Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures

Abstract

The assessment of rocking response is a challenging task due to its high nonlinearity. The present study investigates two methodologies to evaluate finite rocking rotations and overturn of three typical rocking systems. In particular, fuzzy linear regression (FLR) with triangular fuzzy numbers and a hybrid model combining logistic regression and fuzzy logic were adopted. To this end, three typical rocking structures were considered, and nonlinear time history analyses were performed to obtain their maximum response. Eighteen seismic intensity measures (IMs) extracted from recorded seismic accelerograms were considered to predict the responses. In the absence of rocking overturn, the finite rocking rotations and similarity ratios were calculated by adopting the FLR method. Moreover, extensive analysis was performed to evaluate the influence of each IM on the model’s predictions. On the other hand, rocking overturn was evaluated by logistic regression to compute the probability of collapse, followed by the FLR method to estimate the similarity between the different rocking-based structural systems. The root mean square error (RMSE) parameter and the log loss function were determined for every model to assess the predictions that emerged from the two fuzzy methods. As indicated, both methods demonstrated satisfactory results, presenting minimal deviations from the observed values. Finally, in the case of finite rocking rotation predictive models, remarkably high similarity ratios were observed among the various structures, with a median value of 0.96.

1. Introduction

Rocking response refers to a distinctive oscillation observed in free-standing structures that can uplift and rock under intense ground motion excitations [1]. Once uplift occurs, the free-standing structure behaves like an inverted pendulum, rotating about its alternative pivot points at the base. Remarkably, rocking systems can undergo significant displacements without damage [2]. The response of rocking systems was initially investigated by Housner [3], who noticed that water towers allowed to rock freely on their foundations survived the 1960 earthquakes in Chile with minimum damage. In contrast, seemingly more stable monolithic water towers experienced severe damage. Housner’s pioneering work emphasized and revealed the highly nonlinear nature of rocking behavior, bringing to light its fundamental aspects. Recognizing the favorable properties of rocking behavior, the scientific community has explored various ways to incorporate rocking into civil engineering structural systems. These include implementing flexible rocking cantilevers [4,5,6], designing multi-story frame structures with rocking foundations [7,8], constructing rocking bridge piers [9,10,11], connecting conventional structures with rocking walls in parallel [12,13], and employing seismic isolation through a rocking floor configuration at the base [14,15,16,17,18,19]. Although rocking response-based structural systems have been studied and developed over the last 60 years, the exploitation of the rocking response can also be observed in ancient temples consisting of free-standing columns connected with an epistyle at their top [20]. Except for structural members, the rocking response can also emerge in anchored free-standing equipment [21].

Due to the high nonlinearity of the rocking response, it is evident that the phenomenon should be probabilistically addressed [22,23,24,25,26,27,28,29,30]. To this end, probabilistic seismic demand models based on a linear regression between seismic intensity measures (IMs) and the maximum rocking behavior have been employed to predict the response in the absence of a rocking overturn. On the other hand, rocking overturning consists of a classification problem commonly treated using binomial or logistic regression.

As the determination of the seismic response of a structural system is computationally demanding, many researchers focused on predicting their behavior by adopting soft computing techniques [31,32,33]. Machine learning methods have been employed to assess the seismic response of RC, steel, and timber structures [34,35,36]. In addition, the structural damage has been estimated using an artificial neural network (ANN), a Mamdani-type fuzzy inference system (FIS), and a Sugeno-type fuzzy inference system (FIS) [37]. It has been reported that all methods lead to reliable results with an accurate classification process, and thus, they could be used effectively in evaluating the damage of a structure through an earthquake occurrence. Furthermore, comparative studies on the efficiency of different ML algorithms in predicting the total damage of a building under successive ground motions have been conducted [38,39]. While there have been numerous studies on the seismic response evaluation of fixed base structures using soft computing methods, only a few examine the response of rigid rocking blocks [40].

Among the soft computing techniques, it is evident that fuzzy logic offers distinct benefits over other ML algorithms by providing a more flexible and interpretable framework that can handle uncertainty and imprecise data more effectively. In particular, Gkountakou et al. [41] compared the effectiveness of multiple linear regression (MLR) and fuzzy linear regression (FLR) methods in predicting the structural damage indices of a steel building. It was demonstrated that the FLR method resulted in the lowest average percentage error level across all sets of accelerograms that the structure was subjected, making it an effective modeling method in the field of structural engineering. A fuzzy logic procedure was also applied to classify seismic damage potential in a concrete structure using the adaptive neuro-fuzzy inference system (ANFIS) [42]. More specifically, the overall damage indices and the maximum interstory drift ratio were used as outputs, and 20 seismic parameters were applied as independent variables for evaluating the earthquake’s impact. The results revealed that the ANFIS algorithm provided an effective modeling tool for predicting and classifying the damage status of a building.

Evaluating the similarity ratio between two or more fuzzy models is also an important process for indicating how close the results obtained from the two models are. For example, Papadopoulos et al. [43] used four different fuzzy linear regression models for calculating the connection between the nominal GNP and money. By evaluating the results, it was concluded that this method aids in assessing how closely the predictions of the two fuzzy models match, offering understanding into the coherence and concurrence among their individual outcomes.

Due to the high nonlinearity of the rocking response, its evaluation is a challenging task. To this end, in the present study, two different approaches, a fuzzy linear regression with triangular fuzzy numbers and a hybrid model of logistic regression and fuzzy logic, were used for evaluating the finite rocking rotations and rocking overturn of three typical rocking systems, respectively. From the obtained results, the similarity ratios of the prediction models were calculated, while the efficiency of the models was evaluated by adopting the root mean square error (RMSE) coefficient, and the log loss function. It was concluded that both methods demonstrate enhanced performance in predicting the dynamic response of the examined structural systems, while they can be used to assess the similarity between the models. Thus, the novelty of this work is to apply two different fuzzy rule-based approaches for evaluating their accuracy and calculating their similarity ratios. The process of computing the similarity ratio between the three typical rocking systems indicates the effectiveness of the methods in providing insights into the consistency and agreement between their respective outcomes.

