An integrated approach to detection of cracks using vibration characteristics

An Integrated Cracks

Approach

Using Vibration

to Detection

0s

Characteristics

by JIALOU HU and ROBERTY.LIANG Department

of Civil Engineering,

University of Akron, Akron, OH 44325-3905

U.S.A.

ABSTRACT : An integrated technique basedon the vibration theory to nondestructively ident!fj multiple discrete cracks in a structure is presented. Two damage modeling techniques, one involving the use of massless, infinitesimal springs to represent discrete cracks and the other one employing the continuum damage concept, are integrated to provide a crack-detection technique that utilizes the global vibration characteristics qf a structure but ojj%rs local information on each individual crack, including location and extent of the cracks. In the spring model, the Castigliano’s theorem and theperturbation technique are used to derive a theoretical relationship between the eigerzfrequency changes and the t’ocation and extent of the discrete cracks. In the continuum damage model, the efective stress concept coupled with the Hamilton’s principle are used to derive the similar relationship that is cast in a continuum ,form. A u@ed g(p) jimction emerges jLom the two model approaches. The g(p) jimction can be determined through the mode shapes of an intact structure by means of the modal strain energy density. In the proposed integrated approach, the continuum damage model can be used$rst to identljj the discretizing elements of a structure that contain cracks. Then, the spring damage model can be used to quantify the location and severity of the discrete crack in each damaged element. An example of a simply-supported beam containing two discrete cracks is given to illustrate the application and accuracy of the proposed approach.

Notation

; k k, k, U u U’ M’,, lc i

D E I L

depth of a crack depth of a beam spring constant k for longitudinal vibration of bar k for bending vibration of beam displacement field in the intact structure displacement field in the damaged structure u-u the nth mode natural frequency of intact structure the n th mode natural frequency of damaged structure cross-sectional area flexural rigidity of plate Young’s modulus moment of inertia length

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J. Hu and R. Y. Liang damage parameter total elastic strain energy in the nth mode strain energy due to a crack w’ in the n th mode the n th mode shape of the intact structure the n th mode shape of the damaged structure stress x/L, dimensionless location of damage Poisson’s ratio strain energy density of the intact structure strain energy density of the damaged structure

I. Introduction The continued development of a vibration theory based upon the Nondestructive Damage Evaluation (NDE) technique stems partially from the fact that only the global vibration characteristics need to be measured in an evaluation. The knowledge of the global vibration characteristics can detect and localize the deterioration, and consequently can guide more detailed inspection of local areas. The NDE techniques based on the vibration theories include methods such as random decrement, modal frequency theory, transmissibility theory and mode strain analysis. On the theoretical side, the relationships between vibration characteristics and structural (material or geometric) changes have been investigated by numerous researchers, such as Gudmundson (l), Chondros and Dimarogonas (2), among others. On the NDE application side, more recent developments include the work of Akgun and Ju (3) and Stubbs et al. (4). In Akgun and Ju’s work, multiple discrete cracks are modeled as springs with vibration response analysed with the help of electric circuit theory. The method, however, requires that the number of cracks be known a priori. In the work of Stubbs et al., a sensitivity matrix was generated using a finite element program such that the damaged elements can be identified. The method appears to be versatile, but requires a large amount of dynamic analysis to generate the sensitivity matrix. Recently, Liang et al. (5) with the help of a symbolic computational program MACSYMA developed the analytical expressions between the eigenfrequency changes and the cracks in a beam structure. A subsequent work of Liang et al. (6) presented a methodology to utilize the natural frequency changes to identify the damaged areas in a beam structure. Although the method has been checked through numerical as well as experimental studies, the theoretical derivation of the equation was not available at that time. In this paper, two damage modeling techniques are utilized to develop an integrated NDE technique for detecting and quantifying the location and extent of multiple discrete cracks in a structure. The first model involves the use of massless springs with infinitesimal length to represent local flexibility introduced by cracks. The Castigliano’s theorem and the perturbation technique are used to theoretically derive the relationship between eigenfrequency changes and the crack charac-

842

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the Franklin lnst~tute Pergamon Press Ltd

