Damage Detection Frm Mode Shape 1

Journal ofSound and Vibration (1991) 145(2), 321-332

DAMAGE DETECTION FROM CHANGES IN CURVATURE MODE SHAPES

A. K. PANDEY, M. BISWAS AND M. M. SAMMAN Transportation and Infrastructure Research Center, School of Engineering, Duke University,

Durham, North Carolina 27706, U.S.A.

(Received 18 September 1989, and in revised form 29 May 1990)

A damage in a structure alters its dynamic characteristics. The change is characterized by changes in the eigenparameters, i.e., natural frequency, damping values and the mode shapes associated with each natural frequency. Considerable effort has been spent in obtaining a relationship between the changes in eigenparameters, the damage location and the damage size. Most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage. In this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure. By using a cantilever and a simply supported analytical beam model, it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure. The changes in the curvature mode shapes increase with increasing size of damage. This information can be used to obtain the amount of damage in the structure. Finite element analysis was used to obtain the displacement mode shapes of the two models. By using a central difference approximation, curvature mode shapes were then calculated from the displacement mode shapes.

1. INTRODUCTION

Any crack or localized damage in a structure reduces the stiffness and increases the damping in the structure. Reduction in stiffness is associated with decreases in the natural frequencies and modification of the modes of vibration of the structure. Many researchers have used one or more of the above characteristics to detect and locate a crack. Most of the emphasis has been on using the decrease in frequency or the increase in damping to detect the crack. Very little work has been done on using the changes in the mode shapes to detect the crack.

Adams et al. [l] used the decrease in natural frequencies and increase in damping to detect cracks in fiber-reinforced plastics. Loland et al. [2] and Vandiver [3] used the same principle to detect damage in offshore structures. From relative changes in the natural frequencies of different modes, Loland et al. could predict the location of the damage. They demonstrated the use of their technique on some platforms in the North Sea. The essence of the methods developed by the other researchers is similar, but different methods of data analysis were used.

Adams et al. [l] developed a theoretical model to predict the damage and its location based on receptance analysis. The analysis was done by using axial modes of vibration and is valid for structures which can be treated as one-dimensional. Adams and Cawley [5] employed sensitivity analysis to deduce the location of damage in two-dimensional structures, based on a finite element analysis method. Flexural modes of vibration were used in this case. The method was applied to the case of a flat plate with the assumption that the modulus of elasticity in the damage area was equal to zero. For each element of

321 0022-460X/91/050321 + 12 %03.00/O @ 1991 Academic Press Limited

322 A. K. PANDEY ET AL.

the model, the sensitivity of the change was evaluated. The results of the analysis agreed well with the experimental results. The drawback of this method is that a lot of computation has to be performed subsequent to data collection to predict the location of the damage.

Very little use has been made of the changes in the mode shapes in detecting a crack or damage in a structure. Yuen [6] in his paper showed for a cantilever beam that there is a systematic change in the first mode shape with respect to the damage location. He used finite element analysis to obtain the natural frequencies and the mode shapes of the damaged structure.

From changes in frequency one can easily determine the presence of crack or damage in a structure. But determining the location of the crack, knowing the changes in the frequencies, is a completely different question. This is because cracks at two different locations associated with certain crack lengths may cause the same amount of frequency change. Other parameters need to be determined which will directly identify the location of crack or damage in a structure.

In this paper, a new parameter called curvature mode shape is introduced. The difference in the curvature mode shapes between the intact and the damaged case is utilized to detect the location of the crack. The changes in the curvature mode shapes are shown to be localized in the region of damage compared to the changes in the displacement mode shapes. A cantilever and a simply supported beam model are used to demonstrate this characteristic of the curvature mode shapes.

Different ways of comparing mode shapes include the Modal Assurance Criterion (MAC) [7] and the Co-ordinate Modal Assurance Criterion (COMAC) [8]. MAC indicates the correlation between two sets of mode shapes. It is used to study overall differences in the mode shapes. COMAC compares mode shapes in a point-wise manner. It indicates the correlation between the mode shapes at a selected measurement point of a structure.

2. METHOD OF CRACK DETECTION

Curvature mode shapes are related to the flexural stiffness of beam cross-sections. Curvature at a point is given by

u= M/(EI), (I) in which t) is the curvature at a section, M is the bending moment at a section, E is the modulus of elasticity and I is the second moment of the cross-sectional area.

If a crack or other damage is introduced in a structure, it reduces the (EI) of the structure at the cracked section or in the damaged region, which increases the magnitude of curvature at that section of the structure. The changes in the curvature are local in nature and hence can be used to detect and locate a crack or damage in the structure.

The change in curvature increases with reduction in the value of (EI), and therefore, the amount of damage can be obtained from the magnitude of change in curvature.

