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SN Applied Sciences

, 1:703 | Cite as

Estimation of rocking capacity of soil-structure systems using a hybrid inverse solver

  • Aria FathiEmail author
  • S. Mohsen Haeri
  • Mehran Mazari
  • Arash Hosseini
  • Saurav Kumar
  • Cheng Zhu
Research Article
  • 141 Downloads
Part of the following topical collections:
  1. Engineering: Artificial Intelligence

Abstract

Among the geotechnical-earthquake community, the rocking concept is being acknowledged as an energy dissipation mechanism that benefits soil–structure systems during strong vibrations. Nevertheless, several involving factors such as geomaterials behavior and superstructure size can make the rocking analysis of soil–foundation–structure systems complicated. The mobilized moment and the dissipated energy can be represented as the two primary performance indicators of the soil–structure systems under strong motions. This study employs an assembled database comprised of a wide range of geomaterials with different stiffness values associated with high-rise structures with different dimensions acquired from the implementation of a dynamic nonlinear elastic-perfect plastic finite element model. This study aims to develop predictive models for the responses mentioned above using the implemented database. To predict the both mobilized moment and dissipated energy, a hybrid gene-expression programming–artificial neural network technique (GEP–ANN) was used. The results show that the GEP model can yield promising predictions with reasonable accuracy. However, the GEP model can be fine-tuned by introducing the hybrid model. The hybrid model decreases the recorded prediction errors by a factor of three for both the mobilized moment and damping ratio as compared to the GEP model. The results show that the predictive models yield a sensible performance power that minimizes the efforts needed for implementation of time-consuming finite element method. This study also deploys a local sensitivity analysis technique to assess how the input parameters are attributed to the target values.

Keywords

Rocking behavior Soil structure interaction (SSI) Dynamic finite element method (DFEM) Gene expression programming (GEP) Hybrid inverse solver Machine learning (ML) 

1 Introduction

A substantial inertial and eccentric forces induced by strong motions such as an earthquake may cause partial separation of a shallow foundation from the supporting soil system. Shear stress enhances at one side of the foundation during rocking due to the partial separation, or uplift, at the opposite side of the foundation. An increase in compiled plastic strain accompanied by shear and normal stresses at either side of the foundation can basically result in the accumulation of permanent deformation of the shallow foundation. Considering the nonlinear behavior of geomaterials as a result of rocking behavior of shallow foundations has been gaining popularity as a performance enhancement tool in geotechnical earthquake engineering design codes, one example of which can be found in FEMA-356 (2000). Therefore, in the past couple of decades, several studies have been dedicated to evaluate the effects of soil plasticity as well as rocking behavior of structures on the overall performance of the superstructure [1, 2, 3, 4, 5, 6, 7, 8, 9].

Performance-based design (PBD)—a robust engineering technique comprised of conventional seismic design methods with substantial progress—is typically established on the basis of ultimate performance and objectives deemed for the designed structure [10, 11]. While rocking may not be of importance for small structures, and other criteria such as settlement may be selected [12], it plays a vital role in tall buildings where it also affects the ultimate vertical displacement. As a transition from limited equilibrium analysis, soil–structure interaction (SSI) in general, and more specifically rocking behavior of foundations have been emphasized in PBD geotechnical codes [13, 14]. PBD is basically justified on cyclic and kinematic components of motions indicating that the foundation–superstructure system can deploy a displacement-controlled rocking under severe motions as proposed in the recent years [3, 14]. The nonlinear load–displacement behavior of geomaterials under both static and dynamic loading conditions have been investigated by several researchers using implemented numerical and experimental studies in geotechnical context [14, 15, 16, 17, 18, 19, 20, 21].

