STRUCTURAL CONTROL AND HEALTH MONITORINGStruct. Control Health Monit. 2014; 21:303–316Published online 2 May 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.1562
Nondestructive damage detection in Euler–Bernoulli beams usingnodal curvatures—Part I: Theory and numerical verification
Selcuk Dincal and Norris Stubbs*,†
Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843, USA
Nondestructive damage evaluation (NDE) offers effective and economically feasible solutionsfor monitoring the integrity of critical civil engineering structures. Periodic inspection performedthroughout the life span of these structures is essential for public safety. The basic idea behind NDEis that changes in the physical properties of a structural system alter the response characteristics. Fromthe inverse perspective, structural response may be used to nondestructively evaluate the physicalproperties of the structure. Structurally deficient regions that need immediate attention can successfullybe identified utilizing NDE. Detecting structural damage in this manner has the advantage of providinginformation throughout the life cycle of a structure and may lead to more effective and economicallyfeasible solutions, compared to local damage evaluation methods such as visual inspection. For thesereasons, numerous NDE methodologies have been proposed over the past few decades.
Rytter [1] was among the first researchers who attempted to classify systematically the existingNDE methodologies on the basis of the output information they provided. He proposed the followingfour general methodological categories: (1) level I: detection of damage; (2) level II: localization of
*Correspondence to: Norris Stubbs, Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, CollegeStation, TX 77843, USA.†E-mail: [email protected]
damage; (3) level III: assessment of the severity of damage; and (4) level IV: performance evaluationafter a level III assessment. Stubbs et al. [2] and Stubbs and Osegueda [3,4] introduced first-ordersensitivity equations that related the measured change in the natural damped frequency of a dynamicalsystem to the changes in modal mass, stiffness, and damping. Vibration tests performed by Mazurek andDeWolf [5] on a bridge girder indicated that the greatest changes in mode shapes occur in the vicinity ofthe structural defect. Pandey et al. [6] used mode shape curvatures to locate damage in beam-typestructures. Pandey and Biswas [7] utilized the change in modal flexibilities before and after damage tolocate defects on analytical and experimental case studies. Zhang and Aktan [8] utilized the curvature ofuniform load flexibility to detect damage. The largest value of the curvature difference computed beforeand after damage indicated the location of damage. Stubbs et al. [9] developed the damage index method,which utilizes the equivalency of the fraction of modal strain energy (also referred to as element sensitivity)before and after damage. The feasibility and practicality of the method have been demonstrated on variousstructures ranging from offshore platforms [9] to beams [10] and frames [11]. Later, Kim and Stubbs [12]proposed an improved damage identification methodology, which utilizes fractional changes in modalparameters in addition to the strain energy stored in the structure.
Osegueda et al. [13] utilized the difference in the modal strain energy between the damaged andundamaged structures. Locations that exhibited an increase in the difference designated the areas ofpossible damage. Guan and Karbhari [14] introduced a damage detection method based on modalstrain energy. Modal displacements and modal rotations in conjunction with Hermite cubic shapefunctions were utilized to compute the modal strain energy.
On the basis of a review of the NDE methodologies proposed to date, there exists a need for afundamental theory of damage detection and severity estimation. So far, the common practice has beento utilize the pre-damage and post-damage response data in an ad hoc manner intended primarily todetect structural damage. For this reason, many well-known NDE methodologies cannot be extendedto a level III NDE evaluation (e.g., [6–8]).
The objective of this paper is to present a level III damage evaluation methodology based on themoment–curvature relations of the Euler–Bernoulli beam theory and the assumption that internal stressresultants for beams subjected to same external loading are invariant before and after damage. Toachieve this objective, structural responses collected prior and subsequent to damage are related tothe changes in physical properties of the structure. Damage is expressed in terms of local decreasesin the flexural stiffness of structural members. These decreases are assumed to cause singularities inthe curvature profile of the beam. Utilizing basic solid mechanics, we relate discontinuities in theflexural stiffness distribution to the pre-damage and post-damage nodal curvatures. The theory ofdamage evaluation and numerical verification of the proposed approach is presented as part I in thispaper. Field verification of the methodology is presented in an accompanying paper.
