Theoretical Research and Shaking Table Test on Nominal Aspect Ratio of the Isolated Step-Terrace Structure

Abstract

With the installation of rubber isolation bearings in the upper and lower ground layers, an isolated step-terrace structure was created. Considering the ultimate bearing capacity of the rubber bearing under tension as the critical condition, a comprehensive framework was established to evaluate the overturning failure mechanisms present in isolated step-terrace structures. The bound of nominal aspect ratio was identified as the principal control index within this framework, which incorporates critical parameters such as height ratio (α), width ratio (β), vertical tensile stiffness to compressive stiffness ratio (γ), seismic coefficient (k), and nominal vertical compressive stress (σ₀) to provide a thorough analysis of the structural responses and potential failure scenarios. In order to further investigate this matter, a scaled model of an isolated step-terrace concrete frame structure featuring two dropped layers and a single span within an 8° seismic fortification zone was meticulously crafted at a 1:10 similarity ratio. Subsequently, a series of shaking table tests were conducted to analyze the structural response under seismic excitation. The findings indicate that: utilizing the bound of nominal aspect ratio as a metric to gauge the anti-overturning capacity of isolated step-terrace structures is a justified approach. In instances where the height ratio remains constant, the bound of nominal aspect ratio for both positive and negative overturning trended upward with an increase in the width ratio. Notably, the bound of nominal aspect ratio for positive overturning consistently registered lower values compared to that of the negative overturning, underscoring the heightened susceptibility of step-terrace structures to positive overturning. Moreover, in scenarios characterized by higher height and width ratios, the structural integrity remained unscathed by any overturning effects arising from insufficient tensile strength in rubber bearings. Furthermore, the bound of nominal aspect ratio exhibited an ascending trend as the seismic coefficient, nominal vertical compressive stress, and vertical tensile stiffness to compressive stiffness ratio decreased. The outcomes derived from the shaking table test not only confirm the impressive seismic performance of the structure, but also, by closely examining the instantaneous stress variations within the upper and lower isolation layers of the model, substantiate the existence of a positive overturning hazard in scenarios marked by higher seismic coefficients (k). This observation aligns seamlessly with the theoretical projections, thereby substantiating the efficacy of the structural overturning failure theory through direct empirical verification.

1. Introduction

In China, land resources are scarce, with vast areas covered by mountainous and semi-mountainous regions. Despite the government’s high regard for the protection of land resources, the country still faces severe land scarcity [1]. The utilization of land resources has reached its limits, and the rational development and utilization of sloping terrain can effectively alleviate the pressure on land use for construction purposes. When constructing buildings on sloping terrain, excavating mountains and filling ravines can not only increase construction costs, but also potentially trigger a series of issues related to transportation, environment, and geological hazards. Mountainous constructions, known for their strong adaptability to terrain and minimal environmental impact, are widely utilized in sloping areas [2]. Simultaneously, earthquakes represent a significant and abrupt natural calamity that humanity must confront. Typically, earthquakes have the potential to swiftly result in casualties, structural damage, including the collapse of buildings, roads, and bridges, as well as trigger secondary disasters such as water-related incidents, fires, and disease outbreaks, resulting in substantial losses for affected populations [3,4,5,6,7]; even many years later, the environment and health are still affected [8]. Unfortunately, China is a country prone to frequent earthquakes, with significant seismic disasters. Due to the inherent vertical irregularities in mountainous constructions, their seismic resistance is relatively poor, posing seismic safety risks in earthquake-prone regions [9].

Isolation technology is a vibration control method that prolongs the structural period and dissipates seismic energy [10,11,12]. Both practical applications and theoretical studies have shown that isolation technology is one of the most effective measures to enhance the seismic performance of structures [13,14,15]. Currently, vibration isolators are typically categorized into five main types: linear natural rubber bearings (LNRs), lead rubber bearings (LRBs), high-damping rubber bearings (HDRs), elastic sliding bearings (ESBs), and friction pendulum isolation bearings (FPSs) [16]. To enhance performance, these isolation devices can be augmented with dampers for improved effectiveness. To enhance the seismic response of reinforced concrete buildings, Belbachir et al. [17] utilized a novel parallel arrangement of HDRs in conjunction with fluid viscous dampers (FVDs). Current research on isolation technology has shifted from low-rise conventional buildings to high-rise and complex structures.

Due to variations in topographical elevation and the pronounced seismic vulnerability of the region, the No. 4 Middle School of Qiaojia County in Yunnan Province ingeniously incorporated seismic isolation technology within the RC frame structure supported by foundations with different elevations [18]. This strategic implementation aimed at realizing a harmonious blend of cost effectiveness, structural integrity, and functional suitability. Zhang et al. [19] introduced the notion of an isolated step-terrace structure and demonstrated its exceptional seismic performance through a shaking table test. Subsequently, Yang et al. [20] employed LY-DYNA to systematically investigate the progressive collapse of isolated step-terrace structure. The findings indicated that the isolated step-terrace structure not only showcases robust adaptability to varied terrains and minimal environmental impact, but also leverages rubber isolation technology to mitigate the vertical irregularities inherent in mountain constructions. Consequently, the seismic resilience of the structure is significantly enhanced. The isolated step-terrace structure is illustrated in Figure 1.

