Dynamic Characteristics and Lateral-Torsional Vibration Response of SRC Frame With Special-Shaped Columns

Abstract

This study investigates the seismic performance and torsional responses of a 5-floor steel-reinforced concrete (SRC) structure with special-shaped columns. A three-dimensional seismic simulation shaking table test was conducted to analyze the dynamic coupling effects in lateral, torsional, and lateral-torsional directions. The results reveal distinct torsion coupling phenomena in the Y- and X-directions during high-order vibration modes. As seismic intensity increases, the free vibration frequency of the frame structure decreases steadily, while the acceleration amplification coefficient shows a tendency toward reduction. The internal damage to the structure and energy dissipation increase with higher seismic intensities. Notably, when subjected to an 0.80 g PHGA earthquake intensity level, the maximum inter-storey drift ratio attributed to lateral-torsional coupling exceeds the required elastic–plastic inter-storey drift ratio threshold (1/39). Additionally, the damage index of the frame structure, calculated using a deformation and energy parameter model, is 0.56. A novel mathematical model for lateral-torsional coupled vibration has been developed based on experimental data. The findings indicate that accidental eccentric torsion has minimal impact on the seismic performance of the model structure, with an accidental relative eccentric distance less than 0.1. These results demonstrate superior seismic performance and high torsional deformation capacity of the special-shaped column design, offering valuable insights for improving earthquake-resistant structural designs in modern buildings.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Due to the promising functionality and practicality of special-shaped column structures, extensive research has been conducted over recent years on the mechanical properties and construction techniques of RC buildings with custom-designed columns. Prior studies have significantly advanced our understanding of the seismic performance of RC constructions [1–8]. However, despite these advancements, there still exists a critical gap in understanding the dynamic characteristics and seismic response laws of steel-reinforced concrete (SRC) special-shaped column frame structures.

While previous research has focused primarily on domestic studies [9], and while planar frames or special-shaped column members from SRC have often been subjects of fictitious static or dynamic tests, there remains a paucity of studies on the actual dynamic properties and seismic response laws of SRC special-shaped column frame structures. This gap is particularly evident in understanding how the stiffness center and frequency ratio of the SRC special-shaped column frame structure change over time under 3D earthquake loads and their influence on the elastoplastic response. Furthermore, for SRC special-shaped columns with cross-shaped, T-shaped, and L-shaped asymmetric sections, the interaction between bending and twisting under horizontal loads acting on the centroid axis of the asymmetric axis remains inadequately explored [10].

The dynamic characteristics, torsional resistance, and dynamic eccentricity of special-shaped column members have been identified as crucial factors influencing the seismic performance of constructions utilizing SRC structures. Recent theoretical advancements in structural mechanics, particularly those related to nonlinear dynamic analysis and 3D earthquake load effects, have provided new insights into the vibration response and dynamic coupling phenomena in SRC special-shaped column frame structures [11]. These studies highlight the importance of considering the time-dependent changes in the stiffness center and frequency ratio when evaluating the seismic performance of such structures. Additionally, recent works on the interaction between bending and twisting in RC beams with special cross-sections have enhanced our understanding of the torsional response under static and cyclic loading [12, 13]. Despite these progressions, however, there remains a lack of comprehensive studies that integrate advanced theoretical frameworks with numerical modeling to predict the vibration response and dynamic coupling effect in SRC special-shaped column frame structures. This gap in knowledge is particularly critical for the development of design guidelines and seismic retrofitting strategies for such structures.

This study aims to address these research gaps by incorporating recent advancements in nonlinear dynamic analysis, 3D earthquake load effects, and the interaction between bending and twisting in RC beams with special cross-sections. The results are expected to provide new insights into the vibration response and dynamic coupling phenomena of SRC frame structures with special-shaped columns, ultimately contributing to the enhancement of their seismic performance and durability.

2. Test Survey

2.1. Model Design

For the purpose of determining the displacement response, acceleration response, and strain response of the SRC frame with a special-shaped column under 3D earthquake, based on the dynamic similarity connection, a structural model for a five-storey solid web SRC spatial frame with a special-shaped column at a similarity ratio of 1:1 is created. The selection of similarity relationships should comprehensively consider the performance parameters of the shaking table, the vibration characteristics of the actual structure, seismic codes, construction conditions, and lifting capabilities. The similarity relationships of the model structure are shown in Table 1. During testing, the similarity relationships of the model were appropriately adjusted based on the measured values of the strength and elastic modulus of the micro-concrete and steel, as well as the additional mass of the model.