Deng et al. [22] developed a multilinear model for the backbone curve of the nonlinear rotation-moment behavior. The authors proposed an empirical relationship correlating the rotational stiffness to the mobilized moment of soil–structure systems governed by the rotation-moment relationship. Hakhamaneshi et al. [23] studied the effects of shape and embedment depth on rocking performance of shallow foundations from results of a series of centrifuge model tests. They found that the permanent settlement cannot be exclusively correlated to the shape and contact area ratio of the foundation as other factors like rotation amplitude and embedment may affect appreciably. In a recent study, a comprehensive database of soil–structure systems subjected to a controlled low frequency lateral cyclic displacement was assembled using a dynamic nonlinear elastic-perfect plastic finite element (FE) model by Fathi et al. [9]. The nonlinear elastic perfect plastic behavior was employed for the supporting geomaterials using a coded subroutine through an iterative process. The assembled database included a wide range of stiffness values for the geomaterials and different dimensions for the structure system. They concluded that the foundation located on the stiffer materials is more susceptible to lose its contact with the underneath soil. The authors also indicated that the triggered input-energy is dissipated more when the foundation is more in contact with the soil system at a certain rotation amplitude. One contribution of these predictions is to provide some guidance in the earthquake geotechnical design codes.

The rotational stiffness, moment capacity and dissipation of energy of a soil–structure system can be assessed attentively by observing the type of geomaterials, evaluating the amplitude of rotation applied to the superstructure, and estimating rocking responses of a foundation (i.e., contact width, permanent displacement, uplift, etc.) through experimental and numerical works. Inappropriately, these specific means of assessment become ineffective and cost-intensive in terms of time, energy, and expense for assessing the input parameters and computational effort for the appropriate application of design codes. Thus, the predictive models have been evolved to forecast the above-mentioned performance indicators of soil–structure systems. Conventional regression analysis has been of interest to many researchers for development of predictive functions in the geotechnical context [22, 24, 25, 26, 27, 28, 29, 30, 31]. Soft computing techniques have also been used for prediction of SSI effects in general, and rocking responses in particular. One example is the use of the Artificial Neural Network (ANN) for prediction of the overturning failure of a rigid block subjected to cyclic loading. Gerolymos et al. [32] implemented numerical and analytical solutions for a rigid block subjected to rocking oscillations. They stated that the ANN analysis was in a good agreement with the results obtained from the closed-form solutions to forecast the overturning failure of a rigid block.

Many studies have been carried out in several earthquake geotechnical related problems in the recent years. A prediction of resonant frequencies and amplitudes of deep foundations for both horizontal and rocking motion was implemented by Das et al. [33] remarked that ANN rocking prediction models were successful in developing the dynamic soil–pile interaction on the nonlinear response under coupled vibration. Later, Shahin [34] developed a prediction model for the load-settlement response of steel driven piles subjected to cyclic loading deploying Recurrent Neural Networks (RNNs). In addition to ANN, extensive practice of fuzzy logic algorithms can be found in the literature to predict shallow foundations responses under static and dynamic loading as reported in [35, 36, 37]. The combination of ANN and Fuzzy system known as Neuro-Fuzzy also refines the results of any prediction. By combining ANN and Fuzzy methods, Adaptive Neuro-Fuzzy Inference Systems (ANFIS) can be derived. This system uses both the ability of Fuzzy Inference Systems for manipulating the imprecise data and the powerful characteristic of ANN for learning [38, 39]. One example of which is the prediction of the liquefaction-induced lateral spreading using ANFIS successfully implemented by Haeri et al. [40]. One of the drawbacks of the aforementioned learning models is that an easy-to-implement function may not be achieved since these learning algorithms are incompetent to evolve the final product in the form of mathematical equations.

