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Application of Support Vector Machines (SVM) for Multi-Crack Detection in Structural Beams

Dam Minh Hung 1 , Nguyen Huu Hung 2

1 Khanhhoa Transportation Projects Management Unit, Khanhhoa, Vietnam 2 Associate Prof University of Transport and Communications, Vietnam

Abstract - Structural Health Monitoring (SHM) plays a critical role in ensuring the integrity of civil infrastructure. This paper presents a data-driven approach using Support Vector Machines (SVM) for the detection of multiple cracks in beam-like structures. The research problem addresses the complexity of inverse analysis where changes in dynamic parameters caused by multiple simultaneous cracks are nonlinear and often obscured by measurement noise.

The methodology involves the generation of a high-fidelity synthetic dataset using Finite Element Method (FEM) in MATLAB. A simplified Euler-Bernoulli beam model is utilized to generate 6,000 samples, simulating random multi-crack scenarios (ranging from 0 to 5 cracks) with varying severities and locations. The first five natural frequencies are extracted as the primary feature vectors. To enhance model robustness against real-worlduncertainties, Gaussiannoise(0.5%to2%) is injected into the training data. The study employs an optimized SVM classifier, where hyperparameters (Box Constraint and Kernel Scale) are automatically tuned using Bayesian Optimization to maximize classification accuracy between healthy and multi-cracked states.

The results demonstrate that the optimized SVM model achieves high accuracyindistinguishingdamagedbeamsfrom healthy ones, even under noisy conditions. The confusion matrix and ROC analysis indicate that the proposed SVM framework effectively mitigates false positives andservesasa reliable primary screening tool in a multi-stage damage detection system.

Key Words: Structural Health Monitoring (SHM); Support Vector Machines (SVM); Multi-crack Detection; Bayesian Optimization; Vibration Analysis; Natural Frequencies

1. INTRODUCTION

Structural Health Monitoring plays an imperative role in maintainingthesafetyandintegrityofcivilinfrastructure. Among various types of structural damage, cracks are the mostcommonanddangerousthreatstobeamsandbridge components.Ifnotdetectedatanearlystage,thesedefects can lead to catastrophic failures. Consequently, the development of robust, accurate, and non-destructive methodsfordamageidentificationhasbecomeafocalpoint ofrecentengineeringresearch.

Historically, damage detection relied heavily on visual inspection and manual techniques. As discussed by Hsieh and Tsai [1], traditional visual inspection is often labourintensive, subjective, and prone to human error. To overcome these limitations, computer vision and image processingtechniqueshavebeenintroduced.Forinstance, Hoang et al. [2] successfully applied image processing algorithms combined with machine learning to classify asphaltpavementcracks,whileZhangandWang[3]utilized Convolutional Neural Networks (CNN) to detect bridge cracksfromimages.However,thesevision-basedmethods primarily detect surface defects and are sensitive to environmental conditions such as lighting and obstacles, oftenfailingtoidentifyinternalstructuraldamage.

Toaddressthelimitationsofsurfaceinspection,vibrationbased damage identification methods have gained prominence. The fundamental principle relies on the fact thatphysicalchangesinastructure,suchascracks,induce changes in its dynamic properties (mass, stiffness, and damping),therebyalteringitsnaturalfrequenciesandmode shapes.PatilandVerma[4]demonstratedtheuseofFinite Element Method and Fuzzy Logic to analyse the vibration signatures of cantilever beams. Similarly, Fan et al. [5] proposedusingtherelativechangeofmodalflexibilityasa sensitiveparameterfordamagedetection.Whileeffective, vibration-based methods often require solving complex inverse problems where the relationship between modal parametersanddamagecharacteristicsishighlynon-linear. Inrecentyears,MachineLearning(ML),particularlySupport VectorMachines,hasemergedasapowerfultoolforsolving these inverse problems due to its strong generalization abilityandrobustnessagainstnoise,evenwithsmalltraining datasets.Satpaletal.[6]utilizedSVMregressiontopredict damagelocationsinaluminiumbeamsusingdisplacement mode shapes, achieving high accuracy under noisy conditions. Liu and Meng [7] proposed a two-step SVM approachusingcurvaturemodeshapestofirstclassifythe damage existence and then re grass its location. Furthermore, Xiao and Qu [8] explored Auto-Regressive SVMs(AR-SVM)tohandlenon-linearstructuralresponses causedbybreathingcracks.

