Contribution of Torsional Vibration Modes and the Influence on Period Ratios in the Seismic Response of Elastic Plate Bent Frame Structures

Abstract

The structural characteristics of large-span structures inherently differ from those of conventional multistorey structures, making it challenging to accurately describe the contribution of various vibration modes to the overall response using traditional dynamic response analysis methods. Based on the response spectrum method, this paper investigates the influence of the first torsional mode on the overall effects of large-span structures. It proposes a new metric, called the torsional mode contribution factor, to characterize the contribution of torsional modes. Focusing primarily on single-span frames, the study explores the impact of factors such as eccentricity ratio, aspect ratio, and roof stiffness on the torsional mode contribution factor. Additionally, the relationship between the period ratio and the torsional mode contribution factor is examined to assess the necessity of controlling the period ratio. The findings reveal that the contribution of torsional modes to the overall seismic response varies significantly under different conditions, such as eccentricity ratio, aspect ratio, roof stiffness, and torsional stiffness. The torsional mode’s contribution is minimal for small eccentricity ratios, with the response primarily driven by translational modes. As eccentricity increases, translational-torsional coupling becomes more pronounced, amplifying the influence of torsional modes on the overall dynamic response. The study also highlights that increasing roof stiffness and aspect ratios can mitigate torsional effects to a certain extent. Still, excessive eccentricity ratios and stiffness may result in higher torsional contributions. Additionally, it is found that increasing torsional stiffness reduces the influence of torsional modes but does not eliminate the overall torsional deformation. The proposed torsional mode contribution factor offers an effective way to quantify these effects, demonstrating that traditional control methods, such as period ratio control, may not fully capture the torsional contributions.

1. Introduction

Seismic damage experiences reveal that buildings experience not only horizontal and vertical vibrations during earthquakes but also torsional vibrations [1]. Investigations have shown that torsional response during an earthquake significantly degrades seismic performance, leading to component failure [2] and structural damage or even collapse [3].

Key aspects of seismic design include criteria for determining structural regularity, controlling torsion, and implementing construction measures. Seismic codes generally restrict structural regularity through indicators such as the drift ratio [4,5,6,7] and eccentricity [8]. Additionally, the Chinese code JGJ3-2010 “Technical Specification for Concrete Structures of Tall Buildings” [9] specifies requirements for the period ratio (the ratio of the fundamental torsional period to the fundamental translational period) and uses both the period ratio and the drift ratio as dual control indicators for the planar regularity of structures.

Noncoupled modes of a building structure are either translational or torsional, and the corresponding noncoupled period ratio reflects the relative magnitude of lateral stiffness and torsional stiffness [10]. However, long-span buildings are complex, with dense periods and significant coupling between modes, resulting in coupled period ratios. In such cases, the period ratio does not have a one-to-one correspondence with torsional stiffness [11].

Currently, the research on torsional regularity indices in seismic design primarily focuses on rigid roofing systems such as concrete roof slabs, steel–concrete composite roof slabs, and prestressed concrete roof slabs. The “Code for Seismic Design of Buildings” [7] stipulates that for rigid roof slabs, horizontal loads are distributed according to the proportion of equivalent stiffness of the lateral force resisting elements; for flexible roof slabs, they are distributed according to the proportion of the representative value of gravity load on the tributary area of the lateral force resisting elements. For semi-rigid roof slabs, the horizontal force distribution can be taken as the average of the two distributions above the results. Previous research has considered the impact of the rigidity of the roof slab or its in-plane deformation on the structure. Ruggieri et al. [12] noted that, compared to rigid roofing, flexible roofing structures have longer resonant periods, and the flexibility of the roof significantly affects the acceleration at the top of the building and the floor response spectrum. Sánchez et al. [13] proposed a numerical calculation method for the optimal design of semi-rigid steel frames, considering the influence of panel stiffness by introducing second-order effects. Ruggieri et al. [14] pointed out that the in-plane deformation stiffness of the floor slab affects the overall behavior of the building, such as changing the torsional coupling effect of the building, thereby affecting the force distribution on columns and shear walls. He et al. [15] quantified the impact of in-plane stiffness on the distribution of horizontal loads by defining the horizontal load transfer coefficient. The study explored the relationship between in-plane stiffness, the lateral stiffness ratio of vertical elements, and the load transfer coefficient. It defined the concept of a rigid floor slab based on the values of the lateral stiffness ratio and load transfer coefficient.

