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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
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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
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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
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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)
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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
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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)
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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)
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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
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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
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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
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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)
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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.
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