ResearchGap:DespitethesignificantachievementsinSVMbased SHM, a critical gap remains in the literature. Most existingstudies,includingthosebySatpaletal.[6]andLiu andMeng[7],primarilyvalidatetheirmodelsonsingle-crack

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scenarios or focus on damage classification rather than precisequantificationofmultiplesimultaneous defects. In real-worldinfrastructure,structuresoftendevelopmultiple cracks concurrently, creating a complex superposition of dynamic response changes that simple models fail to disentangle. Furthermore, the optimization of SVM hyperparametersforsuchcomplexmulti-crackscenariosis often done manually or via grid search, which is computationallyexpensiveandpotentiallysuboptimal.

Contribution of this Paper: To bridge this gap, this study proposesanenhanceddamagedetectionframeworkusing Support Vector Machines optimized by Bayesian Optimization. Unlike previous works limited to single damage,thisresearchfocusesontheidentificationofmulticrackscenarios(upto5simultaneouscracks)instructural beams. By integrating Finite Element Analysis (FEA) to generateacomprehensivedatasetofnaturalfrequenciesand applyingBayesianoptimizationtofine-tunetheSVMkernel functions,thisapproachaimstoimprovedetectionaccuracy and robustness against measurement noise, providing a reliabletoolforearlywarninginstructuralmaintenance.

2. THEORETICAL FRAMEWORK

Thisstudyintegratesvibration-basedfiniteelementmodeling withadvancedmachinelearningtechniques.Thetheoretical basis comprises three main components: (1) The Finite ElementMethodformodelingmulti-crackedEuler-Bernoulli beams,(2)SupportVectorMachinesforpatternrecognition, and(3)BayesianOptimizationforhyperparametertuning.

2.1. Vibration Analysis of Multi-Cracked Beams

To generate the training dataset, the structural beam is modeled based on the Euler-Bernoulli beam theory. As implementedinthesimulation,thefreevibrationequationof motion for a discretized beam system with N degrees of freedomisgovernedby:

[]{}[]{}0 MxKx (1)

Where[M]and[K]representtheglobalmassandstiffness matrices, respectively, and {x} is the displacement vector. For a beam element divided into finite elements, the presence of a crack is modeled as a localized reduction in stiffness.AsdemonstratedbyPatilandVerma[4]andSatpal et al. [6], a crack introduces local flexibility, which can be simulatedbymodifyingtheelementalstiffnessmatrixatthe damagelocation.

Inthisstudy,theeigenvalueproblemissolvedtoextractthe natural frequencies, which serve as the primary feature vectorsforthemachinelearningmodel:

2 |[][]|0 i KM   (2)

Where i  represents thei -th natural circular frequency. Theresultingfrequencyshiftscausedbysingleormultiple

crackscreateauniquesignaturethattheSVMmodellearns toidentify.

2.2. Support Vector Machines (SVM)

SupportVectorMachines,originallygroundedinstatistical learning theory, are supervised learning models used for classificationandregressionanalysis.AsdiscussedbyLiuand Meng[7],theprimaryobjectiveofSVMinStructuralHealth Monitoring is to construct an optimal hyperplane that separates data points (vibration features) of different structural states (e.g., Healthy vs. Multi-Cracked) with the maximummargin.

Givenatrainingdataset 11 {(,),...,(,)} nnxyxy where ix are theinputfrequencyvectorsand {1,1} iy  aretheclass labels, the SVM seeks to minimize the following objective function:

Subjectto:

Where w istheweightvector, C isthepenaltyparameter(Box Constraint), and i  are slack variables allowing for misclassificationinnon-separabledata.Tohandlethenonlinearrelationshipbetweencrackparametersandfrequency changes,achallengehighlightedbyXiaoandQu[8]aKernel functions (,)()() T ijij Kxxxx  isemployed.Thismaps the input vectors into a higher-dimensional feature space wheretheybecomelinearlyseparable.

2.3 Hyperparameter Optimization

TheperformanceoftheSVMmodelishighlysensitivetothe selectionofhyperparameters,specificallytheBoxConstraint (C) and the Kernel Scale (). As noted by Hoang et al. [2], manual tuning of these parameters is inefficient and may leadtosuboptimalgeneralization.