For long-span buildings, controlling the period ratio depends on the contribution of torsional modes to the overall structural response. If torsional modes minimally contribute to the total effect under seismic action, controlling the period ratio will have little impact on reducing the dynamic response. Conversely, if torsional modes significantly contribute, examining the relationship between torsional mode contribution and the period ratio becomes necessary to determine how to adjust the period ratio to mitigate the overall structural response.

Several methods have been proposed to reflect the contribution of vibration modes. Wang et al. [16] conducted modal analysis on frame structures using Ritz vectors to describe the dynamic characteristics of the structure, pointing out that the closer the spatial distribution of dynamic loads is to the mode shape, the more easily the mode is excited. Oscar et al. [17], through nonlinear static and dynamic analysis of two 21-storey steel buildings located in soft soil regions of Mexico City, noted that higher order modes have a significant impact on the overall nonlinear dynamic response of the steel structures. Feng et al. [18] proposed the mode contribution ratio as a new criterion for identifying the dominant modes. The method is validated using 18 typical cases of single-layer reticulated domes, demonstrating the accuracy of the mode contribution ratio in predicting seismic responses compared to the traditional method of combining the first thirty modes. Sun et al. [19] studied the spectral characteristics of single-layer reticulated domes, finding that higher order modes contribute more to the structural response. Liao et al. [20], through numerical analysis of single-layer reticulated domes, pointed out that the size of a mode’s participation factor does not necessarily correlate with its contribution to the dynamic response of the structure. In some cases, modes with large participation factors contribute little to the response, while modes with smaller participation factors contribute more.

This study focuses on the contribution of the first torsional mode to the seismic response of single-span elastic plate bent frame structures commonly found in public buildings, introducing the torsional mode contribution factor. By examining factors such as eccentricity, aspect ratio, and panel stiffness, the study reveals the mechanisms by which the first torsional mode affects elastic plate bent frame structures and identifies the lateral force resisting components significantly impacted by the torsional response. Additionally, the study explores the relationship between the torsional mode contribution factor and the period ratio to assess the necessity of controlling the period ratio. This research provides valuable reference points for identifying critical structural components in the seismic conceptual design of long-span buildings, thereby shortening the design cycle for structural optimization or reinforcement.

2. Torsional Mode Contribution Factor

2.1. Determination of Torsional Vibration Mode

According to the Chinese code JGJ3-2010 Technical Specification for Concrete Structures of Tall Buildings [9], “The principal vibration mode of torsional coupling can be identified by calculating the modal direction factor. When the torsional direction factor exceeds 0.5, the mode can be considered predominantly torsional”. At present, the modal direction factor for each mode of a structure [21] can be computed as follows:

For a linear structural system discretized into n degrees of freedom, the equation of motion under seismic acceleration time history is ${\ddot{u}}_{\mathrm{g}}(t)$ [22]:

$$[M]\ddot{u}+[C]\dot{u}+[K]u=-[M]r_e{\ddot{u}}_g(t)$$

(1)

where M, C, and K are the mass, damping, and stiffness matrices, respectively; u is the displacement vector of order n, and re represents the seismic direction vector.

The damping matrix C satisfies the conditions for classical damping. Therefore, Equation (1) can be decoupled in the space spanned by the undamped modal shapes, resulting in n independent single-degree-of-freedom equations of motion:

$${\ddot{q}}_i(t)+2{\xi }_i{\omega }_i{\dot{q}}_i(t)+{\omega }_i^2{q}_i(t)=-{\gamma }_i{\ddot{u}}_s(t)$$

(2)

where $q_i(t)$ presents the generalized coordinate corresponding to the i-th undamped mode shape ${\varphi }_i$ (normalized with respect to the mass matrix); ${\xi }_{\mathrm{i}}$ is the damping ratio of the i-th mode; ${\omega }_i$ denotes the natural circular frequency of the i-th undamped mode; ${\gamma }_i$ is the mode participation factor, defined as ${\gamma }_i={\varphi }_i^{\mathrm{T}}Mr$, which reflects the distribution of dynamic loads across the various modes.