Detection

of Cracks

teristics (location and severity). The second model incorporates the effective stress concept in the continuum damage mechanics and the Hamilton’s principle, leading to the derivation of a similar relationship to that derived from the spring model. A combined use of both models allows one to use global vibration characteristics to identify multiple discrete cracks with sufficient local details such as the location and depth of cracks. A beam example is given at the end of the paper to illustrate the application and accuracy of the developed method. II. Method Based on Elastic Spring Concept Local,flexibility due to damage We consider a cracked structure in which the local flexibility due to a crack is modeled as a massless spring with an infinitesimal length as shown in Fig. 1. The numerical value of the spring constant k (k,v for bars in longitudinal vibration and k, for beams in pure bending vibration) represents the severity of cracking. Due to the fact that the direction of both displacement u and internal force p is assumed to be one-dimensional, the value of the spring constant can be simply determined from the crack strain energy function, as shown below. Evoke the Castigliano’s theorem, the additional displacement U’ due to the internal force p can be expressed as

where W’ is the strain energy due to a crack. For example, for a crack of depth a shown in Fig. 2, w’ can be computed as W’ = sg J(a) da, where J(a) is the energy density function due to the crack. The flexibility influence coefficient c can be determined as

ad

(2)

c=dp.

k

FIG.

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1993

1.

Crack modeled

as spring.

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J. Hu and R. Y. Liung

FIG. 2. A crack in a BernaulliCEuler beam.

Since k = l/c, substituting constant :

(1) into

(2), we have an expression

1 -= k

for the spring

d2W’ (3)

T’

When k is infinite, there is no crack. A decreasing extent of the crack.

value of k indicates

an increasing

Elastic vibration of damaged structure Using the perturbation technique, coupled with an assumption that mass as well as volume in the cracked region of a structure remains unchanged, Gudmundson (1) derived the following relationship for the nth mode vibration : (4) where w, and KJ, are the natural frequencies of the structure for pre- and postcracking conditions, respectively ; W,‘,is the n th mode strain energy of the additional displacement due to geometry changes; and Wo, is the total strain energy of the intact structure in the n th mode. The first-order approximation of Eq. (4), yields AM’,

1 W,: _ -~. ..~~~ W,T 2 W”,

(5)

where Abr, = IV,- rP,,. Integration of (3), gives

where pn is the internal force under the nth modal vibration. Note that pi is proportional to the mode strain energy density of a structure in the nth mode T,, : y,, = p,f/(2E), where i? is the stiffness coefficient, i.e. _I?= EZ for pure bending vibration, and ,!? = EA for longitudinal vibration. Thus, Eq. (6) can be expressed as

where p is the dimensionless

location

of the crack. Journal

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Detection

of Cracks

The total strain energy of the structure in the nth mode, Wall, can be determined via integration of the strain energy density over the volume of the structure. For a one-dimensional structure, it becomes

won =L

s’

Ym(B)dP

(8)

0

where L is the length of a structure. In the present study, it is assumed that ‘3, is approximately the same as ‘I”,. It implies that change of the mode shapes due to the presence of a crack is negligible. Combining (5), (7) and (8), the relationship between the change in eigenfrequencies and the location and severity of a crack is obtained as follows : (9) where

and

&. The value of the dimensionless n th mode strain energy density

constant K represents the severity of a crack. The Y,, of an intact structure takes the following form :

E( /3)Z(fi)[cj:( /J)]’ y”(B)

(11)

= i E( @A( p) [& (p)]’

for beam structures for bar structures

where @a(p) and @i(p) are the first and second derivatives of the n th mode shape, respectively. Extension of Eq. (9) to a structure containing more than one crack can be accomplished ria a linear superposition principle, i.e.

(12) where /I, is the location

of the ith crack.

Case qf beam structures Consider a beam structure with a uniform cross-section; the g(B) function (10) can be expressed in terms of the mode shape 4(p) of the intact structure follows : Vol. 330, No. 5, pp. W-853, Printed in Great Britain

1993

in as

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J. Hu and R. Y. Liung

(13)

