Detection of structural damages using the vibration test data has been investigated by many researchers in the last decades. Identification of the defect in the early stages can prevent the unpredicted failure of structure. For beam-like structures, curvature techniques, e.g., mode shape curvature and flexibility curvature have been applied to localize the damage. In this paper, root mean square error using operational deflection shape curvature is used for damage detection. Differential Quadrature Method (DQM) is implemented to obtain the curvatures of the operational deflection shape. To demonstrate the effectiveness of method, the finite element model of a cantilever beam is used. The model is excited and the responses are measured in the simulated test. Enhanced frequency domain decomposition (EFDD) is used to extract the modal parameters. Also Mass change method is used to scale the mode shapes through which the scaled ODS parameters are computed. Using the RMSE the location of the damage is identified. The results show that this approach is more effective than the previous method

Ali Bakhshizade

Mohammad Reza Ashory

English

Mathematical Models in Engineering

 96 MATHEMATICAL MODELS IN ENGINEERING. DECEMBER 2015, VOLUME 1, ISSUE 2  
12. Root mean square error criterion using operational 
deflection shape curvature for structural damage 
detection 
Ali Bakhshizade1, Mohammad Reza Ashory2 
1Department of Mechanical Engineering, Roshdedanesh Institute of Higher Education, Semnan, Iran 
2Modal Analysis Lab, School of Mechanical Engineering, Semnan University, Semnan, Iran 
1Corresponding author 
E-mail: 1alimrm2000@gmail.com, 2mashoori@semnan.ac.ir 
(Received 17 September 2015; accepted 18 October 2015) 
Abstract. Detection of structural damages using the vibration test data has been investigated by 
many researchers in the last decades. Identification of the defect in the early stages can prevent 
the unpredicted failure of structure. For beam-like structures, curvature techniques, e.g., mode 
shape curvature and flexibility curvature have been applied to localize the damage. In this paper, 
root mean square error using operational deflection shape curvature is used for damage detection. 
Differential Quadrature Method (DQM) is implemented to obtain the curvatures of the operational 
deflection shape. To demonstrate the effectiveness of method, the finite element model of a 
cantilever beam is used. The model is excited and the responses are measured in the simulated 
test. Enhanced frequency domain decomposition (EFDD) is used to extract the modal parameters. 
Also Mass change method is used to scale the mode shapes through which the scaled ODS 
parameters are computed. Using the RMSE the location of the damage is identified. The results 
show that this approach is more effective than the previous methods 
Keywords: operational deflection shape, differential quadrature, RMSE. 
1. Introduction 
Damage in the structural and mechanical systems is defined as changes in the material and/or 
geometric properties of these systems, including changes in the boundary conditions and system 
connectivity, which adversely affect the current or future performance of these systems” [1]. 
Vibration based techniques have been widely expanded in few decades. These methods localize 
the flaws and defects in structures based on changes in the modal parameters (natural frequencies, 
mode shapes and damping ratio). 
These methods can be classified in many ways but the damage classification system presented 
by Rytter [2] is adopted here, which defines four levels of damage assessment: 
1) (detection): determination that whether the damage is present in the structure; 
2) (localization): determination of the geometric location of the damage; 
3) (quantification): determination of the severity of damage; 
4) (prediction): prediction of the remaining service life of the structure. 
The aim of investigations in the last 20 years has been the global and on-line nondestructive 
evaluation. Amongst the techniques under development one can outline those using mechanical 
vibrations, smart materials, image processing, optical fibers, novelty based and statistical 
information. Vibration-based methods have been applied successfully on large and complicated 
structures such as civil engineering structures (mostly bridges), aerospace structures and 
composite structures. During the last decades, research on vibration-based structural damage 
identification methods has been expanding rapidly.  