The corresponding seismic force vector for the i-th order mode is:

$$\begin{array}{l}{\mathit{\{F\}}}_i=[K]{\mathit{\{u\}}}_i=-{\displaystyle \frac{{\gamma }_i}{{\omega }_i^2}}[K]{\mathit{\{\varphi \}}}_i=\\ -{\displaystyle \frac{{\gamma }_i}{{\omega }_i^2}}{\omega }_i^2[M]{\mathit{\{\varphi \}}}_i=-{\gamma }_i[M]{\mathit{\{\varphi \}}}_i\end{array}$$

(3)

The seismic force applied to j-th node by vibration i-th mode is:

$$F_{ji}=-{\gamma }_i{m}_j{\varphi }_{ji}$$

(4)

The base shear for the i-th order vibration pattern is:

$$V_i=-{\displaystyle \sum _{j=1}^n{F}_{ji}}={\gamma }_i{\displaystyle \sum _{j=1}^n{m}_j}{\varphi }_{ji}={\gamma }_i{\left\{\varphi \right\}}_i[M]\{I\}={\gamma }_i^2$$

(5)

For unit seismic acceleration loads, the base shear of the entire structure is numerically equal to the total mass of the structure in a given direction and can be expressed by the following equation:

$$F_j=-m_j$$

(6)

$$V=-{\displaystyle \sum _{j=1}^n{F}_j}={\displaystyle \sum _{j=1}^n{m}_j}={\{I\}}^T[M]\left\{I\right\}$$

(7)

The mass participation coefficient of the i-th order vibration mode under seismic action in the k-th direction is:

$${\gamma }_{mi}={\displaystyle \frac{{\gamma }_i^2}{m_k}}$$

(8)

where ${\gamma }_{mi}$ presents the modal mass participation factor used in GB 50011-2010 [7]. The torsional direction factor for the i-th mode can be expressed as:

$${\alpha }_{\mathrm{T}}={\displaystyle \frac{{\gamma }_{mT}}{{\gamma }_{mX}+{\gamma }_{mY}+{\gamma }_{mZ}+{\gamma }_{mT}}}$$

(9)

where ${\gamma }_{mX}$, ${\gamma }_{mY}$, ${\gamma }_{mZ}$ are the modal mass participation factors in the X, Y, and Z directions of the structure; ${\gamma }_{mT}$ is the modal mass participation factor for torsional mode around the Z axis; α_T is the torsional direction factor. If α_T > 0.5, the mode is determined to be a torsional mode.

2.2. Definition of Torsional Mode Contribution Factor

The response spectrum method was used to calculate the torsional response of the structure under coupled translational and torsional actions. The response spectrum curve defined in the “Code for Seismic Design of Buildings” GB 50011-2010 (2016 edition) [7] is given by:

$${\alpha }_j={\left({\displaystyle \frac{T_g}{T_j}}\right)}^{0.9}{\alpha }_{\max }g$$

(10)

where T_g is the characteristic period of the site; ${\alpha }_{\max }$ is the maximum seismic influence coefficient; g is the gravitational constant. The subscript j-th indicates the mode number. The response spectrum curve is shown in Figure 1.

The modal displacement due to an earthquake is given by:

$$U_j={\displaystyle \frac{{\gamma }_{tj}}{{\omega }_j^2}}{\alpha }_j{\phi }_j$$

(11)

where ${\omega }_j$ is the coupling frequency; ${\phi }_j$ is the modal shape vector; ${\gamma }_{tj}$ is the participation factor for the j-th mode, defined as ${\gamma }_{tj}={\phi }_j\left\{M\right\}{\phi }_j^T$, where $\left\{M\right\}$ is the mass matrix of the structure.

Applying the fully quadratic combination to the modal combination of U_j, the torsional coupling displacements under horizontal seismic action are:

$$U_{\mathrm{EK}}=\sqrt{{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^m{\rho }_{jk}{U}_j{U}_k}}}$$

(12)

where $U_{\mathrm{EK}}$ is the displacement of the standard value of seismic action; ${\rho }_{jk}$ is the coupling coefficient of the j-th mode and the k-th mode, which is calculated by:

$${\rho }_{jk}={\displaystyle \frac{8\sqrt{{\zeta }_j{\zeta }_k}({\zeta }_j+{\lambda }_T{\zeta }_k){\lambda }_T^{1.5}}{{(1-{\lambda }_T^2)}^2+4{\zeta }_j{\zeta }_k(1+{\lambda }_T^2){\lambda }_T+4({\zeta }_j^2+{\zeta }_k^2){\lambda }_T^2}}$$

(13)

where ${\zeta }_j$ is the damping ratio of the mode; ${\lambda }_T$ is the ratio of the self-oscillating period of the k-th mode and the vibration j-th mode, and the subscripts denote the ordinal numbers of the k-th mode.