From elementary beam theory, the mode shapes of beams with typical homogeneous boundary conditions can be easily calculated. For example, the mode shape of a simply supported beam is &, = sin (nrrfi). Therefore, for a beam with simple supports at both ends, the relationship between the changes in eigenfrequencies and the crack location and severity of a crack, based on Eqs (9), (11) and (13), can be expressed as (14) Note that the same analytical expression of Eq. (14) has been obtained previously through a different approach by Liang et al. (6). Basically, the elementary beam theory was used to derive a high-order characteristic matrix. Through the use of a symbolic computation package MACSYMA, the functional relationship in Eq. (14) was numerically obtained. Case qf bar structures For a bar with uniform be simply written as :

cross-section,

the g(p) function

according

For a free-free bar, the mode shape is cos (nrcfi). Thus, by combining and (15), the following relation can be obtained :

to (10) can

(9), (11)

(16)

It is interesting to note that receptance, derived an expression crack location and the equivalent

Adams and Cawley (7), using the concept of relating the changes in natural frequencies to the crack-induced spring constant as follows :

-=-

k,

Through some algebraic are identical.

846

manipulation,

it can be shown that Eqs (16) and (17)

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Detection

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Case of plate structures Equation (10) can be applied to calculate the g( /?) function of a two-dimensional structure when the model strain energy density of the structure Y is known. For a two-dimensional plate with uniform thickness, the g( 8) function (expanded to two dimensions) takes the form

Sn(AY)

(v’4J*_q\

_v)

(V2~,)2__2(]

-“)

=

4

where fi and y are dimensionless

dB dy

coordinates

in x-y plane.

Remark 1 In this section, a previous numerically constructed relationship between the eigenfrequency changes and the location and extent of the cracks is theoretically derived based on the perturbation method of Gudmundson (1) and the Castigliano’s theorem. In addition, the validity of the derived relationship is confirmed by comparing it with the equations derived by Adams and Cawley (7) in which the concept of receptance was the main starting point. Thus, a theoretical foundation is provided for the working equation (9) in detecting the location and extent of cracks.

III. Method Based on Continuum Damage Concept Vibration analysis of damaged structure In continuum damage mechanics, damage caused by distributed microcrackings is usually quantified by a damage parameter S, see for example DiPasquale et al. (8). The damage parameter for a case of isotropic damage can be defined via a concept of effective stress, i.e.

where 6 is the effective stress. Thus, S = 0 corresponds to the undamaged state, while S = 1 signifies a complete failure. When the structure is vibrating under a harmonic motion u = 4 sin (wt), the total elastic strain energy can be computed through integration of its density function : W = l,,Y(u) d V. Furthermore, the kinetic energy can be computed as T = St,w’T(q5) cos’ wt d V, where T = pq5 * 412. Thus, the natural frequency of the structure can be obtained via the Hamilton principle :

,I

w,,

Vol. 330, No. 5, pp. 841-853, Printed in Great Britain

y’(4,) dV I

I-

(19

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J. Hu and R. Y. Liung

It is customary to refer to the ratio lp Y( 4) dV’/JV r(4) dV as the Rayleigh quotient. If r$ happened to be the eigenvector, then the ratio is the eigenvalue. When damage exists in a structure, DiPasquale et al. (8) derive the modified elastic strain energy as follows : W =

(1 -S)‘I’(&)

s (1

-dV

(20)

where cP,?is assumed to be the same mode shape as that of the intact structure. The kinetic energy remains the same. By using the Hamilton principle, the natural frequencies of the damaged structure become

(21)

By combining (19) and (2 1) and taking a first-order relationship is obtained :

approximation,

the following

(22)

W’, 2

sI

Wh>

dV

In a discretized form, the structure is divided into m elements with each element assumed to have a respective damage index Si, then Eq. (22) can be expanded as follows : (23)

where e, represents (23) becomes

the domain

of the ith element.

For a one-dimensional

“’ s,,(8) dP*S,. - 2 c K’, j=, sc,

case, Eq.

Aw,

(24)

For application, considering a beam structure that is divided into m elements and that a total of n modes of vibration characteristics are known, then Eq. (24) can be expanded to a set of simultaneous equations as follows : =

2[W,.,,z{S>w.I

where the [H] matrix has element hi, calculated as /z,, = SC,g, (p) dp. The numerical values of the elements in the {S} matrix obtained Journal