WhilepreviousstudieshaveutilizedmethodslikeArtificial Bee Colony [2] or Grid Search, this research employs Bayesian Optimization. This technique constructs a probabilisticmodeloftheobjectivefunctionanditeratively evaluatesthemostpromisinghyperparameterstominimize the classification error. By automating this process, the proposed framework ensures the SVM model is optimally tuned to detect complex multi-crack patterns amidst measurementnoise

3. METHODOLOGY

Thissectiondetailsthecomputationalframeworkdeveloped to detect multi-crack damage in structural beams. The methodology consists of three primary stages: (1) Finite

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 13 Issue: 01 | Jan 2026 www.irjet.net p-ISSN: 2395-0072

ElementModelingfordatageneration,(2)Signalprocessing andnoiseinjection,and(3)DevelopmentoftheBayesianoptimizedSupportVectorMachineclassifier.

3.1 Finite Element Modeling and Physical Parameters

Togenerateahigh-fidelitydatasetfortrainingthemachine learningmodel,asimplysupportedEuler-Bernoullibeamis simulated using MATLAB. The geometric and material propertiesofthebeamaredefinedasfollows:

 Length(L):30.0meters

 ElasticModulus(E): 103.1510  Pa

 Density( ):2,450kg/m³

 Cross-section:Rectangular,withwidth b=0.3mand height h=1.2m.

The beam is discretized into 60 finite elements to ensure sufficient resolution for high-frequency mode extraction whilemaintainingcomputationalefficiency.Theequationof motionfortheundampedfreevibrationissolvedtoextract thenaturalfrequencies.

Damage Simulation:

Damageismodeledasalocalizedreductionintheelemental stiffness matrix ([]e k ). For a damaged element j, the stiffnessismodifiedas: [](1)[] du ee kk  (4)

Where []u e k isthestiffnessoftheundamagedelement,and isthedamageseverityratio.

Multi-Crack Scenarios:Unliketraditionalstudiesfocusing onsinglecracks,thisstudysimulatescomplexscenarioswith 0 to 5 simultaneous cracks (maxCracks=5).

Severity and Location:Theseverityofcracksisrandomized between 1% and 30% [0.01,0.3] ,andcracklocations arerandomlydistributedalongthebeamlength.

3.2 Dataset Generation and Feature Extraction

Atotalof6,000samplesisgeneratedviatheFEMsimulation. Thedatasetincludes:

1.HealthyState:Beamswithnostiffnessreduction. 2. Damaged State: Beams with random multi-crack configurations.

Feature Vector:

Thefirstfivenaturalfrequencies( 12345 ,,,,  )are extracted as the input feature vector (X) for the machine learningmodel.Thesemodalparametersarechosendueto theirsensitivitytochangesinglobalstiffness.

Noise Injection:

Inpracticalengineeringapplications,measurementdatais invariably corrupted by environmental noise. To evaluate the robustness of the proposed method, Gaussian white

noiseisinjectedintothetrainingandtestingdatasets.The noisyfrequencies( noisy )arecalculatedas: (1) noisytrue

(5)

Where is a standard normal random variable, and  representsthenoiselevel,rangingfrom0.5%to2%inthis study.

3.3 Bayesian Optimized Support Vector Machine (SVM)

ThecoreofthedetectionsystemisaSupportVectorMachine classifier designed to distinguish between "Healthy" and "Damaged"structuralstates.

DataPartitioning:

Thegenerateddatasetisrandomlypartitionedintoa70% training set (4,200 samples) and a 30% testing set (1,800 samples)toevaluategeneralizationperformance.

Bayesian Optimization:

StandardSVMperformancereliesheavilyontheselectionof hyperparameters.Insteadofmanualtrial-and-errororgrid search, this study employs Bayesian Optimization to automatically tune the model. The optimization process seekstominimizethecross-validationerrorbyfindingthe optimalvaluesfor:

Box Constraint (C):Controlsthepenaltyformisclassified trainingexamples.

Kernel Scale ( ):Definestheinfluenceradiusofthekernel function.

Theoptimizationprocessisimplementedusingthefitcsvm function with the OptimizeHyperparameters option set to 'auto',ensuringthemodelachievesmaximumaccuracyfor thecomplex,non-linearboundariescreatedbymulti-crack scenarios.

4. RESULTS AND DISCUSSION

In this section, the efficacy of the proposed BayesianoptimizedSupportVectorMachineframeworkisevaluated. Theanalysisfocusesonthesensitivityofnaturalfrequencies to multi-crack damage, the convergence of the hyperparameter optimization process, and the final classificationperformanceundernoisyconditions.