This study mainly attempts to predict the rocking responses of shallow foundations, i.e., moment capacity and energy dissipation of soil–structure systems under slow cyclic loading, utilizing a coupled use of Gene-Expression Programming (GEP) and Artificial Neural Network (ANN). Mazari and Rodrigues [41] used GEP–ANN model to fine-tune the prediction power of models for pavement roughness. This study aids an assembled FE database generated for rocking behavior of shallow foundations by Fathi et al. [9]. This paper is comprised of four major sections. The authors, first, discuss the simulated FE model used for evaluation of rocking behavior of shallow foundations. Then, the constructed architecture of the GEP model used for prediction of the rocking responses of the soil–structure system is explained, and third, the authors describe the hybrid GEP–ANN model which is deployed to fine-tune the predictive models. Thereafter, the influence of the input parameters on the target values (being moment or damping ratio) is studied using Spearman’s correlation. An ideal flowchart of the proposed steps would be in the form of Fig. 1.
Fig. 1

A generic flowchart of developing a hybrid inverse solver for predicting the rocking responses of a soil–structure system

2 Finite element model

The gathered database of rocking responses of soil–structure systems subjected to a low frequency lateral cyclic loading was employed in this study. A wide range of sandy soils (i.e., very loose to very dense sandy materials) was selected as the supporting system beneath an integrated foundation–structure system. The superstructure system was modeled from a comprehensive range of sizes. For simulating the two systems, i.e., the soil and the superstructure, a dynamic FE program was utilized. Through an iterative process, more realistic behavior for the soil system under the application of static and cyclic loading was attempted.

The geosystem can, indeed, behave nonlinearly before the defined Mohr–Coulomb failure criteria by adjusting the soil modulus with the stress and strain levels during the FE analysis, and perfectly plastic when the geomaterial starts yielding (by reaching the Mohr–Coulomb failure envelope). The low frequency lateral cyclic loading—displacement controlled—was applied to the center of gravity of the integrated foundation–structure system as shown in Fig. 2. The time history includes three clusters of loading with three rotations (0.0015, 0.005, and 0.015 radians) applied to the superstructure.
Fig. 2

Assembled FE model of the soil–structure system as well as the low frequency slow cyclic loading applied to the centre of gravity of the superstructure [9]

Figure 3 represents a schematic of the soil–foundation–structure system during the lateral slow cyclic load in conjunction with the free diagram of the forces over the center of the foundation. Hence, different rocking responses including the damping ratio acquired from rotation-moment hysteresis loops and moment of the soil–structure system mobilized during rocking of the superstructure were recorded at different rotation amplitudes.
Fig. 3

Schematic of the soil–structure system and corresponding free body diagram of forces [9]

To simulate the dynamic shear moduli of granular soils, a nonlinear model proposed by Seed et al. [42] was employed:
$$G_{max} = 218.82 \;\;K_{{2\left( { \hbox{max} } \right)}} \left( {\sigma^{{\prime }} } \right)^{0.5} ,$$
(1)
where Gmax is maximum shear modulus, K2(max) is a laboratory shear modulus coefficient measured at low strain level which varies between 30 and 75 for sandy material, and σ’ is mean effective principal stress which is defined as:
$$\sigma^{\prime} = \frac{{\sigma_{1}^{'} + \sigma_{2}^{'} + \sigma_{3}^{'} }}{3},$$
(2)
For the purpose of this study, 400 FE scenarios were randomly chosen using a wide range of properties proposed for the soil and structure systems. The range of variables taken into consideration for the simulated superstructures is reported in Table 1.
Table 1

Range of variables representing structure systems

Structure features

Symbol

Range of values

Weight

W

19.6–58.8 MN

Height

h s

30–60 m

Length

L

20 m

A feasible range of nonlinear K2(max) parameters for the soil system representing sandy material was considered as recommended by Seed et al. [42]. The higher the parameter K2(max) is, the denser the sandy material would be. Table 2 summarizes the range of material properties used for simulating the geomaterials properties.
Table 2

Geomaterial properties used for simulation of soil system

Geomaterials properties

Range of values

K2(max)

30–75

Cohesion (C)

15 kPa

Angle of internal friction (φ)

