ICDVC2010 China Gillich Final

8/3/2019 ICDVC2010 China Gillich Final

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Considerations Regarding the Behavior of Damaged StructuresG.R. Gillich1, D. Amariei2, E.D. Birdeanu1 and V. Iancu1

1Faculty of Engineering, “Eftimie Murgu” University, Resita 320085, Romania

[email protected] (G.R. Gillich), [email protected] (E.D. Birdeanu), [email protected] (V. Iancu)

2Center of Advanced Research, Design and Technology, “Eftimie Murgu” University, Resita 320085, Romania

[email protected] (D. Amariei)

Abstract. Researches concerning structure’s safety improvement have increased significantly in the

past decades, structural health monitoring becoming an important support in taking decision

regarding the need of these structures’ remediation. Various techniques, from visual inspection to

nondestructive evaluation or vibration-based damage detection are used. The last ones use

information regarding changes of natural frequencies, damping ratios, flexibility or mode shapes.

Even if numerous results have been achieved in this field, several problems still have to be solved

for specific applications before their implementation in practice.

Our study approaches the method’s viability of damage detection in simple structures using only

one accelerometer by analytical method, finite element method and experimental test. The obtained

results by all three methods showed that this approach can be used only in high damage rates cases.

Finally, are pointed some issues which have to be further worked out.

1 Introduction

Non-destructive testing, including monitoring and diagnostics, is nowadays widely practiced in

mechanical and civil engineering. It influences our approach of designing of new structures andmonitoring and maintenance of existing ones. From the series of now available non-destructive testing

methods, the dynamic ones grow in importance comparing with conventional diagnostic methods, like

the once based on visual inspection or ultrasonic analysis. This is motivated by the global perspective

offered by dynamic techniques (tests developed over time are able to indicate the appearance of

possible damages in the structure), comparing with the local nature of conventional methods (which

require good preliminary knowledge about the position of the area where the damage is located).

Another advantage of the dynamic methods is linked to the fact that these techniques do not requested

access to the damaged area, while conventional methods require direct accessibility to the area to be

inspected. However, the two methods can be used complementary; they do not exclude each other [1].

Numerous researches have been performed in order to develop and prove the viability of dynamic

methods. In this context, important progresses made on the technological field have generated accurate

and reliable experimental methods, while the interpretation of measurement results remains a bit

behind. Morassi and Vestroni [2] and Zhou [3] presented a series of vibration based damage detection

techniques, for structures and simple elements, based on: natural frequency shifts, damping, change of

mode shapes, flexibility of structures and model updating. Some authors focused their researches on

simple elements, like Choi et al. [4] on simple supported beams or Rucka [5], Barbosa et al. [6] and Xia

et al. [7] on cantilever beams. It has to be mentioned that all methods base on processing data of change

of natural frequency and shape modes.

Similar researches have been done by the authors [8] in order to show that it is possible to detectdamages in beams, without concentrating on the localization or characterization of the damage. The

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researches developed last time, presented in this paper, had the intention to determine the sensitivity of

various types of structures (beams) by different types and locations of damages.

2 Theoretical Investigations

The aim of our research is to determine the natural frequencies for a steel beam in undamaged state,

supported in different ways, and to compare it with the corresponding harmed beam. This helps us tofind out if, for the analyzed cases, the shift in frequency provides enough information to identify and

locate damages. In our theoretical investigations the analytical method and the finite element method

were used.

2.1 Analytical Method

At first a steel cantilever beam, clamped on one end (point - as shown in figure 1) and free on the

other (point in figure 1), is considered as testing structure. The beam has the length l = 1000 mm,

wide b = 50 mm and height h = 5 mm and consequently a cross-section 610250 −⋅= A m2 and

moment of inertia 1210833.520 −⋅= I m4 respectively. The producer indicates following material

parameters: mass density 7850=ρ kg/m3, Young’s modulus E = 2.0·1011 N/m2 and the Poisson’s

ratio µ= 0.3.

Fig. 1. Configuration of the cantilever beam

Afterwards similar beams, simple supported and clamped on both ends (points and) respectively,

are considered, like it is presented in figure 2. The accelerometer was mounted in both this cases close

at the middle of the beam (point ).

a.

b.

Fig. 2. Scheme of the simple supported beam (a.) and double clamped beam (b.)



To find out the analytical calculated natural frequencies for the undamaged cantilever beam, we started

from the beam equations [9]:

02

2

4

4

=∂

∂⋅

ρ+

∂

∂

t

v

EI

A

x

v(1)

where v is the vertical displacement of the beam at distance x measured from the free end. Considering

that v depends on distance x and time t , and the evolution in time is harmonically, the expression of

displacement is:

t X t T x X v ω⋅=⋅= sin)()( (2)

After derivation and substitution in relation (1), one obtains:

0

2

=ωρ

− X EI

A X IV (3)

with the solution:

x D xC x B x A X α+α+α+α= coshsinhcossin (4)

where we noted

4

2

α=ωρ

EI

A. (5)

After three derivations of the solution (4) results the system:

α+α+α+α−α=′′′

α+α+α−α−α=′′

α+α+α−αα=′

α+α+α+α=

x D xC x B x A X

x D xC x B x A X

x D xC x B x A X

x D xC x B x A X

s in hco shs inc o s(

)c o shs in hc o ss in(

)s in hco shs inc o s(

c o s hs in hc o ss in

3

2(6)

Putting the boundary condition for the clamped cantilever beam (to find the constants A, B, C and D)

0)0()0( =′= X X , and 0)()( =′′′=′′ l X l X , one obtains the equation

0coshcos1 =λ⋅λ+ , (7)

with l α=λ , which permits to calculate the iλ values for i vibrations modes. Multiplying relation

(5) with 4l and substituting the values of iλ , results the analytic calculated angular frequencies iω

, and consequently the natural frequencies of the undamaged cantilever beam:

4

2

2 Al

EI f i

iρπ

λ= , (8)

the first six being presented in table 1.

Similar, by putting the corresponding boundary conditions for the undamaged simple supported and

double clamped beam respectively, one obtains the natural frequencies presented also in table 1.

Table 1. Natural frequencies of undamaged beams, differently supported, calculated analytically

Mode Natural frequency f i [Hz]

Cantilever beam Simple supported beam Double clamped beam1 4.077 11.444 26.021



2 25.549 45.776 71.727

3 71.539 102.996 140.614

4 140.189 183.105 232.441

5 231.742 286.101 347.227

6 346.182 411.985 484.970

2.2 Finite Element MethodThe FEM computation was made on 3D model beams, using the first ten modal structural

analysis types, because some bending vibrations modes on orthogonal direction and torsion

vibrations modes are interfered. For meshing, it was used a 5 mm element size; the weight of the

accelerometer was neglected.

The first and fifth mode shapes for the undamaged beams (cantilever, simple supported and

double clamped) are presented in figure 3, while their first six natural frequencies in vertical

direction for the intact and damaged state are presented in table 2. Following notations are used:

- U for the undamaged beam

- D1 for the beam with one damage at point - figure 1

- D2 for the beam with two damages, at point and point - figure 1

a. b.

c. d.

e. f.



Fig. 3. Mode shapes for undamaged cantilever beam (a. first mode and b. fifth mode), simple supported beam (c.

first mode and d. fifth mode) and double clamped beam (e. first mode and f. fifth mode)

Comparing tables 1 and 2, it is obvious that the natural frequencies of beams determined by the FEM

are close to them determined analytically, so we concluded that the used methodology was correct.

Table 2. Natural frequencies of undamaged and damaged beams, differently supported, determined with FEM


Cantilever beam Simple supported Double clamped

U D1 D2 U D1 U D2

1 4.091 4.079 4.0548 11.444 11.289 26.241 25.990

2 25.634 25.302 25.100 45.779 45.775 72.317 72.036

3 71.780 71.767 71.360 103.010 101.650 141.770 139.850

4 140.690 138.940 138.620 183.140 183.070 234.380 234.260

5 232.650 232.530 229.720 286.150 282.450 350.220 345.660

6 347.690 343.440 343.070 411.960 411.640 489.330 488.840

3 Experimental Researches

The experimental tests performed an analysis of the steel beam’s dynamic behavior for two situations:

undamaged and with damage on its middle (point in figure 4). The cantilever beam described in

chapter 2 was mounted in a rigid support. The measurement system (figure 4), composed by a Toshiba

laptop, a NI cDAQ-9172 compact chassis with NI 9234 four-channel dynamic signal acquisition

modules and a Kistler 8772 accelerometer has been used for signal acquisition.

Fig. 3. Experimental stand for frequency analysis of cantilever beams

The LabVIEW programming environment was used to develop the virtual instrument which acquired

the time history of acceleration and realized the spectral analysis. A window presenting the acquired

and processed signal is shown in figure 5.

2

3

1



To measure the accelerations on vertical direction for the free end of the beam (point in figure 4) a

vertical force was applied to bring the mechanical system out of its equilibrium position. Suppressing

the force, the beam started to vibrate. We recorded and stored the acceleration values using the

acquisition system, for the undamaged beam; repeating the process for nine times.

Afterwards damages on the middle (point in figure 4) of the beam were produced by saw cuts, likethose presented in detail A in figure 1, and a new series of measurement realized.

Fig. 5. Acquired and processed signal (time history of acceleration and spectral analysis)

All signals were processed in order to obtain their frequency spectrum. Using the pick detector we

extracted the natural frequencies of the undamaged and damaged steel cantilever beam respectively. The

mean values for the first six natural frequencies are presented in table 3.

Table 3. Natural frequencies of undamaged and damaged cantilever beam


Cantilever beam

U D1

1 4.000 4.0002 24.000 24.000

3 70.000 69.000

4 134.000 131.000

5 226.000 222.000

6 335.000 329.000

4 Centralized Results

Results, obtained by simulations and measurements, are centralized in tables 4 to 7, in order to provide

an overview about the behaviour of damaged beams and the utility of the investigation methods applied.

Because the methodologies are focused on the shift of the natural frequency, also presented in these

tables, we tried to find out relevancy for detection, characterisation and localization of damages.



