Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=umcm20

Mechanics of Advanced Materials and Structures

ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20

Krill herd algorithm-based neural network instructural seismic reliability evaluation

Panagiotis G. Asteris, Saeed Nozhati, Mehdi Nikoo, Liborio Cavaleri &Mohammad Nikoo

To cite this article: Panagiotis G. Asteris, Saeed Nozhati, Mehdi Nikoo, Liborio Cavaleri& Mohammad Nikoo (2018): Krill herd algorithm-based neural network in structuralseismic reliability evaluation, Mechanics of Advanced Materials and Structures, DOI:10.1080/15376494.2018.1430874

To link to this article: https://doi.org/10.1080/15376494.2018.1430874

Published online: 02 Feb 2018.

Submit your article to this journal

View related articles

View Crossmark data

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES, VOL. , NO. , –https://doi.org/./..

ORIGINAL ARTICLE

Krill herd algorithm-based neural network in structural seismic reliability evaluation

Panagiotis G. Asteris a, Saeed Nozhatib, Mehdi Nikooc, Liborio Cavalerid, and Mohammad Nikooe

aComputational Mechanics Laboratory, School of Pedagogical and Technological Education, Athens, Greece; bDepartment of Mathematics, Statisticsand Computer Science, Marquette University, Milwaukee, Wisconsin, USA; cYoung Researchers and Elite Club, Ahvaz Branch, Islamic Azad University,Ahvaz, Iran; dDepartment of Civil, Environmental, Aerospace andMaterials Engineering (DICAM), University of Palermo, Palermo, Italy; eSAMA Technicaland Vocational Training College, Islamic Azad University, Ahvaz Branch, Ahvaz, Iran

ARTICLE HISTORYReceived December Accepted January

KEYWORDSArtificial intelligencetechniques; artificial krillherd algorithm; artificialneural networks; krill herd;optimization; regressionmodels; seismic reliabilityassessment of structures

ABSTRACTIn this research work, the relative displacement of the stories has been determined by means of a feedfor-ward Artificial Neural Network (ANN) model, which employs one of the novel methods for the optimizationof the artificial neural network weights, namely the krill herd algorithm. For the purpose of this work,the area, elasticity, and load parameters were the input parameters and the relative displacement of thestories was the output parameter. To assess the precision of the feedforward (FF) model optimized usingthe Krill Herd Optimization (FF-KH) algorithm, comparison of results has been performed relative to theresults obtained by the linear regression model, the Genetic Algorithm (GA), and the back propagationneural network model. The comparison of results has been carried out in the training and test phases. Ithas been revealed that the artificial neural network optimized with the krill herd algorithm supersedes theafore-mentioned models in potential, flexibility, and precision.

1. Introduction

During the last three decades, nonconventional methodsare becoming an important class of efficient tools providingsolutions to complicated engineering problems. Among thesemethods, soft computing has to bementioned as one of themosteminent approaches. Neural networks (NNs), fuzzy logic, andevolutionary algorithms are the most popular soft-computingtechniques for the solution of structural engineering problems.Such an important significant issue in structural engineeringwith great interest for engineering of practice is the seismic vul-nerability assessment of constructions. Detailed and in-depthstate-of-the-art reports can be found in [1]–[9] and [10].

The protection of critical structures and infrastructuresystem has recently garnered attention with an emphasis ondamage localization and quantification of such systems. Withthe abundance of new algorithms and data, researchers andengineers should be able to better inform preparedness andplan to avoid the unexpected failure of structural components.In this regard, many researchers have been developing andutilizing various algorithms to predict or detect structuraldamages ([11]–[13] and [14]).

The application of the optimization algorithms to the arti-ficial neural networks (ANNs) has increased in the civil engi-neering domain and it includes the use of the genetic algorithm([15], Khademi et al. [16]–[19]), the imperialist competitivealgorithm [20], [21], and the social-based algorithm [22], [23].

This study aims to use the krill herd algorithm to optimize theweights of the artificial neural networks as a new optimizationalgorithm for the seismic reliability assessment of structures.

CONTACT Panagiotis G. Asteris asteris@aspete.gr; panagiotisasteris@gmail.com Computational Mechanics Laboratory, School of Pedagogical and TechnologicalEducation, Heraklion, GR , Athens, Greece.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umcm.