2. Methods

2.1. Examined Structural Systems

Various types of structural and nonstructural elements, such as bridge piers, ancient columns, and electrical equipment, can exhibit a rocking response when subjected to seismic excitation. The rocking response is described by the following equation of motion [9]:

$$\ddot{\mathsf{\theta }}=-{\mathrm{p}}^2\left[\sin \left(\mathrm{sgn}\left(\mathsf{\theta }\right)\mathsf{\alpha }-\mathsf{\theta }\right)+\frac{{\mathsf{\alpha }}_{\mathrm{g}}}{\mathrm{g}}\cos \left(\mathrm{sgn}\left(\mathsf{\theta }\right)\mathsf{\alpha }-\mathsf{\theta }\right)\right]$$

(1)

where α_g represents seismic acceleration, g is the gravitational acceleration, and α is the slenderness of the rocking blocks. The sgn() represents the signum function applied to the ground acceleration. Finally, the frequency parameter p is defined as follows:

$$\mathrm{p}=\sqrt{\frac{3\mathrm{g}}{4\mathrm{R}}}\sqrt{\frac{1+2\mathsf{\gamma }}{1+3\mathsf{\gamma }}}$$

(2)

where R represents the semi-diagonal length of the piers, and the variable γ = m_b/Nm_c indicates the relative mass of the deck, where m_b is the mass of the deck, N is the number of rocking piers, and m_c is the mass of an individual rocking block in the absence of a cap beam, i.e., the case of a single rocking block γ = 0.

Throughout the rocking response, energy dissipation is instantaneous and takes place solely at impact moments when the tilt angle of the rocking piers changes sign. Solving the impact problem yields the ratio between the post and pre-impact rotational velocity. For the rocking frame, this ratio, denoted as the coefficient of restitution (COR), is expressed as follows [18]:

$$\mathrm{C}\mathrm{O}\mathrm{R}=1-2{\sin }^2\mathsf{\alpha }\frac{1+4\mathsf{\gamma }}{\frac{4}{3}+4\mathsf{\gamma }}$$

(3)

In the present study, three representative structures of each category are examined. “Structure 1” represents a bridge pier consisting of rocking columns with a height (H) of 9.6 m and a width (B) of 1.6 m [9]. “Structure 2” corresponds to the columns of the Temple of Aphaia, located in Aegina, with H = 5.27 m and B = 0.99 m [44]. Lastly, “structure 3” refers to a refrigerator with H = 2.13 m and B = 0.635 m [45]. In Figure 1, the examined free-standing structures are illustrated. Moreover, the model parameters are provided in the same figure.

It is worth noting that if the rocking rotation exceeds the slenderness of the rocking block, the system overturns. The ratio of the maximum developed rocking rotation to the slenderness of the rigid block consists of a representative and interpretable damage index.

2.2. Seismic Intensity Measures

The seismic excitation records are considered to be described using 18 intensity measures to assess the maximum rocking response of the examined structures. Specifically, the following parameters were considered: peak ground acceleration (PGA), peak ground velocity (PGV) and peak ground displacement (PGD), the mean period (Tm), the ratios PGV/PGA and PGD/PGV, acceleration’s root mean square (ARMS), velocity’s root mean square (VRMS), displacement’s root mean square (DRMS), characteristic length scale of acceleration (Lc_a) and velocity (Lc_v), Arias intensity (I_A), characteristic intensity (I_C), Fajfar index (I_F), specific energy density (SED), significant duration (t_sig), and cumulative absolute velocity (CAV) and displacement (CAD). To result in a robust predictive model, dimensionless, structure-specific IMs are calculated. To this end, the gravitational acceleration g, the frequency parameter p, and the slenderness α of each structural system are appropriately used according to Bantilas et al. [18].

Table 1. Dimensionless IMs.

i	IMi	Description
1	PGA/(g tan(α))	PGA = max(|αg(t)|)	[46]
2	p PGV/(g tan(α))	PGV = max(|vg(t)|)	[46]
3	p2 PGD/(g tan(α))	PGD = max(|dg(t)|)	[47]
4	p Tm	Tm = ∑(Ci2/fi)/∑Ci2	[47]
5	p PGV/PGA	-	[48]
6	p PGD/PGV	-	[49]
7	ARMS/(g tan(α))	${\mathrm{A}}_{\mathrm{R}\mathrm{M}\mathrm{S}}=\sqrt{1/{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{\mathsf{\alpha }}_{\mathrm{g}}{(\mathrm{t})}^2\mathrm{d}\mathrm{t}}}$	[50]
8	p VRMS/(g tan(α))	${\mathrm{V}}_{\mathrm{R}\mathrm{M}\mathrm{S}}=\sqrt{1/{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{\mathrm{v}}_{\mathrm{g}}{(\mathrm{t})}^2\mathrm{d}\mathrm{t}}}$	[50]
9	p2 DRMS/(g tan(α))	${\mathrm{D}}_{\mathrm{R}\mathrm{M}\mathrm{S}}=\sqrt{1/{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{\mathrm{d}}_{\mathrm{g}}{(\mathrm{t})}^2\mathrm{d}\mathrm{t}}}$	[50]
10	p2 Lca/(g tan(α))	Lca = PGA Tm2	[51]
11	p2 Lcv/(g tan(α))	Lcv = PGV Tm	[24]
12	p IA/(g tan(α))	${\mathrm{I}}_{\mathrm{A}}=\mathsf{\pi }/\left(2\mathrm{g}\right){\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{\mathsf{\alpha }}_{\mathrm{g}}{(\mathrm{t})}^2\mathrm{d}\mathrm{t}}$	[52]
13	p0.50 IC/(g tan(α)) 3/2	Ic = ARMS1.5 ttot0.5	[53]
14	p1.25 IF/(g tan(α))	IF = PGV tsig0.25	[54]
15	p3 SED/(g tan(α)) 2	$\mathrm{S}\mathrm{E}\mathrm{D}={\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{\mathrm{v}}_{\mathrm{g}}{(\mathrm{t})}^2\mathrm{d}\mathrm{t}}$	[46]
16	p tsig	-	[55]
17	p CAV/(g tan(α))	$\mathrm{C}\mathrm{A}\mathrm{V}={\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}\left|{\mathrm{a}}_{\mathrm{g}}(\mathrm{t})\right|}\mathrm{d}\mathrm{t}$	[56]
18	p2 CAD/(g tan(α))	$\mathrm{C}\mathrm{A}\mathrm{D}={\displaystyle {\int }_0^{{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}}}\left|{\mathrm{d}}_{\mathrm{g}}(\mathrm{t})\right|}\mathrm{d}\mathrm{t}$	[56]

presents the dimensionless IMs along with their respective definitions. The IMs are calculated using the time histories of acceleration (a(t)), velocity (v(t)), and displacement (d(t)). The seismic excitation’s total duration is denoted as t_tot. The Fourier amplitudes of the accelerograms are represented by C_i, and the corresponding Fourier transform frequencies between 0.25 and 20 Hz are denoted as f_i.

