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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/260052106 Characterisation of vibration and damage in masonry structures: Experimental and numerical analysis Article in European Journal of Environmental and Civil Engineering · November 2014 DOI: 10.1080/19648189.2014.883335 CITATIONS READS 13 491 3 authors: Trung Bui Ali Limam Institut National des Sciences Appliquées de Lyon Institut National des Sciences Appliquées de Lyon 31 PUBLICATIONS 201 CITATIONS 160 PUBLICATIONS 1,170 CITATIONS SEE PROFILE Quoc-Bao Bui Ton Duc Thang University 63 PUBLICATIONS 635 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Generating Guided Waves for Detection of Transverse Type-Defects in Rails View project STRUCTURE View project All content following this page was uploaded by Trung Bui on 19 June 2016. The user has requested enhancement of the downloaded file. SEE PROFILE This article was downloaded by: [INSA de Lyon - DOC INSA] On: 06 February 2014, At: 02:25 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK European Journal of Environmental and Civil Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tece20 Characterisation of vibration and damage in masonry structures: experimental and numerical analysis a a T.T. Bui , A. Limam & Q.B. Bui b a Université de Lyon, INSA de Lyon, Laboratoire LGCIE, Villeurbanne Cedex, France b Université de Savoie, POLYTECH Annecy-Chambéry, LOCIE - CNRS UMR 5271, Le Bourget du Lac, France Published online: 05 Feb 2014. To cite this article: T.T. Bui, A. Limam & Q.B. Bui , European Journal of Environmental and Civil Engineering (2014): Characterisation of vibration and damage in masonry structures: experimental and numerical analysis, European Journal of Environmental and Civil Engineering, DOI: 10.1080/19648189.2014.883335 To link to this article: http://dx.doi.org/10.1080/19648189.2014.883335 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions European Journal of Environmental and Civil Engineering, 2014 http://dx.doi.org/10.1080/19648189.2014.883335 Characterisation of vibration and damage in masonry structures: experimental and numerical analysis Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 T.T. Buia*, A. Limama and Q.B. Buib a Université de Lyon, INSA de Lyon, Laboratoire LGCIE, Villeurbanne Cedex, France; bUniversité de Savoie, POLYTECH Annecy-Chambéry, LOCIE - CNRS UMR 5271, Le Bourget du Lac, France (Received 30 September 2013; accepted 9 January 2014) For masonry structures, due to their heterogeneity and the complexity of the interface’s behaviour between the blocks and mortar, the discrete element method (DEM) seems to be the best adapted to model this kind of structure, in particular for reproducing complex nonlinear post-elastic behaviour. However, unlike the software using the ﬁnite element method, the DEM does not directly obtain the natural frequencies and the mode shapes of the structure via a classic vibrational analysis. On one hand, these dynamic characteristics are needed for the structural design in the case of dynamic loads (seismic, impact). On the other hand, these dynamic characteristics can be useful for the assessment of existing structures by in situ dynamic tests. Therefore, the ﬁrst objective of this study was to propose a technique that could indirectly identify dynamic characteristics of masonry structures using the DEM. The study began on a simple brick column structure. The sensitivity of the interface parameters following the dynamic characteristics was studied. Then, a masonry arch structure was studied. The technique proposed to identify the dynamic characteristics was validated by comparison with the experimental data. Based on vibrational analysis, the damage to the structure was assessed. Keywords: masonry; DEM; vibration; arch 1. Introduction Due to the heterogeneity of masonry walls (bricks, joints and interfaces), the discrete element method (DEM) is the best adapted tool available today to study this type of structure, especially to reproduce the nonlinear behaviours that appear beyond the elastic phase. An explicit DEM based on ﬁnite difference principles, originated in the early 1970s by a landmark work on the progressive movements of rock masses as 2D rigid block assemblages (Cundall, 1971). This technique was then extended to the modelling of masonry structures. Our previous studies (Bui & Limam, 