The coupling coefficient ${\rho }_{jk}$ of the first-order torsional vibration mode is set to zero, and then a fully quadratic combination of U_j is performed. Under horizontal seismic action, the displacement, excluding the torsional effect, is:

$$U_{REM}=\sqrt{{\displaystyle \sum _{\begin{array}{l}j=1\\ j\ne j_t\end{array}}^m{\displaystyle \sum _{\begin{array}{l}k=1\\ k\ne k_t\end{array}}^m{\rho }_{jk}{U}_j{U}_k}}}$$

(14)

where $U_{REM}$ is the displacement excluding the torsional modes, and $j_t$, $k_t$ are the modes primarily characterized by torsion. The torsional mode contribution factor can be defined as:

$$C_{\mathrm{tU}}=\max \left\{{\displaystyle \frac{U_{EK}-U_{REM}}{U_{EK}}}\right\}$$

(15)

When calculating the torsional mode contribution factor, the principal directional displacement is extracted. For rectangular single-span frames, which typically have a symmetric planar layout, the mode participation factors in the principal direction are mainly contributed by the first-order translational mode and the first-order torsional mode. The principal directional displacement, calculated using the response spectrum method, can be considered by performing a complete quadratic combination (CQC) of the first-order translational mode and the first-order torsional mode. According to Equations (11) and (12), the CQC combination can be expanded as follows:

$$\begin{array}{l}{U}_{1k}={\displaystyle \frac{{\gamma }_{t1}}{{\omega }_1^2}}{\alpha }_1{\phi }_1\\ U_{3k}={\displaystyle \frac{{\gamma }_{t3}}{{\omega }_3^2}}{\alpha }_3{\phi }_3\end{array}$$

(16)

$$U_{Yk}=\sqrt{{\rho }_{11}{U_{1k}}^2+{\rho }_{33}{U_{3k}}^2+2{\rho }_{13}{U}_{1k}{U}_{3k}}$$

(17)

where U_Yk represents the principal directional displacement at node k, and U_1k and U_3k are the principal directional displacements at node k contributed by the first-order translational mode and the first-order torsional mode, respectively. ρ₁₁, ρ₃₃ and ρ₁₃ represent the autocorrelation coupling coefficients of the principal direction mode, the torsional mode, and the coupling between the principal direction mode and the torsional mode, respectively.

$$\begin{array}{l}{C}_{\mathrm{tU}}=\max \left\{{\displaystyle \frac{U_{EK}-U_{REM}}{U_{EK}}}\right\}\\ C_{\mathrm{tUy}}=\max \left\{{\displaystyle \frac{\sqrt{{U_{1k}}^2+{U_{3k}}^2+2{\rho }_{13}{U}_{1k}{U}_{3k}}-U_{1k}}{\sqrt{{U_{1k}}^2+{U_{3k}}^2+2{\rho }_{13}{U}_{1k}{U}_{3k}}}}\right\}\\ C_{\mathrm{tUy}}=\max \left\{{\displaystyle \frac{\sqrt{1+{(\frac{U_{3k}}{U_{1k}})}^2+2{\rho }_{13}\frac{U_{3k}}{U_{1k}}}-1}{\sqrt{1+{(\frac{U_{3k}}{U_{1k}})}^2+2{\rho }_{13}\frac{U_{3k}}{U_{1k}}}}}\right\}\end{array}$$

(18)

Let $\alpha ={\displaystyle \frac{U_{3k}}{U_{1k}}}$, then the following can be obtained:

$$C_{\mathrm{tUy}}=\max \left\{1-{\displaystyle \frac{1}{\sqrt{1+{\alpha }^2+2{\rho }_{13}\alpha }}}\right\}$$

(19)

As the seismic response of elastic plate bent frame structures is relatively minor, more focus should be placed on the lateral force resisting components. Thus, the connections between these components and the roof panels are used as reference points for calculating the torsional mode contribution factor.