Therefore, first the modeling and measurement methods wereused to assess the seismic reliability of the structures andthe results were recorded. Second, different parameters wereassessed to identify the input parameters, and the krill herdalgorithm was applied to the feedforward network for theoptimization purposes. To validate the model, a comparisonwas made using the back propagation artificial neural networkmodel [24] as well as linear and nonlinear regression models.

The remainder of this article is organized as follows. Sec-tion 2 is devoted to the introduction of theKrill HerdAlgorithm.Sections 3 and 4 present, respectively, the case study and modelwe used for this study. Section 5 represents the final values ofweights of the Neural Network. Finally, Section 6 presents ourputative conclusions.

2. Introducing the krill herd algorithm (KH)

The Krill-herd algorithm (KH) is a swarm intelligence opti-mization algorithm, based on the behavior of a krill herd[25], and it used to improve the efficiency of artificial neu-ral networks. In the KH algorithm, the krill populationsearches for the krill food sources in a multidimensionalsearch space and then different decisions are proposed. How-ever, the target is the distance between the krill individualsand the excess food associated with the costs. Accordingly,the time-dependent position of a krill individual is measuredby the three operational processes: (i) motion induced bythe presence of other individuals, (ii) foraging movementand (iii) random physical diffusion [25]–[35]. The most

© Taylor & Francis Group, LLC

2 P. G. ASTERIS ET AL.

Yes

Initialization(Number of populations, Maximum number of iterations)

Population of individual fitnessevaluation

Convergence

Motion calculation(i) motion induced by the presence of other

individuals, (ii) foraging movement and (iii) randomphysical diffusion

Implement the genetic operators

Updating krills positions

Output(Best Solution)

No

New

Population

Figure . The flowchart of the krill herd algorithm.

prominent advantages of KH algorithm are as follows; eachagent plays a role in the process, the derivative informationis not necessary, it takes advantage of crossover and mutationoperators. On the other hand, it needs an optimal approachfor determining the primary krill distribution and parametersand the basic motions in the KH algorithm should be morecomprehensive. The flowchart of the krill herd algorithm ispresented in Figure 1.

2.1. The process inducingmotion

The speed of a krill individual is influenced by the movementsof other krill individuals in the multidimensional search space,where speed drastically and dynamically varies by the localimpact, the target herd, and the repulsive impact. The move-ments of a krill individual are expressed using the followingEqs. (1)– (6) [26]:

θnewi = εiθ

maxi + μnθ

oldi (1)

εi = εlocali + εtargeti (2)

εlocali =Ns−1∑i=0

fi jxi j (3)

fi j = fi − f jfw − fb

(4)

Figure . A scheme of herd sensing around a krill individual [].

xi j = xi − x j

| fw − fb|rand(0.1)(5)

εtargeti = 2

(rand(0.1) + i

imax

)f besti xbesti (6)

In the above equations, θmaxi represents the maximum

induced motion, and θoldi denotes the induced motion. In addi-

tion, μn is the algebraic size of the induced motion, whereasthe target impacts are shown by εlocali and ε

targeti . Besides fw and

fb denote the worst and best population positions, respectively.fi and f j are the fitness values of the ith and jth krill individ-uals, respectively. Finally, imax shows the current and maxi-mum quantities [26]. The sensing distance parameter was usedto identify the neighbors of each krill individual (Figure 2). Notethat if the distance between two krill individuals is shorter thanthe sensing distance, that given krill individual is considered theneighbor of the other krill individual. Eq. (7) shows the formulafor calculating the sensing distance [26].

SDi = 15np

np−1∑i=0

|xi − x j| (7)

As shown inEq. (7),np stands for the number of the krill indi-viduals in the population, while xi and x j denote the positionsof the ith and jth krill individuals, respectively [26].

2.2. Foragingmotion

The foraging motion of each krill individual is formulated asEqs. (8) and (9) under the conditions of the existing position ofthe food and the previous knowledge of the food position [26].

Fm = Vf ai + μ f Foldm (8)

ai = afoodi + abesti (9)

In the above equations, Fm is the initial motion,Vf is the for-aging speed, and μ f is the algebraic size of foraging in the [0, 1]range. In addition, Fold

m denotes the previous foraging motion,afoodi is food absorption, and abesti shows the best fitness of eachkrill individual [26].