848

in Eq. (25)

of the Frankhn lnst~tute Pergamon Press Ltd

Detection

of Cracks

furnish the information about the state of damage of that particular element. The {S} can be solved by a normal inversion procedure when m = n, or by a pseudoinverse technique when it < m. Furthermore, either lower modes or higher modes can be used in the equation. Application of Eq. (25) to various numerical and experimental case studies has been given in Liang et al. (9). Remark 2 In this section, the Hamilton principle in conjunction with the work of DiPasquale et al. (8) is used to derive Eq. (24). Notice that Eq. (24) derived from the continuum damage mechanics takes the same form as Eq. (12) which was derived from the elastic spring concept. However, Eq. (24) is cast in a continuum form such that the damaged area can be identified. Equation (12), on the other hand, is only applicable to identification of discrete, individual cracks. Remark 3 The g( /I) function possesses a very unique feature. As can be seen in the following equation, the integration of the g(p) function over the entire domain of a structure gives a constant numerical value, l/4 : i.e.

(26)

IV. Illustrative

Example

In this section, an example illustrates the integrated approach, utilizing both damage models, to identify the location and extent of multiple discrete cracks in a simply-supported beam structure. Due to the nature of symmetry, one half of the beam will be analysed. Figure 3 shows the geometric and dimensional properties of the beam example and the discretization scheme used in carrying out the NDE solutions. The beam is assumed to consist of two major discrete cracks with location and crack depth quantified by the parameters fi and a, respectively. These assumed values for the case study are given in Table 1. The analytical characteristics equations from the fundamental beam theory are solved via a symbolic computational package MACSYMA to compute the natural frequencies of the beam with and without the simulated cracks. To facilitate the use of Eq. (24), half of the beam structure is divided into five elements, as shown in Fig. 3. Therefore, only five modes of natural frequencies need to be determined. The computed natural frequencies are summarized in Table II. The g(p) function for the simply-supported beam, as discussed previously, can be calculated as gn(p) = ‘, sin’ (m$). A set of simultaneous

equations

can be generated

from (25), (27)

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J. Hu and R. Y. Liang

10

2

E = 2.6x 10

4

N/m

p = 2350 kg/m3

FIG.3. Schematics

of a beam example containing

I = 0.0036m b = 0.2 m

two discrete cracks.

TABLE I. Comparison between predicted and simulated, /I : crack location, a : crack depth Crack

I

#

Parameter Simulated Predicted

2

B

a (mm)

B

a (mm)

0.2500 0.2550

47.83 46.44

0.4500 0.4433

59.17 59.22

TABLE II. NaturLll,frc~rluencies ojthe intact and damaged beam (rad so- ‘) State Intact Damaged

850

)I‘, 59.007 58.625

)1‘2 236.029 235.142

‘L’i 531.065 528.096

Li’d 944.116 942.515

k‘5 1475.182 1469.103

Journal

of the Franklin Institute Pergamon Press Ltd

Solution of the above matrix :

equation

gives the following

nonzero

Detection

of Cracks

elements

in the [S]

S3 = 3.25336% S, = 5.01816%. That is, elements 3 and 5 are predicted to contain the crack-induced damage. Next, the eigenfrequency changes due to each crack are calculated via the simultaneous Eq. (27). It can be shown that the changes of the first two modes of natural frequency due to the crack in element 3 are : (Aw,/w,) = 0.162668%) and (AwJwJ = 0.314842%. Finally, to find the exact location of the crack, the elastic spring approach is used. Substituting the obtained eigenfrequency changes into Eq. (14), the numerical value of k, (or dimensionless K) can then be solved as a function of fi. The computed K vs /I based on the eigenfrequency changes from element 3 is shown in Fig. 4(a). Since there is only one unique value for K regardless of which mode of vibration is investigated, this leads to the identification of crack location as the intersection

(a) 620 l-----T (b)

FIG.4. Crack location and severity determined by the intersection of curves, (4 : crack 1. (b) : crack 2. Vol. 330. No. 5, PP. W-853, Printed m Great Britain

1993

851

J. Hu and R. Y. Liung of these two curves in Fig. 4(a). The following equation provides between the depth a of the crack and the value of k,. (10) :

Z(a/h) is computed

from the following

equation

the relationship

:

Using Eq. (28), the depth or severity of the crack can be calculated. Similarly, the eigenfrequency changes due to the crack in element 5 are found to be: (AI~,/H~,) = 0.048567%, and (A~vJH~J = 0.061020%. Following the same approach, Fig. 4(b) is obtained, which gives the location of the crack and the numerical value of K. The predicted location and depth of two cracks are summarized in Table I, along with the simulated condition. It can be seen that the proposed integrated approach provides a viable technique to identify the discrete cracks based on the global vibration characteristics of the structure.