Table 1

Similarity coefficient.

Type	Physical parameter	Similarity coefficient		Type	Physical parameter	Similarity coefficient
Geometric properties	l	C l	1/4	Dynamical properties	m	C m  = CρCl3	1/36
	θ	C θ  = 1	1		k	C k  = CECl	1/6
	δ	C δ = Cl	1/4		c	C c  = CECl3/2	1/12
Material properties	E	C E	1/1.5		T	C t  = Cl1/2	1/2
	ρ	C ρ	1/0.56		f	C f  = Cl−1/2	2
	ε	C ε  = 1	1		v	C v  = Cl1/2	1/2
	σ	C σ  = CE	1/1.5		a	C a  = CE/ClCρ	1/0.67
Load properties	F	C F  = CECl2	1/24	Model mass			17.33 t
	q	C q  = CECl	1/6
	p	C p  = CE	1/1.5	Model height			4.2 m

The model structure is symmetrically arranged in the plane. To be specific, the frame has a span length of 1000 mm, the ground floor has a height of 900 mm, and the height of the second to fifth floors is 750 mm, as presented in Figure 1. The lateral force resisting members of the structure comprises L-, T-, and cross-shaped SRC special-shaped columns. As depicted in Figure 2, Q235 steel serves as the column steel, HPB300 grade reinforcement is the stressed reinforcement, the stirrup is made of 10#3.0 galvanized iron wire, while the concrete is made of C30 particle concrete. The frame beam is a reinforced concrete rectangular beam with a section size of 60 mm × 140 mm. Moreover, the slab is a 40 mm thick reinforced concrete cast-in-situ slab. The model is fully weighted, and the weight of the respective floor is 1.6 t. The axial compression ratio of the middle column on the bottom floor under vertical load is 0.11. Mechanical properties of steel are shown in Table 2. After the frame model is fabricated and cured, as shown in Figure 3, it is raised into position and fastened to the shaking table.

[figure(s) omitted; refer to PDF]

Table 2

Steel performance.

t (mm)	E/ × 105 (MPa)	f y (MPa)	ε y /10−6	f u (MPa)	ε u/10−6	δ (%)
10	1.72	409.8	2382	566.2	3292	53.0
8	2.08	328.0	1577	477.3	2294	53.7
6	2.09	332.9	1593	487.6	2333	53.6
3	2.07	321.6	1553	457.2	2209	46.0

2.2. Shaking Table Test

In the test, the El Centro wave, Taft wave, and Lanzhou artificial wave are selected as the excitation waveform of the mesa and the peak acceleration input of the mesa is adjusted at the 1:0.67 similarity ratio. There are 60 working conditions (e.g., white noise) after adjustment, and the peak acceleration is set as follows: 0.07, 0.14, 0.20, 0.40, 0.62, 0.80, 0.90, 1.0, and 1.1 g. In the test of the same earthquake grade, at a 1:2 similarity ratio, the time period is changed. The seismic waves in the respective region are input in one direction, two directions, and three directions, respectively. The X, Y, and Z three-way acceleration ratio is 1:0.85:0.65. During the test, the dynamic response of the model structure is examined using the accelerometer, the displacement pickup, and the strain sensor. The arrangement of measuring points is illustrated in Figure 4.

[figure(s) omitted; refer to PDF]

2.3. Test Result

The acceleration response and displacement response of the top layer of the model in X-, Y-, and Z-directions are presented in Figures 5 and 6.

[figure(s) omitted; refer to PDF]

Three phases make up the model structure’s reaction to a seismic wave acting continuously. Before 0.20 g, it is the elastic stage. 0.40–0.80 g represents the elastic–plastic stage. At the elastic–plastic stage, long and thin cracks appear on the flange of the upper column ends with the A1 axis and C3 axis L-shaped corner columns on the 1^st floor, and the cracks at the beam ends of the KL9 and KL10 frames develop upward and downward, respectively, and small diagonal cracks appear at other beam ends (Figure 7a). Multiple horizontal cracks of unequal length appear evenly from top to bottom on the webs with L-shaped corner columns of Axis A1 and C3 (Figure 7b). 45° diagonal crack was found at the connection between the A3 axis L-shaped corner column and beam on the 3^rd floor, and the reinforcement compressive strain at the end of the KL10# beam on the 3^rd-floor side span reaches the measuring limit. After 0.90 g, it represents the plastic stage. At this stage, the intersection of the B1 axis T-shaped side column and beam on the first floor has microcracks developing towards the panel point (Figure 7c). A cross crack is formed at the joint of A3 axis L-shaped corner column and beam on the third floor. The L-shaped corner column of axis A1, A3, C1, and C3 on the 1^st and 2^nd floors is connected with the crack (Figure 7d). After 1.0 g, the cracks at the junction of corner column and side column and beam on the 1^st to 3^rd floors intersect and connect; stress failure of the reinforcement at KL10 beam end on the 3^rd floor occurs.