2.3. Formal Analysis

Fuzzy logic is a widely recognized concept in the field of soft computing, encompassing methods that create practical algorithms by incorporating structured human expertise. It represents a logical framework tailored to encapsulate modes of human understanding that lean towards approximations rather than strict precision [57]. Fuzzy logic involves managing fuzzy sets, where the elements consist of varying degrees of membership functions. The degrees of membership within fuzzy sets are between the interval of [0, 1]. When an axiom is absolutely accurate, the membership value within fuzzy sets assumes a value of 1. Conversely, when it is entirely untrue, the membership value is 0. The outcome yielded by a membership function is referred to as an antecedent. While the input parameters are crisp, they undergo transformation into fuzzy variables through the membership functions. The use of fuzzy rule-based modeling has many applications in predicting the seismic response of a structure [58].

A fuzzy linear regression model may be suitable for predicting the maximum seismic demand for systems where damage continuously evolves during the response. However, this approach is not applicable when dealing with a rocking response. As long as rocking overturning occurs, the damage index tends to infinity, introducing a discontinuity in the response. As a result, the finite rocking response should be examined using fuzzy linear regression models, while rocking overturning can be addressed using logistic regression combined with fuzzy logic.

2.3.1. Fuzzy Linear Regression

Classical regression analysis is a widely used statistical method for modeling the variation of the dependent output Y regarding the variation of one or more input variables X [59,60]. Its utilization is vital for minimizing the difference between the predicted and observed values of the dependent variable. The general form of classical regression with many explanatory variables is [61]:

$$\mathrm{Y}={\mathsf{\alpha }}_0+{\mathsf{\alpha }}_1{\mathsf{{\rm X}}}_1+\dots +{\mathsf{\alpha }}_{\mathrm{n}}{\mathrm{X}}_{\mathrm{n}}+\mathrm{u}$$

(4)

where u is the disturbance term that represents the error between the variables. The utilization of classical regression can be applied in many cases where a predictive model of the output is required. Regression analysis was particularly used for the hotel industry’s quality analysis [62]. The results were compared with other methods, such as rough set and cluster analysis, and it was concluded that the fitted values closely matched the actual values of the regression model. Also, Sabab et al. [63] used multiple linear regression to evaluate the soil strength in Bangladesh. According to the results, the prediction model proficiently captured the inherent patterns within the training data and generated precise forecasts. However, classical regression has some limitations if the variables are highly correlated or the relationship between the parameters is nonlinear. These assumptions can lead to inaccurate results and, thus, poor predictions.

The fuzzy linear regression (FLR) method extends classical logic in which the input and output parameters are expressed in fuzzy sets. More specifically, it manages the perceptual vagueness associated with the prediction process by applying fuzzy numbers. In contrast with the classical regression method, which assigns crisp values to variables, FLR introduces the concept of membership functions to evaluate the degree of association of an element to a set. The equation of the fuzzy linear regression method has the following form [64,65,66]:

$$\mathrm{Y}={\mathrm{A}}_0+{\mathrm{A}}_1{\mathsf{{\rm X}}}_1+\dots +{\mathrm{A}}_{\mathrm{n}}{\mathrm{X}}_{\mathrm{n}}$$

(5)

where ${\mathrm{A}}_{\mathrm{n}}=\left({\mathrm{r}}_{\mathrm{n}},{\mathrm{c}}_{\mathrm{n}}\right)$ express the fuzzy numbers in which ${\mathrm{r}}_{\mathrm{n}}$ denotes the central point that the membership function attains a value of 1 and ${\mathrm{c}}_{\mathrm{n}}$ represents the extent of the values. The function that determines the degree of membership is described as [67]:

$${\mathsf{\mu }}_{\mathsf{A}}\left(\mathrm{x}\right)=\mathrm{L}\left(\frac{\mathrm{x}-\mathrm{r}}{\mathrm{c}}\right)$$

(6)

where L(x) represents the reference function of a fuzzy number that adheres to the following conditions [68]:

$$\mathrm{L}\left(\mathrm{x}\right)=\mathrm{L}\left(-\mathrm{x}\right)$$

(7)

$$\mathrm{L}\left(0\right)=1$$

(8)

$$\mathrm{L}\left(1\right)=0$$

(9)

$$\mathrm{L}\left(\mathrm{x}\right)\mathrm{is}\mathrm{decreasing}\mathrm{in}[0,\infty )$$

(10)

$$\mathrm{L}\left(\mathrm{x}\right)\mathrm{possesses}\mathrm{invertibility}\mathrm{across}\mathrm{the}\mathrm{interval}[0,1]$$

(11)

Hence, the construction of the membership function for the linear possibility Equation (6) is achieved as follows [69]:

$${\mathsf{\mu }}_{{\mathrm{Y}}_{\mathrm{j}}}\left({\mathrm{y}}_{\mathrm{j}}\right)=\mathrm{L}\left[\frac{{\mathrm{y}}_{\mathrm{j}}-({\mathrm{r}}_0+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}\mathrm{j}})}{{\mathrm{c}}_0+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{i}}\left|{\mathrm{x}}_{\mathrm{i}\mathrm{j}}\right|}\right]$$

