Force-displacement behavior of a beam with butterfly-shaped dampers implementing GE programming

Alireza Farzampour 1, Iman Mansouri 2, Seyed Javad Mortazavi 3, Jong Wan Hu 4,5, *

1 Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, United States

2 Department of Civil Engineering, Birjand University of Technology, Birjand, Iran

3 Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

4 Department of Civil and Environmental Engineering, Incheon National University, Incheon 22012, South Korea

5 Incheon Disaster Prevention Research Center, Incheon National University, Incheon 22012, South Korea

* (Corresponding author, E-mail: jongp24@inu.ac.kr)

Abstract

Structural steel plates having engineered cut-outs to exhibit controlled yielding is recently proposed for desirable performance compared to conventional systems. Butterfly-shaped beams with hexagonal cut-outs inside of the beam’s web is studied to better align the bending strength diagram along the link length with the corresponding demand shape of the applied moment diagram. In previous studies, it has been reported that these links have a substantial energy dissipation capability and sufficient ductility which necessities further investigations. In this study, a set of 240 nonlinear finite element models are developed for creation of a database and subsequently calibrated with finite element software packages. The capability of the gene expression programming (GEP) is explored for prediction of force-displacement relationship of a butterfly-shaped beam. Two new models are developed based on the reliable generated database. Subsequently, the proposed models are validated with several conducted analysis and statistical parameters, for which the comparisons are shown in details. The results represent that the proposed models are able to predict the force-displacement relationship of a butterfly-shaped beam with satisfactory accuracy.

Keywords: Structural fuses, Butterfly-shaped beam, Finite element analysis, Gene expression programming.

1. Introduction

Under severe earthquakes the ductile behavior of the structural elements allows inelastic drifts leading to the overall system’s energy dissipation capability (Farzampour and Eatherton (2018b; Luth et al. (2008; Saffari et al. (2013)). In various structural applications, shear links could be implemented by strategically removing of the material to concentrate the inelasticity and damages in one part of the structure while the remaining parts are intact and undamaged (Atasever et al. (2018; Farzampour and Eatherton (2018a; Kim et al. (2016)). Along the same lines, a promising type of a structural fuse for use in different structural applications is butterfly-shaped beams in which the steel web plate have cutouts inside leaving butterfly-shaped links to better align the capacity diagram with corresponding demand diagram (Lim and Kim (2017); Castaldo et al. (2016)). Figure 1 shows the implementation and details of the butterfly-shaped beam in different structural applications.

Previous studies indicated that using butterfly-shaped shear links within the structures leads to reduction in inelasticity concentration at the critical areas (Farzampour and Eatherton (2018a, (2018b)). This concept could be implemented for designing various high-rise and low-rise buildings as well as the fortification of the existing structures. Butterfly-shaped dampers are studied previously (Farzampour and Eatherton (2019; Lee et al. (2015)) for beam-column connection protection from significant damages under earthquakes. This fairly new dampers are able to concentrate the plastic deformations in structural shear links while the columns and beams remain almost elastic. In addition, some studies showed that these fuses are able to withstand more than 30% shear angle ratios if the possibility of the buckling is prevented (Farzampour and Eatherton (2017; Tsai et al. (1993)).

a) Examples of the implementation of a butterfly-shaped fuse

b) The gemoetrical shapes, and the laoding condition of a typical buttrefly-shaped link

Figure 1. The detailed information of a butterfluy-shaped link and applciations

Along the same lines, different studies showed that straight shear links and butterfly-shaped shear links are capable of controlling structural response of multistory buildings under earthquakes, and then be accessible for replacement purposes (Farzampour and Eatherton (2019)). These links usually represent high ductility and stiffness; however, they are subject to brittle limit states, especially buckling (Lee et al. (2015)). The structural shear links employed in various applications could act as structural fuses since they are able to yield and limit the demand force on the surrounding structural elements. For addressing various limit state issues, the effect of various geometrical properties are studied previously to indicate appropriate design ranges for specific applications (Farzampour and Eatherton (2018b)). It is shown that compared to other types of links, the butterfly shaped links are capable of having hinges far from the sharper areas, full hysteric behavior, significant buckling resistance, and applicable for space-constrained areas.

