IMPROVEMENT OF STRAIN ENERGY DAMAGE DETECTION METHOD BY USING INTERPOLATION FUNCTION

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1 INTERNACIONALNI NAUČNO-STRUČNI SKUP GRAĐEVINARSTVO - NAUKA I PRAKSA ŽABLJAK, -. FEBRUARA. Ivana Štimac Grandić, Tatjana Pecak IMPROVEMENT OF STRAIN ENERGY DAMAGE DETECTION METHOD BY USING INTERPOLATION FUNCTION Summary In this paper, the strain energy method using deflection influence surfaces of plate-like structures is used to detect and locate the damage. The use of numerical differentiation procedures is identified as the main cause for the poor performance of the strain energy method under sparse and noisy measurement. The bicubic interpolation is employed to interpolate measured values of the deflection influence surface. The strain energy damage detection method by using interpolated values has been compared by the same method which uses only the measured values of the deflection influence surface. Key words Interpolation, strain energy, damage detection, deflection influence surface UPORABA INTERPOLACIJSKE FUNKCIJE ZA POBOLJŠANJE LOCIRANJA OŠTEĆENJA ENERGETSKOM METODOM Rezime U radu je uporabljena metoda energe deformiranja utjecajnih ploha progiba za otkrivanje i lociranje oštećenja na pločastim konstrukcama. Glavni razlog loših rezultata primjene energetske metode je uporaba numeričke diferencace u slučaju malog broja izmjerenih podataka koji u sebi sadrže mjernu grešku. Da bi se izbjegao navedeni problem upotrebljena je bikubična interpolacska funkca za interpolacu podataka između izmjerenih vrednosti. Napravljena je usporedba metode energe deformiranja korištenjem: (i) samo izmjerenih vrednosti, (ii) interpoliranih vrednosti utjecajnih ploha progiba. Ključne reči Interpolaca, energa deformiranja, detekca oštećenja, utjecajne plohe progiba Ph.D, C.E., Faculty of Civil Engineering, University of Reka, Viktora Cara Emina, Reka, Croatia, ivana.stimac@gradri.hr M.Sc, math.eng., Faculty of Civil Engineering, University of Reka, Viktora Cara Emina, Reka, Croatia, tatjana.pecak@gradri.hr

2 Građevinarstvo - nauka i praksa. INTRODUCTION In many areas of engineering applications such as aerospace, civil and mechanical engineering, car industry, plate elements are widely used as an important structural component. Therefore, structural health monitoring of plate-like elements is an important aspect of global structural health assessment. Especially, structural health monitoring becomes crucial after some extreme event had happened (e.g. earthquake) for avoiding the undetermined damages and preventing the possible human life losses. Recent works deal with damage detection in plate-like structures using changes in strain energy of mode shapes [- ]. In this paper, the strain energy method using deflection influence surface of platelike structures is used. The method is based on static test which have many advantages compared to modal shape identification []. The main problem in static test is incomplete static displacement information due to a limited number of the measurement instruments []. This limitation is overcome in methods presented in papers [, -]. Theoretically, displacement influence line and displacement influence surface can be obtained from measurements at only one point in the structure therefore this method avoids the need for many measurement points and measuring instruments.. DAMAGE LOCALIZATION ALGORITHM The damage localization scheme described below utilises the paradigm of pattern recognitions. In the problem in hand the static response of the structure represents the physical data and the displacements represent the pattern space (segments). The formed damage index for every segment is based on pre-damaged and damaged displacement influence surfaces. y b m,n y j+ y j i,j x i x i+ a x Figure Division of the plate into segments A procedure [] is adapted for using the deflection influence surfaces of predamaged and damaged state. A value of the change in the flexural stiffness for every segment of plate by using deflection influence surface (damage index) is represented by next equation: D D f = = β () f

3 GNP f f = = y x i+ j+ yi x j b a y x i+ j+ yi x j b a [ ρ + ρ + ν ρ ρ yy + ( ν) ρ ] yy dx dy [ ρ + ρ + ν ρ ρ yy + ( ν) ρ ] dx dy yy [ ρ + ρ + ν ρ ρ + ( ν) ρ ] yy dx dy [ ρ ( ) ] + ρ yy + ν ρ ρyy + ν ρ dx dy yy () () Where x ρ ; y ρ = ; = yy ρ = () x y x ρ ; y ρ = ; = yy w ρ = () x y η w (x, y) = w is the displacement influence surface for the pre-damaged state, η w ( x, y) = w is the displacement influence surface for the damaged state. Assuming that the collection of the damage indices, β, represent a sample population of a normally distributed random variable, a normalized damage index is obtained using β β z = () σ where β and σ represent the mean and standard deviation of the damage indices, respectively. Stubbs et al. [] suggested that normalized damage indices with values greater than two are associated with potential damage locations. Thus, the values of normalized damage indices smaller than two will be truncate.. APPLICATION OF THE PRESENTED METHOD The finite element method has been used to calculate the displacement influence surfaces for a measurement point in the middle of the bay. The plate is divided into m= segments in x direction and n= segments in y direction. Numerical model has fournode finite elements, of the size,x, m each. The thickness of non-damaged plate is d=, m. Young's modulus is E=, kn/m and Poisson's ratio is ν=,. The damage has been simulated by reducing the thickness of four finite elements (,,, and ) in the middle of the bay to d d =,m. The values of the displacement influence surface can be calculated in every finite element node. The applied force is F= kn.