38

Dilatation angle

3

Poisson’s ratio

0.3

3 Soft computing methodology

The two rocking responses, i.e., mobilized moment and damping ratio, can exhibit the performance of the soil–structure system under severe loading. Considering the free body diagram of the exerted loads as shown in Fig. 3, the mobilized moment (MMobilized) about the center of the foundation base can be measured using the following equation:
$$M_{Mobilized} = F_{react} \cdot h_{cg} \cdot \cos \left( \theta \right) - W_{s} \cdot h_{cg} \cdot \sin \left( \theta \right) ,$$
(3)
While the damping ratio can be measured using the rotation-moment hysteresis loop developed during the rocking analysis for each of the soil–structure systems; Fig. 4 shows a representative rotation-moment hysteresis loop. Equation 4 accounts for the damping ratio obtained from the hysteresis loop.
Fig. 4

A representative hysteresis loop for rotation moment relationship

$$\xi = \frac{1}{4\pi }\left( {\frac{Area\; of \;Hysteresis \;loop}{Area\; of\; Traingle \;OAB}} \right) ,$$
(4)

It is noted that each of the rocking responses, i.e., the mobilized moment and damping ratio, can be calculated for the soil–structure system at different rotation amplitude applied to the superstructure. The higher the rotation is applied, the greater the rocking responses would become.

In this study, for a better and more reliable prediction of the two responses, a hybrid inverse solver based on the coupled use of Gene-Expression Programming (GEP) and Artificial Neural Network (ANN) is deployed.

3.1 Artificial neural network (ANN)

ANN as a learning method is theoretically developed based on the simulation of the biological nervous system. The application of machine learning methods in general and ANN in particular in solving complex nonlinear models in geotechnical-related problems have been reported extensively in the literature [43, 44, 45]. Multi-layer perceptron (MLP) is one of the common ANN techniques that is used in this study. This technique is built on different layers; an input layer, a hidden layer and an output layer are the main parts of this structure. In each layer, several interconnected variables are connecting to appropriate weighted links. Forward feeding the initial solutions, back-propagating the errors throughout the entire network and fine-tuning the connection weights are the processes that can assist the network to meet the optimized solution [46].

3.2 Gene-expression programming (GEP)

Genetic Programming (GP) as a mathematical evolutionary computation (EC) technique is used to solve the complex problems in dynamic environments [47]. This approach includes an iterative process to optimize and generate the best predictive models. There are several researchers in the field that have used GP technique to solve complex problems [48, 49]. Gene-Expression Programming (GEP) is the specialized form of genetic programming (GP) that was introduced for producing practical solutions for prediction models by Ferreira [50]. This method considered as a type of genetic algorithm since it is composed of a variety of different mathematical solutions that ultimately evolves the selection of the best solution by going through an optimization process. GEP has a tree shape, top-down structure of the genetic algorithm including several subtrees, the ordering of the tree comprises several functions and terminals [51]. The optimization process as the final function of the tree is conducted applying fitness function using the training data. The process of regenerating the data continues until the fitness criterion produced satisfactory results. The regeneration process consists of replication, mutation, transposition and insertion, recombination, and many other methods [50]. In this paper, the Root Means Squared Error (RMSE), was deployed as the fitness function. RMSE is as:
$$RMSE = \sqrt {\sum\limits_{i = 1}^{n} {\frac{{(x_{i} - y_{i} )^{2} }}{n}} }$$
(5)
where xi and yi are measured and predicted rocking responses, respectively, and n is the number of scenarios.
Generally, primary arithmetic operations, i.e., +, −, ×, /, etc.; Boolean logic functions, i.e., AND, OR, NOT, etc.; or other mathematic functions are used to connect the subtrees. The terminal sets consist of the arguments for the functions and numerical constants, logical constants, variables, etc. These functions and terminals are randomly chosen to build a top-sown tree-like structure with root points with ending in a terminal node [49]. Indeed, the terminal nodes are taken into account as the final predictions. Figure 5 shows an example of an expression tree. In this example, the terminal node symbolizes independent variables or constant values. This example can be expressed in the following mathematical form as:
$${ \exp }\left( {X_{1} \cdot \exp \left( {\frac{{X_{2} \cdot C_{1} }}{{\sqrt[3]{{C_{2} }}}}} \right)} \right),$$
(6)
where X1 and X2 are independent variables and C1 and C2 are the constants.
Fig. 5