Some of the existing methods in vibration are based on the comparison between the parameters 
extracted from experimental observations and numerical models of the structure; there are also 
methods based on the comparison between parameters identified from experimental-only 
observations [3]. 
The method presented in this paper is in the category of methods that use the change in the 
12. ROOT MEAN SQUARE ERROR CRITERION USING OPERATIONAL DEFLECTION SHAPE CURVATURE FOR STRUCTURAL DAMAGE DETECTION.  
ALI BAKHSHIZADE, MOHAMMAD REZA ASHORY 
 © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 97 
operational deflection shapes (ODSs) to detect, locate and relatively quantify the damage. 
2. Theory 
2.1. Frequency domain decomposition (FDD) method 
The Frequency Domain Decomposition (FDD) method is proposed by Brincker et al. [4, 5]. In 
this method first the Power Spectral Density (PSD) of the response is computed. Then the singular 
value decomposition is applied to obtain the modal parameters of system. The relation between 
the inputs and outputs is given by [5]: 
ܩ௬௬(݆ω) = ܪഥ(݆߱)ܩ௫௫(݆߱)ܪ்(݆߱), (1)
where ܩ௫௫ is the PSD matrix of input, ܩ௬௬ is the PSD matrix of output and ܪ(݆߱) is the FRF 
matrix. If the input force is assumed to be a white signal, the PSD of the output can be given by: 
ܩ௬௬(݆߱) = ෍
݀௞߶௞߶௞௧
݆߱ − ߣ௞
௡
௄ୀଵ
+ ݀̅௞߶
ത௞߶ത௞௧
݆߱ − ̅ߣ௞ , (2)
where ݀௞ is a scalar, ߶௞ is the ݇th mode shape vector, ߛ௞ is the modal participation vector, ߣ௞ is 
the ݇th complex resonance frequency, (∙)̅ sign stands for the complex conjugate and ݐ indicates 
the transpose of a matrix. 
The PSD of response in each frequency can be decomposed to the singular values and singular 
vectors using the following equation: 
ܩ෠௬௬(݆߱௜) = ௜ܷ ௜ܵ ௜ܷு, (3)
where ௜ܷ  is the ݅th singular vector, ௜ܵ  is the ݅th singular value, ߱௜  is the ݅th frequency and ܪ 
indicates the complex conjugate of a matrix. 
The peaks of first singular values of system correspond to the natural frequencies of system. 
The singular vectors corresponding to the peaks of first singular values approximate the mode 
shapes.  
2.2. Mass change method 
In output only modal analysis, the mode shapes are not scaled. Mass change method is a 
method for scaling of mode shapes. 
For a natural frequency ߱௡ଵ , and the corresponding mode shape ሼ߶ଵሽ of a MDOF system, if 
the masses of vibrating system at some or all of the degrees of freedom are changed, the ݎth natural 
frequency and the corresponding mode shapes are changed to ߱௡ଶ  and ሼ߶ଶሽ  respectively. 
Considering small changes of mass matrix, it can be assumed that the mode shapes change 
negligibly [6]. Therefore: 
ሼ߶ଶሽ ≅ ሼ߶ଵሽ = ሼ߶ሽ. (4)
The relation between the unscaled mode shape ሼ߰ሽ and the scaled mode shape ሼ߮ሽ is: 
ሼ߶ሽ = ߙሼ߰ሽ, (5)
where ߙ is the scaling factor.  
It is shown in [6], that the best scaling factor is obtained when the mode shapes are scaled to 
the length and both the modified and unmodified mode shapes are used: 
12. ROOT MEAN SQUARE ERROR CRITERION USING OPERATIONAL DEFLECTION SHAPE CURVATURE FOR STRUCTURAL DAMAGE DETECTION.  
ALI BAKHSHIZADE, MOHAMMAD REZA ASHORY 
98 MATHEMATICAL MODELS IN ENGINEERING. DECEMBER 2015, VOLUME 1, ISSUE 2  
ߙଵଶ = ඨ
(߱௡ଵଶ − ߱௡ଶଶ )
߱௡ଶଶ ሼ߰ଵሽ் ⋅ ሾΔܯሿ ⋅ ሼ߰ଶሽ,
(6)
where ሾΔܯሿ is the mass change matrix, ሼ߰ଵሽ and ሼ߰ଶሽ are the unscaled mode shapes before and 
after mass change.  
2.3. ODS estimation 
Frequency response function can be obtained using the following equation [7]: 
ߙ௝௞(߱) = ෍
(߮௝௥)(߮௞௥)
߱௥ଶ − ߱ଶ + ݅ߟ௥߱௥ଶ
ே
௥ୀଵ
, (7)
where ߟ is the damping ratio and ߙ the receptance matrix. Damping ratio can be obtained from 
EFDD method. The column vector, k, of the receptance matrix, ൛ߙ௞(ఠ)ൟ, is the ODS at frequency 
߱. Therefore, the ODS describes the shape exhibited by the structure at each excitation frequency 
߱, given by the responses and normalized by the applied forces. 
3. Damage detection 
3.1. Theory 
When a structure is damaged its stiffness and damping alter and, as a result does the receptance 
matrix: 
ሾߙௗ(߱)ሿ = (ሾܭௗሿ + ݅߱ሾܥௗሿ − ߱ଶሾܯሿ)ିଵ, (8)
where the superscript ݀ stands for “damaged”. It is reasonable to assume that the smaller the 
degree of correlation between the column vectors (ODSs), ሼߙ݆(߱)ሽ and ሼ݀ߙ݆(߱)ሽ, the larger the 
damage. Root mean square error using the spectral strain energy was first studied by Bayissa and 
Haritos [8]. In this paper is used the curvature of ODS instead of spectral strain energy in root 
mean square error criterion: 
ܴܯܵܧ(߱) =  ඩ
1
ܰ ∑ ቀߙ௜ௗ
ᇲᇲ(߱) − ߙ௜ᇱᇱ(߱)ቁ
ଶே௜ୀଵ
1
ܰ ∑ ቀߙ௜ᇱᇱ(߱)ቁ
ଶே௜ୀଵ
 