[figure(s) omitted; refer to PDF]

3. Dynamic Characteristics

3.1. Natural Vibration Characteristic

The frequency response function (transfer function) of each floor is determined through frequency domain analysis of the excitation signal and the output signal. The amplitude–frequency curve and phase frequency curve of the model can be generated using the transfer function. Figures 8 and 9 illustrates the X-direction and XY-direction acceleration amplitude–frequency curve at the top measurement point with respect to the base table after all prior white noise inputs.

[figure(s) omitted; refer to PDF]

The distribution of the lateral-torsional coupled frequency with the model structure is presented in Figure 10. According to the analysis of the model structure’s amplitude–frequency curve, the natural frequencies of the model in the X, Y, Z, and torsional directions decrease as the seismic input acceleration peak value rises. In contrast, each order’s frequency values of the model structure tend to move forward. Besides, the model structure is Y and X’s lateral-torsional coupled successively in the higher-order vibration modes. The inherent frequency of the structure gradually decreases after the 0.20 g PHGA seismic wave injection. Specifically, the 0.40 g PHGA seismic wave causes a sharp decline in the model’s natural vibration frequency. First- and second-order frequencies in the X- and Y-directions decline by 18.18%, 19.42%, 18.95%, and 12.61%, respectively, in comparison to those before the earthquake. First- and second-order frequencies in the Z-direction and torsion declined by 23.60%, 19.02%, 16.01%, and 18.31%, respectively, in comparison to those before the earthquake. The crack of the beam and column develops rapidly, and the damage degree of the structure increases significantly. Last, the natural frequencies of the model in X, Y, Z, and torsion slowly decrease after the 0.8, 0.9, and 1.0 g PHGA seismic wave input in turn. After the test, the overall stiffness only accounts for 45.43% and 47.36% of that before the earthquake. Furthermore, the structure is damaged seriously.

[figure(s) omitted; refer to PDF]

3.2. Lateral-Torsional Vibration Characteristic

The transfer function curve’s half-power approach is used to determine the model structure’s damping ratio [14], which reflects the energy dissipation characteristics of the structure. And the transfer function curve is used to determine the model structure’s vibration shape, the amplitude–frequency curve and phase–frequency curve of each floor relative to the base platform can be obtained by using the transfer function [14]. After the normalization of the amplitude ratio, the vibration mode corresponding to the natural frequency can be obtained. The entire distribution of the natural frequency, damping ratio, and vibration mode for the model of a SRC structure with a custom-shaped column subjected to a 3D seismic event is presented in Table 3. Where ζ denotes the damping ratio, Φ(i) expresses the vibration form.

Table 3

The model structure’s naturally occurring frequency, damping ratio, and vibration mode shape.

Mode order		1st	2nd	3rd	4th	5th
Before test1 WN	f (Hz)	7.305	8.047	9.337	14.537	31.289
	ζ	0.037	0.032	0.012	0.012	0.030
	Φ(i)	X translation	Y lateral-torsional	Z-direction	Torsion	Y translation
After 0.20 g31 WN	f (Hz)	6.875	7.305	7.59	12.525	28.793
	ζ	0.061	0.051	0.050	0.088	0.047
	Φ(i)	X translation	Y lateral-torsiona	Z-direction	Torsion	Z-direction
After 0.40 g41 WN	f (Hz)	5.977	6.484	7.134	11.875	26.055
	ζ	0.088	0.066	0.060	0.096	0.083
	Φ(i)	X translation	Y lateral-torsiona	Z-direction	Torsion	X lateral-torsiona
After 0.62 g51 WN	f (Hz)	5.430	5.977	6.706	10.012	23.555
	ζ	0.086	0.065	0.083	0.134	0.077
	Φ(i)	X translation	Y lateral-torsiona	Z-direction	Torsion	X translation
After 1.00 g59 WN	f (Hz)	5.039	5.625	5.629	7.984	21.914
	ζ	0.155	0.109	0.222	0.231	0.097
	Φ(i)	X translation	Y lateral-torsiona	Z-direction	Torsion	X translation