(12)

in which i represents the count of inputs, and j represents the count of sets. Thus, the degree h is computed to encompass the data within the estimated output Y, denoted as [70]:

$${\mathsf{\mu }}_{\mathrm{Y}\mathrm{j}}\left({\mathrm{y}}_{\mathrm{j}}\right)\ge \mathrm{h}$$

(13)

In our study, given that the fuzzy numbers were triangular, the reference function L(x) exhibited the following structure [71]:

$$\mathrm{L}\left(\mathrm{x}\right)=\mathrm{m}\mathrm{a}\mathrm{x}(0,1-\left|\mathrm{x}\right|)$$

(14)

and consequently, it led to the formulation of the following linear programming problem [42]:

$$\mathrm{J}=\mathrm{m}\mathrm{i}\mathrm{n}\{\mathrm{m}{\mathrm{c}}_0+\sum _{\mathrm{j}=1}^{\mathrm{m}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{i}}{|\mathrm{x}}_{\mathrm{i}\mathrm{j}}|\}$$

(15)

$${\mathrm{y}}_{\mathrm{j}}\ge {\mathrm{r}}_0+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}\mathrm{j}}-(1-\mathrm{h})\left({\mathrm{c}}_0+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{i}}{|\mathrm{x}}_{\mathrm{i}\mathrm{j}}|\right)$$

(16)

$${\mathrm{y}}_{\mathrm{j}}\le {\mathrm{r}}_0+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}\mathrm{j}}+(1-\mathrm{h})\left({\mathrm{c}}_0+\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{i}}{|\mathrm{x}}_{\mathrm{i}\mathrm{j}}|\right)$$

(17)

$${\mathrm{c}}_{\mathrm{i}}\ge 0,\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}$$

(18)

the Equations (16)–(18) were employed to facilitate the minimization process of the objective function (15).

2.3.2. Logistic Regression

Logistic regression (LR) analysis establishes a functional connection between a binary response variable and one or more explanatory variables [72]. This statistical technique enables the modeling and understanding the relationship between these variables in a binary classification context [73]. The outcomes that emerged mostly referred to two categories, “1” or “0” [74]. A logistic regression aims to capture the probability of an outcome based on individual characteristics. In logistic regression, the equation among the input and the output variables is not linear. Rather than, the logarithmic transformation of the probability, denoted as π, is considered as [75]

$$\mathsf{\pi }=\frac{{\mathrm{e}}^{{\mathsf{{\rm B}}}_0+{\mathsf{{\rm B}}}_1{\mathrm{x}}_1+\dots {\mathsf{{\rm B}}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}}{1+{\mathrm{e}}^{{\mathsf{{\rm B}}}_0+{\mathsf{{\rm B}}}_1{\mathrm{x}}_1+\dots {\mathsf{{\rm B}}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}}$$

(19)

Therefore, the equation of the model is employed [76]:

$$\ln \left(\frac{\mathsf{\pi }}{1-\mathsf{\pi }}\right)={\mathsf{{\rm B}}}_0+{\mathsf{{\rm B}}}_1{\mathsf{{\rm X}}}_1+{\mathsf{{\rm B}}}_2{\mathsf{{\rm X}}}_2+\dots +{\mathsf{{\rm B}}}_{\mathrm{n}}{\mathsf{{\rm X}}}_{\mathrm{n}}$$

(20)

in which π represents the probability of an event occurring, ${\mathsf{{\rm B}}}_0$ are the coefficients of the analysis, and ${\mathsf{{\rm X}}}_{\mathrm{n}}$ are the explanatory variables. The significance of each input is evaluated by conducting statistical tests to determine the importance of the coefficients. In logistic regression, the output is determined by applying a function that classifies the input data. The most commonly used function for determining the probability of the logistic regression method is the sigmoid function, in which if the probability is greater than 0.5, the outcome of the function tends to approach 1, so it belongs to class 1, and if the probability is less than 0.5, the outcome of the function tends to approach 0, and as a result, it belongs to class 0 [77].

Logistic regression is an accurate predictive approach for modeling binary outputs. Many researchers were interested in combining LR with fuzzy logic and claimed that this approach led to a more effective predictive system. For example, Tomar et al. [78] used logistic regression and fuzzy inference system (FIS) to manage traffic on different routes. Also, logistic regression analysis and fuzzy logic were integrated to identify gully erosion susceptibility [79]. This task was accomplished by evaluating the fuzzy membership values of every coefficient. Another model for evaluating binary data was proposed by Pourahmad [80], who used fuzzy logistic regression, in which the output belonged to a fuzzy set. In our study, we applied a hybrid model in three structures in which LR analysis was implemented for evaluating the probability π of structural collapse. Then, the probability values of every set were subjected to the fuzzy linear regression method to determine the similarity ratio of the structures.

3. Results

This research aimed to evaluate the similarity ratio of three different rocking systems by predicting their finite rocking rotations and rocking overturning under natural ground motions. This goal was accomplished by employing two regression techniques: fuzzy linear regression and a hybrid method with combined logistic regression and fuzzy logic.

Three typical rocking structural systems were subjected to 5000 ground motion records [81]. The ground motion records database encompasses a comprehensive collection of seismic data, capturing a diverse range of scenarios. With V_s,30 values spanning from 100 m/s to 2000 m/s, corresponding to soil classes ranging from A to D, the database accounts for various soil conditions. Moreover, R_rup ranges from 0.07 km to 185 km, while the magnitude M_w ranges from 3.2 to 7.9. The database includes records for all faulting types, providing a holistic view of seismic activity. A comprehensive collection of 18 seismic parameters was carefully chosen to characterize the seismic intensity of an earthquake event. These parameters were extracted from the recorded seismic accelerograms. Among the examined IMs, the acceleration-based ones determine the uplift of the rocking blocks, while the velocity and frequency-based IMs are highly correlated with rocking overturn [23,24,25]. It should be mentioned that elastic-response-spectrum-based IMs are not included since rocking motion cannot be described by equivalent elastic oscillators [45]. As long as rocking overturning implies an infinite rocking rotation, the response data has to be treated separately. In the absence of a rocking overturn, the finite damage index was predicted using fuzzy linear regression models. However, the rocking overturn prediction has to be treated as a classification problem due to the discontinuity of the damage index. To this end, logistic regression combined with fuzzy logic was employed. It has to be mentioned that both the IMs and response values are given on a logarithmic scale.