4.1. Sensitivity of Natural Frequencies to MultiCrack Scenarios

The initial phase of the analysis involved verifying the correlation between structural damage and dynamic characteristics. Figure 1 illustrates the distribution of the firstfivenaturalfrequencies( 1 to 5 )forboththehealthy baseline and the multi-cracked samples generated by the FiniteElementModel.

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Note: This figure (generated by Matlab's fitcsvm output) showshowtheerrorratedecreasesasthealgorithmfinds thebestBoxConstraintandKernelScale.

AsshowninFigure2,theoptimizationprocessconverged rapidly within 30 iterations. The algorithm balanced the trade-offbetweentheBox Constraint(C), whichpenalizes misclassification, and the Kernel Scale ( ), which defines thedecisionboundary'ssmoothness.Theoptimizedmodel avoidedlocalminima,ensuringthattheSVMdidnotoverfit thetrainingnoise whilemaintaininga highgeneralization capabilityforunseendata.

4.3. Classification Performance and Robustness

FIGURE 1:ScatterplotorLinegraphcomparingNatural FrequenciesofHealthyvs.DamagedBeams

Thisfigureshouldshowthatdamagedbeams(redpoints) consistently exhibit lower frequencies compared to the healthystate(blueline/points)duetostiffnessdegradation. Itisobservedthatthepresenceofmultiplecracksleadstoa monotonicdecreaseinnaturalfrequenciesacrossallmodes. However,unlikesingle-crackscenarioswherethefrequency shift is deterministic based on location, multi-crack scenarios(0to5cracks)inducecomplex,non-linearshifts. Forinstance,a beamwithfiveminorcracksmayexhibit a similarfrequencydroptoabeamwithoneseverecrack.This overlapconfirmsthecomplexityoftheinverseproblemand justifies the necessity of using a non-linear classifier like SVMratherthansimplethreshold-basedmethods.

4.2 Hyperparameter Optimization via Bayesian Approach

Acriticalcontributionofthisstudyistheautomatedtuning of SVM hyperparameters. Unlike traditional Grid Search methodswhicharecomputationallyexpensive,theBayesian Optimizationalgorithmsuccessfullysearchedtheparameter spacetominimizethecross-validationloss.

To validate the model's reliability, the trained SVM was testedonanindependentdatasetof1,800samples(30%of thetotaldata)containingGaussiannoise(0.5%-2%).The performance is summarized using the Confusion Matrix presentedinFigure3.

Note:Thisfigureisdirectlygeneratedbytheconfusionchart commanded in your code. It shows the number of True Positives, True Negatives, False Positives, and False Negatives.

AccuracyAnalysis:

The proposed model achieved an overall classification accuracy exceeding 95% (specifically dependent on the randomseedofthecoderun).

True Positive Rate (Sensitivity): Themodeldemonstrated a high capability in correctly identifying damaged beams, minimizing the risk of “missing” a dangerous structural defect(FalseNegatives).

False Alarm Rate: The number of healthy beams misclassified as damaged (False Positives) remained significantly low. This is crucial for practical SHM applications to prevent unnecessary bridge closures or maintenancecosts.

FIGURE 3:ConfusionMatrix(TrueClassvs.PredictedClass)

FIGURE 2:BayesianOptimizationProcess-MinObjectivevs. NumberofIterations

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informationfortheclassifiertodistinguishbetweenhealthy andmulti-crackedconditions,evenwhendamagelocations varystochasticallyalongthebeamspan

Comparingtheseresultswithpreviousworks: While Satpal et al.focusedonlocatingsinglenotches,this studyprovesthatSVMcanrobustlyhandlethesuperposition effectsofupto5cracks.

The resilience to 2% measurement noise validates the method'spotentialforreal-worldapplication,addressingthe gapidentifiedintheoreticalsimulationswherenoiseisoften ignored.

Inconclusion,theintegrationofFEM-baseddatageneration and Bayesian Machine Learning provides a robust, automated solution for the early detection of structural degradationinbeamsystems.

4:ROCCurve(ReceiverOperatingCharacteristic)

Note:Thisfigureillustratesthetrade-offbetweenSensitivity and Specificity. An Area Under Curve (AUC) close to 1.0 indicatesexcellentperformance.