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 3

Table . Building parameters.

Story height . mSpan length . mNumber of floors Modulus of elasticity of steel (Es) ∼N(∗ kN/m, ∗ kN/m)Steel yield strength mPaSize of beams / mmColumns’area of the first story (A) ∼N( cm, . cm)Columns’area of the second story (A) ∼N( cm, . cm)Columns’area of the third story (A) ∼N( cm, . cm)Gravity load (G) ∼N( kN, kN)Method of analysis Nonlinear time history analysis based

on the Tabas earthquake []

Note: X∼N(a,b)means distribution of randomvariable X follows normal distributionby the mean of a and standard deviation of b.

2.3. Randomphysical diffusion

In the KH algorithm, the population diversity is improved bymeans of the diffusion function, which is integrated into the krillindividuals. Eq. (10) shows the mathematical expression of therandom physical process with the highest diffusion speed andthe random direction factor [26].

RDi = RDmaxυ (10)

where RDmax shows the highest diffusion speed, and υ is a ran-dom vector in the [−1, 1] range.

3. Materials andmethods

3.1. Model description

The present work has applied the model in the case of a 3-storysteel frame building (Figure 3) modeled by OpenSees [36]. Thebuilding parameters used for the development of the model arelisted in Table 1. In total, 1000 different cases were analyzed inorder to investigate the influence of several parameters on themaximum story relative horizontal displacements of the 3-storysteel frame building. Five random variables are presented toconsider different sources of uncertainties in load and structuralmodeling. The random variables are columns’ area of the first(A1), second (A2), and third floor (A3), elasticity modulus(E), and gravity loads (G). The distribution of each randomvariable is selected based on Haldar & Mahadevan (Haldar &Mahadeva, 2000) [40]. 1000 samples randomnumbers provided1000 structural models. The nonlinear time history analysis of

Figure . Cross-section details and themaximum relative horizontal displacementsof the three-story steel frame.

the 1978 Tabas earthquake [37] has been performed for eachmodel by OpenSees. The random variables are used as the inputparameters and the maximum relative horizontal displacementof the first (δ1), second (δ2), and third (δ3) story are consideredas the output parameters. Their mean values together with theminimum and maximum values are listed in Table 2.

3.2. Researchmethodology

Thenovel feed-forward artificial neural networkwas used in thisresearch. The network input parameters were the cross-sectionarea of the columns (A1, A2, A3), the modulus of elasticity (Es)and the gravity load (G), and as output parameters were used thethree maximum relative horizontal displacements of the threestories. This network is illustrated in Figure 4. Of the 1000 datapatterns, 70% (700 patterns)were used for training and 30% (300patterns) were used for testing in the network.

The Krill Herd optimization algorithm (KH) was used asa new metaheuristic algorithm for civil engineering to opti-mize the weights of each of the ANN models. Tables 3 and4 show the optimum structure of each model along with thecharacteristics of the krill herd algorithm. In addition, Table 5presents the analytical results of the training and testing phasesusing the different FF-KH models with the optimum structureshown in Tables 3 and 4. The mean absolute error (MAE), meanerror (ME), age absolute error (AAE), root mean square error

Table . The specifications of the input and output parameters.

No. Parameters Units Parameter type Max Min Average STD

Cross-section area of the columns of the ststory(A)

cm Input . . . .

Cross-section area of the columns of the ndstory (A)

cm Input . . . .

Cross-section area of the columns of the rdstory (A)

cm Input . . . .

Elasticity Modulus kN/m Input .E+ .E+ .E+ .E+ Gravity Load (Es) kN Input . . . . Maximum relative horizontal displacement of

the st story (δ1)cm Output . . . .

Maximum relative horizontal displacement ofthe nd story (δ2)

cm Output . . . .

Maximum relative horizontal displacement ofthe rd story (δ3)

cm Output . . . .

4 P. G. ASTERIS ET AL.

Figure . The feedforward artificial neural network with the --- structure.

(RMSE), and mean square error (MSE) statistical parameters,and the convergence coefficient (R2) are used in Table 5. Accord-ing to these results, the FF-KH yielded the best results with the5-5-4-3 structure. The hyperbolic tangent sigmoid (tansig) acti-vation function has been used in the final model.

