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VIBRATION ANALYSIS ON A

COMPOSITE BEAM TO IDENTIFY

DAMAGE AND DAMAGE SEVERITY

USING FINITE ELEMENT METHOD

E.V.V.Ramanamurthy1, K.Chandrasekaran2

1

Research scholar, JNTU, Hyderabad 2

Dean, R.M.K.Engineering college

ABSTRACT

The objective of this paper is to develop a damage detection method in a composite cantilever beam

with an edge crack has been studied using finite element method. A number of analytical, numerical and

experimental techniques are available for the study of damage identification in beams.Studies were carried out for

three different types of analysis on a composite cantilever beam with an edge crack as damage. The material used in this

analysis is glass-epoxy composite material. The finite element formulation was carried out in the analysis section of the

package, known as ANSYS. The types of vibration analysis studied on a composite beam are Modal, Harmonic and

Transient analysis. The crack is modeled such that the cantilever beam is replaced with two intact beams with the crack

as additional boundary condition. Damage algorithms are used to identify and locate the damage. Damage index method

is also used to find the severity of the damage. The results obtained from modal analysis were compared with the

transient analysis results.The vibration-based damage detection methods are based on the fact that changes of

physical properties (stiffness, mass and damping) due to damage will manifest themselves as changes in the

structural modal parameters (natural frequencies, mode shapes and modal damping). The task is then to monitor

the selected indicators derived from modal parameters to distinguish between undamaged and damaged states.

However, the quantitative changes of global modal parameters are not sufficiently sensitive to a local

damage. The proposed approach, on the other hand, interprets the dynamic changes caused by damage in a

different way. Although the basis for vibration-based damage detection appears intuitive, the implementation in

real structures may encounter many significant challenges. The most fundamental issue is the fact that damage

typically is a local phenomenon and may not dramatically influence the global dynamic response of a structure.

For example, it is a common view that the frequency reduction has significant limitations for damage indication

although the sensitivity varies from one structure to another. Therefore, damage may be difficult to detect until it

becomes significantly severe.

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1. INTRODUCTION

Structural damage can be broadly defined as any geometrical deviation or material property change

that may degrade the integrity of the structure or result in unintentional dynamic response of the structure.

Damage may be formed at different stages of a structure’s service life. Major concerns in the operation of

in-service structures are the reliability of the structure and the cost associated with the maintenance program.

Unpredictable occurrence of damage may cause catastrophic failure and hereby possesses a potential threat to

human lives. The importance of damage detection has led to continuous efforts to develop more efficient and

rigorous technologies. It is highly desirable to pursue effective engineering solutions to detect, locate, quantify and

predict the damage situation at the earliest possible stage. In an effort to address some of these issues, the

present study attempts to develop a methodology based on coupled response measurements. This phenomenon

only appears when the structural member contains damage. This implies that one can monitor the coupled

response to determine the existence of damage. Through vibration analysis, it is shown that one possible

method to observe the coupling property is by frequency response function plots. The new resonance peaks on

the plots are correlated to the coupled modes. Hence, the emergence of these peaks is the indication of damage.

There are several reasons for the majority of the attention given to vibration-based damage detection

methods. For instance, the modal parameters contain both global and local information of the structural

dynamic properties. The global nature of the parameters (such as natural frequencies) implies that tests can

be conducted at virtually arbitrary points. A priori knowledge of the damage location is not necessary. The

availability of high-speed portable computers and data acquisition systems has made it feasible to integrate

vibration monitoring into the routine maintenance program at a relatively low cost.

Accurate mode shapes, which are difficult to obtain, can be used for damage identification but such an

approach requires high spatial resolution. A large number of transducers are needed to obtain meaningful mode

shapes. Furthermore, the full-scale modal analysis is not an easy task, especially for large structures with

high-density modes.

Due to its importance, damage detection has attracted considerable attention from the engineering

research community in the past several decades. For structures that require high reliability such as aerospace

vehicles, transportation systems, offshore platforms and various civil structures, etc., it is vital to maintain

systems at an optimally healthy condition. T h i s was normally achieved through localised point-by-point

inspection techniques. Many local damage detection methods have been developed and are routinely

applied to a variety of structures.

The continuous demand for effective, fast and economical damage detection methods has led to the

development of vibration-based approaches that examine changes in the vibration characteristics of the structure.