2. THEORY OF DAMAGE EVALUATION
2.1. Basic assumptions
Figure 1 shows the undamaged and damaged beams composed of NE sub-elements and NN nodes. Thedeformed beams obey the Euler–Bernoulli assumptions. Let M(li) be the bending moment at x= li, andalso let the beam be subjected to an external load P(x). kL lið Þ is the curvature at x = li, as x approaches li
Figure 1. Graphical representation of the undamaged and damaged beams.
NDE USING NODAL CURVATURES—PART I: THEORY AND NUMERICAL VERIFICATION 305
from the left; kR lið Þ is the curvature at x = li, as x approaches li from the right; EIj is the flexural stiffnessof the jth element; and EIj� 1 is the flexural stiffness of the (j� 1)th element. Then, the followingconditions hold:
M lið Þ ¼ EIj�1kL lið Þ ¼ EIjkR lið Þ (1)
Similarly, for the damaged beam subjected to the same loading P(x),
M� lið Þ ¼ EIj�1�kL lið Þ� ¼ EIj
�kR lið Þ� (2)
where the asterisk denotes the damaged beam.Also, at the ith node of the beam, where EIi is the flexural stiffness and ki is the curvature,
M lið Þ ¼ EIiki (3)
From calculus, the Fourier series representation of any function converges to a value that is theaverage of the values immediately to the left and right of the discontinuity [15]. Therefore, even thoughthe curvature is discontinuous at x = li, its value converges to
ki ¼ 12
kL lið Þ þ kR lið Þ� �
(4)
Using Equations (1), (3), and (4) gives (li)
M lið Þ ¼ EIiki ¼ EIi12
M lið ÞEIj�1
þM lið ÞEIj
� �(5)
or simply
EIi ¼ 2EIj EIj�1
EIj þ EIj�1
� �(6)
Using similar reasoning, we can write the following expression for the damaged structure as
EIi� ¼ 2
EIj� EIj�1�
EIj� þ EIj�1�
� �(7)
2.2. Proposed damage evaluation methodology
Assuming that the internal stress resultants in the beam are not affected by the inflicted damage (whichis always true for statically determinate structures and approximately true for statically indeterminatestructures subjected to small damages as long as the undamaged and damaged structures are subjectedto identical loading), the following condition holds at the ith node of the beam:
M lið Þ ¼ M� lið Þ (8)
Then, from Equations (3), (6), and (7), it follows that at the ith node,
2EIj EIj�1
EIj þ EIj�1
� �ki ¼ 2
EIj� EIj�1�
EIj� þ EIj�1�
� �ki� (9)
Assuming a constant flexural stiffness distribution for the undamaged beam (i.e., for j= 1, . . .,NE,EIj=EI) and collecting element stiffnesses on the right-hand side of Equation (9) gives
EIEIj�1
� þ EIj�
EIj�1� EIj�
� �ki ¼ 2 ki� (10)
Simplifying Equation (10) leads to the following result:
Using the recursive scheme established in Equation (13), we find that the relation between thedamage indices for the jth and (j+ 1)th beam elements at the (i+ 1)th node becomes
bjþ1 þ bj� �
kiþ1 ¼ 2 kiþ1� (14)
Thus, for a beam with NN nodes and NE sub-elements, where NE =NN� 1, NN� 2 linear equationscan be written using the scheme defined by Equations (13) and (14). This process results in a system ofunderdetermined linear equations. Moore–Penrose pseudoinverse may then be used to obtain thegeneralized inverse of the matrix built by rearranging the system of linear equations. This processyields the unknown element damage ratios.
The system of equations for damage evaluation can be written in the form
A NN�2ð Þ� NEð Þ b NE�1ð Þ ¼ B NN�2ð Þ�1 (15)
The NE� 1 (i.e., NE by 1) vector b denotes the unknown damage ratio vector. The (NN� 2)� (NE)matrix A contains the measured curvatures for the undamaged beam, and the (NN� 2)� 1 vector Bcontains the measured curvatures for the damaged beam.
The solution to Equation (15) is given by
b ¼ A�1P B (16)
where A�1P is the pseudoinverse of A. The physical meaning of A�1
P is a collection of the radius ofcurvatures of the undamaged beam. SVD is used to compute the generalized (or pseudoinverse) ofmatrix A [16]. This procedure may be described as follows.