Figure 5 depicts the comparison of the “exact” computationalvalues with the predicted values of the optimum FF-KH modelwith topology 5-5-4-3 for the three maximum relative horizon-tal displacements. These results clearly show that the three max-imum relative horizontal displacements predicted from the KrillHerd Algorithm-Based multilayer feed-forward neural networkare very close to “exact” computational results.

4. Model validation

The statistical models used in this research were the multiplelinear regression and the nonlinear regression models. In themultiple linear regression, two or several independent variablessubstantially influence the dependent variable as shown in thefollowing equation [38].

y = f (x1, x2, . . .) → y = a0 +∑i=1

aixi (11)

In the above equation, y is the dependent variable, xi are theindependent variables, and ai are the regression coefficients [38].In this research, different linear regression models were studiedin MINITAB 14.0 [41] for the input and output variables. Thebest multiple linear regressionmodel that mostly complied withthe maximum relative horizontal displacements of the stories is

Table . The characteristics of the feedforward artificial neural network combinedwith the krill herd algorithm.

Features neural network

ModelNumberof input

numberof output

NeuralNetwork

HiddenLayers Nodes

TransferFunction

FF_KH newff _ Tansig

expressed by

y1 = −0.733∗x1 + 0.100∗x2 + 0.050∗x3 + 0∗x4+ 0∗x5 + 0.048 (12)

y2 = −0.620∗x1 − 0.475∗x2 + 0.078∗x3 + 0∗x4+ 0∗x5 + 0.085 (13)

y3 = −0.566∗x1 − 0.438∗x2 − 0.323∗x3 + 0∗x4+ 0∗x5 + 0.104 (14)

where y1, y2, and y3 denote the maximum relative horizontaldisplacements of the first, second, and third story, respectively.Moreover, x1, x2, x3, x4 and x5 stand for A1, A2, A3, Es, and G,respectively (see Tables 1 and 2).

Themean absolute error (MAE), mean error (ME), age abso-lute error (AAE), root mean square error (RMSE), and meansquare error (MSE) statistical parameters, and the convergencecoefficient (R2) are used in Table 6.

Figure 6 depicts the comparison of the “exact” computationalvalues with the predicted values of the linear regression modelfor the three maximum relative horizontal displacements.

In addition, in the nonlinear regression model, two or sev-eral independent variables substantially influence the dependentvariable as shown in Eq. (15) [39].

y = e(a∗x1+b∗x2+c∗x3+d∗x4+e∗x5+ f ∗x6+g∗x7+h) (15)

Table . The statistical results of the artificial neural network combinedwith the krillherd algorithm to determine the relative displacement of each story.

Statistics index

Model Data Story number ME MSE RMSE MAE AAE

KH_ANN All Data . . . . . . . . . . . . . . .

Train . . . . . . . . . . . . . . .

Test . . . . . . . . . . . . . . .

Table . The characteristics of the krill herd algorithm used in the feedforward network.

Features Krill Herd Algorithm

Model Number of Krillsminimum number of

Krill Herdmaximum number of

Krill HerdMaximumIteration

number ofclusters

maximum inducedspeed

FF_KH .

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 5

Figure . Comparison of the computational (‘Exact’) with predicted values of themaximum relative-story displacement using KHA-ANNmodel (All data).

Table . The statistical results of linear regression model in determining the maxi-mum relative horizontal displacement of each story.

Statistics index

Data Story number ME MSE RMSE MAE AAE

All Data . . . . . . . . . . . . . . .

Train . . . . . . . . . . . . . . .

Test . . . . . . . . . . . . . . .

Figure . Comparison of the computational (‘Exact’) with predicted values of themaximum relative-story displacement using LR model (All data).

where y is the dependent variable, xi are the independent vari-ables, and a, b, c,… are the nonlinear regression coefficients. Inthis research, different nonlinear regression models were exam-ined inDatafit 9.0 for the input and output variables. In addition,the best nonlinear regression model, which complied the mostwith the data on themaximum relative horizontal displacementsof the stories, is defined by

y3 = e(−66.689∗x1+13.467∗x2+5.112∗x3+0∗x4+0.001∗x5−2.480) (16)

y4 = e(−30.292∗x1−23.845∗x2+4.655∗x3+0∗x4+0∗x5−0.942) (17)

y5 = e(−22.037∗x1−17.431∗x2−13.460∗x3+0∗x4+0∗x5−0.683) (18)

6 P. G. ASTERIS ET AL.

Table . The statistical results of the nonlinear regressionmodel in determining therelative displacement of each story.