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considered in these approaches. The modal parameters are related to the physical properties of a structure:

mass, damping and stiffness. The changes in the physical properties due to damage can then be expected to

manifest themselves as changes in modal parameters. During the last several decades, extensive analytical,

numerical and experimental investigations have been conducted for various applications.

In general, vibration-based damage detection methods can be divided into non-model and model-based

approaches. Non-model methods detect damage via the direct use of such quantities as natural frequencies,

mode shapes, damping ratios, stiffness and flexibility matrices or information derived from these quantities.

Model-based methods use a pre-selected set of parameters to define the model of the structure under

investigation and the assumed damage mechanisms. The damage state of the structure is then determined by

means of changes in the values of these parameters relating to the model. Typically, finite element models are

often used in such occasions.

A prevalent method is based on identifying changes in modal frequencies resulting from local stiffness

changes or mass loss.

Lifshitz and Rotem [10] (1969) proposed vibration measurements as a damage detection method. They

looked at the change in the dynamic moduli, which could be related to the frequency shift, as indicating damage in

particle-filled elastomers.

Cawley and Adams [2] (1979) gave a formulation to detect damage in composite materials from

frequency shifts. An error term was constructed that relates the measured frequency shifts to those

predicted by a model based on a local stiffness reduction. Special considerations were given to the

anisotropic behaviour of the composite materials.

Chondros and Dimarogonas [4](1980) investigated the influence of a crack in welded joints. They

modelled a cantilever beam with a mass at the free end and a crack at the welded root. The change of the

bending stiffness due to the crack was measured experimentally and then used in the mathematical model for

other boundary conditions.

Dimarogonas and Massouros [5](1981) studied the torsional vibration of a shaft with a circumferential

crack and modelled the cracked section as a torsional spring. The stiffness change due to a crack was also

modelled as reduction in the local elastic modulus by some researchers.

Gudmundson [7] (1982) presented a first order perturbation method which predicts changes in natural

frequencies of a structure resulting from cracks, notches or other geometrical changes. The method was applied

to an edge-cracked rectangular beam.

Yuen [25](1985) reported a finite element modelling technique to establish the relationship between

damage and the change of modal parameters when a uniform cross-sectional cantilever beam was subject to

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was shown that the changes of the eigenparameters demonstrated a definite trend in relation to the location and

extent of damage.

Ju and Mimovich [9] (1988) used changes in natural frequencies to locate damage at sections of a

beam. The damage was modelled as a “fracture hinge” and the built-in end of the beam was represented by a

torsional spring. In addition to the existence of the damage, the authors predicted the location of the damage

within the accuracy of 1% of the beam length.

Gounaris and Dimarogonas [ 6 ] (1988) investigated a cracked prismatic beam element for structural

analysis. Similar to above discussed local flexibility matrix, the relationship between the displacement and force

vectors was formulated with respect to the element containing the crack. One could then combine this flexibility

matrix with any finite element program for further analysis.

Rizos, et al. [18](1990) developed a method of measuring the amplitude of a steel beam at two points

during forced vibration at one of its natural frequencies. Analytical results were used to relate the measured

vibration modes to the crack length and depth. The identification method was based on the assumption of a

transverse surface crack, extending uniformly along the width of the structure. Mode shapes were measured

by using two calibrated accelerometers mounted on the beam. With the measurement points confined within

tenth of beam length from the location of maximum amplitude at each mode, the maximum error in location and

severity of the crack was no large than 8% and in most cases less than 5%.

Pandey, et al. [ 1 4 ] (1991) introduced the curvature mode shapes as a possible candidate to detect and

locate damage in a structure. By using a cantilever and a simply supported analytical beam model, it was

demonstrated that the absolute changes in mode shape curvature were localised in the region of damage and

hence can be a good indicator of damage in the structure. Finite element analysis was used to obtain the

displacement mode shapes of the two models. The curvature values were computed from the displacement mode

shapes using a central difference approximation.

Armon, et al. [ 1 ] (1994) used the idea of rank-ordering of the modes according to the fractional

reduction of natural frequencies. It was shown both by simulation and experiment that the rank-ordering of

the modes based on frequency shifts was useful for determining the crack location. The method claimed to be

not sensitive to small uncertainties in the boundary conditions and changes in the surrounding temperature.