2.3. Analytical Model

The study focuses on single-storey elastic plate bent frame structures with rectangular or quasi-rectangular building plans. These structures are widely used in public buildings, such as airport concourses and exhibition halls in convention centers. Additionally, as shown in

Table 1. Comparison of building plane failure rates under earthquake action.

Plan Shape	Rectangular	L-Shaped	Π-Shaped	Cross-Shaped	H-Shaped	Others
Damage ratio (%)	85.0	4.0	8.5	0.05	0.15	2.4

, rectangular plans exhibit more pronounced torsional seismic damage under seismic action [23,24].

Long-span buildings have complex forms, with connection methods including rigid connections and semi-rigid connections [25]. According to the “Code for Seismic Design of Buildings” (GB 50011-2010, 2016 edition) [7] and the “Code for Design of Steel Structures” (GB 50017-2017) [26], a rectangular-plan elastic plate bent frame structure was designed and analyzed, as illustrated in Figure 2. The structure is constructed using Q355B and has isotropic material properties. The spacing between transverse columns is 24 m, with a single span. The cross-section of all roof support columns measures 290 × 14, with a height of 5 m. The roof load is simplified to a permanent load of 1.5 kN/m² and a live load of 0.5 kN/m². The bottom of the support columns is rigidly connected to the ground, while the top is hinged to the roof panel.

The seismic precautionary intensity is 8 degrees (0.20 g), with a site category of type III, Group I seismic design, a seismic influence coefficient of 0.16, and a site period of 0.45 s. A bidirectional seismic analysis was conducted, and the results used the envelope of the bidirectional seismic response spectrum.

The structure’s aspect ratio, ranging from 1.5 to 3, was adjusted by modifying the number of longitudinal spans or column spacing. Roof stiffness was evaluated by varying panel thickness, assuming a 120 mm thick concrete slab with infinite in-plane stiffness set at 1.00EA. In practical applications, eccentricity typically remains below 5%, but for this analysis, the upper limit was set at 20%. Detailed parameter variations are shown in

Table 2. Parameters of the calculation model.

Calculation Parameters	Parameter
Longitudinal span × column spacing (Aspect ratio η)	3 × 12 m/4 × 12 m/5 × 12 m/6 × 12 m structures corresponding to aspect ratios of 1.5/2/2.5/3 respectively.
Panel stiffness	1.00EA (Rigid plate, 120 mm thick concrete plate)
	0.05EA (6 mm thick concrete plate)
	0.01EA (1.2 mm thick concrete plate)
	0.005EA (0.6 mm thick concrete plate)
Eccentricity	2%, 4%, 6%, 8%, 10%, 12%, 14%, 16%, 18%, 20%.

.

The Rayleigh–Ritz method was employed to determine the vibration modes of the structure, ensuring the accuracy of the period ratios. The rigid center of the case study was located at the geometric center of the slab. By applying additional surface loads to specified areas, the distribution of the floor mass was adjusted, causing the centroid to deviate from its geometric center. This setup allowed the case study to simulate torsional effects under different eccentricities, as illustrated in Figure 3. The additional load was a multiple of the original structural mass m₀. The calculation formula for the eccentric load is as follows:

$$F_{Zi}={\displaystyle \frac{8\beta m_0}{(1-4\beta )S_h}}$$

(20)

$$\beta ={\displaystyle \frac{e}{L}}$$

(21)

where $F_{Zi}$ is the distributed eccentric force additionally applied to half of the side panels; e is the eccentric distance, which is the distance between the centroid and the rigid center within the plane of the structure; $\beta$ is the eccentricity ratio, defined as the ratio of the static eccentric distance to the main dimension of the structure; L is the main dimension of the structure; $m_0$ is the total mass of the structure; and $S_h$ is half the area of the roof slab.

3. Properties of Torsional Mode Contribution Factor

3.1. Torsional Mode Contribution Factor Under Different Eccentricities

The eccentricity ratio affects the vibration characteristics of building structures and increases the torsional effect of the structure [27]. To examine the impact of the eccentricity ratio on the torsional mode contribution factor, the example of a panel stiffness of 0.01EA is used. Figure 4 shows the effect of the eccentricity ratio on the torsional mode contribution factor in elastic plate bent frame structures with different aspect ratios.