3.1. Fuzzy Linear Regression Results

Fuzzy linear regression analysis was applied to predict the rotation of every rocking structure using 18 seismic parameters as inputs. Particularly, the fuzzy linear regression equation with triangular fuzzy numbers was derived as follows:

$$\mathrm{Y}=A_0+A_1{\mathsf{{\rm X}}}_1+\dots +A_{18}{\mathrm{X}}_{18}$$

(21)

in which ${\mathsf{{\rm X}}}_{\mathrm{i}}$ were the seismic parameters, and Y was the rotation of the structure. The results of triangular fuzzy coefficients obtained through the method for every specific case were represented in

Table A1. The results of triangular fuzzy coefficients ${\mathrm{A}}_{\mathrm{i}}$ for every case of the FLR method.

	Structure 1		Structure 2		Structure 3
Ai	ri	ci	ri	ci	ri	ci
A0	−12.05	5.75	−12.02	6.62	−11.73	5.62
A1	0.98	0.00	0.61	0.00	0.68	0.00
A2	7.56	0.00	10.78	0.00	19.96	0.00
A3	1.24	0.00	0.27	0.00	0.20	0.00
A4	2.00	0.75	2.64	0.00	−0.36	0.09
A5	−7.58	0.00	−10.86	1.54	−14.83	1.37
A6	−0.02	0.00	0.02	0.00	−0.01	0.00
A7	3.90	0.00	−4.39	0.00	5.36	0.00
A8	7.15	0.00	4.29	0.00	−16.23	0.00
A9	−2.66	0.07	−0.58	0.00	−0.26	0.00
A10	−1.03	0.00	−0.80	0.00	0.27	0.00
A11	−0.05	0.00	−0.02	0.00	−0.09	0.00
A12	−0.54	0.00	−1.27	0.00	−1.15	0.02
A13	−0.95	0.00	3.20	0.00	1.20	0.00
A14	1.95	0.00	1.01	0.00	−2.89	0.00
A15	−0.03	0.00	0.00	0.00	−0.01	0.00
A16	−0.04	0.03	−0.01	0.01	−0.03	0.02
A17	0.22	0.00	0.05	0.00	0.42	0.00
A18	0.21	0.00	0.01	0.00	−0.02	0.00

in the Appendix A.

In order to evaluate the similarity ratio of the three fuzzy linear regression models containing the same input variables, we used the following equation [67]:

$${\mathsf{\lambda }}^2(\mathrm{d},\mathrm{d}\mathrm{\prime })=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathsf{\kappa }}^2,\frac{1}{{\mathsf{\kappa }}^2}\right\}$$

(22)

in which $\mathrm{d}$ and $\mathrm{d}\mathrm{\prime }$ were different sets of data and κ was a ratio expressed as

$${\mathsf{\kappa }}^2=\raisebox{1ex}{$\left[{\sum }_{\mathrm{i}=0}^{\mathrm{n}}{{\left({\mathrm{c}}_{\mathrm{i}}\right)}^2}_{\mathrm{d}}\right]$}\!\left/ \!\raisebox{-1ex}{$\left[{\sum }_{\mathrm{i}=0}^{\mathrm{n}}{{\left({\mathrm{c}}_{\mathrm{i}}\right)}^2}_{\mathrm{d}\mathrm{\prime }}\right]$}\right.$$

(23)

The results of the similarity ratio λ_i,j of the three models mentioned above are summarized in

Table 2. Estimated similarity ratios for FLR analysis.

λ	Value
λ1,2	0.853
λ1,3	0.99
λ2,3	0.851

. The subscript notation represents the parameter calculated with respect to structure i and j.

By evaluating the results, it was concluded that the models of the first and third structures had the highest similarity ratio value, and thus, the regression models were closer in contrast to the other sets.

For determining the accuracy of the predicted outputs that emerged from the fuzzy linear regression method, the root mean square error (RMSE) parameter was evaluated with the following equation:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{{\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left({\mathrm{e}}_{\mathrm{t}}\right)}^2}{\mathrm{n}}}=\sqrt{\frac{{\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left({\mathrm{A}}_{\mathrm{t}}-{\mathrm{F}}_{\mathrm{t}}\right)}^2}{\mathrm{n}}}$$

(24)

in which ${\mathrm{A}}_{\mathrm{t}}$ and ${\mathrm{F}}_{\mathrm{t}}$ represented the observed and predicted values, respectively, and n corresponded to the number of data sets. The results of the RMSE parameter for the finite rocking response of every structure are depicted in

Table 3. The results of the RMSE parameter for FLR analysis.

Structure	RMSE
1	3.88
2	4.14
3	3.54

.

In order to assess the impact of individual input variables on the similarity ratio of our model, we conducted a comprehensive analysis by constructing a total of 18 fuzzy linear regression models. Each model was formulated to include a distinct input variable, while the resulting output corresponded to the rotation of the structure. This approach allowed us to thoroughly evaluate each input variable’s impact on our models’ similarity ratio. The results of the triangular fuzzy coefficients are demonstrated in

Table A2. The results of triangular fuzzy coefficients ${\mathrm{A}}_{\mathrm{i}}$ through the FLR method.