Statistics index

Data Story number ME MSE RMSE MAE AAE

All Data . . . . . . . . . . . . . . .

Train . . . . . . . . . . . . . . .

Test . . . . . . . . . . . . . . .

where y3, y4, and y5 denote the relative displacements of the first,second, and third stories, respectively. Moreover, x1, x2, x3, x4and x5 stand for A1, A2, A3, area, Elasticity Modulus Es, andGravity Loads G, respectively. The results obtained by meansof the multiple linear regression model are listed in Table 7.Figure 7 depicts the comparison of the ‘exact’ computational val-ues with the predicted values of the nonlinear regression modelfor the three maximum relative horizontal displacements.

The analytic and statistical criteria listed in Tables 6 and 7were used to assess the linear and nonlinear regression mod-els. The analysis results indicated that the LR model shown inEqs. (12)–(14) yielded the best results in determining the relativedisplacements of the stories. In other words, the linear regres-sion model even led to more suitable results than the nonlinearregression model.

To assess the precision of the FF model with the krill herdoptimization algorithm, the back propagation neural networkmodel was used along with the linear and nonlinear regres-sion models (as two statistical models) [24]. In addition, theRMSE statistical criterion was used to assess all models againsta single criterion, which was also used by [24]. The results ofall of the model comparisons are shown in Table 8. For thefinal assessment of the models, the best state of each modelwas selected, and eventually four models, namely the KH-ANN,back propagation neural network [24], nonLR, and LR models,were selected as the best models. The results of determining therelative displacement of the stories are also shown in Table 8.Considering the observed and calculated RMSE values of eachmodel and the related results it is concluded that the artificialneural network optimized using the krill herd algorithm deter-mines the relative displacement of each story with more preci-sion than the other models.

5. Final values of weights of the NNmodel

It is common practice, in the majority of the published articleson NNs Models, for authors to present the architecture of the

Figure . Comparison of the computational (‘Exact’) with predicted values of themaximum relative-story displacement using NonLR model (All data).

optimum NN model without any information about the finalvalues of NN weights. Any architecture without the values offinal values of NN model weights has very little value for oth-ers researchers and practicing engineers. In order to be useful, aproposedNNarchitecture should be accompanied by the (quan-titative) values of weights. In such a case, the NN model can

Table . The statistical results of the different models in determining the relative displacement of each story.

RMSE (All Data) RMSE (Test Data)

StoryNeural Network BackPropagation [] KH_ANN

NonLinearRegression Linear Regression

Neural Network BackPropagation [] KH_ANN

NonLinearRegression Linear Regression

. . . . . . . . . . . . . . . . . . . . . . . .

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 7

Table . Final weights and bias values of the optimum KH-ANNmodel ---.

iw{1,1} b{1,1}

− . − . . − . . − .. . . . − . .. . . − . − . .

− . . . . . .− . − . . − . . − .

iw{2,1} b({2,1}

. . . . . − .− . − . − . . . .

. . − . − . . .. . − . . − . .

iw{3,2} b{3,1}

. − . . . .. . . − . .

− . − . . − . .

iw{1,1}=Weights values for Input Layer; iw{2,1}=Weights values for First HiddenLayer; iw{3,2}=Weights values for Second Hidden Layer;b{1,1}= Bias values forFirst Hidden Layer; b{2,1} = Bias values for Second Hidden Layer; b{3,1} = Biasvalues for Output Layer.

be readily implemented in an MS-Excel file, available to anyoneinterested in the problem of modeling.

To this end, in Table 9 the final weights for both hidden lay-ers and bias are presented. By using the weights and bias valuesbetween different layers of ANN the value of the maximum dis-placements for each one story can be estimated (predicted).

6. Conclusions

� The results of testing the proposed method on the struc-ture of the artificial neural network revealed the notablesuccess of the krill herd algorithm in finding the optimumpoint of the functions.