As seen in Figure 4, for small eccentricities, the torsional mode contribution factor increases with an increasing eccentricity ratio. However, for large eccentricities, the torsional mode contribution factor decreases as the eccentricity ratio increases. This indicates that when the eccentricity ratio is small, the torsional effect of the structure is relatively weak, and the seismic response is dominated by translational motion. As the eccentricity ratio increases, the center of mass and center of stiffness gradually deviate, leading to a significant increase in the torsional effect of the structure under seismic action, and the torsional mode contribution factor increases accordingly. According to Figure 5, when the eccentricity increases beyond a certain point, the coupling effect between the translational and torsional modes becomes stronger, resulting in a reduction in the mode participation factor of the torsional mode. At the same time, the mode participation factors of the other modes, primarily governed by the translational mode, increase, and the torsional component of the torsional mode shifts to other modes.

3.2. Torsional Mode Contribution Factor Under Different Aspect Ratios

According to Clause 3 of the seismic design code GB50011-2010 Code for Seismic Design of Buildings [7], “the dynamic characteristics of a structure should be similar in the two principal axes”. However, in practical engineering, when the building plan is rectangular or elliptical, the vibration periods of the two principal axes often differ due to one axis being longer than the other, resulting in significant differences in lateral stiffness in the two directions. To study the impact of the aspect ratio on the torsional mode contribution factor, the case with a panel stiffness of 0.01EA and a longitudinal column spacing of 12 m was used, with the longitudinal length varied by changing the number of longitudinal spans n. Figure 6 shows the influence of the aspect ratio on the torsional mode contribution factor of elastic plate bent frame structures under different eccentricity ratios.

From Figure 6, it can be seen that for small eccentricity ratios, the torsional mode contribution factor increases with an increasing aspect ratio, and the torsional mode contribution factor for structures with larger aspect ratios reaches its peak earlier. This indicates that the larger the aspect ratio, the smaller the lateral stiffness in the shorter direction, making the structure more susceptible to torsion.

From Figure 7, it can be observed that at large eccentricities, the torsional mode contribution factor initially increases and then decreases with the increase in eccentricity, with structures having larger aspect ratios exhibiting this decrease earlier. The reason is that as the aspect ratio increases, the mass distribution of the structure becomes more uneven under the same eccentricity ratio, resulting in a larger eccentric distance. This, in turn, enhances the coupling effect between translational and torsional modes, causing the torsional component of the torsional mode to migrate to other modes earlier (as shown in Figure 5), leading to a reduction in the torsional mode contribution factor earlier than in other cases.

3.3. Torsional Mode Contribution Factor Under Different Panel Stiffnesses

The roof panel, as a horizontal load transmitting system in spatial structures, coordinates the work of the vertical, lateral force resisting members [28]. For example, a structure with six longitudinal spans and a column spacing of 12 m (aspect ratio of 3) is used to illustrate this. Figure 8 and Figure 9 show the influence of roof panel stiffness on the torsional mode contribution factor under different eccentricity ratios.

As shown in Figure 8, for small eccentricities with a constant eccentricity ratio, the torsional mode contribution factor generally decreases as the panel stiffness increases. This indicates that at small eccentricity ratios, the flexible slab provides weaker constraints on the lower lateral force resisting members, making the structure more prone to torsion.

From Figure 9, it can be observed that for large eccentricity ratios, the torsional mode contribution factor generally increases with increasing panel stiffness, showing an overall inverse relationship between the torsional mode contribution factor and panel stiffness as the eccentricity ratio increases. This suggests that at large eccentricity ratios, the mass of the floors tends to concentrate on one side of the plane, making the translational-torsional coupling effect more significant.

As shown in Figure 10 and Figure 11, even though flexible slabs provide weaker constraints on the lower lateral force resisting members at large eccentricity ratios, the translational-torsional coupling in flexible slab structures becomes more severe. The mode participation factors for both translational and torsional components in the torsional mode are significantly smaller than those in rigid slab structures, resulting in a smaller torsional mode contribution factor for flexible slabs compared to rigid slabs at large eccentricity ratios.

3.4. Torsional Mode Contribution Factor Under Different Torsional Stiffnesses

When the structure undergoes overall torsion, its members inevitably participate in resisting torsion [29]. Taking the example of a frame structure with six longitudinal spans and a column spacing of 12 m (aspect ratio of 3), the overall torsional stiffness of the structure is adjusted by changing the corner column cross-section specifications to investigate the effect of different torsional stiffnesses on the torsional mode contribution factor. Considering the practical engineering scenario, where accidental eccentricity is often included, the eccentricity ratio in this example is set to 6%. Figure 12 shows the influence of torsional stiffness on the Torsional Mode Contribution Factor of elastic plate bent frame structures with different panel stiffnesses under an eccentricity ratio of 6%.