The occurrence of overturning failure represents a critical form of global instability and collapse in structures, leading to severe consequences upon manifestation. As such, it is imperative to prevent overturning failure in the design of isolated step-terrace structures. The aspect ratio limit serves as a crucial parameter for evaluating the anti-overturning capacity of isolated structures [21]. Li and Wu [22,23] comprehensively investigated the influence of vertical seismic motion, site characteristics, seismic intensity, isolation duration, stiffness of isolation layers, stiffness of upper structures, and positioning of rubber bearings on the aspect ratio limit of isolated structures and they developed a simplified calculation formula to determine the aspect ratio limit of such structures. Wang et al. [24,25] conducted shaking table tests on a base-isolation model with a similarity ratio of 1:5 and an aspect ratio of 3.1, and the results showed that rubber isolation structures in high-intensity areas may experience overturning. Hino et al. [26] found that the bound of the aspect ratio is related to the ultimate state of the tensile stress of rubber bearings, ultimate state of drift of the base-isolation story, and the prediction of the overturning moment under seismic conditions, as well as to the ultimate state of the axial compressive stress of rubber bearings under dead loads. Based on the Eurocode eight response spectra and seven near-fault ground motion records, Petrol et al. [27] obtained maximum allowable aspect ratios for different vibration periods, ground conditions, and design ground accelerations. Qi et al. [28] developed an explicit calculation formula for the aspect ratio limit of isolated structures and provided aspect ratio limits for different buildings. He et al. [29] conducted shaking table tests on a large aspect ratio base-isolation structure with a similarity ratio of 1:4, and the results showed that high-intensity seismic motion may cause the rubber bearings to buckle under tension. On the basis of complex modal method, Lai et al. [30], considering the effect of the overturning kinetic energy of the superstructure, derived a dynamic calculation formula of the limit value of the aspect ratio limit.

Although many scholars have conducted extensive theoretical and experimental research on the overturning effects of isolated structures and have achieved some significant research outcomes, particularly in the context of foundation isolation structures, the significance of guidance on preventing overturning effects for isolated step-terrace structures is limited. As such, this research is dedicated to the in-depth analysis of isolated step-terrace structures, with a comprehensive investigation into the underlying mechanisms that contribute to their overturning failure. This examination involves the introduction of parameters such as the vertical tensile stiffness to compressive stiffness ratio, height ratio, width ratio, and the length ratio of the tension zone to the compression zone. Subsequently, the study establishes a control index for the nominal aspect ratio of the isolated step-terrace structure. The efficacy of the overturning failure theory for isolated step-terrace structures is then empirically validated through a series of rigorous shaking table tests.

2. Mechanism of Overturning Failure of Isolated Step-Terrace Structures

2.1. Assumptions

To streamline the theoretical derivation process, the following assumptions were posited prior to investigating the overturning failure mechanism of isolated step-terrace structures:

(i) The slope is considered to exhibit stability akin to that of a rock slope;

(ii) Uniform distribution of mass and stiffness characterizes the structure;

(iii) Factors such as ground motion amplification induced by the slope and vertical seismic effects are excluded from consideration.

According to the above assumptions, the drop layer isolation structure can be simplified into a two-dimensional isolation system, as shown in Figure 2.

In Figure 2, the upper isolation layer and its extension line divide the whole structure into the upper storeys and the step-terrace storeys. Let m₁, m₂ denote the mass of the upper and lower storeys, respectively, and it is stipulated that the slope direction is positive. Based on findings from shake table experiments [24,25,27], the overturning of the isolated structure is attributed to the tensile buckling of the rubber bearing. Consequently, this study considers the ultimate bearing capacity of the rubber bearing under tension as the pivotal criterion for the overturning failure of the isolated step-terrace structure.

The literature review [31] indicates that the vertical tensile stiffness of a rubber bearing is inferior to its compressive stiffness, leading to the introduction of the vertical tensile stiffness to compressive stiffness ratio, denoted as γ, for the isolation layer. Assuming a seismic load distribution in a rectangular manner along the structure’s height, the resultant point of the seismic force aligns with the centroid position of the upper and lower storeys. If it is hypothesized that the structural overturning effect is positive in the direction of the terrain reduction, then, it follows that the overturning effect in the opposite direction would be negative. As per references [32], the most critical scenarios for overturning failure in mountainous structures encompass positive and negative overturning, respectively. By incorporating the rotation center position’s influence, the stress analysis of the isolated step-terrace structure can be simplified into four distinct conditions, as illustrated in Figure 3 and Figure 4.

In Figure 3 and Figure 4, the symbols H, h, b, and a denote the total height, height of step-terrace storeys, total width, and width of step-terrace storeys in the isolated step-terrace structure, respectively. The rotation center O, in the presence of the overturning moment M_ov, serves as the pivotal point, with O distinctly delineating the transition between the tension and compression states of the rubber bearings on either side. Within this context, F₁ and V_1r signify the horizontal seismic force acting upon the upper storeys and the horizontal reaction from the upper isolation layer, respectively. Similarly, F₂ and V_2r represent the horizontal seismic force affecting the step-terrace storeys and the horizontal reaction from the lower isolation layer, respectively. The parameters σ₁ and σ₂ denote the maximum tensile stress and compressive stress experienced by the rubber bearing under the influence of the overturning moment M_ov.

The scenario postulates the absolute acceleration of the upper structure floor as a₁ and that of the descending floor segment as a₂ during horizontal seismic events. Citing reference [19], the acceleration distribution across floors typically exhibits uniformity concerning height, thereby warranting the approximation a₁ = a₂. Herein, designating H/b as the nominal aspect ratio of the isolated step-terrace structure and l as the tension zone length, the introduction of height ratio α (h/H), width ratio β (a/b), and the length ratio of the tension zone to the compression zone φ (l/(b − l)) is imperative for a comprehensive analysis.