In recent years Gene expression programming (GEP) has been employed to engineering problems (Ebrahimzade et al. (2018; Gandomi and Alavi (2012; Gandomi et al. (2014b; Güllü (2012; Mahdavi Jafari and Khayati (2018; Mansouri et al. (2016)). This method is implemented in this study to investigate the butterfly-shaped beams in details. This new data-driven method uses individual population to extract the best solution according to one or several gene operations (Mansouri et al. (2017a)). It is noted that the tangible discrepancy in previously employed method of Genetic Programming (GP) and GEP is related to the nature of their programs. The GP considers parse tree programs length, while in GEP, the programs are in the format of linear strings with fixed length Mansouri et al. (2017b), which are represented as the chromosomes.

In this paper, the GEP method is employed to assess the force-displacement relationship of butterfly-shaped beams. The suggested models based on the validated reduced-order models are developed and a database is generated using OpenSees platform. A total of 240 models are used to establish the database for further analysis related to the various geometrical properties. The results of the developed models are culminated in deriving force-displacement relationships for a butterfly-shaped beam with general geometrical properties, and subsequently investigated in details.

2. Gene expression programming

As a progressive algorithm, gene expression programming (GEP) is competent of determining functions from the vast datasets via computer programs evolvement that adopt the role of a living organism. It is generally accepted that they are able to ascertain the links among variables and adapt them by varying their shapes and sizes (Bingöl and Kılıçgedik (2018; Mansosuri and Farzampour (2018)).

By using genetic operators, population evolution occurs upon the selection of a population consisting of individuals (Gandomi et al. (2014a)). The algorithm is terminated if an acceptable fitness level is achieved. A comparison of the predicted and actual values distinguishes minimal errors and such procedure is repeated until a satisfactory solution is obtained (Nie et al. (2013)). There are two GEP components; chromosomes and expression trees (ETs). Chromosomes represent a mathematical expression consisting of one or more genes. The head and the tail are two main components of the genes. Upon transforming every chromosome into an ET, a mathematical relation is deduced.

Numerous parameters should be taken into account amidst the GEP optimization process. The range of varying parameters include fitness function, genetic operators and relevant rates, linking function, random numerical constant status, function set, size of genes, quantity of genes, size of population and learning algorithm, etc. Despite some available guidelines on optimal parameter selection, the simplest and most efficient approach is to apply a trial and error process (Onen (2014)).

According to the literature review, GEP generates more favorable results if there is an increase in the head size. Nevertheless, a head that is too large can hinder GEP performance. Several researchers recommended a head size of 5-30 (Azamathulla (2012; Bingöl and Kılıçgedik (2018; Zakaria et al. (2010)). Evidently, greater head size entails higher computation costs. Thus, a balance is required between efficiency and accuracy. In this study, initially, a single gene with five head sizes is implemented. Then, gene numbers and head sizes are increased upon every run. There were no substantial improvements by over three genes and head sizes of over eight. On the basis of priori knowledge (Azamathulla (2012)), linking functions are selected for specific problems. Here, we applied subtraction, addition, division and multiplication as linking functions. The most favorable fitness values were obtained by linking sub-expression trees (sub-ETs) with addition.

The head is responsible for encoding variables and functions in GEP. Numerous function sets may be utilized. It is noted that choosing the appropriate function is not simple and appropriate guessing is beneficial. In this study, basic arithmetic operators (+, –, ×, /), various mathematical functions (natural logarithm, exponential, xy, square root) were used in addition to various trigonometric functions (arctangent, tangent, cosine, sin). Lastly, in order to configure the chromosomes, genetic operators were required. With the increasing mutation and transposition probability, GEP performance is typically improved. The sufficient mutation probability is 0.03-0.10. Every genetic operator was combined to achieve genetic diversity. Table 1 lists the GEP optimized parameters.

Table 1. Genetic parameters

Parameter

Parameter description

Parameter setting

P1

Chromosomes

100

P2

Fitness function error type

RMSE

P3

Number of the genes

3

P4

Head size

20

P5

Linking function

Multiplication

P6

Function set

+, –, ×, /, Pow, Sqrt, Exp, Ln

P7

Mutation rate

0.00138

P8

One-point recombination

0.00277

P9

Two-point recombination

0.00277

P10

Inversion rate

0.00546

P11

Transposition rate

0.00546

3. Data preparation and reduced order models

The models are developed according to a comprehensive database obtained from the finite element analysis (FEA) (OpenSees software). The 240 models are generated subjected to the cyclic analysis for modeling purposes to establish a general database, based on which the force-displacement equations are derived accordingly. Table 2 shows the parameters of interest ranges. It is noted that a couple of material yielding stresses are considered for the corresponding yielding stress.