3.3. Spectral Characteristic

Based on the vibration pattern of the multistory SRC frame structure with a special-shaped column during a horizontal earthquake, as depicted in Figure 11, the multistory SRC frame structure with a special-shaped column is simplified as a damping shear structure with multiple free degrees. The motion equation is:$$\begin{array}{}\text{(1)}& M\ddot{u}+C\dot{u}+Ku=-M{\ddot{u}}_g\delta ,\end{array}$$where [M], [C], and [K] are the mass, damping, and stiffness matrices of the structure, respectively; u is the response time history of X-direction floor displacement; ${\ddot{u}}_g$ represents the ground acceleration time history; ${\ddot{u}}_g\delta ={1\cdots 1}^T$ is the unit column vector.

[figure(s) omitted; refer to PDF]

Equation (1) is transformed by using Laplace transform. According to the orthogonality of vibration modes and the introduction of proportional damping, the frequency response function matrix $Hs=-M{s}^2+cs+k^{-1}M\delta$ is converted as follows [15]:$$\begin{array}{}\text{(2)}& Hs=\Phi \text{diag}{\Delta }_i{\Phi }^{-1}\delta ,\end{array}$$where [$\Phi$] is the vibration mode matrix; S is a complex variable; α represents the mass damping coefficient; β denotes the stiffness damping coefficient; ω expresses the circular frequency; j is an imaginary unit; ${\Delta }_i={\omega }^2-{\omega }_i^2+\alpha +\beta {\omega }_i^{2}j\omega /{{\omega }^2-{\omega }_i^2}^2-{\alpha +\beta {\omega }_i^2}^2{\omega }^2$.

Equation (2) suggests that the frequency response function curve of the test model can be obtained by Laplace transformation with the displacement from the base to the top layer of the test model. Figure 12 depicts the real part, imaginary part, and amplitude curve of the frequency response function in the X-direction of the model structure under a variety of working conditions. With the extension of the loading duration, the zero point of the real part curve, the peak point of the imaginary part, and the amplitude curve tend to reach a low frequency, thus, revealing that the continuous loading leads to the gradual accumulation of structural damage, the continuous degradation of stiffness, and the continuous reduction of the natural frequency. To facilitate the observation, the frequency response function curve is standardized by MATLAB. Figure 13 depicts the 1st-order standardized natural frequency of the structure under 1 WN to 57 WN working conditions.

[figure(s) omitted; refer to PDF]

4. Lateral-Torsional Vibration Response

4.1. Lateral Vibration

Figure 14 depicts the model’s acceleration amplification coefficient envelope diagram and inter-storey displacement envelope diagram. As shown in the graph, with the increase of the acceleration input peak value in the shaking table, the acceleration amplification coefficient continues to decrease. The inter-storey displacement is shown as the second floor, third floor, first floor, fourth floor, and fifth floor, with the inter-storey displacement rising in decreasing order. The first and second floors of the model enter the elastic–plastic stage at 0.40 g PHGA seismic wave input and because the inter-storey displacement increases more than that of the third to fifth floors, S-shaped is the overall transverse displacement trajectory of the structure. The structure reaches the elastic–plastic stage with a seismic wave input of 0.80 g PHGA. The second level’s second floor has a maximum value of 1/39 for the inter-storey displacement angle. The model’s inter-storey rigidity progressively deteriorates when plastic hinges form at the third floor’s beam ends.

[figure(s) omitted; refer to PDF]

4.2. Torsional Vibration

Figure 15 presents the torsion angle envelope diagram of the respective floor relative to the base. As depicted in the figure, with the increase of the peak acceleration, the total torsional angle of each layer increases evenly. Until the end of loading, the total torsional angle of the top layer is 22.4 × 10⁻⁷ rad. In general, the model has a significantly small inter-storey torsional angle, demonstrating the great torsional rigidity of the model construction. For the SRC special-shaped column structure with symmetrical arrangement and height meeting the specification requirements, the accidental eccentric torsion of the model is not significant.