3. ANALYTICAL MODEL

A cantilever beam and a simply supported beam of uniform square cross-section were used for this study. The dimensions of the two beams are shown in Figures 1 and 2. These two cases were chosen because, for a simply supported beam both curvature mode shapes and displacement mode shapes are of the same form, whereas, for a cantilever beam, they are different. For example, in an uniformly loaded simply supported beam, maximum curvature occurs at the mid-span where displacement is also maximum whereas, in a cantilever beam, maximum curvature occurs at the support, where displacement is minimum.

CURVATURE MODE SHAPE DAMAGE DETECTION 323

Y

t 500

12.7

I

Figure 1. Cantilever beam model. (a) Finite element model; (b) cross-section of the cantilever. Dimensions in mm.

Y

t _ 500

254

o~o~,o,o,o,o,o,o,o,o,o,o,qo,o,o,o,o,e 1 ,2, 3 4 5 6 7 8 9 10 11 12 13 14 1 16 17 (8 19

A2 3 4 5 6 7 8 91011121314151617161920 - 1 21

la)

12.7

I

(bl

Figure 2. Simply supported beam model. (a) Finite element model; (b) cross-section of the simply supported beam. Dimensions in mm.

The finite element model for each of the beam consisted of 20 equal length two- dimensional beam elements. Three degrees of freedom, translations along the X and Y axes and rotation along the 2 axis, were used at each node in the finite element analysis.

It was assumed that damage in a structure will affect only the stiffness matrix and not the inertia matrix in the eigenvalue problem formulation. This assumption is consistent with those used by Adams and Cawley [5]. The eigenvalue problem for the intact case can be written as

(K - AjM)xj = 0, (2)

and for the cracked case as

(K- hjM)x; = 0, (3) in which K is the stiffness matrix of the intact structure, K is the stiffness matrix of the damaged structure, Aj is the jth eigenvalue of the intact structure, Ai is the jth eigenvalue

324 A. K. PANDEY ET Al

of the damaged structure, M is the inertia matrix of the structure, x, is thejth displacement eigenvector of the intact structure, and x: is the jth displacement eigenvector of the damaged structure.

The change in the stiffness due to damage was modeled by a reduction in the modulus of elasticity of the section [6]. The degree of damage is then related to the extent of reduction in the modulus of elasticity, E. This method requires only a simple modification in the finite element analysis data and no new element is needed.

A study was conducted with a 50 percent reduction of the modulus of elasticity imposed in turn to all 20 elements for the two beams. This simulated cases of damage located in each of the sections, respectively. For each of the damage locations, the first five natural frequencies and corresponding mode shapes were calculated.

A further study was carried out in which damage was prescribed in the element number 10 of each of the two beams. The intensity of damage was varied by changing the modulus of elasticity over the range of 0.1 to 0.9 of the full value in steps of 0.2. This represented the case of a varying degree of damage at a particular location.

4. ANALYSIS OF RESULTS

The mode shapes calculated were orthonormalized against the inertia matrix, i.e.,

xfMxi = 1. (4)

Only the translation degree of freedom along the Y axis was considered in the analysis. This was done because, in any experimental work, in general, rotations are not measured because of difficulty in their measurement. Moreover, since we are interested only in flexural modes of vibration, translation along the X axis can also be neglected.

From the displacement mode shapes, obtained from the finite element analysis, curvature mode shapes were obtained numerically by using a central difference approximation as

Vr=(Ui+I-2Ui+vi_,)/h2,

where h is the length of the elements.

(5)

For each damage location, the percentage change in the natural frequency from the intact case to the damaged case, MAC between the intact and the damaged mode shapes, COMAC between the intact and the damaged mode shapes, absolute difference in the displacement mode shapes and absolute difference in the curvature mode shapes were calculated.

5. NUMERICAL RESULTS

5.1. DAMAGE LOCATED IN TURN AT EACH ELEMENT

5.1 .l. Cantilever beam model The first five natural frequencies of the intact cantilever and of the cantilever damaged

at element 13 (50 percent reduction in E) are listed in Table 1. The first five displacement mode shapes for the cantilever are shown in Figure 3. From the percentage change in the frequency from the intact to the damaged case a state of damage is discernible, but no indication of the location of damage is obtained from this, without further analysis.

The Modal Assurance Criterion (MAC) and the Co-ordinate Modal Assurance Criterion (COMAC) values for the intact and the damaged (element 13) displacement mode shapes


TABLE 1

Naturalfrequencies of thejrstjve modes for the intact cantilever and the cantilever damaged at element 13

Mode no.

Natural frequency (Hz) Percentage , L- \ change in

Intact Damaged frequency

1 39.63 1 39463 042 2 247.110 237.181 4.02 3 687.663 665.472 3.23 4 1336.898 1331.641 0.39 5 2188.828 2108.587 3.67

32.61

16.36

; 6.24 E

8 0 11 % 0

-6 01 -

-16.14-

-24.26 -

Point no.