Pre-multiplying both sides of Equation (15) by the transpose of A (designated as A ’) leads to
A0Ab ¼ A0B (17)
where, for convenience, the aforementioned quantities can be written as
Aþ ¼ A0ABþ ¼ A0B
(18)
Using SVD, we can express matrix A+ in the following form:
Aþ ¼ U S V 0 (19)
where the diagonal matrix S contains scalars referred to as the singular values of A+ and U and V arereferred to as left singular vectors and right singular vectors, respectively. The inverse of A+ may beeasily computed as
Aþð Þ�1 ¼ V Sð Þ�1 U0 (20)
where
Sð Þ�1 ¼ diag S�1i
� �(21)
and
S�1i ¼ 1=Si for Si > 0
0 for Si ¼ 0
�(22)
Finally, the unknown damage vector b given in Equation (16) can be explicitly written as
b ¼ V Sð Þ�1 U0Bþ (23)
The predicted severity of the localized damage can be expressed in terms of the pristine anddamaged flexural stiffnesses
ajP ¼ ΔEIjEIj
¼ EIj� � EIjEIj
¼ 1bj
� 1 (24)
Equations (23) and (24) constitute the proposed level III damage evaluation methodology forEuler–Bernoulli beams.
NDE USING NODAL CURVATURES—PART I: THEORY AND NUMERICAL VERIFICATION 307
3. NUMERICAL VERIFICATION
To validate the proposed methodology, damage in Euler–Bernoulli beams was simulated usingtwo-dimensional finite element (FE) models. These numerical experiments were intended tosimulate the more costly real-world experimental studies.
3.1. Physical description of the simulated test beam
The selected test structure was a rectangular cantilever steel beam. The designated width, depth, andlength of the beam were 6.35 cm (2.5 in.), 30.5 cm (12 in.), and 365.8 cm (144 in.), respectively.Figure 2 shows the elevation and cross-sectional views; Table I summarizes the material and sectionalspecifications of the beam.
The FE model of the test beam was constructed using bilinear quadrilateral plane elements. Eachquadrilateral consists of eight DOFs [17]. In addition to the physical DOFs, each element containsinternal DOFs at the element level, which are not associated with any node. These nodeless DOFs wereused to circumvent shear locking and improve the bending behavior of the beam.
Convergence tests were performed to identify an FE model that could provide an accurateprediction of the deformation of the beam. The free end of the cantilever beam was subjected to a staticload of 4448.2N (1 kip). The vertical displacement at the tip of the beam was computed to be 0.244 cm,using the theory of elasticity solution given in [18]. The solution obtained by the FE model rapidlyconverged to the exact solution given by the theory of elasticity when the beam was subdivided intoat least 120 plane elements. The error between the numerical and exact solutions was less than0.5%. Figure 3 depicts the results of convergence tests performed on the trial FE models.
Figure 4 shows the chosen FE mesh for the simulated beam. Each plane element was 12.2 cm(4.8 in.) wide, 7.62 cm (3 in.) deep, and 6.35 cm (2.5 in.) thick in the perpendicular z direction. Notethat the plane stress assumption was adopted in the FE model.
3.2. Proposed damage scenarios
Damage was simulated by reducing Young’s moduli of the individual plane elements in the FE modelof the test beam. Three different parameters were varied while simulating damage, namely, the damagelocation, the size of the damaged region (i.e., damage extent), and the damage severity (i.e., relativepercent reduction in the elastic modulus). The damage location corresponds to the center of theinflicted damage. The damage extent is represented by the area of the damaged region. Lastly, thedamage severity is defined as the percent reduction in Young’s modulus within the area defined bythe damage extent. Figures 5–7 show the damage scenarios inflicted on the simulated beam. Elastic
Figure 2. Schematic of the test beam.
Table I. Material and sectional specifications of the test beam.
Description Magnitude
Span length (cm) 365.8Beam thickness (cm) 6.35Beam depth (cm) 30.5Cross-sectional area (cm2) 193.55Moment of inertia (cm4) 14, 984.33Mass density (kg/m3) 7850Modulus of elasticity (N/m2) 20� 1010
NDE USING NODAL CURVATURES—PART I: THEORY AND NUMERICAL VERIFICATION 309
moduli of various plane elements are decreased for simulating the prescribed damage scenarios. Table IIsummarizes the geometric and elastic details of the inflicted damages. Note that the damage parametersindicated in Table II are based on the FE mesh presented in Figure 4.