As can be seen from Figure 12, regardless of the panel stiffness, the torsional mode contribution factor decreases as the torsional stiffness of the structure increases (i.e., as the corner column cross-sections increase). Furthermore, the larger the panel stiffness (as seen in Section 3.3), the smaller the torsional mode contribution factor.

Extracting the first three translational modes of the frame structure, with corner column cross-section specifications of 290 × 14 and an eccentricity ratio of 6% due to accidental eccentricity, reveals that the first-order translational mode in the Y direction exhibits a noticeable translational-torsional coupling, as shown in Figure 13.

Figure 14 and Figure 15 show, respectively, the calculation of node α(U_3k/U_1k) for the torsional mode contribution factor under different torsional stiffnesses in elastic plate bent frame structures with accidental eccentricity, and the coupling coefficient between the first-order translational mode (Y direction) and the first-order torsional mode. From Figure 14, it can be seen that α decreases as the torsional stiffness increases, indicating that when the torsional stiffness is low, the displacement in the principal direction (Y direction) is mainly contributed by the first-order torsional mode. However, as the torsional stiffness increases, the displacement contribution from the first-order translational mode (translational-torsional coupling) increases. Figure 15 shows that the coupling coefficient between the torsional mode and the first-order translational mode is relatively small, and the coefficient decreases as the torsional stiffness increases.

In summary, as the torsional stiffness increases, both α and the coupling coefficient decrease, which, according to Equation (18), leads to a reduction in the torsional contribution. This indicates that when there is eccentricity in the structure, enhancing the structure’s torsional stiffness can mitigate the impact of torsional vibration modes, but it cannot eliminate the structural torsional deformation. The reason is that when the frame structure has high torsional stiffness, the displacement response in the principal direction is mainly contributed by the first-order translational vibration mode (coupling of translation and torsion), and the contribution of other modes to the displacement response is limited.

4. The Relationship Between Torsional Mode Contribution Factor and Period Ratio

When discussing the relationship between the period ratio and the torsional mode contribution factor, cases where the eccentricity ratio is zero need to be excluded. This is because, in practical engineering, it is common for structural mass and stiffness to be distributed unevenly due to construction or usage, leading to a misalignment between the center of mass and the center of stiffness, making cases with an eccentricity ratio of zero rare. Additionally, when the eccentricity ratio is zero, the structure is in a fully symmetrical state, and the torsional effect is relatively weak or negligible. Using the example of a panel stiffness of 0.01EA, the aspect ratio is varied by changing the number of longitudinal spans n. Figure 16 and Figure 17 show the relationship between the period ratio and the torsional mode contribution factor for elastic plate bent frame structures with different aspect ratios under varying eccentricity ratios.

As shown in Figure 16, for small eccentricity ratios with consistent values, the torsional mode contribution factor is positively correlated with the period ratio. Additionally, when the eccentricity ratio is less than 6%, the torsional mode contribution factor is generally below 10%, indicating that the contribution of the torsional mode to the overall effect is limited. From Figure 17, it can be observed that for large eccentricity ratios with consistent values, the torsional mode contribution factor is not necessarily positively correlated with the period ratio. Moreover, although the torsional mode contribution factor decreases as the eccentricity ratio increases, it remains above 10%, indicating that the torsional mode’s impact on the overall effect cannot be ignored.

In summary, when the eccentricity ratio is small, the contribution of the torsional mode to the overall effect is not significant, and the dynamic response of the structure is mainly contributed by the first-order translational mode (translational-torsional coupling), with limited contributions from other modes to the displacement response. Even if the period ratio meets the code limits, it may not effectively limit the structure’s torsional effect. On the other hand, when the eccentricity ratio is large, even if the period ratio meets the requirements, the contribution of the torsional mode to the overall dynamic response of the structure remains significant, which may not necessarily indicate that the structure has sufficient torsional stiffness to resist seismic loads.