� The optimization potential of the krill herd algorithm canbe used as a powerful tool for the optimization of theweights of the artificial neural networks. The comparisonbetween the training and testing results of the differentANN models optimized with the KH algorithm indicatedthat the artificial neural network with the 4-5-4-3 struc-ture, 10 krill individuals, 2 minimum krill herds, 5 maxi-mum krill herds, and 50 maximum iterations offered thebest results as compared to the other models under study.

� The precision of the maximum relative story horizontaldisplacement determined by the KH-optimized artificialnetwork model was higher than the other models of thistype. In addition, the MSE, ME, MAE, and RMSE statis-tical coefficients were smaller in this model; this reflectedthe lower error rate of this model.

� To assess the artificial neural network model optimizedusing the krill herd algorithm, the model was compared tothe multiple linear and nonlinear regression models andthe back propagation neural network model. The com-parison results indicated that the KH optimized artificialneural network determined the relative displacement ofeach story with more precision and flexibility than theother models.

Conflict of interestThe authors confirm that this article content has no conflict of interest.

Acknowledgments

The authors would like to thank Dr. Panayiotis Roussis, Assistant Professorat the Department of Civil and Environmental Engineering of the Uni-versity of Cyprus, for his valuable comments and discussions. Moreover,we gratefully acknowledge the anonymous reviewers for their insightfulcomments and suggestions.

ORCID

Panagiotis G. Asteris http://orcid.org/0000-0002-7142-4981

References

[1] H. Adeli, “Neural networks in civil engineering: 1989–2000,”Comput.-Aided Civil Infrastructure Eng., vol. 16, no. 2, pp. 126–142,2001. Doi: 10.1111/0885-9507.00219.

[2] P. G. Asteris, K. G. Kolovos, M. G. Douvika, and K. Roinos, “Predic-tion of self-compacting concrete strength using artificial neural net-works,” Eur. J. Environ. Civil Eng., vol. 20, pp. s102–s122, 2016. Doi:10.1080/19648189.2016.1246693.

[3] P. G. Asteris and V. Plevris, “Anisotropic masonry failure criterionusing artificial neural networks,”Neural Comput. Appl., vol. 28, no. 8,pp. 2207–2229, 2017. Doi: 10.1007/s00521-016-2181-3.

[4] P. G. Asteris and K. G. Kolovos, “Self-compacting concrete strengthprediction using surrogate models,” Neural Comput. Appl., pp. 1–16,2017. DOI: 10.1007/s00521-017-3007-7.

[5] I. Mansouri and O. Kisi, “Prediction of debonding strength formasonry elements retrofitted with FRP composites using neuro fuzzyand neural network approaches,” Compos. Part B: Eng., vol. 70, pp.247–255, 2015. Doi: 10.1016/j.compositesb.2014.11.023.

[6] I. Mansouri, A. Gholampour, O. Kisi, and T. Ozbakkaloglu, “Evalu-ation of peak and residual conditions of actively confined concreteusing neuro-fuzzy and neural computing techniques,” Neural Com-put. Appl., pp. 1–16, 2016.

[7] V. Plevris and P. G. Asteris, “Modeling of masonry failuresurface under biaxial compressive stress using neural net-works,” Constr. Build. Mater., vol. 55, pp. 447–461, 2014. Doi:10.1016/j.conbuildmat.2014.01.041.

[8] M. Y. Rafiq, G. Bugmann, and D. J. Easterbrook, “Neural networkdesign for engineering applications,” Comput. Struct., vol. 79, no. 17,pp. 1541–1552, 2001. Doi: 10.1016/S0045-7949(01)00039-6.

[9] M. Safiuddin, S.N. Raman,M.A. Salam, andM.Z. Jumaat, “Modelingof compressive strength for self-consolidating high-strength concreteincorporating palm oil fuel ash,”Materials, vol. 9, no. 5, p. 396, 2016.Doi: 10.3390/ma9050396.

[10] P. G. Asteris, P. C. Roussis, and M. G. Douvika, “Feed-forward neu-ral network prediction of the mechanical properties of sandcretematerials,” Sens. (Switzerland), vol. 17, no. 6, p. 1344, 2017. Doi:10.3390/s17061344.

[11] M. I. Friswell and J. E. T. Penny, “A simple nonlinear model of acracked beam,” in Proc. Int. Modal Anal. Conf., vol. 1992, 1992, p. 516.