V. Concluding Remarks The theoretical basis for a vibration based NDE technique for an assessment of structure integrity has been presented in this paper. Two different types of models for representing the crack-induced damage were investigated, one involving the use of massless, infinitesimal springs and the other one employing the continuum damage concept. A unified g(p) function emerged in both models to provide a link between the eigenfrequency change and the location and magnitude of cracks in a structure. When these two models are used together, a versatile NDE technique emerges, which enables identification and quantification of multiple discrete cracks in a structure. An example of a simply-supported beam with two discrete cracks was used to illustrate the application and accuracy of the developed integrated approach. The developed NDE method relies on measurements of eigenfrequencies of existing structure and the knowledge of the baseline eigenfrequencies of the sound structure. Modern technology in modal analysis hardware and software should enable measurements of natural frequencies to be performed rather routinely. The baseline eigenfrequencies can be the ones measured before damage incurs or could be computed with the aid of a dynamic analysis technique along with field verifications. Additionally, the method requires a calculation of the g(p) function in forming a set of simultaneous linear equations. Computation of the g( /?) function nevertheless can be carried out once the mode shapes of an intact structure are Journal

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of the Franklin Institute Pergamon Press Ltd

Detection

qf Cracks

known. Again, either modern modal testing techniques or advanced structure dynamics theories can lend support in this area. Finally, although the developed method has been verified to some extent via comparisons with numerical and experimental results for simple structures, such as beams and bars, there is a need to investigate the sensitivity and accuracy of the method when applied to more complex structures such as space frames, bridge truss, etc. Investigation into full-scale experiments and the attendant sensitivity study is currently under way. Acknowledgement The work reported in this paper is in part supported FRG # 1167, from The University of Akron.

by a Faculty

Research

Grant,

References (1) P. Gudmudson, (2) (3) (4) (5)

(6)

(7)

(8)

(9)

(10)

“Eigenfrequency changes of structures due to cracks, notches or other geometrical changes”, J. Mech. Phys. Solids, Vol. 30, pp. 339-352, 1982. T. G. Chondros and A. D. Dimarogonas, “Dynamic sensitivity of structures to cracks”, J. Vib. Acoustics, Stress Reliubility in Design, Vol. 111, pp. 25 l-256, 1989. M. Akgun and F. D. Ju, “Diagnosis of multiple cracks on a beam structure”, ht. J. Analyt. Exp. Modal Analysis, Vol. 2, No. 4. pp. 149-154, 1987. construction error N. Stubbs, T. H. Broome and R. Osequeda, “Nondestructive detection in large space structures”, AIAA JI, Vol. 28, No. 1, pp. 146152, 1990. R. Y. Liang, F. K. Choy and J. Hu, “Detection of cracks in beam structures using measurements of natural frequencies”, J. Franklin Inst., Vol. 328, No. 4, pp. 505518, 1991. R. Y. Liang, J. Hu and F. K. Choy, “A theoretical study of crack-induced eigenfrequency changes in beams”, J. Engng Mech., ASCE, Vol. 118, No. 2, pp. 384 396. 1992. R. D. Adams, P. Cawley, C. J. Pye and B. J. Stone, “A vibration technique for nondestructively assessing the integrity of structures”, J. Me&. Engng Sci., Vol. 20. No. 2, pp. 933100, 1978. E. DiPasquale, J. W. Ju, A. Askar and A. S. Cakmak, “Relation between global damage indices and local stiffness degradation”, J. Struct. Engng, Vol. 116, No. 5, pp. 144&1456, 1990. R. Y. Liang, J. Hu and F. K. Choy, “A quantitative NDE technique for assessing damage in beam structures”, J. Engng Mech., ASCE. Vol. 118, No. 7, pp. 14681487, 1992. P. F. Rizos and N. Aspragathos, “Identification of crack location and magnitude in a cantilever beam from the vibration modes”, J. Sound Vibv., Vol. 138, No. 3, pp. 381-388, 1990.

Received : 15 January 1993 Accepted : 29 March 1993

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