		Structure 1		Structure 2		Structure 3
IM	Ai	ri	ci	ri	ci	ri	ci
PGA/(g tan(α))	A0	−9.64	8.57	−9.49	7.99	−10.83	8.16
	A1	0.85	0.00	1.03	0.00	1.76	0.00
p PGV/(g tan(α))	A0	−10.27	7.43	−9.66	7.14	−9.29	6.61
	A1	8.54	2.31	5.78	2.54	1.18	4.88
p2 PGD/(g tan(α))	A0	−8.26	8.24	−8.23	8.22	−8.62	8.61
	A1	−0.52	0.52	0.01	0.00	0.00	0.00
p Tm	A0	−9.36	8.36	−8.27	8.21	−7.95	7.94
	A1	1.97	0.00	0.04	0.00	−0.36	0.35
p PGV/PGA	A0	−9.39	8.54	−8.32	8.21	−7.96	7.94
	A1	10.34	0.00	0.69	0.00	−1.88	1.82
p PGD/PGV	A0	−8.25	8.24	−8.24	8.21	−8.62	8.61
	A1	−0.03	0.03	0.02	0.00	0.00	0.00
ARMS/(g tan(α))	A0	−9.92	8.42	−9.81	7.88	−9.83	8.33
	A1	8.23	0.00	8.95	0.00	8.60	0.00
p VRMS/(g tan(α))	A0	−10.22	7.22	−9.55	6.98	−8.48	7.49
	A1	35.14	28.16	29.49	24.60	−3.19	16.31
p2 DRMS/(g tan(α))	A0	−8.26	8.24	−8.23	8.22	−8.63	8.61
	A1	−0.70	0.69	0.02	0.00	0.00	0.00
p2 Lca/(g tan(α))	A0	−9.38	8.02	−8.89	7.61	−8.23	7.89
	A1	1.60	0.00	0.45	0.32	−0.11	0.20
p2 Lcv/(g tan(α))	A0	−9.55	7.87	−8.58	7.97	−8.19	7.95
	A1	0.73	0.09	0.16	0.02	−0.07	0.09
p IA/(g tan(α))	A0	−8.97	8.56	−8.58	8.07	−9.48	8.34
	A1	0.33	0.00	0.45	0.00	0.47	0.00
p0.50 IC/(g tan(α)) 3/2	A0	−9.26	8.51	−9.06	7.98	−10.26	8.12
	A1	0.86	0.00	1.33	0.00	2.15	0.00
p1.25 IF/(g tan(α))	A0	−10.51	7.16	−9.74	6.97	−8.45	7.45
	A1	4.59	2.06	2.78	1.65	−0.35	1.12
p3 SED/(g tan(α)) 2	A0	−8.68	8.67	−8.23	8.22	−8.62	8.61
	A1	0.01	0.00	0.003	0.00	0.002	0.00
p tsig	A0	−8.15	8.14	−8.30	8.22	−7.94	7.95
	A1	−0.02	0.02	0.004	0.00	−0.02	0.02
p CAV/(g tan(α))	A0	−8.48	7.89	−8.95	8.15	−9.04	8.57
	A1	−0.04	0.09	0.12	0.00	0.04	0.00
p2 CAD/(g tan(α))	A0	−9.31	8.07	−8.28	8.17	−8.62	8.61
	A1	0.31	0.29	0.02	0.008	0.00	0.00

in the Appendix A section, and the values of the similarity ratio λ and the RMSE parameter for every model are represented in

Table 4. The results of estimated similarity ratios for every model of FLR analysis.

IM	λ1,2	λ1,3	λ2,3
PGA/(g tan(α))	0.93	0.95	0.98
p PGV/(g tan(α))	0.97	0.95	0.92
p2 PGD/(g tan(α))	0.99	0.96	0.95
p Tm	0.98	0.95	0.97
p PGV/PGA	0.96	0.95	0.99
p PGD/PGV	0.99	0.96	0.95
ARMS/(g tan(α))	0.93	0.99	0.94
p VRMS/(g tan(α))	0.88	0.62	0.70
p2 DRMS/(g tan(α))	0.99	0.92	0.91
p2 Lca/(g tan(α))	0.95	0.98	0.96
p2 Lcv/(g tan(α))	0.97	0.98	0.99
p IA/(g tan(α))	0.94	0.97	0.96
p0.50 IC/(g tan(α)) 3/2	0.93	0.95	0.98
p1.25 IF/(g tan(α))	0.96	0.99	0.95
p3 SED/(g tan(α)) 2	0.95	0.99	0.96
p tsig	0.99	0.98	0.97
p CAV/(g tan(α))	0.97	0.92	0.95
p2 CAD/(g tan(α))	0.99	0.94	0.95

and

Table 5. The results of the RMSE parameter for every model of FLR analysis.

IM	RMSE1	RMSE2	RMSE3
PGA/(g tan(α))	4.17	4.23	4.25
p PGV/(g tan(α))	4.49	4.82	5.39
p2 PGD/(g tan(α))	8.30	5.07	5.27
p Tm	4.82	5.08	5.09
p PGV/PGA	4.89	5.07	5.09
p PGD/PGV	4.87	5.06	5.27
ARMS/(g tan(α))	4.11	4.40	4.61
p VRMS/(g tan(α))	5.34	5.24	5.35
p2 DRMS/(g tan(α))	6.97	5.07	5.27
p2 Lca/(g tan(α))	4.63	5.00	5.28
p2 Lcv/(g tan(α))	4.93	5.00	5.27
p IA/(g tan(α))	4.90	4.66	4.82
p0.50 IC/(g tan(α)) 3/2	4.69	4.44	4.61
p1.25 IF/(g tan(α))	5.02	4.93	5.41
p3 SED/(g tan(α)) 2	5.09	5.08	5.26
p tsig	4.91	5.08	5.12
p CAV/(g tan(α))	5.16	4.78	5.24
p2 CAD/(g tan(α))	6.41	5.10	5.27

, respectively. The subscript notation of RMSE indicates the structures under examination for which the parameter is calculated.

It has to be highlighted that due to the normalization of IMs, a self-similar response between seemingly different rocking systems is achieved. That fact justifies the observed high similarity ratios between the models. Furthermore, the significant similarity indicates the effectiveness of dimensionless IMs in accurately predicting the response of rocking systems with different dynamic properties.

3.2. Hybrid Method Results

Logistic regression analysis was used to model the relationship between the 18 seismic parameters and the binary output, representing whether the structure collapsed. More specifically, the response results obtained by time history analyses are divided into two categories, i.e., the nonoverturning and overturning classes. The equation of logistic regression was deduced in the following manner:

$$\ln \left(\frac{\mathsf{\pi }}{1-\mathsf{\pi }}\right)={{\rm B}}_0+{{\rm B}}_1{\mathsf{{\rm X}}}_1+{{\rm B}}_2{\mathsf{{\rm X}}}_2+\dots +{{\rm B}}_{18}{\mathsf{{\rm X}}}_{18}$$

(25)

The results of the model parameters are demonstrated in

Table A3. The results of logistic regression parameters for every case.