2.2. Theoretical Derivation

Under the premise that the horizontal shear force within the isolation layer aligns with its equivalent horizontal stiffness, and the absolute acceleration of the structure is denoted as a₀, the relationships F₁ = m₁a₀ and F₂ = m₂a₀ emerge. Introducing the seismic coefficient k = a₀/g, where g signifies the acceleration due to gravity, the stated conditions can be derived accordingly:

$$\left\{\begin{array}{c}\eta =\frac{K_{\mathrm{h}1}}{K_{\mathrm{h}2}}\\ V_{1r}=\left(m_1+m_2\right)a_0·\frac{1}{1+\eta }\\ V_{2r}=\left(m_1+m_2\right)a_0·\frac{\eta }{1+\eta }\end{array}\right.$$

(1)

Here, η represents the equivalent horizontal stiffness ratio between the upper and lower isolation layers, with K_h1 denoting the equivalent horizontal stiffness of the upper isolation layer and K_h2 denoting that of the lower isolation layer. Given the uniform mass distribution across the superstructure, the following relationship can be established:

$$\left\{\begin{array}{l}\frac{m_1}{m_1+m_2}=\frac{b(H-h)}{Hb-(b-a)h}=\frac{1-\alpha }{1-\alpha (1-\beta )}\\ \frac{m_2}{m_1+m_2}=1-\frac{m_1}{m_1+m_2}=\frac{\alpha \beta }{1-\alpha (1-\beta )}\end{array}\right.$$

(2)

The subsequent discourse entails a theoretical derivation concerning the overturning failure mechanism of the isolated step-terrace structure, focusing on the distinct scenarios of positive overturning failure.

When the rotation center is situated within the upper isolation layer, depicted in the force schematic diagram 3a, the fulfillment of the condition l < b − a is observed, leading to a simplified representation, as follows:

$$\frac{\phi }{1+\phi }<1-\beta$$

(3)

Considering the overturning moment at the rotation center O, and adhering to the equilibrium condition ΣM = 0, a deduction can be drawn as follows:

$$F_1\left(\frac{H-\mathrm{h}}{2}\right)+m_{1}g\left(\frac{\mathrm{b}}{2}-l\right)-F_2\frac{h}{2}+m_{2}g\left(b-l-\frac{a}{2}\right)+V_{2r}h-\frac{1}{3}\left[{\sigma }_1{l}^2+{\sigma }_2{\left(b-l\right)}^2\right]=0$$

(4)

When the isolated step-terrace structure is exposed to the concurrent influence of a horizontal seismic force and an overturning moment, predicated on the plane section assumption, the condition ΣF_v = 0 yields the ensuing outcome:

$${\displaystyle {\int }_0^l{K}_{Vt}}\theta xdx+\left(m_1+m_2\right)g={\displaystyle {\int }_0^{b-l}{K}_{Vp}}\theta xdx$$

(5)

Here, θ is the angle of rotation of the upper and lower isolation layers around the rotation center O, K_vt is the vertical tensile stiffness of the isolation layer, and K_vp is the vertical compression stiffness of the isolation layer.

Equation (5) can also be expressed as:

$$\phi =\frac{l}{b-l}=\sqrt{\frac{K_{\mathrm{vp}}}{K_{\mathrm{vt}}}-\frac{2\left(m_1+m_2\right)g}{\left(b-l\right)K_{\mathrm{vt}}\theta }}$$

(6)

Utilizing the relationships provided by equation σ₂ = (b − l)K_Vpθ and by equation γ = K_vt/K_vp, Equation (6) is simplified to:

$$\phi =\sqrt{\frac{K_{\mathrm{vp}}}{K_{\mathrm{vt}}}-\frac{2\left(m_1+m_2\right)g}{\gamma \left(b-l\right){\sigma }_2}}=\sqrt{\frac{1}{\gamma }-\frac{2\left(m_1+m_2\right)g}{\gamma \left(b-l\right){\sigma }_2}}$$

(7)

Further simplification of Equation (7) leads to:

$${\phi }^2=\frac{1}{\gamma }-\frac{2\left(1+\phi \right)\left(m_1+m_2\right)g}{\gamma {\sigma }_{2}b}=\frac{1}{\gamma }\left(1-\frac{2\left(1+\phi \right)}{{\sigma }_2}·{\sigma }_0\right)$$

(8)

where σ₀ is the nominal vertical compressive stress of the isolation layer under the action of self-weight, and, by definition, σ₀ can be expressed as: σ₀ = (m₁ + m₂)g/b.

From Equations (1)–(4) and (8), the following relationship can be obtained:

$$\left\{\begin{array}{c}\frac{\phi }{1+\phi }<1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\alpha \eta }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{{\sigma }_1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\alpha \eta }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$$

(9)

In accordance with the guidelines outlined in the Standard for Seismic Isolation Design of Buildings [33] (National Standard of the People’s Republic of China (NSPRC) 2021), it is stipulated that the tensile stress experienced by rubber bearings should not surpass 1 MPa, denoted as σ₁ = 1. Consequently, the critical criteria for the overturning failure of the isolated step-terrace structure are as follows:

$$\left\{\begin{array}{c}\frac{\phi }{1+\phi }<1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\alpha \eta }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\alpha \eta }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$$

(10)