3.3. Sensor placement scheme and basic measurements required by the theory
The proposed damage evaluation methodology utilizes the pre-damage and post-damage curvatureprofiles to identify damage in Euler–Bernoulli beams. Static deflection data are directly utilized toobtain point curvatures along the length of the beam. Note that the undamaged and damaged beams weresubjected to the same external loading condition. Although an infinite number of data points are available intheory, only a limited number of sensors can be used to collect data, in practice. To simulate this morerealistic case, the sensor layout given in Figure 8 is proposed. The recorded responses are the simulatedstatic displacements measured resulting from 4448.2N (1kip) load applied at the tip of the beam.
Once the pre-damage and post-damage deformed shapes are available, cubic spline interpolationwith 1.524 cm (0.6 in.) uniform intervals was used to generate a finer sensor layout along the lengththe beam. This process leads to 241 simulated nodal points (including the node that corresponds tothe clamped support). Ideal supports (i.e., those that exhibit no settlements) were utilized in this study.The resulting curvature profiles were estimated from the deflected shapes of the undamaged anddamaged beams by using the central difference approximation.
3.4. Assessment of damage prediction accuracy
The performance of the proposed NDE methodology is evaluated using three criteria, namely, theaccuracy of damage localization, the accuracy of identifying the damage extent, and the accuracy ofdamage severity estimation.
Damage localization accuracy is quantified by the relative distance between the true and predicteddamage locations, normalized by the length of the beam. This error can be expressed in terms of apercentage as
eL ¼ xT � xPL
� 100 (25)
where xT and xP correspond to the true and predicted damage locations and L is the total length of thebeam. Note that the proposed methodology is limited to a one-dimensional Euler–Bernoulli beamtheory. Therefore, possible damage locations can only be identified along the longitudinal axis ofthe beam; here, the distribution of damage along the beam depth is not addressed.
Table II. Details of the prescribed damage scenarios.
Damagescenario
Damage location Damage sizeDamageseverity
Damagedelementsx (cm) y (cm) Δx (cm) Δy (cm) ΔA (cm2)
1 250.0 15.2 12.2 30.5 371.6 �5% of E 21, 51, 81, 1112 6.1 11.4 12.2 22.9 278.7 �11% of E 1, 31, 61
256.0 11.4 24.4 7.6 185.8 �7% of E 51, 523 85.3 22.9 24.4 15.2 371.6 �3% of E 67, 68, 97, 98
189.0 19.1 12.2 22.9 278.7 �5% of E 46, 76, 106286.5 19.1 12.2 7.6 92.9 �6% of E 84
Figure 8. Proposed sensor layout for the test beam.
Damage extent was initially defined as the area of the damaged region. In order to be consistent with theone-dimensional damage evaluation methodology proposed here, damage extent is expressed in terms ofthe axial length of the damaged region. Consequently, the accuracy of identifying the damage extent isquantified by dividing the difference between the axial length of the true and predicted damaged regionsto the total length of the beam. This dimensionless error can be expressed in terms of a percentage as
eE ¼ LT � LPL
� 100 (26)
where LT refers to the true length of the inflicted damage (Δx), LP represents the length of the predicteddamage extent, and L is a metric that defines the total length of the beam.
The error in damage severity estimation is quantified by utilizing the fractional error in stiffnessprediction. Using the true, predicted, and undamaged element stiffnesses, we can express in terms ofa percentage a dimensionless error in stiffness prediction as
eK ¼ kjT � kjPk
� 100 (27)
where kjT and kjP correspond to the true and predicted stiffnesses of the jth beam element, respectively.The term k represents the undamaged element stiffness.
Utilizing the true and predicted damage severities, we can rewrite Equation (27) as
eK ¼ k 1þ ajT� �� k 1þ ajP
� �k
� 100 (28)
where ajT and ajP correspond to the true and predicted damage severities of the jth beam element,respectively. Equation (28) simplifies to
eK ¼ ajT � ajP� �� 100 (29)
The true damage severity, denoted by ajT in Equation (29), may not correspond to the actual reductionimposed on the modulus of elasticity of the beam elements. For example, in damage scenario 2 where theelastic moduli of three of the four plane elements located in the vicinity of the clamped support were reducedby 11%, the true damage severity does not coincide with the 11% stiffness reduction in the one-dimensionalbeam. Damage distribution throughout the beam’s depth cannot be addressed by a one-dimensional damagedetection theory. Thus, as described later, equivalent one-dimensional flexural stiffnesses for the damagedand undamaged beams were estimated from the two-dimensional damage scenarios.