[figure(s) omitted; refer to PDF]

4.3. Lateral-Torsional Coupled Vibration

According to the analysis’s findings for the model’s dynamic properties, the structure’s higher-order mode is where the lateral-torsional linked vibration occurs. Based on the results of the shaking table test, the eccentric rotation angle component is substituted into the displacement vector [u] in the motion Equation (1) of the model structure, and the rotation component is substituted into the seismic acceleration vector ${\ddot{u}}_g$. The secular equation, $K-{\omega }^{2}m=0$, is solved to simplify the calculation when only the seismic action in the main vibration direction of X-direction and the eccentricity in Y-direction are considered.

Subsequently, the natural frequency of the eccentric structure is expressed as follows:$$\begin{array}{}\text{(3)}& {\omega }_{1,2}^2=\frac{{\omega }_x^2+{\omega }_{\phi }^2}{2}\pm \sqrt{\frac{{\omega }_x^2-{\omega }_{\phi }^2}{2}+\frac{e_y^2}{r^2}{\omega }_x^4},\end{array}$$where ω_x and ω_φ denote the natural frequencies of the structural system considering the effect of eccentricity, ${\omega }_x=\sqrt{k_x/m}$ and ${\omega }_{\phi }=\sqrt{k_{\phi }/J}$. Where K_φ represents the combined torsional stiffness considering the effects of ground rotation and eccentricity, which is written as follows:$$\begin{array}{}& K_{\phi }=\sum k_{yj}{x}_j^2+\sum k_{xi}{y}_i^2+\sum G{I}_t/h=k_{\phi }^e+\sum G{I}_t/h=k_{\phi }^e+k_{\phi }^t.\end{array}$$

The lateral-torsional coupling coefficient is written as follows:$$\begin{array}{}& {\rho }_{12}=\frac{0.02{{\omega }_1/{\omega }_2}^{1.5}}{1+{\omega }_1/{\omega }_2{1-{\omega }_1/{\omega }_2}^2+0.01{\omega }_1/{\omega }_2},\end{array}$$$$\begin{array}{}\text{(4)}& \frac{{\omega }_1^2}{{\omega }_2^2}=\frac{{T_{\phi }/T_u}^2+1+{e_y/r}^2{T_{\phi }/T_u}^2-\sqrt{{{T_{\phi }/T_u}^2+1+{e_y/r}^2{T_{\phi }/T_u}^2}^2-4{T_{\phi }/T_u}^2}}{{T_{\phi }/T_u}^2+1+{e_y/r}^2{T_{\phi }/T_u}^2+\sqrt{{{T_{\phi }/T_u}^2+1+{e_y/r}^2{T_{\phi }/T_u}^2}^2-4{T_{\phi }/T_u}^2}},\end{array}$$where $T_{\phi }^2/T_u^2=1/T_u^2/T_{\phi }^{e^2}+T_u^2/T_{\phi }^{t^2}$, T_φ^e/T_u is the period ratio of uncoupled translation-torsion considering only the effect of eccentricity; T_φ^t/T_u expresses the period ratio of translation and torsion caused by ground rotation.

The relative torsional effect index, or φr/u, is used to assess the structure’s torsional impact. The combined φr/u is stated as follows when the period is in the falling portion of the response spectrum:$$\begin{array}{}\text{(5)}& \frac{\phi r}{u}=r\frac{\sqrt{{\phi }_1^2+{\phi }_2^2+2{\rho }_{12}{\phi }_1{\phi }_2}}{\sqrt{u_1^2+u_2^2+2{\rho }_{12}{u}_1{u}_2}}=\Phi e_y/r,T_{\phi }/T_u.\end{array}$$

The variation of the 1st and 2nd-order structural natural frequency ratio ω₁/ω₂ is shown in Figure 16. As depicted in Figure 10a, when T_φ^t/T_u is higher than or equal to 2.0 and T_φ^e/T_u is equal to 0.8, T_φ^t/T_u is lower than or equal to 1.0, and T_φ^e/T_u is equal to 1.25, the 1st and 2nd order natural frequencies ratio of the structure appears a peak point. As depicted in Figure 10b, when the eccentricity is fixed and the translation-torsion period ratio caused by ground rotation is constant, the peak point of the natural frequency ratio of the structure considering the effect of ground rotation is significantly delayed compared with the peak point only considering the effect of eccentricity.