(2) When the rotation center is situated within the lower isolation layer, depicted in the force schematic diagram 3b, the fulfillment of the condition l > b − a is observed, leading to a simplified representation, as follows:

$$\frac{\phi }{1+\phi }>1-\beta$$

(11)

Considering the bending moment at the rotation center O and adhering to the equilibrium condition ΣM = 0, a deduction can be drawn as follows:

$$F_1\left(\frac{H+\mathrm{h}}{2}\right)+m_{1}g\left(\frac{\mathrm{b}}{2}-l\right)+F_2\frac{h}{2}+m_{2}g\left(b-l-\frac{a}{2}\right)-V_{1r}h-\frac{1}{3}\left[{\sigma }_1{l}^2+{\sigma }_2{\left(b-l\right)}^2\right]=0$$

(12)

From Equations (1), (2), (8), (11), and (12), the following relationship can be obtained:

$$\left\{\begin{array}{c}\frac{\phi }{1+\phi }>1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\frac{H}{b}-\frac{1}{2}·\frac{1-\alpha \left(1+3\beta \right)\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{\phi }{1+\phi }}\right\}\\ \frac{{\sigma }_1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}-\frac{1}{2}·\frac{1-\alpha \left(1+3\beta \right)\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-1\end{array}\right.$$

(13)

When σ₁ = 1, the critical criteria for the overturning failure of the isolated step-terrace structure are as follows:

$$\left\{\begin{array}{c}\frac{\phi }{1+\phi }>1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\frac{H}{b}-\frac{1}{2}·\frac{1-\alpha \left(1+3\beta \right)\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}-\frac{1}{2}·\frac{1-\alpha \left(1+3\beta \right)\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-1\end{array}\right.$$

(14)

The aforementioned derivation elucidates the critical threshold for positive overturning failure. Analogously, the critical criteria for negative overturning failure can be deduced. Subsequently, the pivotal conditions for overturning failure of the isolated step-terrace structure are consolidated and presented in

Table 1. Overturning mechanisms of the isolated step-terrace structure.

	Rotation Center within the Upper Isolation Layer	Rotation Center within the Lower Isolation Layer
Positive overturning	$\left\{\begin{array}{c}\frac{\phi }{1+\phi }<1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\eta \alpha }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\eta \alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$(i)	$\left\{\begin{array}{c}\frac{\phi }{1+\phi }>1-\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha {\left(1-\beta \right)}^2}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$(ii)
Negative overturning	$\left\{\begin{array}{c}\frac{\phi }{1+\phi }>\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\eta \alpha }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha \left(1-{\beta }^2\right)}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-2\alpha +{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}+\frac{2\eta \alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha \left(1-{\beta }^2\right)}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$(iii)	$\left\{\begin{array}{c}\frac{\phi }{1+\phi }<\beta \\ {\phi }^2=\frac{1}{\gamma }\left\{1-\frac{2}{\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha \left(1-{\beta }^2\right)}{1-\alpha \left(1-\beta \right)}-\frac{\phi }{1+\phi }}\right\}\\ \frac{1}{{\sigma }_0}=\frac{3}{2}·k\left[\frac{1-{\alpha }^2\left(1-\beta \right)}{1-\alpha \left(1-\beta \right)}-\frac{2\alpha }{1+\eta }\right]\left(1+\frac{1}{\phi }\right)\frac{H}{b}+\frac{3}{2}·\frac{1-\alpha \left(1-{\beta }^2\right)}{1-\alpha \left(1-\beta \right)}\left(1+\frac{1}{\phi }\right)-\frac{2}{\phi }-3\end{array}\right.$(iv)

σ₀―nominal vertical compressive stress of the isolation layer under the action of self-weight; k―seismic coefficient, k = a₀/g, where a₀ is the absolute acceleration of the structure and g is the acceleration due to gravity; α―height ratio, α = h/H, where h and H refer to the height of step-terrace storeys and the total height, respectively; β―width ratio, β = a/b, where a and b refer to the width of step-terrace storeys and the total width, respectively; γ―vertical tensile stiffness to compressive stiffness ratio, γ = K_vt/K_vp, where K_vt and K_vp refer to the tensile and compressive stiffness, respectively, for the isolation layer; η―equivalent horizontal stiffness ratio, η = K_h1/K_h2, where K_h1 and K_h2 refer to the equivalent horizontal stiffness of the upper and lower isolation layer, respectively; φ―length ratio of the tension zone to the compression zone; $\frac{H}{b}$―nominal aspect ratio of the isolated step-terrace structure.

.

As depicted in Table 1, the bound of nominal aspect ratio of an isolated step-terrace structure can be delineated as a function of the nominal vertical compressive stress (σ₀), seismic coefficient (k), width ratio (α), height ratio (β), vertical tensile stiffness to compression stiffness ratio (γ), equivalent horizontal stiffness ratio (η), and the length ratio (φ) of the tension zone to the compression zone. This function is expressed as:

$$\left[\frac{H}{b}\right]=f\left({\sigma }_0,k,\alpha ,\beta ,\gamma ,\eta ,\phi \right)$$

(15)

where $\left[\frac{H}{b}\right]$ is the bound of nominal aspect ratio of an isolated step-terrace structure.

Typically, the nominal vertical compressive stress (σ₀) experienced by the isolation layer due to self-weight can be determined based on the guidelines provided in

Table 2. Nominal vertical compressive stress.