2012; Bui, Limam, Bertrand, Ferrier, & Brun, 2010) were conducted on masonry walls submitted to inplane and out-of-plane static loads on a wall, which indeed conﬁrmed the effectiveness of the DEM. The objective of this article is to study the dynamic behaviour of masonry structures using the DEM. Unlike the software using the ﬁnite element method (FEM), the DEM does not directly obtain the structure’s natural frequencies and mode shapes via a classic vibrational analysis. On one hand, these dynamic characteristics are needed for the structural design in the case of dynamic loads (seismic, impact). On the other *Corresponding author. Email: [email protected] © 2014 Taylor & Francis Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 2 T.T. Bui et al. hand, they can be useful for assessment of existing structures using in situ tests (Anzani, Binda, Carpinteri, Invernizzi, & Lacidogna, 2010; Bui, Morel, Hans, & Meunier, 2009; Carpinteri, Invernizzi, & Lacidogna, 2009). Dynamic behaviour of masonry structures was studied by using FEM in several previous researches (see e.g. Peña, Lourenço, Mendes, & Oliveira, 2010). However, by DEM’s approach, the studies in literature about this topic are still limited. In a recent paper, Dimitri, De Lorenzis, and Zavarise (2011) realised a study on the dynamic behaviour of masonry structure but the study concentrated on temporal analysis, no information about dynamic characteristic of the structures was presented. To our knowledge, identiﬁcation of masonry’s dynamic characteristics by DEM is not yet addressed in the literature. That is why, the ﬁrst objective of our study is to propose a technique that can identify dynamic characteristics of masonry structures using DEM. Then, based on dynamic analysis, an assessment of the damage of the structure can be given. 2. Modal analysis using the DEM applied to a masonry column 2.1. DEM validation based on the FEM result 2.1.1. Technique proposed To verify the capacities of the DEM in modal analysis (mode shapes and natural frequencies), a column made of bricks with dry joints built at its base was studied. To validate the DEM, a calculation using the FEM and based on homogenisation was made. The column was made of bricks without mortar. The idea was to study several states of cohesion in the column: from the state of dry bricks alone with weak interaction between them (no transmission of rigidity, total discontinuity) to the state of a monolithic system. In the monolithic system, the bricks develop a strong interaction, making it possible to create ﬂexural and torsional rigidities. In other words, as far as the interfaces behave linearly, the behaviour of modal by FEM and DEM are expected to be identical. The 1.47 m column consists of 30 blocks measuring 220 × 103 × 49 mm3 (length × width × height) (Figure 1). The base of the column was assumed to be perfectly embedded. Modal analysis was ﬁrst carried out using the FEM with the ABAQUS code. The FEM provided a direct resolution of the eigenvalue problem by solving the classical equation in dynamic of structures (det [K0 −ω2 M]=0). Solide C3D8R elements were used. The column consisted of 32,340 elements and was studied in the elastic part: the Young modulus, 9300 MPa; density, 2200 kg/m3 and Poisson’s ratio, .2. For the DEM, the numerical simulation was carried out using the 3DEC code (ITASCA, 2002). The principle was based on the discretisation of the bricks and the interaction between them was governed by the interface laws. For deformable bricks, the elastic behaviour was adopted. The masonry wall is simulated by a simpliﬁed micro-modelling (Figure 2(b)). The joints were not directly modelled as elements, but indirectly through an interface law between the blocks. The Mohr-Coulomb criterion was chosen for the interface law (Figure 2(c)). However, for the modal analysis, the joints were assumed to remain within the elastic range. The elastic behaviour (details can be seen in Bui, 2013) depended on normal stiffness (kn) and shear stiffness (ks): ss ks 0 us frg ¼ ½Kfug or ¼ 0 k n un rn 3 where: σn: normal loading; un: normal displacement; τs: shear stress and us: shear displacement. Assuming that the theory of elasticity was applicable, the relation was: ks = kn /[2(1+ν)] with ν = .2. For laboratory dynamic tests, the excitation is often a hammer impact (Bui et al., 2009). With the same principle, in the DEM modal analysis, the structure was