Example of an expression tree in the form of mathematical equation

4 Predictive GEP-models

The descriptive statistics of the input parameters as well as the dependent variables (being moment or damping ratio) are listed in Table 3. The input dataset contains rotation (θ) which is achieved using the controlled displacement during rocking of the superstructure, stiffness parameter (k2(max)) which is laboratory coefficient used to estimate modulus of geomaterials as defined in Eq. (1) and varies from 30 to 75 for sand (very loose to very dense material), the height of the structure (h), the weight of the structure (W), and the ratio of the contact area (η) which is defined as the contact area of the foundation during rocking to the actual area of the foundation.
Table 3

Summary of descriptive statistics of the predictor and dependent variables

Descriptive statistics

Predictor variables

Parameter (k2(max))

Height (h), m

Weight (W), MN

(a) Variables that are constant during different applied rotations

 Mean

47.78

62.10

41.03

 Median

47.34

61.86

40.76

 SD

10.72

15.95

10.56

 25 percentile

38.72

49.36

32.93

 75 Percentile

56.51

75.97

50.07

Descriptive statistics

Predictor variables

Dependent variables

Contact area ratio (η)

Mmobilized (MN m)

Damping ratio (ξ) (%)

(b) The responses that vary during different applied rotations

At θ = 0.0015 rad rotation

 Mean

89.16

218.40

6.93

 Median

88.59

216.07

7.02

 SD

3.67

36.84

1.93

 25 percentile

86.67

190.89

5.50

 75 Percentile

91.71

242.75

8.10

At θ = 0.005 rad rotation

 Mean

54.49

248.10

10.23

 Median

55.90

243.61

9.84

 SD

11.98

64.83

2.51

 25 percentile

46.65

196.99

8.28

 75 Percentile

64.08

289.26

12.29

At θ = 0.015 rad rotation

 Mean

39.35

319.11

19.42

 Median

39.50

305.19

18.05

 SD

10.46

83.94

4.85

 25 percentile

32.23

250.31

16.09

 75 Percentile

48.40

367.22

22.41

The mobilized moment (Mmobilized, MN m) was defined as a function of rotation (θ, radian), stiffness parameter (k2(max)), height of structure (h, m), weight of structure (W, MN), and ratio of contact area (η); the general form of the predictive function is expressed as:
$$M_{mobilized} = f\left( {\theta ,k_{{2\left( {max} \right)}} , h, W, \eta } \right) ,$$
(7)

4.1 Prediction of rocking response using GEP model: mobilized moment

To predict the mobilized moment the computed database including 374 cases at different rotations (0.0015, 0.005, and 0.015 rad)—1122 FE cases in total—was utilized. As the first step, the database was randomly divided into the training and testing datasets. For this purpose, 80% of the database (896 cases) was assigned to the training set and the rest (226 FE models) to the testing set. The GEP structure was built using twenty chromosomes with three genes and the head size of ten. It is noteworthy to mention that the selection of the parameters mentioned above can profoundly influence the optimized model. The size and number of GEP structure parameters were thus optimized through an iterative process. To develop the best predictive model, RMSE was selected as the fitness function. The following equation was developed as the best predictive function for measuring mobilized moment during foundation rocking:
$$M_{mobilized} = \left( {k_{{2\left( {max} \right)}} - C_{1} } \right) - h^{{\frac{1}{3}}} \cdot \ln \left( {C_{2} + h} \right) + \ln \left( {W + \eta } \right)^{2} \cdot \ln \left( {\frac{{C_{3} \theta }}{{e^{{C_{4} }} }}} \right) + \theta + \left( {h + \left( {h + C_{5} } \right)\theta \cdot h} \right)C_{6}^{{\frac{1}{3}}}$$
(8)
where Ci are the constants; C1 = 0.709, C2 = 0.4503, C3 = 6.25, C4 = − 7.95, C5 = 3.21, and C6 = 0.372.
Figure 6 compares the predicted mobilized moment determined by the GEP-developed Eq. (8) with the corresponding data from the FE model. A promising estimate of the mobilized moment can be governed by GEP model at different rotations, as most of the cases (training data) fall inside the ± 10% uncertainty bounds, with an R2 of 0.89 and root mean squared error (RMSE) of 30.51 MN m. Figure 6b demonstrates the comparison of the GEP and FE models using the testing dataset; the results show less but the acceptable correlation with R2 of 0.87 and RMSE of 36.59 MN m.
Fig. 6

