributes significantly less than the second mode. Consequently, the base bending moment for the coupled system is reduced compared to that for the sole billboard system.

Figure 10.

The Fourier spectra of billboard’s dimensionless base moment for different story numbers of the building (WL1).

Figure 11 shows the Fourier spectra of billboard’s dimensionless base moment for different stiffness of the building ( $N_{S} = 4$ , WL1). In this picture, E0.80, E0.90, E1.00, E1.10, and E1.20 indicate that the lateral rigidity of the building was multiplied by 0.80, 0.90, 1.00, 1.10, and 1.20, respectively. It is evident that an increase in the lateral stiffness of the building results in an upward trend in the first order frequency of the billboard with building system along the wind load direction, approaching the second order frequency. As the first and second order frequencies converge, the coupling between the two modes becomes increasingly pronounced with the amplitude of the first order mode increases and that of the second mode decreases. For the case E1.10, the amplitudes of the first and second order modes are almost the same, indicating that both modes contribute equally to the billboard’s base bending moment. As the lateral stiffness of the building continues to increase, the amplitude of the first order mode continues to grow while that of the second order mode continues to decrease. Consequently, the first mode becomes the dominating mode governing the billboard’s base bending moment.

Figure 11.

The Fourier spectra of billboard’s dimensionless base moment for different stiffness of the building (N_S = 4, WL1).

It is important to note that the mode governing the base bending moment shifts: the first mode dominates when the building’s lateral stiffness is high, while the second mode dominates when the stiffness is low. However, the governing mode shape itself remains unchanged: it is consistently characterized by lateral bending and shear deformation of the billboard with negligible deformation of the frame structure, as shown in Figure 12. The variation in lateral stiffness of the frame structure merely alters the position of this mode in the sequence of modes.

Figure 12.

The mode shape.

When the story number is 2, 6, 8, 10, or 12 ( $N_{S} = 2 / 6 / 8 / 10 / 12$ ), the coupling between the first and second modes is minimal, and their mutual influence is negligible, due to the significant frequency gap between the first and second order frequencies of the billboard with building system along the wind load direction. The contribution of the modal shape controlling the billboard’s base bending moment to the billboard’s base bending moment is basically the same in both the sole billboard system and the billboard with building system. This is evident in the Fourier spectra, which display almost identical amplitudes for both the sole billboard system and the billboard with building system as shown in Figure 10. Therefore, for these cases, the influence coefficients are not uniformly negative (as for $N_{S} = 4$ ) but exhibit a mixed distribution of positive and negative values clustered around zero.

In summary, compared to the sole billboard, the natural frequency of the billboard in the coupled system is slightly lower due to the finite stiffness of its base (i.e., the roof slab). When the first order frequency of the sole billboard system is marginally higher than that of the sole building system along the wind load direction, the reducing natural frequency of billboard, the aforementioned reduction for the billboard’s natural frequency results in a proximity of the billboard’s natural frequency to that of the building and a coupling between the corresponding mode. Consequently, compared to the sole system, the amplitude in the Fourier spectrum for the coupled system is reduced, leading to a smaller dynamic response.

In the following section, the cases of $N_{S} = 4$ and $N_{S} = 8$ are used to illustrate how key parameters of wind, billboards, and buildings exert influence on the interaction between billboards and buildings.

5. Discussion

This study employs time-history analysis method to systematically investigate the dynamic response of rooftop billboards under fluctuating wind loads and the dynamic interaction between the billboard and the host building. The computational results reveal critical phenomena whose mechanisms warrant further discussion.

The most significant finding is that the host building exerts a non-negligible influence on the dynamic response of the rooftop billboard, with the influence coefficient ( $e_{M}$ ) exceeding ±20% (Figure 9). This finding demonstrates numerically significant influence and implies substantial engineering implications, challenging the conventional design approach. It suggests that the conventional approach—treating rooftop billboards as isolated, base-fixed structures—may be non-conservative or uneconomical. The traditional method overlooks the integrated behavior of the coupled billboard-building system, and this study’s results challenge this common practice.