Table 4. Shift of natural frequencies for the cantilever beam, determined by using the FEM


U D1 Change U-D1 D2 Change D1-D2 Change U-D2

1 4.091 4.079 -0,293 % 4.0548 -0.588 % -0,880 %

2 25.634 25.302 -1,295 % 25.100 -0,798 % -2,083 %

3 71.780 71.767 -0,018 % 71.360 -0,576 % -0,585 %

4 140.690 138.940 -1,244 % 138.620 -0,230 % -1,471 %

5 232.650 232.530 -0,052 % 229.720 -1,208 % -1,259 %

6 347.690 343.440 -1,222 % 343.070 -0,108 % -1,329 %

Table 5. Shift of natural frequencies for the cantilever beam, determined by measurements

Mode Natural frequency f i [Hz]U D1 Change U-D1

1 4.000 3.975 -0,625 %

2 24.000 24.000 0,000 %

3 70.000 69.000 -1,429 %

4 134.000 131.000 -2,239 %

5 226.000 222.000 -1,770 %

6 335.000 329.000 -1,791 %

Table 6. Shift of natural frequencies for the simple supported beam, determined by using the FEM


U D1 Change U-D1

1 11.444 11.289 -1,354 %

2 45.779 45.775 -0,009 %

3 103.010 101.650 -1,320 %

4 183.140 183.070 -0,038 %

5 286.150 282.450 -1,293 %

6 411.960 411.640 -0,078 %

Table 7. Shift of natural frequencies for the simple supported beam, determined by using the FEM


U D1 Change U-D1

1 26.241 25.990 -0,957 %

2 72.317 72.036 -0,389 %3 141.770 139.850 -1,354 %

4 234.380 234.260 -0,051 %

5 350.220 345.660 -1,302 %

6 489.330 488.840 -0,100 %

5 Conclusions

Previous studies present various methods to the structures integrity dynamic evaluation, without

specifying the conditions for which the methods are recommended to be applied. This implies, by

structures’ dynamic analysis, a previous good experience for engineers in the field of structural health

monitoring. Our research, partially presented in the actual paper, intends to bring novelty in this field,



providing recommendations about the methods to be used for different cases in dynamical evaluation of

structures.

Examining the results for the cantilever beam, presented in tables 1 to 3, we remark a good

concordance between the natural frequency values obtained by analytical calculus, deduced with the

FEM and the experimental ones. This proves that correct theoretical algorithms and high-accuracymeasurement equipment and methodologies have been used.

On the other side the results presented in tables 4 to 7 show that the shifts in frequency for the

analyzed structures, even for damages which reduce dramatically the moment of inertia, don’t exceed

3%, being for almost the large majority of the cases lower than 1.5%. It is also to be mentioned that

increased number of damages does not increase dramatically the shift in frequency. This makes

methods based alone on the shift of frequency not suitable, the results have to be completed with other

information about the structure’s behaviour.

As next researches the authors plan to extend the experiments on trusses as well as on plates too.

Acknowledgement

The authors acknowledge the support of the Romanian Ministry of Labor, Family and Social

Protection by the Managing Authority of Sectorial Operational Programme for Human Resources

Development (AM POSDRU) for creating the possibility to perform these researches by Grant POS

DRU 5159 - “PhDStudents for Innovation and Competitiveness support”, co-financed by structural

funds.

References

1. Balageas D., Fritzen C.P., Güemes, A.: Structural health monitoring. London: ISTE Ltd, 2006

2. Morassi, A., Vestroni, F.: Dynamic Methods for Damage Detection in Structures, CISM Courses and Lectures,

Vol. 499, Springer Wien New York (2008)

3. Zhou, Z.: Vibration-based damage detection of simple bridge superstructures, Doctoral Thesis, University of

Saskatchewan, Saskatoon (2006)

4. Choi, F.C., Li, J., Samali, S., Crews, K.: An Experimental Study on Damage Detection of Structures Using a

Timber Beam, Journal of Mechanical Science and Technology 21 (2007) 903-907

5. Rucka, M.: Experimental and numerical studies of guided wave damage detection in bars with structural

discontinuities, Arch Appl Mech. (2009) Published online

6. Barbosa, F.S., Cury, A.A., Vilela, A.R., Borges, C.C.H., Cremona, C.: Damage evaluation of structures using

dynamic measurements, Proceedings of the Experimental Vibration Analysis for Civil Engineering Structures

EVACES 05 Bordeaux (2005)

7. Xia, Y., Hao, H., Brownjohn, J., Xia, PQ.: Damage identification of structures with uncertain frequency and

mode shape data, Earthquake Engineering and Structural Dynamics 31 (2002) 1053–1066

8. Gillich, G.-R., Birdeanu, E.D., Gillich, N., Amariei, D., Iancu, V., Jurcau, C.S.: Detection of damages in simple

elements, Proceedings of the 20th International DAAAM Symposium, Volume 20 No. 1 (2009) 0623-0624

9. Buzdugan, Gh.: Material strenght (Rezisten a materialelor in Romanian in original). 10ţ th edition, Editura Tehnica

Bucuresti (1974) 448-453

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