Pandey and Biswas [13] (1994) studied the flexibility approach based on the computation of changes in

the flexibility matrix of the structure using experimentally obtained modal parameters. It was shown that the

flexibility of a structure converged rapidly with increasing frequency and a good estimate of the flexibility

matrix was obtained from just the first two modes. A beam under three types of boundary conditions was

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amount of damage. The author stated that the proposed method worked best when damage is located at a

section where high bending moments occur.

Park, et al. [ 1 5 ] (1998) compared three methods to obtain the flexibility matrices in their

investigation on structural damage: a free-free substructural flexibility method, a deformation-based

flexibility methods and a strain-based flexibility method. These methods were used on a ten-story building, a

bridge and an engine structure. The results suggested that these techniques could correctly identify the damage

locations.

Vestroni and Capecchi [23](2000) represented damage by a more or less concentrated decrease in

stiffness. Attention was focused on the basic aspects of the problem, by discussing the amount of

frequencies necessary to locate and quantify the damage uniquely.

Xia and Hao[[24] (2003) developed a statistical damage identification algorithm based on frequency

changes to account for the effects of random noise in both the vibration data and finite element model. The

structural stiffness parameters in the intact state and damaged state were, respectively, derived with a two-stage

model updating process. The technique was applied to detect damage in a numerical cantilever beam and a

laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and

damage level on damage identification results were also discussed.

Maia,et al. [11](2003) suggested a new version of mode shape-based method by generalising them

to the whole frequency ranges of measurement. A series of numerical simulations on a simple beam were made in

order to compare various damage detection methods based on mode shape changes. The experimental results

showed a certain degree of agreement.

Parloo, et al. [16](2003) presented, a damage assessment method based on the use of mode shape

sensitivities.The sensitivities were calculated on the basis of the experimentally determined mode shapes. A

comparison with other damage localisation techniques showed that the sensitivity-based approach could localised

the damage in an earlier state than the other techniques. One of the advantages claimed by the author was that

there was no need for a prior finite element model of the test structure.

Cerri and Ruta [ 3 ] (2004) investigated the detection of the structural damage affecting a narrow

zone of a doubly hinged plane circular arch by means of a few measured natural frequencies. The localised

damage was modelled as a torsion spring joining two adjacent sections and characterised by the location

and the stiffness of the spring. Two different procedures to identify the damage parameters were introduced:

the first one was based on the search of an intersection point of curves obtained by the modal equation;

the second is based on the comparison between the analytical and experimental values of the

variation of frequencies passing from the undamaged to the damaged state. The identification of damage

parameters was satisfactory in both techniques based on the pseudo-experimental data.

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Shang, et al. [20](2003)) have attempted to detect and localise damage by using changes in natural

frequencies. An intensive review of literature on this category was made by Salawu [ 1 9 ] (1997). The most

appealing feature associated with using natural frequencies is that natural frequencies are relatively cheap to

obtain and easy to extract. The frequency measurement can be quickly conducted and confident accuracy is

often achievable. Another advantage is its global nature of the identified frequencies; thus allowing the

measurement points to be customised at will.

However, it is a common view that the frequency shift is not sensitive to the local damage. Significant

damage may cause very small changes in natural frequencies, particularly for large structures, and these

changes may be undetectable due to measurement or processing errors. Also, the natural frequency changes

alone may not be sufficient for a unique localisation of structural damage. This is because cracks associated

with similar crack lengths but at two different locations may cause the same amount of frequency change. Other

factors such as boundary conditions can also lead to frequency shift but without damage. Variations in the

mass of the structure or measurement temperatures may introduce uncertainties in the measured

frequency changes. Currently, using frequency shifts to detect damage appears to be more practical in applications

Used in conjunction with natural frequencies, mode shape approach has also received considerable

attention in the literature. Deterioration of a structure can alter local stiffness and cause discontinuity near

the region. The consequent change of mode shapes depends on both the severity and the location of the

damage. The magnitude of change with respect to each mode may vary from one to another. This can be used as

a tool to predict the location of the damage.

However, there are some drawbacks on using mode shape methods. First, it is not an easy task to

obtain the meaningful mode shapes, especially for large structures. Confident identification of the damage

requires proper number of sensors and right choice of sensor coordinates. Another concern is that damage is a

local phenomenon and may not significantly influence mode shapes of the lower modes that are usually

measured from vibration tests of large structures. Also, extracted mode shapes were affected by environmental

noises from such sources as ambient loads or inconsistent sensor locations.