Βi	Structure 1	Structure 2	Structure 3
Β0	−8.216	−9.019	−7.315
Β1	0.205	−0.933	−0.993
Β2	−2.753	11.699	0.167
Β3	0.878	−0.531	1.278
Β4	2.143	3.254	−14.401
Β5	−14.308	−11.970	0.122
Β6	−0.670	0.011	−0.026
Β7	−0.966	8.000	−0.078
Β8	57.348	0.001	0.001
Β9	−0.776	0.494	−0.962
Β10	−0.889	−0.033	1.181
Β11	0.489	−0.284	−0.239
Β12	0.228	0.118	−0.029
Β13	−1.428	−1.694	0.129
Β14	2.891	1.651	0.124
Β15	−1.814	0.015	8.024
Β16	−0.054	−0.055	−0.249
Β17	0.178	0.171	9.175
Β18	0.504	0.177	−0.094

in the Appendix A. For assessing the accuracy of the logistic regression model and quantifying the extent of errors, the log loss function was determined according to the following equation:

$$\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}=-\frac{1}{\mathrm{n}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{y}}_{\mathrm{i}}\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{p}}_{\mathrm{i}}\right)+(1-{\mathrm{y}}_{\mathrm{i}})\mathrm{l}\mathrm{o}\mathrm{g}(1-{\mathrm{p}}_{\mathrm{i}})\right)$$

(26)

in which y_i were the observed binary outputs and p_i were the probabilities of every case. The results of the log loss function are illustrated in

Table 6. The results of the log loss function for logistic regression.

Structure	Log Loss
1	0.008
2	0.018
3	0.019

.

Then, based on the aforementioned logistic regression parameters, the probability π of every case was computed according to Equation (19). The results that emerged were subjected to three models of fuzzy linear regression analysis, in which the inputs were the 18 seismic parameters, and the output was the probability for every structure. This hybrid approach assessed the degree of similarity among the three structures. By employing this method, it became possible to integrate the principles of fuzzy logic with logistic regression, thereby facilitating their combined utilization. The results of the similarity ratio λ are presented in

Table 7. Estimated similarity ratios for hybrid analysis.

λ	Value
λ1,2	0.108
λ1,3	0.203
λ2,3	0.533

.

Through the examination of the outcomes, it was determined that the second and third structural models exhibited the highest degree of similarity ratio value.

The same procedure was followed to evaluate the influence of individual input variables on the similarity ratio of our model. For that purpose, 18 logistic regression models comprised of one input, were analyzed. The results of the log loss function are summarized in

Table 8. The results of the log loss function for every model of logistic regression.

IM	Log Loss1	Log Loss2	Log Loss3
PGA/(g tan(α))	0.013	0.035	0.04
p PGV/(g tan(α))	0.014	0.022	0.027
p2 PGD/(g tan(α))	0.013	0.037	0.045
p Tm	0.014	0.036	0.044
p PGV/PGA	0.012	0.034	0.044
p PGD/PGV	0.013	0.037	0.050
ARMS/(g tan(α))	0.051	0.032	0.035
p VRMS /(g tan(α))	0.011	0.042	0.036
p2 DRMS /(g tan(α))	0.013	0.040	0.045
p2 Lca/(g tan(α))	0.011	0.020	0.031
p2 Lcv/(g tan(α))	0.011	0.023	0.034
p IA/(g tan(α))	0.012	0.031	0.052
p0.50 IC/(g tan(α)) 3/2	0.012	0.031	0.042
p1.25 IF/(g tan(α))	0.010	0.021	0.032
p3 SED/(g tan(α)) 2	0.013	0.037	0.045
p tsig	0.013	0.037	0.046
p CAV/(g tan(α))	0.019	0.030	0.040
p2 CAD/(g tan(α))	0.015	0.037	0.020

. The subscript notation in Log Loss signifies the structure for which the log loss function is applied.

Then, the probability π of every case was subjected to fuzzy linear regression. The outcomes pertaining to the triangular fuzzy coefficients are presented in Appendix A 

Table A4. The results of triangular fuzzy coefficients ${\mathrm{B}}_{\mathrm{i}}$ through the hybrid method.

		Structure 1		Structure 2		Structure 3
IM	Bi	ri	ci	ri	ci	ri	ci
PGA/(g tan(α))	B0	−0.03	0.00	−0.04	0.00	−0.11	0.00
	B1	0.03	0.014	0.06	0.017	0.12	0.03
p PGV/(g tan(α))	B0	0.42	0.42	−0.02	0.005	−0.03	0.01
	B1	0.00	0.00	0.47	0.33	0.42	0.26
p2 PGD/(g tan(α))	B0	0.03	0.00	0.08	0.00	0.10	0.00
	B1	0.00	0.00	0.00	0.00	0.00	0.00
p Tm	B0	−0.04	0.00	−0.04	0.004	−0.04	0.008
	B1	0.25	0.17	0.20	0.1	−0.15	0.06
p PGV/PGA	B0	−0.017	0.00	−0.01	0.004	−0.009	0.005
	B1	1.03	0.70	0.89	0.42	0.64	0.24
p PGD/PGV	B0	0.03	0.00	0.085	0.00	0.102	0.00
	B1	0.00	0.00	0.00	0.00	0.00	0.00
ARMS/(g tan(α))	B0	0.00	0.00	0.00	0.00	0.097	0.00
	B1	0.25	0.17	0.49	0.23	0.46	0.46
p VRMS/(g tan(α))	B0	0.50	0.49	0.004	0.00	0.02	0.00
	B1	0.00	0.00	2.05	1.75	1.37	1.08
p2 DRMS/(g tan(α))	B0	0.031	0.00	0.08	0.00	0.10	0.00
	B1	0.00	0.00	0.00	0.00	0.00	0.00
p2 Lca/(g tan(α))	B0	0.004	0.00	0.00	0.008	−0.003	0.012
	B1	0.066	0.05	0.05	0.03	−0.03	0.016
p2 Lcv/(g tan(α))	B0	0.002	0.00	−0.001	0.007	0.008	0.005
	B1	0.04	0.03	0.03	0.02	0.02	0.009
p IA/(g tan(α))	B0	−0.017	0.00	0.035	0.00	0.036	0.00
	B1	0.018	0.013	0.028	0.014	0.03	0.031
p0.50 IC/(g tan(α)) 3/2	B0	0.009	0.00	0.012	0.002	0.014	0.00
	B1	0.04	0.03	0.07	0.04	0.076	0.072
p1.25 IF/(g tan(α))	B0	−0.01	0.00	−0.017	0.005	−0.008	0.00
	B1	0.22	0.18	0.21	0.15	0.13	0.13
p3 SED/(g tan(α)) 2	B0	0.51	0.49	0.08	0.00	0.01	0.00
	B1	0.00	0.00	0.00	0.00	0.00	0.00
p tsig	B0	0.02	0.00	0.051	0.00	0.08	0.00
	B1	0.002	0.001	0.002	0.00	0.00	0.00
p CAV/(g tan(α))	B0	−0.06	0.00	−0.02	0.007	−0.01	0.01
	B1	0.01	0.01	0.01	0.008	0.01	0.005
p2 CAD/(g tan(α))	B0	0.03	0.00	0.08	0.00	0.10	0.00
	B1	0.00	0.00	0.00	0.00	0.00	0.00