Comparison of GEP-predicted model and FE model for the mobilized moment: a training dataset and b testing data

4.2 Prediction of rocking response using GEP model: damping ratio

Similar to the function developed for the mobilized moment, through an exhaustive process, a GEP-equation was evolved to predict the amount of dissipated energy for the soil–structure system during foundation rocking. The general form of the predictive function for damping ratio, ξ, is as follows:
$$\xi = f\left( {\theta ,k_{{2\left( {max} \right)}} , h, W, \eta } \right),$$
(9)
The following equation was found to be as a good predictor for the damping ratio for the soil–structure system.
$$\xi = C_{1} + C_{2} \theta \cdot k_{{2\left( {max} \right)}} + C_{3} W^{2} + \frac{{C_{4} \theta }}{{k_{{2\left( {max} \right)}} }} + \frac{{C_{5} h}}{ \eta } + C_{6} k_{{2\left( {max} \right)}} + C_{7} \theta ,$$
(10)
where Ci are the constants; C1 = 10.1, C2 = 29.1, C3 = 0.0011, C4 = 1.01e05, C5 = − 0.00137, C6 = − 0.137, and C7 = − 2.68e03.
The damping ratio determined using the GEP model agrees well with the results gathered from the FE model as shown in Fig. 7a. The proposed relationship predicts the dissipated energy with an R2 of 0.90 and an RMSE of 1.94. To further validate the GEP-predicted model, the testing dataset was deployed (Fig. 7b). The results obtained from the testing dataset show a reasonable prediction power with R2 of 0.88 and an RMSE of 2.03.
Fig. 7

Comparison of GEP-predicted model and FE model for damping ratio: a training dataset and b testing data

5 Hybrid GEP–ANN model

As shown in Sect. 4, the rocking responses (mobilized moment and energy dissipation) were predicted with promising accuracy. This section, indeed, attempts to fine-tune the model by introducing a hybrid GEP–ANN approach. The first step was to create a dataset of the absolute errors, ΔErr—defined as the absolute difference between GEP-predicted model and FE model—which was selected as the target dataset for the ANN model. The target datasets for both rocking responses were trained using the same predictor variables used for the GEP model. In this study, the ANN model was built using an input layer containing the predictor variables, hidden layers with 10 neurons, and an output layer comprised of the absolute error values, ΔErr, as shown in Fig. 8.
Fig. 8

Constructed architecture of multilayer perceptron neural network

For the training purpose, the Levenberge–Marquardt algorithm with a multilayer feed-forward neural network and utilizing backpropagation of errors were employed. Figure 9a, b demonstrate the comparison of ANN-predicted error with ANN target error for the mobilized moment for both training and testing datasets.
Fig. 9

Performance of ANN predicted errors from GEP model for mobilized moment; a Training dataset and b testing dataset

The results indicate that the ANN model can reasonably predict the targeted error with an R2 of 0.98, RMSE of 2.86; and R2 of 0.98, RMSE of 3.82 for the training and testing datasets, respectively.