Suppose that a system of linear springs connected in parallel (Figure 10) models the plane elementslocated at an arbitrary distance x0 from the clamped end (Figure 9).
The equivalent bending stiffness of the linear spring model, for the jth location, given in Figure 10,can then be written as
kjMequ ¼ kBN hNð Þ2 þ kBi hið Þ2 þ⋯þ kAi hið Þ2 þ kAN hNð Þ2 (30)
where hi represents the height of the ith spring measured from the neutral axis. The subscripts A and Brepresent the springs located above and below the neutral axis, respectively. The stiffness of eachspring that appears in Equation (30) can be estimated from the following equation:
ki ¼ EiAi
li(31)
Figure 9. Plane elements centered at the distance x0.
Figure 10. The deformed shape of the linear springs model subjected to bending.
NDE USING NODAL CURVATURES—PART I: THEORY AND NUMERICAL VERIFICATION 311
where the terms E, A, and l denote the elastic modulus, the area, and the length of the individual planeelements, respectively.
Similarly, for the damaged structure (where the moduli of elasticity of one or more plane elementsare reduced), the equivalent bending stiffness given in Equation (30) can be written as
kjMequ
� ¼ kBN� hNð Þ2 þ kBi
� hið Þ2 þ⋯þ kAi� hið Þ2 þ kAN
� hNð Þ2 (32)
where the asterisk represents the stiffness of the damaged springs. Then, using Equations (24), (30), and(32), we can compute the true equivalent one-dimensional flexural stiffness damage severity, ajT, as
ajT ¼ kjMequ�
kjMequ� 1 (33)
3.5. Damage prediction results for the case studies
3.5.1. Damage scenario 1. Damage localization result for damage scenario 1 is depicted in Figure 11.Table III summarizes the method’s damage localization accuracy. Damage localization results arereported in terms of the element damage ratios (Equation (16) or Equation (23)). Note that LP and
Figure 11. Damage localization result for damage scenario 1.
Table III. Assessment of the damage localization accuracy for damage scenario 1.
Figure 12. Damage extent and severity estimate for damage scenario 1.
Table IV. Assessment of the damage extent and severity accuracy for damage scenario 1.
Damage extent (cm)
Error (%)
Damage severity (%)
Error (%)True Predicted True Predicted
12.2 27.4 �4.2 �5.0 �5.3 0.3
312 S. DINCAL AND N. STUBBS
LT denote the axial length (Δx) of the predicted and true damage extents along the x-coordinate,respectively. The term LT ⊂ LP shows whether the predicted damage extent (LP) contains the truedamage extent (LT). For instance, 100% indicates that the predicted region covers 100% of thesimulated (true) damage region. Figure 12 shows the damage extent and damage severity estimatesfor damage scenario 1. The severity of the predicted damage is computed using Equation (24).Table IV demonstrates the assessment of damage extent prediction and damage severity estimation.Note that the extent and the severity of damage are estimated after enhancing the sensor resolutionat the predicted damage location. The enhanced sensor resolution corresponds to the nodes of theFE mesh at the centerline of the beam. The reader is referred to the subsequent section for a morethorough discussion of the damage prediction results.
3.5.2. Damage scenario 2. Figures 13 and 14 depict the damage prediction results for damagescenario 2. Tables V and VI summarize the performance of the damage evaluation methodology forthe damage scenario.
3.5.3. Damage scenario 3. Figures 15 and 16 depict the damage prediction results for damagescenario 3. Tables VII and VIII summarize the performance of the damage evaluation methodologyfor the damage scenario.
3.6. Discussion of results
Discussion of the damage prediction results is presented in this section. Accuracy of the proposed NDEmethodology is evaluated on the basis of the criteria listed in Section 3.4. The predicted damagelocation, xP, in Equation (25) was determined by tracing the peak value of the damage indicator, bj,within a possible damaged region. Although, in theory, any damage indicator greater than 1 indicatesa possible location of damage, the most unambiguous damage locations are unarguably positionedwithin the vicinity of the dominating peaks in damage localization plots. In view of the case studiesperformed, excluding the false negative prediction (i.e., the true damage location is not predicted) indamage scenario 3, the largest error between the central location of the true and predicted damages,
NDE USING NODAL CURVATURES—PART I: THEORY AND NUMERICAL VERIFICATION 315
eL, was 3.1%. The proposed NDE methodology did not unambiguously identify the location of thedamaged region centered at 286.5 cm from the clamped end of the beam in damage scenario 3. Thisfalse negative prediction was due to the fact that the relatively small damage within this region wasmasked by the other two more dominant damage locations.