excited by an impact (impact on the column with a relatively weak force). The dynamic response at 10 points of the column (Figure 1(a)) was recorded and analysed. For the dynamic analysis, the damping was assumed to be zero. The structure’s damping was due to the friction between blocks. The natural frequencies were identiﬁed by transforming data from the time domain to the frequency domain using the FFT (Fast Fourier Transform). The mode shapes were identiﬁed using the FDD technique (Frequency Domain Decomposition), (see details in Andersen, Brincker, Goursat, & Mevel, 2007). Two elastic stiffness values of the interfaces (kn and ks) were interpolated so that the ﬁrst natural frequency would be close to that obtained by the FEM. These values were kn = 62 GPa/m, ks = 25.83 GPa/m. The natural frequencies obtained by the FFT of 10 measured responses (in two horizontal directions X and Z, Figure 3) are presented in Figure 4. The peaks in the frequency domain correspond to the natural frequencies of the structure. Each natural frequency corresponds to a vibrational mode. Figure 5 shows the mode shapes of six modes obtained by FDD (DEM) and FEM. The natural frequencies of the ﬁrst six Measurement points (X11,Y11,Z11) Point impact (X10,Y10,Z10) (X9,Y9,Z9) (X8,Y8,Z8) (X7,Y7,Z7) 1470 (X6,Y6,Z6) (X5,Y5,Z5) (X4,Y4,Z4) 147 294 (X3,Y3,Z3) (X2,Y2,Z2) Y 49 Y Y 103 220,5 X Z X (a) (b) 1.2 Diagonal force (N) Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 European Journal of Environmental and Civil Engineering 1 0.8 0.6 0.4 0.2 0 0 0.5 Time (ms) 1 (c) Figure 1. force. (a) Impact point and measurement points; DEM mesh; (b) FEM mesh; and (c) impact 4 T.T. Bui et al. (a) (b) (c) Vibration of the point P2(X2, Y2, Z2) after impact. Power spectrum Figure 3. 0.0005 15,93Hz 0.0004 0.0003 0.0002 96,49Hz 0.0001 164,9Hz 260,4Hz 0 0 50 100 150 200 250 300 Frequency (Hz) Power spectrum Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 Figure 2. (a) Detailed micro-modelling masonry wall; (b) Simpliﬁed micro-modelling masonry wall; and (c) Mohr-Coulomb model of joint with tension cut-off. 0.00025 29,98Hz 0.0002 0.00015 0.0001 0.00005 164,9Hz 172,4Hz 0 0 50 100 150 200 250 Frequency (Hz) Figure 4. 300 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 Natural frequencies of the column, obtained by FFT. modes obtained by two methods (FEM and DEM) are reported in Figure 6. The similarity of natural frequencies and mode shapes obtained by the two methods conﬁrm the relevance of the DEM dynamic analysis. European Journal of Environmental and Civil Engineering 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 FEM -1 0 Z 1 FEM -1 -0.5 0 0.5 1 FEM DEM -0.5 0 0.5 DEM f=172.4Hz FEM f=191.23Hz (f) FEM -0.5 0 0.5 0 0.5 1 FEM f=156.77Hz FEM -0.5 0 1 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 FEM f=263.62Hz 0.5 1 DEM f=260.4Hz DEM FEM -2 -1 0 1 Z X Figure 5. -0.5 Z DEM -1 FEM DEM -1 1 Y (m) Y (m) 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 DEM -1 DEM f=164.9Hz 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 X (e) 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 Z (d2) FEM f=156.77Hz FEM f=97.29Hz DEM f=96.49Hz (c) X DEM f=164.9Hz 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 FEM f=33.45Hz DEM Y (m) Y (m) (d1) -1 Mode shapes obtained by FDD and FEM. (a) Frequency (Hz) Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 1.47 1.323 1.176 1.029 0.882 0.735 0.588 0.441 0.294 0.147 0 -1.5 DEM -2 DEM f=29.98Hz (b) Y (m) Y (m) DEM f=15.93Hz Y (m) FEM f=15.88Hz (a) 5 (b) 300 FEM DEM 250 200 150 100 50 0 0 1 2 3 4 5 Mode FEM (Hz) DEM (Hz) Difference (%) Mode FEM (Hz) DEM (Hz) Difference (%) 1 15.88 15.93 -0.31 4 156.77 164.9 -5.19 2 33.45 29.98 10.37 5 191.23 172.4 9.85 3 97.29 96.49 0.82 6 263.62 260.4 1.22 6 Mode Figure 6. Comparison of natural frequencies obtained by the DEM and FEM. 2.1.2. Sensitivity of dynamic characteristics as a function of interface parameters The two elastic stiffnesses, kn and ks, were studied. kn varied from a fairly low value (10 GPa/m) to a relatively high value (1000 GPa/m). The kn/ks = 2.4 ratio was kept constant. Figure 7 shows the inﬂuence of kn (or ks) on the natural frequencies of the ﬁrst six modes (1–6). 6 T.T. Bui et al. 302 Frequency (Hz) 252 202 152 102 52 2 Figure 7. 