Similarly, the prediction power of the ANN model for damping ratio is shown in Fig. 10 with an R2 of 0.90 for both the training and testing datasets, and RMSE of 0.217 and 0.342 for the training and testing subsets, respectively.
Fig. 10

Performance of ANN predicted errors from GEP model for damping ratio; a training dataset and b testing dataset

The final GEP–ANN predictive models were then constructed by introducing the ANN residual errors, ΔANN_Err, as a predictor variable to the primary model variables. The general form of the constructed GEP–ANN models for the mobilized moment and damping ratio are, respectively, as follows:
$$M_{mobilized} = f\left( {\theta ,k_{{2\left( {max} \right)}} , h, W, \eta ,\Delta _{ANN\_Err} } \right) ,$$
(11)
$$\xi = f\left( {\theta ,k_{{2\left( {max} \right)}} , h, W, \eta ,\Delta _{ANN\_Err} } \right) ,$$
(12)
Equations (13) and (14) were derived as the best products of the GEP–ANN models for the mobilized moment and damping ratio, respectively.
$$M_{mobilized} = C_{1} + C_{2} W + C_{3} \theta^{2} + C_{4} W^{2} + \frac{{C_{5} W}}{\theta } + C_{6} \exp \left( {C_{7} k_{{2\left( {max} \right)}} } \right) + \frac{{C_{8} h\Delta _{ANN\_Err} }}{\eta } + C_{9}$$
(13)
where Ci are the constants; C1 = 292, C2 = 2.93, C3 = 3.43e06, C4 = 0.0321, C5 = − 0.0094, C6 = 15.9, C7 = − 0.0375, C8 = − 0.369, and C9 = 6.7e04.
$$\xi = C_{1} + C_{2} \theta + C_{3} W + \frac{{C_{4} h \cdot\Delta _{{ANN_{Err} }} }}{\eta } + k_{{2\left( {max} \right)}}^{2} \cdot \left( {\frac{{C_{5} }}{{k_{{2\left( {max} \right)}} }} + \frac{{C_{6} \theta }}{{k_{{2\left( {max} \right)}} }} + C_{7} + C_{8} \theta + C_{9} \theta \cdot k_{{2\left( {max} \right)}}^{2} } \right)$$
(14)
where Ci are the constants; C1 = 6.19, C2 = 7.88e03, C3 = 0.0891, C4 = − 0.631, C5 = − 0.0352, C6 = − 341, C7 = − 0.00115, C8 = 4.6, and C9 = − 0.000244.
Figure 11 shows that both performances were meaningfully improved compared to the initial GEP model, due to the number of outlying cases exceeding the ± 10% uncertainty bounds. As a result, the coefficient of determination (R2) and RMSE scores higher and lower numbers, respectively, in comparison to the initial values obtained from the GEP model. The suggested equations are predicting the rocking responses with R2 of 0.976 and 0.991; and RMSE of 10.551 and 0.724 for both the mobilized moment and damping ratio, respectively (shown in Fig. 11a, b).
Fig. 11

Comparison of GEP–ANN predicted model and FE model for a moment data and b damping ratio data

Table 4 summarizes the results for both GEP and hybrid GEP–ANN models for the rocking responses. The results indicate that RMSE decreased by the factor of three and the coefficient of determination (R2) increased by about 9% for the hybrid GEP–ANN models as compared to the GEP models.
Table 4

Comparison of prediction models for the rocking responses

Comparison of prediction models

Rocking response

Coefficient of determination (R2)

Root mean squared error (RMSE)

GEP

Mobilized moment

0.891

30.511

Hybrid GEP–ANN

0.976

10.551

GEP

Damping ratio

0.902

1.943

Hybrid GEP–ANN

0.991

0.724

The predicted data over the corresponding measured FE results for both GEP and hybrid inverse solvers are shown in Fig. 12. GEP-predicted over measured values are having ratios of less than 30% and 40% for moment and damping ratio, respectively. Whereas, the ratio of the hybrid model decreased to less than 10% for the rocking responses.
Fig. 12

The ratio of predicted models to measured FE models for a moment data, and b damping ratio data

6 Sensitivity analysis

Given the results explained in the Sects. 4 and 5, and the desire to utilize rocking responses as performance indicators for soil–structure systems, it is imperative to quantify the effects of the predictor parameters on the rocking responses. To get a better understanding about the rocking capacity of soil–structure systems, the relationships between involving variables and the rocking responses were evaluated using Spearman’s correlation [52]. The results of such activities are discussed for mobilized moment and damping ratio.