[figure(s) omitted; refer to PDF]

In Figure 17, the theoretical torsion effect and test value of the Y-direction lateral-torsional coupling of the SRC special-shaped column frame structure are compared. In contrast, the SRC special-shaped column frame structures have a maximum ratio of 0.0007 between the maximum torsional displacement at the bottom level and the horizontal displacement in the Y-direction. Furthermore, the spatial symmetric frame structures with SRC special-shaped columns have a relative dynamic eccentricity in the Y-direction that is smaller than 0.1.

[figure(s) omitted; refer to PDF]

5. Mechanical Behavior

5.1. The Law of Seismic Response

While the model structure’s largest total displacement angle is 1/23 rad under the effect of 0.40 g PHGA seismic. According to the Technical Specification for Concrete Special Shaped Column Structures (JGJ149-2006), the inter-storey displacement angle induced by lateral-torsional coupling may be as large as 1/67 rad and the inter-storey displacement angle caused by lateral torsional coupling can be as large as 1/67 rad. This satisfies the requirements of the elastic–plastic inter-storey displacement angle limit of 1/60 rad. The inter-storey displacement angle distribution in descending order is 2nd floor, 3rd floor, 1st floor, 4th floor, as well as 5th floor. According to Figure 18, under the effect of 0.80 g PHGA seismic, the maximum unintentional eccentric torsional angle and total displacement angle of the model structure are 1/12 and 17.0 × 10⁻⁷ rad, respectively. A model structure enters the completely plastic stage when the inter-storey displacement angle generated by lateral-torsional coupling reaches a maximum value of 1/39 rad. The deterioration of the floor’s torsional rigidity is not appreciably accompanied by an increase in horizontal lateral displacement, according to the study of the model structure’s stiffness and seismic force.

[figure(s) omitted; refer to PDF]

5.2. Seismic Force

The shear force of the floor is the sum of the inertial force with all floors above this floor, which is written as follows:$$\begin{array}{}\text{(6)}& V_j=\sum _{i=j}^n{m}_i{a}_i,\end{array}$$where V_j is the floor shear force of the jth floor; m_i is the equivalent mass of the ith floor; α_i expresses the acceleration response of the ith floor; n represents the floor number of the model.

The shear envelope diagram of the floor is shown in Figure 19. As depicted in the figure, the first floor’s maximum shear force, measured between 0.07 and 0.20 g PHGA seismic, is close to 50 kN, and the structure is at its elastic stage. With a rise in peak acceleration, the floor shear force drops dramatically. Following a 0.80 g PHGA seismic input, the model structure’s plastic deformation gradually deepens as a result of an increase in floor shear force and the total amount of damage steadily rises.

[figure(s) omitted; refer to PDF]

5.3. Energy Dissipation

The inter-storey hysteretic energy dissipation of the model structure under the seismic action of jth working condition is calculated as follows [16]:$$\begin{array}{}\text{(7)}& E_{jk}{t}_i=\sum _{i=1}^m\frac{1}{2}{V}_k{t}_i+V_k{t}_{i-1}{x}_k{t}_i+x_k{t}_{i-1},\end{array}$$where E_jk (t_i) denotes the cumulative hysteretic energy consumption of the kth floor at time t_i under jth working condition; V_k(t_i), V_k(t_i−1) is the inter-storey shear force at time t_i and t_i−1 under jth working condition; x_k(t_i), x_k(t_i−1) represents the inter-storey displacement at time t_i and t_i−1 under jth working; m is the total number of sampling points.

Figures 20 and 21 depict the time history curves of the cumulative hysteretic energy dissipation between tales and the overall cumulative hysteretic energy dissipation for the model structure, respectively. As shown in the figures, the unrecoverable plastic deformation energy with a substantial transition in a very little time period makes up the majority of the hysteretic energy loss when the model structure reaches the elastic–plastic stage. The total hysteretic energy dissipation time history curve’s transition time advances as the peak acceleration rises, aggravating the cumulative damage and reducing the amount of time needed for the structure to reach the plastic stage.