[12] E. Özkaya and H. R. Öz, “Determination of natural frequencies andstability regions of axially moving beams using artificial neural net-works method,” J. Sound Vib., vol. 252, no. 4, pp. 782–789, 2002. Doi:10.1006/jsvi.2001.3991.

[13] O. S. Salawu, “Detection of structural damage through changes in fre-quency: A review,” Eng. Struct., vol. 19, no. 9, pp. 718–723, 1997. Doi:10.1016/S0141-0296(96)00149-6.

[14] S. Suresh, S. N. Omkar, R. Ganguli, and V. Mani, “Identification ofcrack location and depth in a cantilever beam using amodular neuralnetwork approach,” Smart Mater. Struct., vol. 13, no. 4, pp. 907–915,2004. Doi: 10.1088/0964-1726/13/4/029.

[15] Ł. Sadowski, “New non-destructive method for linear polarisationresistance corrosion rate measurement,” Arch. Civil Mech. Eng., vol.10, no. 2, pp. 109–116, 2010. Doi: 10.1016/S1644-9665(12)60053-3.

[16] F. Khademi, M. Akbari, S. M. Jamal, and M. Nikoo, “Multiple lin-ear regression, artificial neural network, and fuzzy logic prediction of28 days compressive strength of concrete,” Front. Struct. Civil Eng., pp.1–10, 8 Nov 2016. Doi: 10.1007/s11709-016-0363-9.

8 P. G. ASTERIS ET AL.

[17] E. Momeni, D. Jahed Armaghani, M. Hajihassani, and M. F. MohdAmin, “Prediction of uniaxial compressive strength of rock samplesusing hybrid particle swarm optimization-based artificial neural net-works,” Meas.: J. Int. Meas. Confed., vol. 60, pp. 50–63, 2015. Doi:10.1016/j.measurement.2014.09.075.

[18] M. Nikoo, L. Sadowski, F. Khademi, and M. Nikoo, “Determinationof damage in reinforced concrete frames with shear walls using self-organizing featuremap,”Appl. Comput. Intell. Soft Comput., vol. 2017,p. 10, 2017. Doi: 10.1155/2017/3508189.

[19] M. Nikoo, Ł. Sadowski, and M. Nikoo, “Prediction of the corro-sion current density in reinforced concrete using a self-organizingfeature map,” Coatings, vol. 7, no. 10, pp. 1–14, 2017b. Doi:10.3390/coatings7100160.

[20] Ł. Sadowski, M. Nikoo, and M. Nikoo, “Hybrid metaheuristic-neural assessment of the adhesion in existing cement com-posites,” Coatings, vol. 7, no. 4, pp. 1–12, 1 Apr. 2017. Doi:10.3390/coatings7040049.

[21] L. Sadowski and M. Nikoo, “Corrosion current density predictionin reinforced concrete by imperialist competitive algorithm,” Neu-ral Comput. Appl., vol. 25, no. 7–8, pp. 1627–1638, Dec. 2014. Doi:10.1007/s00521-014-1645-6.

[22] M. Nikoo, F. Ramezani, M. Hadzima-Nyarko, E. Karlo Nyarko, andM. Nikoo, “Flood-routing modeling with neural network optimizedby social-based algorithm,” Nat. Hazards, vol. 82, no. 1, pp. 1–24, 02Feb 2016. Doi: 10.1007/s11069-016-2176-5.

[23] F. Ramezani, M. Nikoo, and M. Nikoo, “Artificial neural networkweights optimization based on social-based algorithm to realize sed-iment over the river,” Soft Comput., vol. 19, no. 2, pp. 375–387, Feb.2015. Doi: 10.1007/s00500-014-1258-0.

[24] S. M. Vazirizade, S. Nozhati, and M. A. Zadeh, “Seismicreliability assessment of structures using artificial neuralnetwork,” J. Build. Eng., vol. 11, pp. 230–235, 2017. Doi:10.1016/j.jobe.2017.04.001.

[25] A. H. Gandomi, S. Talatahari, F. Tadbiri, and A. H. Alavi, “Krillherd algorithm for optimum design of truss structures,” Int. J.Bio-Inspired Comput Rev, vol. 5, no. 5, pp. 281–288, 2013. Doi:10.1504/IJBIC.2013.057191.