Building Category	The 1st Class Buildings	The 2nd Class Buildings	The 3rd Class Buildings
σ0 (MPa)	10	12	15

of reference [33]. Additionally, the seismic coefficient (k) serves as a direct indicator of the effectiveness of the isolation layer. As outlined in reference [34], the vertical tensile stiffness to compressive stiffness ratio (γ) of the isolation layer may vary within the range of 1/5 to 1/15.

Assuming that the equivalent horizontal stiffness of the isolation layer is directly proportional to its load mass, the parameter η can be simplified as:

$$\eta =\frac{\left(b-a\right)\left(H-h\right)}{aH}=\frac{\left(1-\beta \right)\left(1-\alpha \right)}{\beta }$$

(16)

Then, the allowable aspect ratio of the isolated step-terrace structure can be simplified as:

$$\left[\frac{H}{b}\right]=f\left({\sigma }_0,k,\alpha ,\beta ,\gamma ,\phi \right)$$

(17)

Further exploration of the four equations delineated in Table 1 elucidates that the length ratio (φ) of the tension zone to the compression zone is exclusively associated with the vertical tensile stiffness to compressive stiffness ratio (γ) and the nominal vertical compressive stress (σ₀). Their interdependence can be articulated as follows:

$$\gamma \left(1+2{\sigma }_0\right){\phi }^2+2\gamma {\sigma }_0\phi -1=0$$

(18)

In accordance with Equation (18), considering that φ is a non-negative parameter, the length ratio (φ) of the tension zone to the compression zone can be distinctly ascertained for a specified nominal vertical compressive stress (σ₀) and vertical tensile stiffness to compressive stiffness ratio (γ). Consequently, the bound nominal aspect ratio can be further streamlined as follows:

$$\left[\frac{H}{b}\right]=f\left({\sigma }_0,k,\alpha ,\beta ,\gamma \right)$$

(19)

According to GB/T 51408-2021 [33] on the peak value of earthquake acceleration, if the isolation efficiency is calculated as 1/2, the standard range of seismic coefficient (k) typically falls between 0.1 and 0.3 based on the seismic fortification category for rare earthquakes in China.

Figure 5 depicts the variations in the bound nominal aspect ratio in relation to height ratio (α) and width ratio (β) under specific conditions wherein the nominal vertical compressive stress (σ₀) stands at 12 MPa, the vertical tensile stiffness to compressive stiffness ratio (γ) is 1/12, and the seismic coefficient (k) is determined to be 0.2.

Evidently, as illustrated in Figure 5, it is apparent that irrespective of positive or negative overturning, the bound of nominal aspect ratio escalates with an increase in the width ratio while maintaining a constant height ratio. Furthermore, in scenarios where both the height ratio and width ratio exhibit significant magnitudes, the bound of nominal aspect ratio indicative of overturning effects becomes non-existent, which implies that the isolated step-terrace structure will not experience overturning due to the tension of the rubber bearing under such circumstances. Additionally, it is observed that, for higher ratio and width ratio configurations, the bound of nominal aspect ratio for positive overturning consistently remains smaller than that for negative overturning, indicating a higher susceptibility to positive overturning for the isolated step-terrace structure under equivalent conditions. Based on the preceding analysis, it is evident that the overturning encountered in isolated step-terrace structures as a result of rubber tension pertains to positive overturning mechanism.

Figure 6 illustrates the correlation between the bound of nominal aspect ratio and key variables including the seismic coefficient, nominal vertical compressive stress, and the vertical tensile stiffness to compressive stiffness ratio concerning the isolated step-terrace structure in instances of positive overturn. It is presumed in this analysis that the height ratio is 0.2.

As depicted in Figure 6, the bound of nominal aspect ratio of the isolated step-terrace structure exhibits a pattern of augmentation corresponding to reductions in the seismic coefficient, nominal vertical compressive stress, and the vertical tensile stiffness to compressive stiffness ratio. This pattern demonstrates a congruent trend across these variables.

3. Shaking Table Tests of Isolated Step-Terrace Structure

3.1. Description of the Prototype Model

The prototype structure with two step-terrace storeys, located in Yunnan Province, has a short side of three spans (X-direction), a long side of seven spans (Y-direction), a storey height of 3.0 m, a column grid dimension of 6 m × 6 m and 6 m × 3 m, and a total height of 19.2 m, and was designed according to GB/T 51408-2021 [33]. The step-terrace storeys have a long side of seven spans and a short side of one span. The main design information of the prototype structure is as follows: a fortification intensity of 8°, a peak acceleration of 0.20 g, site classification II, a characteristic period of 0.45 s, a standard live load of the floor of 3.5 kN/m², a main frame column size of 600 mm × 600 mm (step-terrace storeys), 550 mm × 550 mm (upper storeys), a frame beam of 250 mm × 500 mm (step-terrace storeys), a frame beam of 300 mm × 600 mm (upper storeys), concrete strength grade C30, an isolation floor thickness of 160 mm, and a thickness of 110 mm for the other floors. Twelve rubber bearings with a diameter of 600 mm are installed on the upper and lower isolation layers, respectively. The rubber shear modulus is 0.392 N/mm², and the vertical surface pressure range is 6–10 MPa.

3.2. Design of the Structural Model

Based on the above prototype structure, a simplified and scaled model was fabricated for the shaking table test. The length similarity ratio was set to 1:10 considering conditions such as the size of the shaking table of 4 m × 4 m and its maximum working load limit of 30 tons.

Table 3. Law of similitude.