The essence of this dynamic interaction is identified as modal coupling. As detailed in Section 4, when the first and second order frequencies of the coupled system are close (e.g., the case of $N_{S} = 4$ ), a strong modal coupling effect occurs. The Fourier spectra in Figures 10 and 11 clearly demonstrate that this coupling redistributes energy among the modes: it reduces the contribution of the mode dominating the base moment while enhancing that of a non-dominant mode. This redistribution can ultimately result in a total system response smaller than that of the sole system, as evidenced by the negative $e_{M}$ and $e_{σ}$ for $N_{S} = 4$ .

It is noteworthy that the modal shape governing the billboard’s base bending moment consistently involves lateral bending and shearing deformation of the billboard with negligible deformation of the building (Figure 12). Alterations in the building’s stiffness (including by changing the story number) do not change this fundamental mode shape but alter its order in the modal sequence, thereby affecting its degree of coupling with adjacent modes. This explains the strong, systematic influence (consistently negative coefficients) observed for $N_{S} = 4$ , in contrast to the random distribution of influence coefficients around zero for other story numbers where greater modal frequency separation results in weak coupling.

Finally, it is necessary to acknowledge the limitations of this study. The analysis relied on linear elastic assumptions and a simplified damping model. Although the reliability of the stiffness-proportional damping model is verified, under strong winds, structures may exhibit nonlinear behavior (e.g., material nonlinearity or slippage in connections), where energy dissipation mechanisms and dynamic interactions might differ. Furthermore, the dynamic response is closely related to the relationship between the system’s natural frequency and the spectral characteristics of the fluctuating wind. In time-history analysis, the simulated fluctuating wind time histories exhibit slight, unavoidable variations in their spectral characteristics. Consequently, the system’s dynamic response varies, which hinders the exploration of the underlying patterns of dynamic interaction. In future research, random vibration analysis could be employed to address this issue.

6. Conclusion

This study investigates the wind-induced dynamic response of rooftop billboards by establishing a finite element model of the coupled billboard-building system and employing time-history analysis. The time-domain dynamic response is compared with that of the sole billboard system to assess the importance of the dynamic interaction. The main conclusions are as follows:

6.1. Significant dynamic interaction

A significant dynamic interaction might exist between the host building and the rooftop billboard, with the peak base bending moment of the billboard varying by up to approximately ±20%. This demonstrates that the conventional practice of analyzing rooftop billboards in isolation is inaccurate and might lead to either safety risks or inefficient design.

6.2. Mechanism of modal coupling

The physical essence of the dynamic interaction is modal coupling within the coupled system. When the natural frequencies of the system are close, strong modal coupling occurs, leading to a redistribution of energy that can significantly amplify or diminish the dynamic response of the billboard. The mode governing the base dynamic response of the billboard is characterized by bending and shear deformation of the billboard with minimal deformation of the building.

This study serves as a preliminary, mechanism-focused investigation. Therefore, a linear-elastic assumption with simplified damping is adopted to concentrate on the core dynamic interaction. This framework is suitable for analyzing the fundamental dynamic behavior under serviceability-level winds but explicitly cannot capture nonlinear behaviors expected under severe or ultimate wind loads. The key findings derived from this linear framework establish a necessary foundation for subsequent, more complex nonlinear studies. It must be emphasized that the results presented in this study are derived from numerical simulations and have not yet been constituted a general design formulation. Experimental validation is still required prior to their adoption as practical engineering design guidelines.

Future studies should explore the effects of material nonlinearity, detailed connection behavior, and cross-wind induced vibrations to provide a more comprehensive understanding of the wind-induced response of such complex systems. Moreover, it is meaningful to develop a correction method for the wind-induced vibration responses of rooftop billboards that takes into account the influence of the host building, and to provide a unified correction formula. It should be noted that while the base moment serves as the primary metric in this study, other response quantities—such as accelerations, maximum displacements, and connection forces—are also critical for a holistic design assessment. Future studies are recommended to investigate these parameters in detail, as damage to specific structural or non-structural elements may occur due to displacement-based or acceleration-based demands rather than purely moment-induced actions.

Footnotes

ORCID iD

Huai-feng Wang

Funding

The authors would like to gratefully acknowledge the support from the Fujian Provincial Key Laboratory of Wind Disaster and Wind Engineering under Grant No. KF20250103 and No. KF20250201, the Science and Technology Project of Fujian Province under Grant No. 2022J011251 and No. 2023Y0077, and the Science and Technology Project of Xiamen under Grant No. 2023CXY0401.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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