Table 2. The value parameters considered for establishing the GEP database

a/b

b/L

L/t

fy, ksi (MPa)

0.1

0.1

10

39 (269)

0.33

0.3

20

52 (358)

0.75

0.5

40

1.0

0.7

60

0.9

80

100

To ensure a successful comparison, a couple of verifications studies are conducted. The first verification study is related to a beam with hourglass-shaped links. FE ABAQUS package is used to model the general fuse system, and obtain the cyclic pushover hysteretic results. Twenty-nodded solid element with reduced integration capability is used (Shin et al. (2017)) as the selected element type to reduce the chance of shear locking and hour-glassing effects. A bilinear material properties with yielding strength of 379 MPa and elastic modulus of 200 GPa, and strain-hardening of 1.38 GPa is considered. According to the geometry of the test, the story shear should be evaluated as the 1.42 times beam shear obtained from FEA, and the story drifts should be estimated by the beam chord rotation divided by 1.34. The beam chord rotation is defined as the transverse displacement by the beam’s clear span length. Figure 2 shows the compared shear-story drifts responses with the corresponding laboratory test of the beam with hourglass-shaped links. In addition, it is reported that that the buckling limit state is occurred at the 2% drift ratio, which is precisely captured from the FEA models. From the FE model, the beam strength before and after buckling were 73.2 kN and 52.7 kN. The reported strength from the laboratory test specimen before and after buckling were 76 KN and 57.4 kN, which shows that the FE modeling methodology was able to capture the actual behavior and limit states of the beam with shear links inside within less than 5% difference.

a) Load vs. deformation b) Test specimen and FE model

Figure 2. Verification of the finite element modeling methodology against laboratory specimens

In the second verification study, the reduced order model from OpenSees is verified with FE package results. The modeling methodology with OpenSees is conducted through validating the cyclic behavior against the FE model. A butterfly-shaped beam is modeled with a set of a flexurally dominated shear links (Farzampour and Eatherton (2018a, (2018b)) with a total length of 120cm. The geometrical properties of the model is shown in Figure 3, the material model is based on the yielding stress of 36 250 MPa (36 ksi)Ksi, modulus of elasticity of 2.0E+5 MPa (229000 ksi) and the strain-hardening ratio of 0.0005.

For verifying the reduced order model with FE model, a cyclic load previously proposed by AISC (AISC 341-16) for EBF behavior investigations is applied at one end of the shown beam in Figure 3, while the other end was fixed. This loading protocol is chosen due to having similarity in behavior of the EBF system with the studied butterfly-shaped beam. The studied FE model is shown in Figure 3.a and Figure 3.b; and the corresponding reduced order model is shown in Figure 4.a, and Figure 4.b, respectively. In addition, the applied cyclic loading protocol is shown in Figure 4c. for verifying the reduced order models with FEA. The hysteretic results of the reduced order model verification study shows that the reduced order model is able to capture the cyclic behavior of a typical butterfly-shaped beam with more than 98% accuracy, which is confirmed with Figure 3.b.

a) The model simulated in FE program

Comment by Eatherton, Matthew: This is not well drawn. Label the dimensions as inches. All dimensions should have leader lines. The dimension line should be moved outside the fuse.

b) Details of the simulated model in FE program

Figure 3. The schematic representation of the butterfly-shaped beam model

a) The schematic representation of the FBF model in OpenSees

b) The verification of the reduced order model in OpenSees with FEA model (120 kips is equal to 530 kN)

c) Loading protocol

Figure 4. The verification of the reduced order beam model

4. Model development

The main purpose of this section is to propose a prediction equation for the force-deformation of butterfly-shaped beam that envelopes the hysteretic behavior subjected to cyclic loading. Figure 5. shows a cyclic envelope, which is specified by connecting the peak force responses at each displacement level. For each one of the 240 models, the envelope curve is derived and stored. The envelope curve subsequently are used for soft computational investigations and the force-displacement prediction equation in the next steps.

Figure 5. Example of a cyclic envelope based on hysteretic response

The considered parameters employed as the predictor variables are a/b, b/L, L/t, fy and d in which the terms a/b, b/L and L/t are the geometrical ratios calculated based on the Figure 1, fy is the yielding stress in ksi, and d is the vertical displacement of linking beam in inches. As a result, formulation of the shear force is formed as follows:

V = f (a/b, b/L, L/t, fy and d) (1)

in which V (pound) is shear force of the butterfly-shaped beam. Here are two proposed prediction models:

(2)

VII = v1v2v3 (3)

in which:

in which, the Eq. (2) is simple and less precise force-displacement equation while Eq. (3) is more accurate and complex. In the next section, the accuracy of the proposed equations are evaluated and studied in details.