Physical Factor	Model/Prototype	Physical Factor	Model/Prototype
Geometry Sl	0.1000	Volume SV	0.0010
Time St	0.3162	Mass Sm	0.0025
Acceleration Sa	1.0000	Stiffness SK	0.0250
Force SF	0.0025	Modulus SE	0.2500

shows the similarity ratios of the structural model to the prototype structure.

The short side of the upper storeys (X-direction) of the testing structure measured three spans and the long side of the upper storeys (Y-direction) measured seven spans. The short side (X-direction) and long side (Y-direction) of the step-terrace storeys were one span and seven spans, respectively, and all of the storeys had the same height of 0.3 m. The structural model was finally constructed using micro concrete with a strength grade of M 7.5. The mass of the entire structural model was approximately 13.11 tons.

In accordance with the equivalence principle proposed by Shao et al. [35], it is suggested that the upper and lower isolation layers could be effectively represented by four lead rubber bearings, each with a diameter of 100 mm (LRB100). The layout of the rubber bearings is delineated in Figure 7. Furthermore, Figure 8 presents both the cross-sectional diagram and a photograph of an isolation rubber bearing.

Eight LRB100s, identical to those utilized in seismic simulation testing were allocated for component testing. Characterization tests included lateral shear, vertical compression, and vertical tension tests aimed at determining the mechanical properties of the bearings. The testing configuration is illustrated in Figure 9. In the lateral shear and vertical compression test, an axial standard compression stress σ₀ (σ₀ = 3 MPa) was applied to each bearing. According to JG/T 118-2018 [36], the horizontal equivalent stiffness K_h, the yielded stiffness K_d, the yield strength Q_d, the equivalent damping ratio h_eq, and the vertical compression stiffness K_vp were determined from the third cycle response. It is widely accepted that rubber isolation supports exhibit elasticity until the tensile strain reaches 10% [33]. Consequently, a displacement loading control methodology was implemented for the tensile stiffness assessment of LRB100s, entailing loading from 0 mm to 2.6 mm and subsequent unloading to 0 mm. The initial stiffness recorded during the third cycle was regarded as the vertical tensile stiffness of the rubber bearing.

Each rubber bearing’s mechanical properties underwent individual testing procedures. Following the consolidation and averaging of the test results, the specification parameters and mechanical properties of the rubber bearings were obtained and are shown in

Table 4. Specification parameters and mechanical properties of the rubber bearings.

Outer diameter D (mm)	100	Vertical tensile stiffness Kvt (kN/mm)	12.01
Inner diameter d (mm)	10	Vertical compression stiffness Kvp (kN/mm)	46.57
Shear modulus G (MPa)	0.29	Horizontal equivalent stiffness Kh (kN/mm)	0.139
Rubber layer thickness Tr (mm)	26	Yielded stiffness Kd (kN/mm)	0.103
First shape factor S1	19.23	Yield force Qy (kN)	0.92
Second shape factor S2	3.85	Equivalent damping ratio heq (%)	9.3

. Figure 10 presents the samples of isolator L2 horizontal and vertical force displacement loops.

3.3. Installation of Model and Instrument

The experimental setup incorporated a range of instrumentation to facilitate comprehensive testing. Key instruments utilized included accelerometers, displacement meters, and triaxial force sensors, essential for evaluating both the dynamic response of the isolated structure and the mechanical characteristics of the isolators. Specifically, the type 4381 acceleration/displacement meter from the B&K Company (Copenhagen, Denmark) was employed to gauge the seismic responses of the structure during the testing phase. This piezoelectric charge accelerometer boasts high sensitivity and minimal susceptibility to environmental influences. Acceleration and displacement sensors were strategically positioned on each floor to capture the corresponding structural data accurately. Additionally, the type JDS-5T tri-axial force sensor from Bengbu Jinnuo Sensor Co., Ltd. (Bengbu, China) was utilized to monitor the horizontal and vertical force responses of the rubber bearings. Each triaxial force sensor was securely affixed to the base of every LRB100 unit using bolts, ensuring precise data collection. The structural model and sensor layout are visually depicted in Figure 11 for reference.

3.4. Cases of Shaking Table Test

According to GB/T 51408-2021 [33], the seismic records of 1952 Taft ground motion and an artificial ground motion were employed to analyze the seismic response of the isolated structure. Furthermore, in order to explore the impact of near-field effects on the isolated step-terrace structure, seismic records from the Ludian ground motion event that occurred in Yunnan Province in 2014, notable for its distinctive velocity pulses [37], were utilized. The characteristics of the three seismic ground motions are shown in

Table 5. Details of the selected seismic ground motions.

Seismic Wave	Station	Direction	PGA/cm/s2
Artificial ground motion	—	—	200
Taft 1952	Taft Lincoln school tunnel	N21E	949.1
Ludian 2014	Longtoushan	EW	152.7

and Figure 12.

It is imperative to highlight that the primary objective of this research is to delve into the issue of overturning failure concerning the isolated step-terrace structure. Therefore, the focus was solely directed toward elucidating the operational parameters associated with the single X-direction ground motion input.

Table 6. Test scheme.