, (9)
where ܰ is the number of measurement points. But central difference can not present the curvature 
accurately, two nodes are missing in the approximation and it needs to many points of 
measurement. In this work, differential quadrature method was used to obtain the curvature.  
3.2. Differential quadrature method (DQM) 
The idea of using differential quadrature method is to quickly compute the derivative of a 
function at any grid point within its bounded domain by estimating a weighted linear sum of values 
of function at a small set of points related to domain. According to this method nth derivative of 
݂(ݔ) can be approximately obtained by: 
12. ROOT MEAN SQUARE ERROR CRITERION USING OPERATIONAL DEFLECTION SHAPE CURVATURE FOR STRUCTURAL DAMAGE DETECTION.  
ALI BAKHSHIZADE, MOHAMMAD REZA ASHORY 
 © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 99 
߲௡݂(ݔ)
߲ݔ௡ ቤ௫ୀ௫೔
= ෍ ܣ௜௝௫(௥)݂൫ݔ௝൯
ேೣ
௝ୀଵ
, ݅ = 1, 2, … , ௫ܰ,   ݎ = 1, 2, … , ௫ܰ − 1. (10)
In order to determine the weighting coefficients a specific test function is required. One of the 
best approaches has been proposed by Quan and Chang [9, 10]. They used the Lagrange 
interpolation function as test function and determined the weighting coefficients on this basis. 
Weighting coefficients for the first-order derivatives are: 
ܣ௜௝௫(ଵ) =
ە
ۖ
۔
ۖ
ۓ ܲ(ݔ௜)൫ݔ௜ − ݔ௝൯ܲ൫ݔ௝൯
, ݅ ≠ ݆,
− ෍ ܣ௜௝௫(ଵ)
ேೣ
௝ୀଵ
௜ஷ௝
, ݅ = ݆, ݅, ݆ = 1, 2, … , ௫ܰ.
  (11)
ܲ(ݔ௜) = ෑ (ݔ௜ − ݔ௝)
ேೣ
௝ୀଵ,௜ஷ௝
. (12)
To evaluate the weighting coefficients of higher order derivatives recurrence formulae are 
derived as: 
ܣ௜௝௫(௥) = ݎ ൥ܣ௜௜௫(௥ିଵ)ܣ௜௝௫(ଵ) −
ܣ௜௝௫(௥ିଵ)
ݔ௜ − ݔ௝ ൩ , ݅, ݆ = 1, 2, … , ௫ܰ, ݅ ≠ ݆, 2 ≤ ݎ ≤ ௫ܰ − 1, (13)
ܣ௜௜௫(௥) = − ෍ ܣ௜௝௫(௥)
ேೣ
௝ୀଵ,௜ஷ௝
, ݅ = 1, 2, … , ௫ܰ, 1 ≤ ݎ ≤ ௫ܰ − 1. (14)
4. Numerical case study 
A study based on finite element model of a cantilever beam was conducted to evaluate the 
feasibility of applying curvature of ODS in RMSE criterion to identify the crack of a 
beam-structure. The geometry of the beam and its material properties are as follows: Young’s 
modulus ܧ = 206 GPa, density ߩ = 7800 kg/m3, length ݈ = 800 mm, Cross sectional area  
ܣ = ݓ × ℎ = 32 mm × 16 mm. The total number of elements is 8 and two damage cases are 
studied. Damage characteristics have been described in Table 1. 
Table 1. Damage scenarios 
 ݈ଵ (mm) Element number ܽ (mm) 
Case I 650 6 0.8 
Case II 450 3 0.6 
 