Damage extent accuracy was initially quantified using the subset notation. For instance, if thepredicted damage extent contained the true damage extent, then (LT ⊂ LP) = 1.0. Here, LP denotes theaxial length of the predicted damage region where the damage indicator bj is greater than 1. The truedamage extent, LT, corresponds to the total length of the damage region along the x-coordinate (Δx).Damage extent was subsequently quantified by enhancing the sensor resolution in the region of thepredicted damage location. In this study, the enhanced sensor resolution corresponds to the nodes of theFE mesh at the centerline of the beam. The damage extent can be expressed as the footprint of damageseverity. Therefore, the extent of damage corresponded to the length of the region where the footprint ofthe predicted damage severity was prescribed to be less than 0. Using Equation (26), largest error betweenthe true and predicted damage extents, eE, was 4.6% in damage scenario 3.
Equation (29) was utilized to compute the error in damage severity estimation. Note that the damageseverities presented in Figures 12, 14, and 16 and Tables 4, 6, and VIII were quantified after enhancingthe sensor resolution (given in Figure 8) in the region of the predicted damage locations. The damagedregions previously identified with the proposed sensor layout in Figure 8 were instrumentedsubsequently to predict the severity of damage. Note that the predicted damage severity correspondsto the peak value of the damage severity computed directly from Equation (24).
The true (inflicted) damage severities presented in Figures 12, 14, and 16 and Tables 4, 6, and VIIIwere computed using Equation (33). For damage scenario 1, in which all the elements centered atx = 250 cm were uniformly damaged with a severity of �5%, the predicted damage severity was�5.3%. This result indicated an error of 0.3% using Equation (29). In damage scenario 2, for the threeelements centered at x= 6.1 cm from the clamped support, the true damage severity was �6.1%,whereas the predicted severity was �5.0%. The resulting error was calculated to be �1.2%. For thesame scenario, and for the two elements centered at x= 256 cm and y= 11.4 cm, the true damage sever-ity was �0.4% and the predicted damage severity was �0.9%. The resulting error in damage severityestimation was 0.5%. In damage scenario 3, for the four elements centered at x = 85.3 cm andy = 22.9 cm, the inflicted damage severity was �1.5% and the predicted damage severity was�2.1%. This result gave an error of 0.6%. For the three elements centered at x = 189 cm andy = 19.1 cm, the inflicted damage severity was �2.8% and the predicted damage severity was�2.9%. The error in damage severity estimation was as low as 0.1% for this case. Finally, for the singledamaged element centered at x = 286.5 cm and y = 19.1 cm, the inflicted damage was�0.3%. However,no damage severity estimation was reported. The following comments can be made regarding this ob-servation: First, the damage localization results in Figure 15 indicated a peak in the neighborhood ofthe damage. Second, the size of the damage is an order of magnitude less than the damages to the left.Third, according to Equation (24), the value of bj at the damage location is equal to 1.003. This value isnot perceptible in Figure 15. And fourth, the inflicted damage was located in the vicinity of the neutralaxis of the beam and thus had an insignificant effect on the bending response. Another way ofinterpreting the result is that with the mesh and sensor outline presented in this study, the resolutionof severity measurement is 0.3% of the bending stiffness of the beam.
4. CONCLUSIONS
A level III damage evaluation methodology that simultaneously identifies the location, the extent, andthe severity of damage in Euler–Bernoulli beams has been developed in this paper. Utilizing the basicprinciples of solid mechanics, we related the ratios of the flexural stiffnesses before and after damage tothe pre-damaged and post-damaged nodal curvatures. The performance of the proposed damagedetection methodology was evaluated using the response data collected from a set of numericalexperiments. These experiments were based on two-dimensional FE models of a cantilever beam.Excluding the relatively small effective stiffness reduction imposed in the vicinity of the neutral axis ofthe beam, the numerically simulated damages for single and multiple damage locations were successfullylocalized and quantified.
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