, and the values indicating the similarity ratio λ are displayed in

Table 9. The results of estimated similarity ratios for every model of hybrid analysis.

IM	λ1,2	λ1,3	λ2,3
PGA/(g tan(α))	0.83	0.54	0.65
p PGV/(g tan(α))	0.78	0.61	0.77
p2 PGD/(g tan(α))	0.004	0.34	0.01
p Tm	0.59	0.35	0.58
p PGV/PGA	0.56	0.34	0.60
p PGD/PGV	0.75	0.97	0.73
ARMS/(g tan(α))	0.76	0.38	0.50
p VRMS/(g tan(α))	0.28	0.46	0.62
p2 DRMS/(g tan(α))	0.02	0.04	0.56
p2 Lca/(g tan(α))	0.61	0.14	0.62
p2 Lcv/(g tan(α))	0.52	0.32	0.61
p IA/(g tan(α))	0.89	0.18	0.22
p0.50 IC/(g tan(α)) 3/2	0.79	0.39	0.49
p1.25 IF/(g tan(α))	0.81	0.68	0.85
p3 SED/(g tan(α)) 2	0.00	0.00	0.03
p tsig	0.86	0.15	0.18
p CAV/(g tan(α))	0.95	0.73	0.81
p2 CAD/(g tan(α))	0.42	0.12	0.27

.

4. Conclusions

The assessment of rocking response presents a formidable challenge due to its significant nonlinearity. In this paper, two distinct methodologies to evaluate the finite rocking rotations and rocking overturn of three typical rocking systems were studied. Specifically, we employed fuzzy linear regression with triangular fuzzy numbers and a hybrid model that combined logistic regression and fuzzy logic. These approaches were chosen to achieve our research objectives effectively.

The fuzzy linear regression method was applied to determine the finite rocking rotations and the similarity ratio of the three structures. The model consisted of 18 seismic parameters, which were extracted from the recorded seismic accelerograms and considered as inputs. Extensive analysis was carried out to evaluate each input variable’s influence on the model’s similarity ratio. This procedure involved creating a set of 18 fuzzy linear regression models. This comprehensive approach highlighted the significance of each parameter in shaping the model’s outcomes. According to the results, high similarity ratios were observed among the various structures in every case. More specifically, in the case where the fuzzy model included all input variables, the similarity ratio of the structures had a value of 0.99 between structures 1 and 3, which indicated that these regression models were closer, in contrast to the other sets.

Similarly, a hybrid model that combined logistic regression and fuzzy logic was applied to evaluate the rocking binary overturn of three typical rocking systems and, thus, their similarity ratio. The application of this model necessitated the computation of probability π of structure failures with logistic regression. Then, the computed probability was subjected to the FLR method to determine the similarity ratio of every system. In this analysis, the similarity ratios in every case were lower, contrary to the first method, mainly when the fuzzy model included all input variables. The highest similarity ratio was observed between the 2 and 3 structures (λ_2,3 = 0.533). The model that exhibited the highest similarity ratio among the structures was the model that included only the variable p PGD/PGV as an input, with a similarity ratio value of λ_1,3 = 0.97. This result demonstrated that these regression models were closer for structures 1 and 3.

In order to assess the outputs that emerged from the fuzzy linear regression and the logistic regression, the root mean square error (RMSE) parameter and the log loss function were determined, respectively. As indicated, both methods demonstrated satisfactory outcomes, showing minimal deviations from the observed values. However, the results revealed that in both methods, in cases where the models were composed of all input variables, the coefficients expressing the error of the methods provided better results compared to the cases where the models consisted of only one input. For example, in the fuzzy linear regression method, the smallest RMSE parameter (RMSE = 3.54) was observed where the model was composed of all input parameters of the third structure. Similarly, in the logistic regression method, the smallest log loss function (long loss = 0.008) was evaluated in the model consisting of all seismic input parameters of the first structure. This result indicated a small deviation between the predicted and observed values, highlighting the method’s accuracy and precision.

In conclusion, the fuzzy linear regression method and the hybrid model that combined logistic regression and fuzzy logic are effective modeling tools that can be used accurately for regression and classification problems, respectively. Also, the parameters of triangular fuzzy numbers can be utilized to determine the applied model’s similarity ratio. Thus, these models result in accurate seismic assessment of rocking structures and serve as valuable modeling tools in the realm of engineering. The high similarity ratios, stemming from the dimensionless IMs, indicate that the proposed methodology may result in a universal assessment tool that can be applied to evaluate the response of different rocking structures. However, a thorough investigation, including systems with various geometries and boundary conditions, should be conducted in order to extend the potential application of the adopted methodology.