2 3 4 Number of mode 5 6 Variation of natural frequencies as a function of kn (and ks). As can be seen in Figure 7, the more kn increases, the more the DEM’s natural frequencies were close to the FEM’s natural frequencies and when kn ≥ 200 GPa/m, the DEM results were very close to the FEM results. Figure 8 shows the inﬂuence of kn on the natural frequency for each mode. For low values of kn (<300 GPa/m), the frequencies were very sensitive with kn but when kn ≥ 300 GPa/m, the frequencies were less sensitive with kn, because in these cases, the column’s behaviour was close to that of a monolithic structure. Theoretically, in the case of dry bricks without any “cohesion”, vibrational modes, especially for higher modes, are difﬁcultly captured. These results show the inﬂuence of contact’s characteristics, on the dynamic behaviour of masonry structures. This point is 18 Mode 2 29 14 Frequency (Hz) Frequency (Hz) 34 Mode 1 16 12 10 8 6 4 2 24 19 14 9 4 0 200 400 600 800 1000 0 1200 200 400 110 100 90 80 70 60 50 40 30 20 Frequency (Hz) Mode 3 0 200 400 600 800 1000 1200 kn (xE9) Figure 8. 600 800 1000 1200 800 1000 1200 kn (xE9) kn (xE9) Frequency (Hz) Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 1 kn=10e9 kn=20e9 kn=40e9 kn=60e9 kn=80e9 kn=100e9 kn=200e9 kn=300e9 kn=400e9 kn=500e9 kn=600e9 kn=700e9 kn=800e9 kn=900e9 kn=1000e9 FEM Natural frequencies according to kn. 200 180 160 140 120 100 80 60 40 20 Mode 4 0 200 400 600 kn (xE9) European Journal of Environmental and Civil Engineering 7 Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 important for discrete modelling (especially for masonry structures with weak bonds such as dry stone masonry) and for assessment of existing structure (cracking corresponds to decrease of effective contact). 2.2. Validation of the DEM based on experimental results The vibrational analysis in the previous section was conducted on a column with dry joints. It was validated by numerical results obtained using the FEM. An experiment was also conducted. In practice, it was difﬁcult to measure the vibrations on a dry brick column (without mortar), so the experiment was conducted on a column of bricks with mortar to experimentally validate the DEM’s dynamic results. The column consisted of 17 bricks measuring 22.05 × 10.3 × 4.9 cm3 stacked vertically (Figure 9(a)). The mortar between the bricks had an average thickness of 1 cm. The column was ﬁxed at its base. Tests were conducted to characterise the Young modulus of mortar and bricks. For mortar, six specimens measuring 4 × 4 × 16 cm3 were manufactured and their frequencies were measured in time using the Grindosonic system (Figure 9(b)). The Young modulus was measured by analysing speciﬁc frequencies (see further details in Boukria & Limam, 2012a, 2012b). The brick’s elastic modulus was also measured using a nondestructive method (the details of other NDT methods can be seen in Invernizzi, Lacidogna, Manuello, & Carpinteri, 2011): an impact was applied to the brick; its dynamic response was measured by uniaxial piezoelectric accelerometers (Figure 9(c)); by analysing the dynamic response, the elastic modulus could be identiﬁed (Bui et al., 2009). To overcome the boundary condition problem, the brick was suspended by two strings. The bricks’ elastic modulus was 11,200 MPa and their density was 2200 kg/m3. An impact was applied at the top of the column, on an angular point, following the diagonal line of the bricks’ section. After the impact, the dynamic response was measured at several points in the column. The same analysis presented in Section 2.1 was used: kn was found to be 450 GPa/m. Figure 10 shows the comparison of the natural frequencies obtained by DEM and by experimentation. The DEM captured the ﬁrst three experimental modes well with an error less than 5%. From the fourth mode, the error between calculation and test increased. This difference can be explained by the limits of current experimental Figure 9. (a) Column; (b) Grindosonic mortar test; and (c) brick test. 8 T.T. Bui et al. 700 DEM Experiment Frequency (Hz) 600 Mode 500 1 2 3 4 5 6 400 300 200 100 0 1 2 3 4 5 DEM (Hz) 40.24 73.54 237.7 284.5 383.9 613.8 Experiment 39.06 75.68 249.02 305.18 439.15 651.25 Error (%) 2.93 -2.91 -4.76 -7.27 -14.39 -6.10 6 Mode Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 Figure 10. Experimental and numerical results. methods in the identiﬁcation of higher modes. Indeed, it is well known that the higher modes are less well captured than the ﬁrst modes, but the ﬁrst modes are more important (Hans, Boutin, Ibraim, & Roussillon, 2005). The DEM therefore gives results with sufﬁcient accuracy for modal analysis. Analysis of the mode shape is not presented here, because the number of the accelerometers used was limited, and therefore the mode shape of the column could not be identiﬁed. 