An alternative to using mode shapes to obtain spatial information about vibration changes is

using mode shape derivatives, such as curvature. The curvature can be obtained by calculation from

displacement modal shapes or measuring the curvatures/strain directly.

The presence of a crack in a structure reduces the stiffness of the structure but it is not practical to

measure this change directly. Some researchers used modal parameters to estimate changes in the static

flexibility of the structure. Theoretically, the synthesis of the complete static flexibility matrix would require the

measurement of all of the mode shapes and frequencies.

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Element (FE) model updating techniques. The basic idea is to modify the objective parameters in a base FE

model so that the model predictions match the measured data in an optimal way. Two steps are often involved

in the damage detection process. The first step is to produce a validated finite element model of the undamaged

state as a baseline. Then the model is updated to obtain a model in damaged state matching the measured

vibration data at the damaged state. Comparisons of the updated matrices to the baseline model provide an

indication of the damage and can be used to quantify the location and extent of damage.

The drive to use vibration-based methods for damage detection is the fact that the presence of a

crack or localised damage in a structure results in the stiffness reduction, mass loss and damping increase,

which affects the dynamic characteristics of the structure. How to quantify such effects is an important

aspect of the research for the successful implementation of these methods. In general, the stiffness change

caused by a crack is more significant than the influence to mass and damping. Hence, large proportions of

the modelling techniques are related to the stiffness (or flexibility). On the other hand, the loss of the mass

due to damage was treated, in most cases, as negligible.

From literature, the vibration-based damage detection methods have received considerable attentions.

Although the basic idea is very similar, there are many different ways to monitor the dynamic change of the

structure. It is natural that some methods have their own area of applications, with advantages and

disadvantages. Each of these methods has its particular benefits making one more suitable in certain applications

than the others. There are constant demands for the investigation of more effective damage detection techniques.

The sensitivity problem is the most common obstacles for the success of the techniques. Another issue is the

dependence on a very accurate analytical model. Obtaining an accurate model in itself is not an easy task.

Even when the analytical model is present, to predict and locate the damage, these methods require excessive

post-processing after data collection.To investigate the alternative methods, it may be possible to study the local

member instead of the whole structure and observe the change of the behaviour rather than just quantitative

change of the parameters. This philosophy is the motivation for the current studies.

2. THE FINITE ELEMENT ANALYSIS

L

1

L W

L

Figure (1) Geometry of cantilever beam with an edge crack

a

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Length = L = 400mm; Width = W = 30 mm; Thickness = t =10 mm; Number of layers = n = 10; Fiber orientation = θ = 45 º; L1 = Damage location from fixed end ;a = Depth of crack at L1

(crack is considered as damage); Properties of Glass-Epoxy composite material for the analysis:

Young’s modulus of fiber = E f = 72.4 GPa; Young’s modulus of matrix = E m = 3.45 GPa; Modulus of rigidity of fiber = G f = 29.67 GPa; Modulus of rigidity of matrix = G m = 1.277 GPa; Poisson’s ratio = f = 0.22;

Poisson’s ratio = m = 0.35; Density of fiber = ρf = 2.6 gr/cm3 ; Density of matrix = ρm = 0.33 gr/cm3 ;

The local parameters like bending and torsional mode deviation and strain mode changes

have insignificant contribution or highly localized contribution which cannot be practically measured

to identify the location of the crack. Therefore, an attempt is made to look into the global parameters,

like frequencies and the change in frequencies, to identify the presence of a crack.

A cantilever composite beam was considered for the numerical studies using ANSYS software. The

geometrical dimensions are shown in figure (1). Only the f irst three bending modes was extracted which

were adequate to identify the crack location. The analysis was carried out for two damage cases.

Damage case 1: L1 = Damage location from fixed end =80 mm; a = Depth of crack at L1 = 6 mm;

Damage case 2: L1 = Damage location from fixed end =80 mm; a = Depth of crack at L1 = 9 mm;

The general purpose finite element computer software ANSYS was used to carry out the

finite element analysis in the frequency domain and obtain natural frequencies, mode shapes and transfer

functions.