5. Results and discussions

Several phases are conducted to select appropriate model, and accordingly the model accuracy is determined. Smith (1986) mentioned that models with correlation coefficient of R2 more than 0.64, would have a strong correlation between the observed and predicted values, meaning that the models’ prediction capability is satisfactory. However, the insensitiveness of the model evaluation capability corresponding to the additive differences is neglected if only R-values are considered. For addressing this issue, root mean square error (RSME) and mean absolute error (MAE) are employed to have more sensitivity to discrepancies between the observed and predicted values. RSME equation is shown in Eq. (4), which considers the summation of the error magnitudes in predictions. Along the same line, MAE equation shown in Eq. (5) indicates how large error values are expected compared to the predicted values on average. It is clear that the model performance would be more desirable if lower RSME and MAE values are achieved.

(4)

(5)

where N, Xoi and Xei are the number of samples, observed and estimated values, respectively.

Figure 6 shows the scatter plot of the target (generated database) and predicted models (GEP). To visualize the correlation between possible variable pair consisting input variables, model and achieved variables, the scatter plots are implemented. Figure 6 shows that the testing and training datasets in which Target is the force values obtained from OpenSees, and model means predicted force value from GEP mode.

It can be observed from Figure 6 that the GEP model with high R values for both testing and training phases predicts the target values to an acceptable degree of accuracy. Having R2 equal to one does not show a prefect prediction and it only indicates a linear correlation between the experimental and predicted data; hence, other statistical indexes (e.g. RMSE and MAE) for evaluation purposes should be implemented for verifying the accuracy of predictions (Sadeghian and Fam (2015)).

(a) Training

(b) Testing

Figure 6. Scatter plot of predicted versus generated shear force using the GEP model for Eq. 3

In addition, the curve-fitting chart shows how the models are fitting the data. For instance, Figure 7 illustrates this chart for both the training and testing phases. It can be figured out that the GEP based shear force model with high fitting status is able to predict the target values with an acceptable degree of accuracy.

a) Curve fitted for training stage

b) Curve fitted for testing stage

Figure 7. Curve fitting of predicted versus generated shear force values using the GEP model for Eq. 3

a) Target sorted fitting for training dataset

b) Target sorted fitting for testing dataset

Figure 8. Target sorted fitting of predicted versus generated shear force values using the GEP model for Eq. (3)

A unique number related to each one of the possible category values are assigned in categorical variable coding. In this process, which is also called dummy variable application, the dummy values are substituted with actual category designations eventually. To indicate how the target and model output data are able to cover the total range, the distributions in parallel scatter plots with dummy random points in Y-axis are developed and shown in Figure 9. Figure 9 also shows dummy values versus the observed values related to the proposed Eq. (3). It is concluded that the target and model outputs are well covered in the entire range of target and model outputs, and the spread as well as overlap of the actual and predicted values are significantly close.

a) Model

b) Target

Figure 9. Training: dummy values versus observed values related to Eq. 3

c) Model

d) Target

Figure 10. Testing: dummy values versus observed values based on Eq. 3

The proposed equations results for training and testing set are represented in Table 3. It is concluded that based on the Table 3 results, the accuracy of the proposed Eq. (3) with more complicated formulization compared to Eq. (2) is high since the R 2 values are close to 1. For Eq. (3) The RSME and MAE for training and testing sets are close with less than 0.5% difference. However, this difference between the testing and training statistical parameters is between 10% and 20% for the model proposed in Eq. (2). The operator is able to decide to use either of the equations based on the application and required accuracy for design purpose.

Table 3. The statistical data of the proposed equation related to the derived equations

Model

Vtarget vs. Vmodel

R2

RMSE

MAE

 

Training

Testing

Training

Testing

Training

Testing

I (Eq. 2)

0.83

0.82

136.37

156.17

72.43

79.32

II (Eq. 3)

 0.99

0.99 

22.28 

23.04 

13.30 

13.78 

6. Conclusions

The application of the relatively new soft computing approach of GEP to predict the force-displacement relationship of butterfly-shaped beams was described in this study. For this purpose, 240 finite element models were generated using OpenSees platform to establish a database. Two GEP models were developed accordingly to predict the force-displacement of butterfly-shaped beams with general geometrical properties based on the generated database.