Test No.	Seismic Wave	PGA (g)	Test No.	Seismic Wave	PGA (g)
1	White noise	0.05	10	Artificial ground motion	0.40
2	Artificial ground motion	0.07	11	Taft 1952	0.40
3	Taft 1952	0.07	12	Ludian 2014	0.40
4	Ludian 2014	0.07	13	White noise	0.05
5	White noise	0.05	14	Artificial ground motion	0.51
6	Artificial ground motion	0.20	15	Taft 1952	0.51
7	Taft 1952	0.20	16	Ludian 2014	0.51
8	Ludian 2014	0.20	17	White noise	0.05
9	White noise	0.05	-	-	-

presents the loading peak ground motions (PGAs) observed during the shake table test. The peak ground accelerations recorded correspond to values of 0.07 g, 0.20 g, 0.40 g, and 0.51 g, respectively, matching an 8° (0.2 g) frequent earthquake, 8° (0.2 g) fortification earthquake, 8° (0.2 g) rare earthquake, and 8° (0.3 g) rare earthquake, respectively [33].

4. Results and Discussion of the Shake Table Tests

The fundamental periods of the non-isolated prototype structure model were measured at 0.589 s (X-direction) and 0.584 s (Y-direction), whereas the corresponding periods of the isolated prototype stood at 2.003 s (X-direction) and 1.994 s (Y-direction), reflecting a lengthening of approximately 3.4 times. Notably, the fundamental periods of the non-isolated model were recalibrated to 0.648 s (X-direction) and 0.540 s (Y-direction), registering a modest 10.0% increase compared to the fundamental period of the prototype structure. This disparity could potentially be ascribed to discrepancies arising from control inaccuracies in material properties and counterweight adjustments during the model fabrication phase.

4.1. Floor Acceleration and Displacement Responses

Figure 13 illustrates the floor acceleration response corresponding to peak input ground motion amplitudes of 0.07 g, 0.20 g, 0.40 g, and 0.51 g, respectively.

As illustrated in Figure 13, the acceleration experienced by each floor of the isolated step-terrace structure was significantly lower than that of the base, emphasizing the efficacy of the isolation system in reducing seismic acceleration responses. Additionally, the isolation effect demonstrated an increasing trend with rising peak ground acceleration (PGA) values, with the top floor displaying a marginal amplification in acceleration compared to the other floors. It is noteworthy that the isolation effect was more pronounced in the Ludian ground motion, distinguished by a velocity pulse, in comparison to artificial and Taft waves. Figure 14 illustrates the horizontal displacement of the floor.

The analysis of Figure 14 reveals that the horizontal deformation of the model structure was predominantly localized within the isolation layers. The relative displacement between storeys remained minimal, with floor displacements evenly distributed throughout the height of the structure, suggesting an overall rigid body movement characteristic. Figure 15 shows the displacement time history curves of the lower isolation layer, the upper isolation layer, and the roof under the artificial ground motion (PGA = 0.20 g).

As depicted in Figure 15, the displacement among the upper isolation layer, the lower isolation layer, and the roof exhibited a high degree of synchronicity. Moreover, the relative displacement differences among the three components at each moment were minimal, offering additional support for the notion that the movement of the vibration model structure closely emulated that of a rigid body.

4.2. Seismic Responses of Isolation Layer

Figure 16 and Figure 17 display the instantaneous maximum surface pressure and the instantaneous minimum surface pressure encountered by individual rubber bearings across diverse ground motion scenarios.

It is evident from Figure 16 that, as the input peak values of different ground motions increased, the instantaneous maximum surface pressure of each rubber bearing exhibited a consistent and gradual rise. However, as depicted in Figure 17, the trend in the instantaneous minimum surface pressure of each rubber bearing varied, except for when PGA = 0.07 g. The instantaneous minimum surface pressure of each rubber bearing demonstrated a decreasing trend with the escalation of input peak values of different ground motions. Moreover, the emergence of tensile stress in specific isolators (U2, U4) signifies a proclivity toward positive overturning within the structural model due to tension within the upper isolation layer. Notably, the recorded tensile stress remained below the critical threshold of 1 MPa, indicating that the model had not reached a critical point where overturning failure was imminent. Concurrently, the compression state observed across all bearings in the lower isolation layer suggests a lack of apparent inclination toward negative overturning within the model. The hysteretic curves of shear force horizontal deformation of the isolation layer subjected to the ground motions of Taft and Ludian at PGA = 0.2 g are shown in Figure 18.

During seismic events, the lead core of rubber bearings undergoes yielding and transitions into the plastic stage, facilitating the dissipation of seismic energy. Figure 18 illustrates that the hysteresis curves of the upper and lower isolation layers in the isolated step-terrace structure model exhibited a considerable degree of completeness, indicative of a notable energy dissipation effect.

4.3. Validation of Overturning Mechanism

Based on test results and model size, the key parameters of the test structural model were obtained and are summarized in

Table 7. Key parameters.

Total height H (m)	1.8	Nominal vertical compressive stress σ0 (MPa)	2.3
Height of step-terrace storeys h (m)	0.6	Ultimate tensile stress σ1 (MPa)	1.0
Total width b (m)	1.5	Vertical tensile stiffness to compressive stiffness ratio γ	0.26
Width of step-terrace storeys a (m)	0.6	Equivalent horizontal stiffness ratio η	1.0

.

Subsequently, the overturning mechanism of the isolated step-terrace structure, as previously derived, was validated.

In the context of analyzing the tension characteristics of rubber bearings, when considering a scenario where the rotation center is positioned within the upper isolation layer, the parameters α, β, and η were obtained based on the values provided in Table 7. Subsequently, upon substitution of these parameters into Equation (18), the resulting value of φ was determined to be 0.514.