An eight node linear layered 3-D shell element with six degrees of freedom at each node (Specified as

Shell 99 element in ANSYS) was selected based on convergence test and used throughout the study. Each node

has six degrees of freedom, making a total forty eight degrees of freedom per element. The Figure (2) represents

the detail and Geometry of element type. The figure (3) represents the layer stacking by using ANSYS

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Figure 3. Layer stacking in ANSYS

A Cantilever composite beam with the above geometry and material properties was considered for the

numerical studies by using ANSYS software. The location of the crack and depth of crack were considered for

two damage cases. First three bending modes were extracted and analyzed. From the numerical analysis, the

mode shape displacements are calculated at these frequencies.

Numerical modal analysis based on the finite element (FE) modeling is performed for studying the

dynamic response of a structure. The natural frequencies and mode shapes are important modal parameters in

designing a structure under dynamic loading conditions. The numerical analysis is carried out by using the

commercial finite element program ANSYS. It is mainly used to verify the effectiveness of damage detection

algorithms used in this study.

In the present study, since the specimen is made of fiberglass/epoxy composite material, the FE

modeling of the composite beam is simulated with the layered element (Shell 99). The composite beam consists

of several orthotropic layers, and it is considered as orthotropic material. The FE analysis software ANSYS was

used in the modal analysis to obtain mode shapes. Only the growth of crack through the thickness is modeled

and also, the crack opening and closing has not been explicitly considered in the analysis. Hence the process has

been taken to be linear.

3. IDENTIFICATION OF CRACK LOCATION

The presence of the crack introduces extra peaks into the FRF plots. Comparing Figures

4(a),4(b) and 4(c)for the first damage case, similarly 7(a),7(b) and 7(c) for the second damage case. It observes

that the presence of the crack influences the FRFs in two ways:

(i) All natural frequencies are reduced because of loss of stiffness at the crack location; and

(ii) An extra peak is introduced as noted on the plot. The natural frequency corresponding to the

new peak is close to the undamaged axial natural frequency. This indicates the coupling

of lateral and axial vibration.

A crack may significantly change the system’s dynamic behaviour. One of the manifestations is the

appearance of extra new peaks on the lateral FRF plots. To use this property to detect damage in field

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normally contaminated with noise it is necessary to demonstrate that the accuracy of FRF obtained through

normal vibration tests is readily achievable for this purpose.

Analytical modal analysis was carried out to determine the global changes of modal parameters that

occur due to increase of crack depth. The Frequency range was chosen to vary from 0 to 2000 Hz, which include

the first seven frequencies.

The identified natural frequencies are listed in Table (1).

It can be seen that:

• The frequencies of the extra new mode in both cracked cases are very close to that of the axial

mode.

• Although the frequencies are reduced by some degree because of the loss of stiffness at the crack

location, the change on lower modes are not sufficiently sensitive to the crack. For complex structures, this

frequency shift could be hard to notice. This reflects the weakness of previously mentioned approaches that aim

to detect cracks by observing the reduction in natural frequencies.

Table1. Identified Natural Frequencies (Hz)

Lateral Mode No. 1 2 3 4

New Peak

5

Un Damage 64 388 1045 1947

Damage Case 1

(Crack depth = 6 mm)

62 388 1036 1303 1914

Damage Case 1

(Crack depth = 9 mm)

60 388 1025 1281 1880

4. EVALUATION OF DAMAGE DETECTION

This detection technique is extended in the present study, and the amplitude difference of curvature mode

shapes is employed to estimate the local stiffness loss due to damage. The first curvature mode is used in the

damage assessment, since the curvature nodal points of damaged structures at high modes may shift

significantly from the original undamaged case thus generating misleading results. It is relatively difficult to

experimentally obtain the mode shapes at the higher modes using surface-bonded sensors; while at the lower

modes, the mode shapes may not be sensitive to small damage. In this study, the first three low curvature mode

shapes of the structures between an undamaged state and a damaged state are extracted from the finite element

analysis data. Then, the differences of these curvature modes are used to determine damage locations and the

corresponding magnitudes. The procedure illustrates the damage state from the original undamaged state;

however, it can also be modified to consider the changes from one damaged state to another damaged state. The

study is conducted by using Glass / Epoxy laminated composite beam, and the curvature mode shapes are

measured by using FEM software ANSYS. The results of damage identification by using damage index

method for the Glass / Epoxy laminated beam are presented.