5. Application of the Torsional Mode Contribution Factor

Unlike the period ratio and the drift ratio, current calculation software cannot compute the torsional mode contribution factor directly. The energy ratio of the structure can be calculated through the following steps:

(1)

Create a three-dimensional model simulating the dynamic behavior of the structure, and then obtain the mass matrix and stiffness matrix of the structure using calculation software.

(2)

Use the stiffness matrix and mass matrix to perform a response spectrum analysis of the mode shapes. Based on Equations (4)–(9), calculate the mode direction factors to determine the principal direction of each mode and identify the torsional modes.

(3)

Set all coupling coefficients related to the torsional modes to zero according to Equation (13).

(4)

Use the complete quadratic combination (CQC) method to combine the mode displacements after excluding the influence of torsional modes, thus determining the Torsional Mode Contribution Factor.

In seismic design, most structural calculation software provides mass matrices, mode vectors, and torsional coefficients. Therefore, designers only need to complete steps (2), (3), and (4).

6. Conclusions

The torsional mode contribution factor proposed in this paper effectively quantifies the contribution of torsional modes to the dynamic (displacement) response of the structure, revealing the effects of various parameters such as eccentricity ratio, aspect ratio, and roof panel stiffness on the contribution of torsional modes. The relationship between the period ratio and the torsional mode contribution factor is also explored, indicating that the period ratio may not fully reflect the torsional effect.

(1)

The contribution of torsional modes to the dynamic response of the structure varies significantly under different parameters. The proposed torsional mode contribution factor can accurately quantify the impact of torsional modes on the overall effect of the structure.

(2)

The eccentricity ratio has a significant impact on the translational-torsional coupling effect of the structure. For small eccentricities, the torsional mode contribution factor increases as the eccentricity ratio increases, but once the eccentricity ratio reaches a certain value, the torsional components of the torsional mode begin to migrate to other modes due to the translational-torsional coupling effect. At this point, the torsional mode contribution factor decreases with further increases in the eccentricity ratio but remains above 20%.

(3)

The torsional mode contribution factor reflects the nonuniformity of stiffness in the two principal directions of the structure. The larger the aspect ratio, the weaker the lateral stiffness in the shorter direction and the more prone the structure is to torsion, resulting in a higher torsional mode contribution factor.

(4)

When the eccentricity ratios are consistent and relatively small, the torsional mode contribution factor of flexible slab structures is greater than that of rigid slab structures, as flexible slabs provide weaker support constraints to the lower members, making the structure more prone to torsion. For larger eccentricity ratios, although flexible slabs provide weaker support constraints, the torsional-translational coupling effect in flexible slab structures becomes more pronounced, with both the torsional and translational components of the torsional mode being smaller than in rigid slab structures, leading to a reduction in the torsional mode contribution factor, although it remains above 10%.

(5)

When eccentricity exists in a structure, selecting an appropriate panel stiffness and aspect ratio ensures that the contribution of torsional modes remains at a lower level. It is not always the case that increasing panel stiffness or reducing the aspect ratio will result in a lower torsional mode contribution during seismic response.

(6)

When considering accidental eccentricity, increasing the torsional stiffness of elastic plate bent frame structures can reduce the influence of the torsional mode. However, at this point, the displacement response of the frame structure is mainly contributed by translational-torsional coupling modes, so it is not possible to eliminate the torsional response of the structure.

(7)

When the eccentricity ratio is small, although the torsional mode contribution factor increases as the eccentricity ratio increases, it remains below 10%, indicating that the torsional mode has a limited impact on the overall effect of the structure. In this case, the dynamic response of the structure is mainly contributed by the first-order translational mode (translational–torsional coupling), and controlling the period ratio has little significance. In cases with large eccentricity ratios, where the translational-torsional coupling is severe, the torsional mode contribution factor exceeds 10%, and the torsional mode’s impact on the overall effect cannot be ignored. However, as the eccentricity ratio increases, the period ratio may decrease, meaning that to meet code requirements, the period ratio must be adjusted by increasing the eccentricity ratio, which may, in turn, increase the dynamic response.

(8)

This study explored the influence mechanisms of various parameters on the torsional mode contribution factor. The analysis was limited to single-span, single-storey frames, and future research could extend to more complex cases, particularly those involving complex plans and multispan structures. Additionally, the current study mainly focused on elastic seismic responses; future research could explore the nonlinear behavior of the torsional mode contribution factor under elastoplastic seismic conditions to optimize seismic design further.