Figure 3. First five displacement mode shapes for the intact cantilever

TABLE 2

Modal Assurance Criterion (MAC) values for the intact and the damaged (element 13) displacement mode shapes

for the cantilever

Intact cantilever Damaged , cantilever 1 2 3 4 5

1 140 0.01 0.01 0.01 0.01 2 0.01 140 0.00 0.01 0.01 3 0.01 0.01 l-00 0.01 0.01 4 0.01 0.01 0.01 140 0.00 5 0.01 o-01 0.00 0.01 140


are listed in Tables 2 and 3, respectively. All the diagonal entries in Table 2 and all entries in Table 3 are equal to 1, which indicates that the mode shapes are almost identical. Hence, MAC and COMAC do not detect the state of damage.

The curvature mode shapes for the intact cantilever are shown in Figure 4. The absolute differences between the curvature mode shapes of the intact and the damaged (element 13) cantilever are plotted in Figure 5. The maximum difference for each curvature mode shape occurs in the damaged region, which is between point 13 and 14 for this case. The differences in the curvature mode shapes are localized near the damaged zone, i.e., it is much smaller outside the damaged region. The changes in the displacement mode shapes are not localized to the damaged zone. The absolute difference in the displacement mode shapes between the intact and damaged (element 13) cantilever is shown in Figure 6. This characteristic of curvature mode shapes can be very useful in locating the damaged area.

TABLE 3 Co-ordinate Modal Assurance Criterion (COMAC) values for the intact and the damaged (element 13)

displacement mode shapes for the cantilever

Point COMAC no. values

Point no.

COMAC values

1 1.00 2 1.00 3 1.00 4 1.00 5 1.00 6 1.00 7 1.00 8 1.00 9 1.00

10 1.00 11 1 .oo

12 1.00 13 1.00 14 1.00 15 1.00 16 1.00 17 1.00 18 1.00 19 1.00 20 1.00 21 1.00

f 3 5 7 9 11 13 15 17 19 21 Point no.

Figure 4. First five curvature mode shapes for the intact cantilever.


Figure 5. Absolute difference between the curvature mode shapes for the intact and the damaged (element 13) cantilever.

Figure 6. Absolute difference between the displacement mode shapes for the intact and the damaged (element 13) cantilever.

Similar results were obtained for damage at other elements.

5.1.2. Simply supported beam model A similar analysis was done for the simply supported beam model. In Table 4 the

natural frequencies for the first five modes of vibration for the intact simply supported beam are listed. Natural frequencies for the simply supported beam damaged at element 13 (50 percent reduction in E) are also listed in Table 4. Again, from the percentage change in the natural frequencies a state of damage is discernible but the damage cannot be located.

The MAC and COMAC values, again do not indicate the presence of damage. For the simply supported beam the curvature mode shapes are of the same form as

the displacement mode shapes and are shown in Figure 7. The absolute difference between

328 A. K. PANDEY ET AL

TABLE 4

Natural frequencies for the first five modes for the intact and the damaged (element 13) simply supported beam

Mode no.

Natural frequency (Hz) Percentage I I change in Intact Damaged frequency

1 111.327 106.857 4.02 2 444.228 434.274 2.24 3 995.420 987.643 0.78 4 1759.279 1690.227 3.93 5 2727.609 2702.741 0.91

l-02-

0.77-

0.51 -

0.26-

1 3 5 7 9 11 13 15 17 19 21

Point no.

Figure 7. First five curvature mode shapes for the simply supported intact beam.

the curvature mode shapes of the intact and the damaged (element 13) case are plotted in Figure 8. For each curvature mode, the maximum difference again occurs in the damaged zone. The difference in the displacement mode shapes, plotted in Figure 9, does not give any idea of the damage location.

5.2. INCREASING DAMAGE AT AN ELEMENT For the case of increasing damage at element 10, the absolute differences in the curvature

mode shapes for the cantilever beam and the simply supported beam are plotted in Figures 10 and 11, respectively. The maximum difference for each of the cases occurs in the damaged zone. The maximum difference increases with the reduction in the stiffness of the damaged zone. Therefore, the amount of damage can be obtained from the maximum difference in the curvature mode shapes.

6. DISCUSSION AND CONCLUSIONS

The numerical results for the cantilever beam model and the simply supported beam model demonstrate the usefulness of curvature mode shapes in detecting and locating a state of damage. It has been shown that changes in the curvature mode shapes are localized


Figure 8. Absolute difference between the curvature mode shapes for the intact and the damaged (element 13) simply supported beam.