A study is made to compare the change in frequencies, obtained due to the presence of a crack in a

composite beam with an undamaged beam. The general purpose FEM analysis software ANSYS was used in the

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The following Damage detection algorithms were used to locate the damage in the composite

beam buy using FE analysis. The modal parameters that change locally are mode shapes, i.e., Bending mode

shape

1. The Curvature Mode Shape Method,

2. The Damage Index Method,

3. The Gapped Smoothing Method.

4. The Laplacian operator Method

The above methods are used in the evaluation of damage in the composite beam for bending and

torsion mode shapes. The results obtained were tabulated, plotted and discussed in the next section

1. The Curvature Mode Shape :

Pandey, et al. (14) Stated that, the mode shapes of a damaged and the corresponding undamaged

structure are identified, the curvature at each location i on the structure is numerically obtained by central

difference approximation.

(1)

2

/

)

j

1),

-(

2

-1),

(

(

''

h

i

ij

j

i

ij

φ

φ

φ

φ

=

+

+

Where, i is the node number, j is the mode shape number and h is the distance between the

measurement points i + 1 and i - 1.

The location of the damage is then identified by the largest computed absolute difference

between the mode shape curvatures of the damaged and undamaged structure, as follows

{ } { } {

}

ly.

respective

structure

undamaged

and

damaged

of

ion

approximat

diffrence

central

the

is

ud

ij,

φ

and

d

ij,

φ

where

(2)

''

ud

ij,

φ

-''

d

ij,

φ

''

ij

φ

Δ

=

2. The Damage Index Method:

Stubbs and Kim (21) stated that, the damage identification based on the changes in the

curvature of the j th mode at location i. The damage parameter formulation of this method is presented as

{ }

{ }

{ }

{ }

{ }

{ }

(3)

max 1 2 , " max 1 , " , 2 " max 1 2 " max 1 , 2 " , 2 "

+

+

=

i j i d i j i ud j i ud i ud i j i d j i d ij

φ

φ

φ

φ

φ

φ

β

where βij is the Damaged Index at location i for mode j.

3. The Gapped Smoothing Method:

Ratcliff and Bagaria (17) proposed the gapped smoothing method for the damage detection in

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shape of the undamaged structure has a smooth curve and it can be approximated by a polynomial in one variable. (4) ) ( 0

= − = n i i n i x a x U

The damage parameter based on this method is defined as the square of the difference between

measured data of the damaged structure and the smoothed fitted value.

The maximum value of this method indicates the damage location. This method does not

require an undamaged structure reference. The procedure mainly depends on the measured mode shape obtained

from damaged structure. The main advantage of this method is, that it can be applied to an existing structure

where there is no prior data of its undamaged structure.

The displacement mode shape of a damaged structure is to be converted to curvature shape by

using central difference approximation. Then using a gaped polynomial at each point to generate mode shapes,

which representing the healthy beam, locally smoothes the curvature mode. The Damage index by this method

was defined as the difference between the curvature and the gapped smooth polynomial at each point. The

largest index represents the location of the damage.

Damage index = D GSM(x) = {(U measured (x) – Ufitting (x)}2 (5)

4 . Laplacian Method:

Cracks and other forms of localized damage in a beam can introduce a reduction in the

flexural stiffness (EI), but minimal change to the mass. Localized changes to EI result in a mode shape that

has a localized change in slope. Experimental mode shape data are discrete in space, and therefore the change

in slope can be estimated using a finite difference approximation. The Laplacian difference equation is a

common method used to calculate an estimate of the second difference of a discrete function.

A beam can be analyzed as a dimensional structure, and in this case the

one-dimensional Laplacian, Li , of the discrete mode shape, yi , is given by

Li = ( yi+1 + yi-1) – 2yi (6)

The Laplacian has a similar shape and identifies damage in a similar fashion to the curvature shapes in

reference [13]. The main difference is that reference [14] considers the difference in curvature between

undamaged and damaged beams, whereas this study only considers the damaged model.