The aforementioned correlation investigates the impact of each parameter on the responses of the soil–structure systems under a displacement-controlled rocking. In other words, the higher the correlation between the input and output is, the more impact on the response is expected. The sensitivity analysis implemented in this section can provide a great insight for the designers to considerably harness the rocking capacity of soil–structure systems during strong vibrations such as an earthquake that can result in an enhancement in the overall integrity as well as performance of the buildings. Table 5 shows the influence of the nonlinear predictor parameters on the rocking responses. Stiffness (k2(max)) and rotation (θ) impact the mobilized moment the most while height (h) and rotation (θ) affect the damping ratio the most. The contribution of each parameter on the rocking responses is also shown using the pie charts in Fig. 13.
Table 5

Impact of input parameters on rocking responses

Rocking responses

Spearman’s correlation coefficients

Rotation (θ)

Stiffness (k2(max))

Height (h)

Weight (W)

Contact area ratio (η)

Mobilized moment

− 0.13

− 0.17

− 0.12

− 0.08

− 0.03

Damping ratio

− 0.28

− 0.22

− 0.30

− 0.19

− 0.07

Fig. 13

The contribution of input parameters on the rocking responses: a moment data; and b damping ratio data

7 Summary and conclusion

Strong seismic motion causes large inertial and eccentric forces to the thin and high-rise structures. The shallow supporting foundations beneath the structures may experience a partial separation from the underneath soil due to the overturning moments. To minimize damage and keep the superstructure safe from the vibrations, a displacement-controlled rocking can be exploited. The rocking responses (e.g., moment capacity of the soil–structure system, the dissipation of energy, contact area of the foundation, etc.) can be closely related to each other. The mobilized moment and the energy dissipation are the two primary performance indicators of the soil–structure systems under severe loading. This study mainly attempted to predict the rocking responses of shallow foundations, i.e., moment capacity and energy dissipation of soil–structure systems under slow cyclic loading. For this purpose, Gene-Expression Programming (GEP) was utilized. For further fine-tuning of the predictive model, a coupled use of Gene-Expression Programming (GEP) and Artificial Neural Network (ANN) was introduced.

This study benefited from an implemented FE database gathered for rocking behavior of shallow foundations. The database included a wide range of geomaterials stiffness values for the soil system and different building sizes for the structure part. To evolve the best predictive, the fitness function was selected as RMSE for both GEP and hybrid GEP–ANN models in this study. A promising estimate of rocking responses was governed by GEP model as judged by the coefficient of determination (R2 = 0.89; R2 = 0.90) and root mean squared error (RMSE = 30.51 MN m; RMSE = 1.94%) for the mobilized moment and damping ratio, respectively.

For further improvement of the prediction power of the GEP model, a hybrid inverse solver was employed. The performances were significantly improved for both rocking responses as compared to the initial GEP models. The performance enhancement of the hybrid models was judged by the number of cases lying outside the ± 10% uncertainty bounds. The results indicated that RMSE decreased by the factor of three and the coefficient of determination (R2) increased by about 9% for the hybrid GEP–ANN models as compared to the GEP models. Finally, the influence of the nonlinear nature of the different input parameters on the rocking responses was studied using Spearman’s correlation. The results show that the stiffness parameter (k2(max)) and rotation (θ) have the most influence on the mobilized moment followed by height, weight, and the contact area of the foundation. While, the height of the structure (h) and the stiffness parameter (k2(max)) were found as the most influential factors on damping ratio followed by rotation (θ), weight (W), and the contact area ratio (η).

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringThe University of Texas at El PasoEl PasoUSA
  2. 2.Department of Civil EngineeringSharif University of TechnologyTehranIran
  3. 3.Department of Civil EngineeringCalifornia State University Los AngelesLos AngelesUSA
  4. 4.Civil and Environmental Engineering DepartmentTemple UniversityPhiladelphiaUSA

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