[figure(s) omitted; refer to PDF]

5.4. Degree of Damage

By comparing the differences in the frequency response (Figure 12) under various working circumstances, the model structure’s level of uniaxial damage is evaluated [14]. The uniaxial damage degree of the model structure can be calculated as:$$\begin{array}{}\text{(8)}& D_j=\frac{\sqrt{{I_j^R-I_0^R}^2+{I_j^I-I_0^I}^2}}{\sqrt{{I_0^R}^2+{I_0^I}^2}},\end{array}$$where $I_j^R={\int }_0^{\omega }{f}_j^R\omega f_0^R\omega d\omega$ is adopted to measure the similarity between the real parts of the two frequency response functions with jth state and structural health. For the imaginary part, it yields $I_j^I={\int }_0^{\omega }{f}_j^I\omega f_0^I\omega d\omega$. Based on the double-parameter of deformation and energy [17], the damage model of the structure under 2D seismic is expressed as follows:$$\begin{array}{}\text{(9)}& D=D_x+D_y-A\min D_x,D_y,\end{array}$$where A denotes the biaxial damage coupling factor (0 < A < 1). According to the analysis with considerable test data, A is taken as 0.58; D_x and D_y represent the damage indicators of the structure in the X- and Y-directions, respectively.

The damage index of the frame structure under various operating circumstances is shown in Table 4. When seen in the table, when the seismic peak acceleration input increases, the damage index of the model structure initially rises quickly and then gradually. The steel completely demonstrates excellent deformation ability in the later loading phase and the plastic deformation of the model structure grows slowly. These three factors are the main causes of this outcome. Plastic deformation of the model structure develops quickly during the early loading stage. The variety rule is consistent with the structural failure state observed under a wide variety of working conditions during the test.

Table 4

Damage index of the frame.

Working condition	Damage index	Working condition	Damage index
Before test	0	After 0.40 g PHGA seismic	0.38
After 0.07 g PHGA seismic	0.05	After 0.62 g PHGA seismic	0.48
After 0.14 g PHGA seismic	0.17	After 0.80 g PHGA seismic	0.56
After 0.20 g PHGA seismic	0.30	After 1.00 g PHGA seismic	0.63

6. Conclusion

The dynamic characteristics and lateral-torsional vibration response of the SRC spatial frame structure with special-shaped columns are investigated using shaking table testing and numerical analysis. The following are the conclusions:

1. The research reveals that SRC frames with special-shaped columns exhibit sequential Y-direction lateral-torsional coupling and X-direction lateral-torsional coupling under high-order vibration modes. The natural frequencies at the first, second, third, and fourth orders are calculated to be 7.305, 8.047, 14.537, and 31.289 Hz, respectively. The frequency response function of the model structure demonstrates that, with increasing load holding time, the zero-point and imaginary part tend to stabilize at a low-frequency state, while the real part curve peaks at intermediate frequencies. These findings suggest that the dynamic characteristics of such structures are influenced by both static and transient loading conditions.

2. The analysis indicates that under strong seismic loads (0.80 g PHGA earthquake), the inter-storey stiffness of the structure rapidly degrades, leading to an S-shaped overall lateral displacement curve. The maximum unintentional eccentric torsional angle is calculated as 17.0 × 10⁻⁷ rad and the corresponding total displacement angle reaches rad. Furthermore, the highest inter-storey displacement angle caused by lateral-torsional coupling is 1/39 rad, exceeding the elastic–plastic inter-storey displacement angle limit, highlighting potential vulnerabilities in the structure’s antisismic design.

3. The research establishes a mathematical model based on theoretical analysis of lateral-torsional coupled vibration and finds that the natural frequency is influenced by relative eccentricity, the ratio of the translation-torsion period induced by ground rotation and the ratio of the translation-torsion period considering only eccentric torsion. Importantly, accidental eccentric torsion has a limited impact on the seismic performance, with an accidental relative eccentric distance less than 0.1 cm. The results also show that as peak acceleration increases, the internal forces at corresponding floors initially increase steadily before stabilizing and subsequently degrading, indicating a complex nonlinear stress distribution.

4. The dual-parameter damage index of reflects the structure’s overall energy and deformation state, while simultaneously revealing significant hysteretic energy dissipation due to increasing plastic deformation during cyclic loading. This analysis demonstrates enhanced torsional deformation resistance and high seismic performance of the tested structure.

Disclosure

This work described was original research that has not been published previously, and not under consideration for publication else where, in whole or in part.

Author Contributions

All the authors listed have approved the manuscript that is enclosed.

Acknowledgments

The financial assistance was provided by the Shaanxi Province Natural Science Basic Research Project under Grant 2023-JC-YB-419, the Postdoctoral Science Foundation of China under Grant 2020M683432, and the Natural Science Foundation of China under Grant 51308444. These supports are gratefully acknowledged.

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