3. DEM and damage characterisation 3.1. The masonry arch In this section, the DEM was used to study the vibration of an arch made in masonry. The numerical results will be compared with the experimental results, which were presented in Ramos, De Roec, Lourenco, and Campos-Costa (2010). This structure had vertical and horizontal joints, which induces anisotropic behaviour. The arch had a span measuring 1500 mm, outer radius 795 mm, inner radius 745 mm, thickness 50 mm and width 450 mm. This arch was made of clay bricks measuring 100 × 50 × 25 mm3. The arch was assembled of 63 rows of bricks. The thickness of the joints was about .5 cm. The arch was built on two concrete bases bolted to the ﬂoor. Damage of masonry structure can be identiﬁed by many approach, for example, a multilevel approach can be described in (Anzani et al., 2010). In our case, Ramos et al. (2010) identiﬁed the damage of this structure with vibrational analysis. The measurements were taken with several accelerometers and strain gauges on the vault. The vault was impacted by a hammer or excited by the ambient solicitations. First, the vibrational analysis was conducted on the virgin structure (without damage). Then, the arch was loaded on the section located at one quarter of the span, up to a load level that generated cracking damage. Then, the vibrational analysis was re-performed. The operation was repeated eight times for which each test had an increment of the load. For the ﬁrst four levels of the load, the structure worked in the elastic range (no cracks visually observed). From the ﬁfth loaded/unloaded test, cracking was observed in the arch at the position of crack 1 in Figure 10 (crack in the joint between the brick rows 16 and 17). The vibrational analysis at this damage level was called the C1 status. The C2 status corresponded to the appearance of cracks 1 and 2 (Figure 11), C3 status corresponded to the appearance of cracks 1, 2 and 3, and C4 status corresponded to the appearance of cracks 1, 2, 3 and 4. European Journal of Environmental and Civil Engineering (a) (b) 17 14 13 12 11 10 16 15 18 19 20 21 22 23 24 25 26 31 32 33 34 35 36 37 27 28 29 30 38 39 40 41 42 43 44 crack 3 45 46 47 48 crack 1 49 50 51 52 9 8 7 6 5 4 crack 4 3 2 Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 53 54 55 56 57 58 59 60 61 62 63 crack 2 1 Figure 11. 9 (a) Geometry of the arch (Ramos et al., 2010); and (b) failure modes of the arch. 3.2. Results of the undamaged state The arch was ﬁrst modelled by the FEM using the ABAQUS code. The arch was divided into seven layers in its thickness. The behaviour of the arch and two concrete bases was elastic. The concrete bases had an elastic modulus of 35 GPa, a density of 2500 kg/m3 and a Poisson’s ratio of .2. Their boundary conditions were the ﬁxed bases. The arch had a density of 1930 kg/m3 and a Poisson’s ratio of .2; its elastic modulus was recalibrated in order to reproduce the ﬁrst experimental natural frequency. A 4-GPa Young modulus was identiﬁed for the vault. It is important to note that the module experimentally measured by the author was 3.28 GPa. This was obtained by means of ﬁve compression tests conducted on three assembled bricks. For the DEM, the arch was modelled by assembling deformable bricks whose elastic modulus was 5 GPa (Figure 12(a)). The contacts between the bricks were modelled by elastic joints without thickness. Joint rigidities were optimised in order to reproduce the experimental frequency of mode 1, by varying from 0 to 1000 GPa/m. The parameters selected were: kn = 62 GPa/m and ks = 25.83 GPa/m. In these calculations, damping was ignored. The same principle used for the column was applied to the arch: the dynamic characteristics were identiﬁed by adding an impact excitation on the structure. The arch was impacted by a relatively small dynamic force to excite the vibration. After impact, the responses of 22 points on the arch (Figure 12(b), points P2–P23 on the outer edge of the vault) were measured. Figures 13 and 14 show the natural frequencies and the mode shapes obtained. The similarity of the mode shapes obtained by the DEM (Figure 