(1)

Positive overturning mechanism

For a given value of φ = 0.514, the ratio φ/(1 + φ) = 0.339 fell below 0.5 (1 − β = 0.5), indicating a potential scenario of positive overturning, with the center of rotation likely located in the upper isolation layer within the model. Furthermore, with φ determined to be 0.514, the seismic coefficient (k) can be derived as 0.581 through the application of formula (i) from Table 1. Subsequently, should the seismic coefficient (k) exceed this threshold of 0.581, the structural model would be at risk of experiencing positive overturning, as the tensile stress surpasses the ultimate capacity of the rubber bearing.

(2)

Negative overturning

For a given value of φ = 0.514, the ratio φ/(1 + φ) = 0.339 was determined to be below 0.5 (β = 0.5). This situation suggests a potential for negative overturning in the model, with the center of rotation likely positioned in the lower isolation layer. Moreover, with φ computed as 0.514, the seismic coefficient (k) can be calculated as 0.790 using formula (iv) from Table 1. Furthermore, if the seismic coefficient (k) exceeds the value of 0.790, the structural model is susceptible to negative overturning due to the tensile stress exceeding the ultimate capacity of the rubber bearing.

In summary, according to theoretical calculations, when the seismic coefficient (k) exceeds 0.581, the model structure experiences positive overturning, with the rotation center positioned in the upper isolation layer. Furthermore, for a seismic coefficient (k) exceeding 0.790, the model structure undergoes negative overturning, with the rotation center in the lower isolation layer. Evidently, the probability of encountering a positive overturning risk surpasses that associated with a negative overturning. This theoretical pattern of overturning is consistent with the experimental observations of the tensile stress recorded in the upper isolation layer and the state of the lower isolation layer under conditions of constant pressure. This alignment serves to affirm the precision and dependability of the overturning theory.

Figure 19 illustrates the comparative analysis between the test and theoretical values regarding the overturning effect of the isolated step-terrace structure model.

As depicted in Figure 19, there was a gradual rise in the seismic coefficient (k) of the model with escalating input peak values of ground motion. This increment aligned incrementally with the theoretically computed threshold for positive overturning, yet it did not surpass it. This correlation is reinforced by the experimental observation that the tensile stress levels detected in the upper isolation later consistently remained below the critical 1 MPa threshold. Such findings not only validate but also provide empirical support for the credibility and precision of the overturning failure theory.

5. Summary and Conclusions

The presence of both upper and lower isolation layers renders the behavior of the isolated step-terrace structure notably intricate in comparison to conventional base isolation structures. This research delves into the theoretical and experimental exploration of the overturning mechanism of the isolated step-terrace structure. The key findings and implications derived from this investigation can be summarized as follows:

(1) To comprehensively evaluate the potential for overturning failure in isolated step-terrace structures, with a focus on the critical criterion of the ultimate bearing capacity of rubber bearings under tension, an evaluation system is established. This system emphasizes the bound of nominal aspect ratio as the primary control parameter to facilitate a comprehensive assessment of structural stability and failure mechanisms under seismic influences.

(2) The parameters influencing the bound of nominal aspect ratio consist of key variables including the height ratio, width ratio, vertical tensile stiffness to compressive stiffness ratio, seismic coefficient, and nominal vertical compressive stress. Keeping the height ratio, vertical tensile stiffness to compressive stiffness ratio, seismic coefficient, and nominal vertical compressive stress constant, the bound of nominal aspect ratio increases proportionally with a rise in the width ratio. Additionally, the bound of nominal aspect ratio increases with a decreasing vertical tensile stiffness to compressive stiffness ratio, seismic coefficient, and nominal vertical compressive stress.

(3) For identical height and width ratio configurations, the bound of nominal aspect ratio for positive overturning consistently remains smaller than that for negative overturning in isolated step-terrace structures, indicating that overturning resulting from rubber tension predominantly aligns with the positive overturning mechanism. Moreover, despite the significant magnitudes of both the height ratio and width ratio, the absence of a bound of nominal aspect ratio suggests that, under such circumstances, the isolated step-terrace structure will not undergo overturning due to rubber bearings tension.

(4) The instantaneous minimum surface pressure directly monitored exhibits a decreasing trend as the ground motion amplifies, leading to the development of tensile stress within the upper isolation layer. This occurrence suggests the possibility of positive overturning failure, in accordance with the theoretically derived threshold.

(5) The isolated step-terrace structure demonstrates favorable seismic performance, as evidenced by the significant attenuation of floor acceleration in comparison to the base. When compared to both artificial ground motion and Taft ground motion, the Ludian ground motion, characterized by a velocity pulse, showcases no apparent displacement response, yet a distinct isolation effect is observable.

(6) The data obtained from the shaking table test serve as direct validation of the failure mechanism of the isolated step-terrace structure, affirming the accuracy of the theoretical framework.

While some progress was made in investigating the overturning failure mechanisms of the isolated step-terrace structure, there remain areas warranting attention in future research endeavors, including:

(1) The theoretical derivation conducted by the authors overlooked the influence of vertical seismic activity on the structural overturning mechanism. Future research may explore the incorporation of vertical earthquake effects to provide a more comprehensive analysis.

(2) The theoretical derivation assumed even distribution of mass and stiffness within the superstructure. To enhance the universality of the research outcomes, subsequent studies may consider uneven distribution coefficients for mass and stiffness in the superstructure.

(3) In the course of both theoretical formulation and experimental investigations, the torsional effects induced by seismic activity were not systematically accounted for, resulting in a notable underestimation of the seismic impact. Consequently, the forthcoming exploration of the impact of torsional effects on the behavior of isolated step-terrace structures emerges as a paramount research direction warranting in-depth examination in our future research endeavors.