The results of the GEP-based models were able to accurately predict the butterfly-shaped beams shear force at a specified displacement value. The model’s validity is tested for parts of the results beyond the training data and it is shown that GEP prediction model satisfies the various criteria related to external validations. Furthermore, the validation phase verified the model efficiency for general shear force estimation of butterfly-shaped beams. Several factors (a/b, b/L, L/t, fy and d) are represented within the model to consider the effect of the different geometrical and material properties of a typical butterfly-shaped beam. The accuracy of the proposed equations are shown to be high due to the close agreement in R2, RMSE and MAE between training and testing data sets. Therefore, it is concluded that the proposed models could be implemented for design and pre-planning purposes.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2017R1A2B2010120).

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00.040.080.120.160.20.240.280000000000000030.320.360.40.440.480.520.560000000000000050.60.640.680.720.760.80.840.880.920.961.041.12000000000000011.21.281.361.441.521.61.681.761.841.9222.082.162.24000000000000022.31999999999999982.42.482.562.642.722.82.8833.123.243.363.483.63.723.843.964.084.24.324.44000000000000044.55999999999999964.684.84.925.045.165.285.45.525.645.765.8866.126.246.46.566.726.887.047.27.367.527.687.8488.168.328.488.648.80000000000000078.96000000000000099.11999999999999929.27999999999999949.449.689.9210.1610.410.6410.8811.1211.3611.611.8412.0812.3212.5612.813.0413.2813.5213.761414.2414.4814.7214.9615.215.5215.8416.1616.4816.817.1217.44000000000000117.76000000000000218.07999999999999818.39999999999999918.7219.0419.3619.682020.3220.6420.9621.2821.621.9222.2422.5622.8823.3623.8424.3224.825.2825.7626.2426.7227.227.6828.1628.6429.1229.630.0830.5631.231.8432.47999999999999733.11999999999999733.7634.435.0435.6836.3236.9637.638.2438.88000000000000339.52000000000000340.15999999999999740.79999999999999741.7642.7243.6844.6445.646.5647.5248.4849.7651.0452.3253.655.256.858.46061.663.264.866.4000000000000066869.59999999999999471.272.874.4000000000000067677.59999999999999479.280.882.48485.687.288.890.49293.695.296.898.4100101.6103.2104.8106.4108109.6111.201.25E-30-1.25E-301.25E-30-1.25E-301.25E-30-1.25E-301.25E-30-1.25E-301.25E-30-1.25E-301.25E-30-1.25E-302.5000000000000001E-30-2.5000000000000001E-302.5000000000000001E-30-2.5000000000000001E-302.5000000000000001E-30-2.5000000000000001E-302.5000000000000001E-30-2.5000000000000001E-302.5000000000000001E-30-2.5000000000000001E-302.5000000000000001E-30-2.5000000000000001E-303.7499999999999999E-30-3.7499999999999999E-303.7499999999999999E-30-3.7499999999999999E-303.7499999999999999E-30-3.7499999999999999E-303.7499999999999999E-30-3.7499999999999999E-303.7499999999999999E-30-3.7499999999999999E-303.7499999999999999E-30-3.7499999999999999E-305.0000000000000001E-30-5.0000000000000001E-305.0000000000000001E-30-5.0000000000000001E-305.0000000000000001E-30-5.0000000000000001E-305.0000000000000001E-30-5.0000000000000001E-305.0000000000000001E-30-5.0000000000000001E-305.0000000000000001E-30-5.0000000000000001E-307.4999999999999997E-30-7.4999999999999997E-307.4999999999999997E-30-7.4999999999999997E-307.4999999999999997E-30-7.4999999999999997E-307.4999999999999997E-30-7.4999999999999997E-307.4999999999999997E-30-7.4999999999999997E-307.4999999999999997E-30-7.4999999999999997E-300.010-0.0100.010-0.0100.010-0.0100.010-0.0100.010-0.0100.010-0.0101.4999999999999999E-20-1.4999999999999999E-201.4999999999999999E-20-1.4999999999999999E-201.4999999999999999E-20-1.4999999999999999E-201.4999999999999999E-20-1.4999999999999999E-200.020-0.0200.020-0.0200.020-0.0200.020-0.0200.030-0.0300.030-0.0300.040-0.0400.050-0.0507.0000000000000007E-20-7.0000000000000007E-200.090-0.0900.110-0.1100.130-0.1300.150-0.1500.170-0.1700.190-0.1900.210-0.210

Time (s)

Drift (%)

Curve fitting

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