Fig. 1. Geometry of cantilever beam with a crack  
Damage is simulated with a method that is proposed by Nahvi and Jabbari [11]. In this method 
the length of damaged element is smaller than the other elements. 
12. ROOT MEAN SQUARE ERROR CRITERION USING OPERATIONAL DEFLECTION SHAPE CURVATURE FOR STRUCTURAL DAMAGE DETECTION.  
ALI BAKHSHIZADE, MOHAMMAD REZA ASHORY 
100 MATHEMATICAL MODELS IN ENGINEERING. DECEMBER 2015, VOLUME 1, ISSUE 2  
The application of ODS curvature using DQM compared to the method presented in [8] for 
the above cases have been shown in the Figs. 2-5. 
 
Fig. 2. Method in [8] (Case I) 
 
Fig. 3. Proposed method (Case I) 
 
 
Fig. 4. Method in [8] (Case II) 
 
Fig. 5. Proposed method (Case II) 
For case (I), the proposed method shows that the damage is between nodes 6 and 7 (Fig. 3). 
The result is concluded from Fig. 2. However, the difference in Fig. 3 is more indicative compared 
to the result in Fig. 2. The same conclusion can be found for case II by comparing Figs. 4 and 5. 
5. Conclusion 
In this study, a new method is developed to improve the damage localization of a beam 
structure which has been numerically investigated. Results show that the proposed approach can 
be used to localize the damage in the structure without any analytical model. When the number of 
measurement points is limited, DQM method has the good accuracy and the less time of 
computation. In comparison with method in [8] that need, a large number of measurement points 
to obtain an accurate result, the proposed method has inherent advantages of finding cracks where 
the method suggested in [8] is relatively weak. 
References 
[1] Sohn H., Farrar C. R., Hemez F. M., Shunk D. D., Stinemates S. W., Nadler B. R., Czarnecki J. J. 
A Review of Structural Health Monitoring Literature Form 1996-2001. Los Alamos National 
Laboratory Report LA-13976-MS, 2004. 
[2] Rytter A. Vibration Based Inspection of Civil Engineering Structures. Ph.D. Thesis, Department of 
Building Technology and Structural Engineering, Aalborg University, 1993. 
[3] Farrar C. R., Doebling S. W. The state of the art in vibration-based structural damage identification. 
2-day Short Course, Madrid, Spain, 2000 
12. ROOT MEAN SQUARE ERROR CRITERION USING OPERATIONAL DEFLECTION SHAPE CURVATURE FOR STRUCTURAL DAMAGE DETECTION.  
ALI BAKHSHIZADE, MOHAMMAD REZA ASHORY 
 © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 101 
[4] Brincker R., Zhang L., Andersen P. Modal identification from ambient responses using frequency 
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[5] Brincker R., Zhang L., Andersen P. Modal identification of output only systems using frequency 
domain decomposition. Journal of Smart Materials and Structures, Vol. 10, 2001, p. 441-445. 
[6] Aenlle M. L., Brincker R., Canteli A. F. Some methods to determine scaled mode shapes in natural 
input modal analysis. Proceedings of 23rd International Modal Analysis Conference, Florida, 2005. 
[7] Maia N. M. M. Theoretical and Experimental Modal Analysis. Research Studies Press Ltd, 1997 
[8] Bayissa W. L., Haritos N. Structural damage identification in plates using spectral strain energy 
analysis. Journal of Sound and Vibration, Vol. 307, 2007, p. 226-249. 
[9] Quan J. R., Chang C. T. New insights in solving distributed system equations by the quadrature 
methods – Part 1. Computers and Chemical Engineering, Vol. 13, 1989, p. 779-788. 
[10] Quan J. R., Chang C. T. New insights in solving distributed system equations by the quadrature 
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[11] Nahvi H., Jabbari M. Crack detection in beams using experimental modal data and finite element 
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Root mean square error criterion using operational deflection shape curvature for structural damage detection

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Root mean square error criterion using operational deflection shape curvature for structural damage detection

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