[26] A. L. Bolaji, M. A. Al-Betar, M. A. Awadallah, A. T. Khader, and L.M. Abualigah, “A comprehensive review: Krill herd algorithm (KH)and its applications,” Appl. Soft Comput., vol. 49, pp. 437–446, 2016.Doi: 10.1016/j.asoc.2016.08.041.

[27] A. H. Gandomi and A. H. Alavi, “Krill herd: a new bio-inspired optimization algorithm,” Commun. Nonlinear Sci.Numer. Simul., vol. 17, no. 12, pp. 4831–4845, 2012. Doi:10.1016/j.cnsns.2012.05.010.

[28] L. Guo, G.-G. Wang, A. H. Gandomi, A. H. Alavi, and H.Duan, “A new improved krill herd algorithm for global numeri-

cal optimization,” Neurocomputing, vol. 138, pp. 392–402, 2014. Doi:10.1016/j.neucom.2014.01.023.

[29] G.-G. Wang, A. Hossein Gandomi, and A. Hossein Alavi, “A chaoticparticle-swarm krill herd algorithm for global numerical optimiza-tion,” Kybernetes, vol. 42, no. 6, pp. 962–978, 2013. Doi: 10.1108/K-11-2012-0108.

[30] G.-G. Wang, L. Guo, A. H. Gandomi, G.-S. Hao, and H. Wang,“Chaotic krill herd algorithm,” Inform. Sci., vol. 274, pp. 17–34, 2014.Doi: 10.1016/j.ins.2014.02.123.

[31] G.-G. Wang, S. Deb, and L. Coelho, “Earthworm optimiza-tion algorithm: A bio-inspired metaheuristic algorithm for globaloptimization problems,” Int J Bio-Inspired Comput., 2015. Doi:10.1504/IJBIC.2015.10004283.

[32] G.-G. Wang, S. Deb, X.-Z. Gao, and L. D. S. Coelho, “A newmetaheuristic optimisation algorithm motivated by elephant herd-ing behaviour,” Int. J. Bio-Inspired Comput., vol. 8, no. 6, pp. 394–409,2016. Doi: 10.1504/IJBIC.2016.081335.

[33] L. M. Abualigah, A. T. Khader, E. S. Hanandeh, and A. H. Gandomi,“A novel hybridization strategy for krill herd algorithm applied toclustering techniques,” Appl. Soft Comput. J., vol. 60, pp. 423–435,2017. Doi: 10.1016/j.asoc.2017.06.059.

[34] G.-G. Wang, A. H. Gandomi, A. H. Alavi, and D. Gong, “A compre-hensive review of krill herd algorithm: variants, hybrids and applica-tions,” Artif. Intell. Rev., pp. 1–30, 2017.

[35] Y. Feng, G.-G. Wang, S. Deb, M. Lu, and X.-J. Zhao, “Solving 0–1 knapsack problem by a novel binary monarch butterfly optimiza-tion,” Neural Comput. Appl., vol. 28, no. 7, pp. 1619–1634, 2017. Doi:10.1007/s00521-015-2135-1.

[36] S. Mazzoni, F. McKenna, M. H. Scott, and G. L. Fenves, “OpenSeescommand language manual,” Pac. Earthq. Eng. Res. Cent.,2006.

[37] A. Mohajer-Ashjai and A. A. A. A. Nowroozi, “The tabas earth-quake of September 16, 1978 in east-central Iran: A preliminary fieldreport,” Geophys. Res. Lett., vol. 6, no. 9, pp. 689–692, 1979. Doi:10.1029/GL006i009p00689.

[38] M. Nikoo, P. Zarfam, and M. Nikoo, “Determining displacementin concrete reinforcement building with using evolutionary artificialneural networks,” World Appl. Sci. J., vol. 16, no. 12, pp. 1699–1708,2012.

[39] M. Nikoo and P. Zarfam, “Determining confidence for evaluation ofvulnerability in reinforced concrete frames with shear wall,” J. BasicAppl. Sci. Res., vol. 2, no. 7, pp. 6605–6614, 2012.

[40] A. Haldar and S. Mahadevan, Probability, Reliability and Statis-tical Method in Engineering Design, John Wiley & Sons, 2000,2000.

[41] Minitab 17 Statistical Software, [Computer software]. State College,PA: Minitab, Inc., 2010 (www.minitab.com).