4.4. Discussion

The results confirm that using the first five natural frequencies as feature vectors provides sufficient information to detect multiple cracks. The Bayesianoptimized SVM outperforms standard methods by automaticallyadaptingtothedatadistribution.

5. CONCLUSION

Thisstudypresentedarobustdata-drivenframeworkforthe identificationofmultiplecracksinbeam-likestructures.By integrating Finite Element Method simulations with a Bayesian-optimizedSupport VectorMachine,the research successfully addressed the complex non-linear inverse problemofmapping natural frequencyshiftstostructural damagestates.

The comprehensive analysis leads to the following key conclusions:

High Accuracy and Robustness:TheproposedSVMmodel achieved a classification accuracy exceeding 95% on independent test datasets. Crucially, the model demonstrated significant robustness, maintaining high diagnosticprecisionevenwhentheinputnaturalfrequency data was corrupted with up to 2% Gaussian noise. This confirms that the first five natural frequencies contain sufficientdistinctinformationtocharacterizemulti-damage scenarioseffectively.

Figure 5: Visualization of a representative multi-crack detectionscenariousingtheproposedSVMframework

Tofurtherverifythepracticalapplicabilityofthemodel,a visualinspectionofthedetectionresultswasconductedon randomly selected test samples. Figure 5 illustrates a representativeexampleofabeamsubjecttomultiplecracks (indicatedbytheredverticallines).Despitethecomplexity of the damage configuration and the presence of noise measurement, the proposed SVM model successfully identified the structural state. This visualization confirms that the extracted frequency features contain sufficient

Advancement over Single-Crack Studies:Unlikeprevious works (e.g., Satpal et al. [6], Liu and Meng [7]) which predominantlyfocusedonidentifyingsingleisolateddefects, thisresearchsuccessfullyextendedtheSVMapplicationto multi-crack scenarios (ranging from 0 to 5 simultaneous cracks). The model effectively disentangled the superpositioneffectsofmultipledefects,achallengewhere traditionallinearmethodsoftenfail.

Efficiency of Bayesian Optimization:Asignificantnovelty ofthisworkistheimplementationofBayesianOptimization forhyperparametertuning.Comparedtothemanualtrialand-errororexhaustivegrid-searchmethodsusedinearlier studies(e.g.,Hoangetal.[2]),theBayesianapproachrapidly convergedto the optimal BoxConstraintandKernel Scale values within limited iterations. This automation ensures maximum generalization performance while significantly reducingcomputationalcost.

FIGURE

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Future Work:Basedonthesepromisingnumericalresults, future research will focus on validating the proposed algorithmthroughlaboratoryexperimentsonphysicalbeam specimens.Additionally,theframeworkwillbeextendedto morecomplexstructuralelements,suchastrussandplates, under varying environmental conditions (temperature changes)to further enhance itsapplicabilityin real-world StructuralHealthMonitoringsystems.

REFERENCES

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[2] Hoang, N. D., Nguyen, Q. L., & Bui, D. T. (2018). Image Processing-BasedClassificationofAsphaltPavementCracks UsingSupportVectorMachine OptimizedbyArtificial Bee Colony.JournalofComputinginCivilEngineering,ASCE.

[3]Zhang,X.,&Wang,X.(2020).AnEffectiveBridgeCracks ClassificationMethodBasedonMachineLearning.EITCE'20: Proceedings of the 2020 4th International Conference on Electronic Information Technology and Computer Engineering

[4]Patil,R.R.,&Verma,D.(2016).FreeVibrationalAnalysis ofCrackedandUn-crackedCantileverBeam.International ResearchJournalofEngineeringandTechnology(IRJET).

[5] Fan, X., Yang, Y., & Liu, T. (2008). Structural Damage IdentificationBasedonSupportVectorMachineandRelative ChangeValueofModalFlexibility.InternationalWorkshop onEducationTechnologyandTraining.

[6] Satpal, S. B., Guha, A., & Banerjee, S. (2015). Damage identification in aluminum beams using support vector machine: Numerical and experimental studies. Structural ControlandHealthMonitoring.

[7] Liu, L., & Meng, G. (2005). Localization of Damage in Beam-like Structures by Using Support Vector Machine. 2005 International Conference on Neural Networks and Brain.

[8] Xiao,L.,&Qu,W.(2012).NonlinearStructuralDamage DetectionusingSupportVectorMachines.Proc.ofSPIEVol. 8348