Figure 9. Absolute difference between the displacement mode shapes for the intact and the damaged (element 13) simply supported beam.

in the region of damage. On the other hand, changes in the displacement mode shapes are not localized and, hence, they do not give any indication of the location of damage.

The Modal Assurance Criterion (MAC) and the Co-ordinate Modal Assurance Criterion (COMAC) are not sensitive enough to detect damage in its earlier stages. This is because in calculating MAC and COMAC, differences become averaged over all the measurement points (MAC) or all the mode shapes (COMAC) [93.

For obtaining curvature mode shape, by using the techniques of experimental modal analysis, one has to take a full set of readings on a structure. To avoid this problem, curvature mode shapes can be used in conjunction with the natural frequencies. Changes in natural frequencies can be used to detect the presence of a state of damage, since this can be done from a single point (e.g., reference point or driving point) measurement.


2.0

_ (e) 14.

12 -

Figure 10. Absolute difference between the curvature mode shapes for the intact and the damaged (element 10) cantilever. (a), Mode 1; (b) mode 2; (c) mode 3; (d) mode 4; (e) mode 5.


1 (b) t (cl 6

1 (d) 14t-- (e)

m ;l3 5 !% 6 *

4 E= O.lE

2 E= 0.3E

0

@ 0

5 Point 10

nun

Figure 11. Absolute difference between the curvature mode shapes for the intact and the damaged (element 10) simply supported beam. (a)-(e) as Figure 10.

332 A. K. PANDEY ET AL

Once the presence of damage is detected, curvature mode shapes can be obtained to locate the damage.

Since curvature is proportional to the bending strain, in experimental tests curvature mode shapes can be obtained directly by measuring strains instead of displacement or acceleration. Recently, many researchers have demonstrated the feasibility and usefulness of measuring strain mode shape with the help of strain gauges instead of measuring acceleration or displacement mode shape [ 10-121.

ACKNOWLEDGMENTS

The work reported here is a part of a continuing research project sponsored by the Pennsylvania Department of Transportation (G. L. Hoffman, Project Manager) and the U.S. Department of Transportation (USDOT), Federal Highway Administration (FHWA).

The contents of this paper reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views of either the USDOT-FHWA, or the Commonwealth of Pennsyl- vania at the time of publication. This paper does not constitute a standard, specification or regulation.

REFERENCES

1. R. D. ADAMS, D. WALTON, J. E. FLITCROFT and D. SHORT 1975 Composite Reliability, ASTM STP 580,159-175. Vibration testing as a nondestructive test tool for composite materials. Philadelphia: American Society for Testing Materials.

2. 0. LOLAND and C. J. DODDS 1976 Proceedings ofthe 8 th Annual O$shore Technology Conference 2,313-319. Experiences in developing and operating integrity monitoring system in North Sea. OTC paper no. 2551.

3. J. K. VANDIVER 1975 Proceedings of rhe 7th Annual O$shore Technology Conference 2,243-252. Detection of structural failure on fixed platforms by measurements of dynamic response. OTC paper no. 2267.

4. R. D. ADAMS, P. CAWLEY, C. J. PYE and B. J. STONE 1978 Journal of Mechanical Engineering Science 20(2), 93-100. A vibration technique for non-destructively assessing the integrity of structures.

5. P. CAWLEY and R. D. ADAMS 1979 Journal of Strain Analysis 14(2), 49-57. The location of defects in structures from measurements of natural frequencies.

6. M. M. F. YUEN 1985 Journal of Sound and Vibration 103, 301-310. A numerical study of the eigenparameters of a damaged cantilever.

7. T. WOLFF and M. RICHARDSON 1989 Proceedings of the 7th International Modal Analysis Conference 1, 87-94. Fault detection in structures from changes in their modal parameters.

8. N. A. J. LIEVEN and D. J. EWINS 1988 Proceedings of the 6th Znrernational Modal Analysis Conference 1, 690-695. Spatial correlation of mode shapes, the Co-ordinate Modal Assurance Criterion (COMAC).

9. W. HEYLEN and T. JANTER 1989 Proceedings of rhe 12th Biennial Conference on Mechanical Vibration and Noise, ASME 1, 289-294. Applications of the Modal Assurance Criterion in dynamic model updating.

10. W. F. TSANG 1990 Proceedings of rhe 8th Internafional Modal Analysis Conference 2, 1246-1251. Use of dynamic strain measurements for the modeling of structures.

11. 0. BERNASCONI and D. J. EWINS 1989 Proceedings of the 7th Znternarional Modal Analysis Conference 2, 1453-1464. Application of strain modal testing to real structures.

12. D. B. LI, H. C. ZHUGE and B. WANG 1989 Proceedings of the 7th Internarional Modal Analysis Conference 2, 1285-1289. The principles and techniques of experimental strain modal analysis.

Damage Detection Frm Mode Shape 1

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