4 Građevinarstvo - nauka i praksa L= m B= m Figure The finite element model of the one-bay plate.. APPLICATION OF THE METHOD USING ONLY THE MEASURED (CALCULATED) VALUES Figure Normalized damage indicies after truncation for only the measured values The numerical differentiation is employed to get damage indices from calculated values displacement influence surfaces of damaged and pre-damaged plate described in previous chapter. Calculated damage indices (potential damaged segments) are shown in D plot on the left side of the figure. On the right side of the figure, the potential damaged segments are marked on the finite element mesh. As it can be seen from figure, the normalized damage indices show potential damaged segments.

GNP.. APPLICATION OF THE METHOD USING INTERPOLATED VALUES In this paper, the bicubic interpolation is employed to interpolate data points between known values of displacement influence surface.

5 GNP.. APPLICATION OF THE METHOD USING INTERPOLATED VALUES In this paper, the bicubic interpolation is employed to interpolate data points between known values of displacement influence surface. In mathematics, bicubic interpolation is an extension of cubic interpolation for interpolating data points on a two dimensional regular grid. The interpolation mesh is chosen as x mm. In the first case, the normalized damage indices were calculated for the same pattern space (x mm) as in chapter.. As it can be seen from figure there are less potential damage segments () than on figure. Figure Normalized damage indicies after truncation for the first interpolated case In the second case, the normalized damage indices were calculated on the finest pattern space (x m) using the same intepolated values as in the first case. Figure Normalized damage indicies after truncation for the secon interpolated case The potential damage location showed in figure gives worse result in damage localization than in the first case in chapter.. (figure ).

6 Građevinarstvo - nauka i praksa. CONCLUSION The strain energy damage detection method by using interpolated values is compared by the same method which uses only the measured values of the deflection influence surface. In the case where the interpolated values were used the normalized damage indices were calculated on two pattern spaces (x mm and x mm). The pattern space of x mm si used to calculate the normalized damage indices using only the measured values. As it can be seen from chapter, the method using interpolated values on the pattern space as pattern of calculated (or measured) values gives the best results in damage localization. LITERATURE [] P. Cornwell, S.W. Doebling, C.R. Farrar: "Application of the strain energy damage detection method to plate-like structures", Journal of Sound and Vibration,, (), p.. [] E. Sazonov, P.Klinkhachorn: "Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes", Journal of Sound and Vibration,, -(), p. -. [] H. Huiwen, W.Bor-Tsuen, L. Cheng-Hsin, S. Jing-Shiang: "Damage detection of surface cracks in composite laminates using modal analysis and strain energy method", Composite Structures,, (), p. -. [] W.L. Bayissa, N. Haritos: "Structural damage identification in plates using spectral strain energy analysis" Journal of Sound and Vibration,, -(), p. -. [] H. Huiwen, W. Chengbo: "Development of scanning damage index for the damage detection of plate structures using modal strain energy method" Mechanical Systems and Signal Processing,, (), p. -. [] I. Štimac, I. Kožar, A. Mihanović: "Beam damage detection by deflection influence lines" Građevinar,, (), p. -. (Croatian) [] M. Sanayei, R. B. Nelson: "Identification of structural element stiffness from incomplete static test data", Society of Automotive Engineering, Technical paper, SAE-,. [] I. Štimac Grandić, A. Mihanović, I. Kožar: "Slab damage detection by comparing curvature of relevant deflection areas", Građevinar,, (), p. -. (Croatian) [] I. Štimac, I. Kožar, A. Mihanović: "Damage Detection from Displacement-Influence-Line", International Conference on Bridges, Dubrovnik: SECON,. p. - [] C. Y. Wang, C. K. Huang, Y. T. Zeng, C. S. Chen, M. H. Chen: "Damage Assessment of Beam by a Quasi-Static Moving Vehicular Load", th International Conference on Structural Safety and Reliability, Osaka, Japan.,. [] N. Stubbs, J.-T. Kim, C.R. Farrar: "Field verification of a nondestructive damage localization and severity estimation algorithm", Proceedings of the Modal Analysis Conferenceth International,. p. -

VIBRATION MONITORING OF COMPOSITE LAMINATES WITH DELAMINATION DAMAGE

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