14(b)), the FEM (Figure 14(a)), and the experiments (Ramos et al. 2010) allow comparison of the natural frequencies obtained. Figure 15 presents this comparison. The DEM, which used the approach proposed, gave results close to the experiment’s results. (a) (b) P8 P7 P6 P5 14 13 12 11 10 P4 9 8 P3 7 6 5 4 P2 16 15 17 18 19 20 21 22 P9 23 24 P12P13 P14 P15 P10 P11 25 26 27 28 29 30 P16 31 32 33 34 35 36 37 38 39 40 41 42 P17 43 44 47 48 R7 95 R7 45 3 2 X (a) DEM mesh; and (b) measurement points. P19 49 50 51 52 P20 P21 53 54 55 56 57 58 59 60 61 62 63 P22 Y 1 Figure 12. P18 45 46 P23 10 T.T. Bui et al. 1.E-04 Power spectrum X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 35,41Hz 1.E-04 8.E-05 6.E-05 130,2Hz 4.E-05 2.E-05 156,2Hz 120,8Hz 57,28Hz 71,86Hz 180,2Hz 201Hz 0.E+00 0 100 Frequency (Hz) 150 200 Frequencies obtained from the vault (FFT). (a) Y (m) 0.6 0.4 0.2 -1 -0.5 0 0.5 1 f=71,86 Hz 1 0.6 0.4 -1 -0.5 0 0.5 0.6 0.4 0.8 0.6 0.4 0.2 -1 1 f=130,2 Hz 1 0.8 0.2 0.2 X (m) Figure 14. f=57,28 Hz 1 0.8 Y (m) f=35,41 Hz 1 0.8 Y (m) (b) Y (m) X (m) -0.5 0 0.5 -1 1 X (m) -0.5 0 0.5 1 X (m) Initial shape and mode shapes: (a) FEM; and (b) DEM. 250 Experimentation DEM FEM 200 Frequency (Hz) Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 Figure 13. 50 150 100 50 Mode 0 0 Figure 15. 2 4 6 8 Natural frequencies: DEM, FEM and experimental test. 3.3. Results of the damaged structure The presence of damage in the structure can lead to changes in frequencies and mode shapes. Important questions today are how to identify the appearance of damage, its location and its severity. A crack was modelled by a joint with lower kn, ks stiffnesses (Figure 16). The rigidities were recalibrated to reproduce the ﬁrst experimental frequency (mode 1) of the deteriorated structure. kn was determined to be 7.2 GPa/m. European Journal of Environmental and Civil Engineering Positions of the cracks of two damage levels (using the DEM). Frequency (Hz) Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 250.00 200.00 150.00 250.00 Experimentation C0 C1 C2 C3 C4 Frequency (Hz) Figure 16. 11 100.00 50.00 150.00 100.00 0.00 (a) Figure 17. DEM C0 C1 C2 C3 C4 200.00 50.00 0.00 0 2 4 Mode 6 8 (b) 0 2 4 6 8 Mode Damage assessment from natural frequencies. For the experiment, the damage level C1 was a very small crack. This “crack 1” appeared more clearly at the C2 damage level. In other words, crack 1 (in C1) and crack 2 (in C2) did not reﬂect the same damage level. Consequently, to translate the experimental result, kn, ks should be changed according to both the position of the crack and its size (opening, depth). In the numerical model, only the cracks’ positions were taken into account; kn, ks were not changed for the C2, C3 and C4 damage levels, which is why, for these states, greater differences between numerical and experimental results can be observed (Figure 17). 4. Conclusions The aim of this paper is to propose a technique which can identify the dynamic characteristics (natural frequencies and mode shapes) of masonry structures by using DEM.This was done not by solving an eigenvalue problem, but indirectly by making an impact analysis. These dynamic characteristics are important for dynamic studies, especially in seismic design or assessment of existing structures by nondestructive tests.This approach demands a calibration to identify the appropriate interface parameters. Thus, the spectral method, which is classical for the FEM but had remained an open question for the DEM, now ﬁnds a solution in the approach proposed here, which opens new prospects for the characterisation of interface elastic parameters (inter-brick contacts). The proposed approach also opens the way for damage assessment of masonry structures using the DEM. References Andersen, P., Brincker, R., Goursat, M., & Mevel, L. (2007). Automated modal parameter estimation for operational modal analysis of large systems. In R. Brincker & N. Møller (Eds.), Downloaded by [INSA de Lyon - DOC INSA] at 02:25 06 February 2014 12 T.T. Bui et al. Proceedings of the 2nd international operational modal analysis conference (pp. 299–308). Copenhagen. Anzani, A., Binda, L., Carpinteri, A., Invernizzi, S., & Lacidogna, G. (2010). A multilevel approach for the damage assessment of historic masonry towers. Journal of Cultural Heritage, 11, 459–470. Boukria, Z., & Limam, A. (2012a). 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