Various methods of enhancing the discontinuity in the graph, such as piecewise linearization and

cubic spline, were tried. The method that was most effective was to fit a cubic polynomial to the

Laplacian, and calculate a difference function between the cubic and Laplacian. A separate cubic was

determined for each element of the Laplacian in turn, with the coefficients being determined from

the data on either side of the element, but excluding the actual element. For example, the cubic calculated for the

ith element of the Laplacian, Li , at position i

x

along the beam, was defined as

) 7 ( 3 3 2 2 1

0 + a xi + a xi + a xi − − −

a

The coefficients a 0 , a 1 ,a 2 and a 3 were determined using Laplacian elements

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The difference function, δi was calculated from the cubic and the Laplacian:

(

i i i i

)

i

i

=

a

+

a

x

+

a

x

+

a

x

L

2 3 2 2 0

δ

---(9)

The above four damage algorithms are used in the evaluation of damage in the composite beam. The results

obtained were plotted and discussed in the next section.To establish a comparative basis for different modes, the

damage index values is transformed into the normal standard normal space, and the normalized damage

indicator Zij is given by,

estimator. severity ij Where (11) ij 1 -1 ij by, d isexpresse j element for severity damage the of estimation The elements. j all for values ij β the of deviation standard the βij σ and mean the being βij with (10) ij ij -ij damage is ij Z

α

β

α

β

σ

β

µ

β

= =

Positive Zij and αij values indicate the possibility of damage and can therefore be utilized to locate and

quantify defects, respectively.

CASE 1: Damage location from fixed end = L1= 80 mm;

Crack depth = a =6 mm;

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(15)
(16)

TRANSIENT ANALYSIS:

(17)
(18)

CASE 2: Damage location from fixed end = L1= 80 mm;

Crack depth = a = 9 mm;

(19)
(20)

TRANSIENT ANALYSIS:

(21)
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5. RESULTS AND DISCUSSIONS

A cantilever composite beam was considered for the numerical studies using ANSYS software. The

geometrical dimensions are shown in figure (1). Only the f irst three bending modes was extracted which

were adequate to identify the crack location. The analysis was carried out for two damage cases.

Damage case 1: L1 = Damage location from fixed end =80 mm; a = Depth of crack at L1 = 6 mm;

Damage case 2: L1 = Damage location from fixed end =80 mm; a = Depth of crack at L1 = 9 mm;

Initially the harmonic analysis was performed on the composite cantilever undamaged and damaged beam

for the above geometry and material properties. The figure 4(a) represents frequency Vs amplitude for the first

four bending modes of undamaged and damaged beam. A new peak mode was observed between third and

fourth modes of vibration. The figure 4(b) represents frequency Vs phase angle and the new peak was also

observed between third and fourth modes of vibration. The figure 4(c) represents the frequency Vs amplitude

between third and fourth modes with a new peak mode. The new peak mode was observed in the damaged

beam for both the damage cases.

Then the modal analysis was carried out on the same composite cantilever beam. The modal

displacements were calculated and by using these values the damage was identified as damage indicator by

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case. From the diagrams the positive value of Z i indicate the damage location by applying four damage algorithms.

The transient analysis was performed on the same geometry for the undamaged and damaged beam. The

figure 6(a) represents time Vs amplitude. It was observed that at t / T = 0.68 , the amplitude was maximum

for the undamaged and damaged beam. Based on the fundamental frequency, the time period (T) was

calculated and applied during transient analysis. From the analysis, the displacement vectors were calculated

and these vectors are used in damage algorithms to find the damage location. The figures 6(b) to 6 (e)

represent the damage location for the first damage case. From the diagrams the positive value of Z i indicate the damage location by applying four damage algorithms.

The same procedure was repeated for the second damage case and the results were plotted. The figures

7(a) to7(c) represents the harmonic analysis. A new peak mode was observed from this analysis. Then the

modal analysis was carried out on the same cantilever beam and the results were plotted .The figures from 8

(a) to 8(d) represents the damage indicators. Finally transient analysis was carried out and the results were

plotted. The figures 9(a) represents the plot between time Vs amplitude. Finally the figures 9(b) to 9(e)

indicate the damage indicators.

DAMAGE SEVERITY:

The damage severity was calculated by using damage index method. The figures 5(e) and 6(f) represent

the damage severity for modal analysis and transient analysis respectively during the first damage case.

Similarly the figures 8(e) and 9(f) represent the damage severity (αi ) for modal analysis and transient analysis respectively during the second damage case.

6. CONCLUSION

It was concluded that by performing FEM analysis on a composite cantilever beam, the damage

identification and damage severity can be found. As the depth of crack increases, the increasing value of

damage severity was observed. The damage algorithms are useful for the identification of location of the

damage and the intensity of the damage as the crack depth increases.

References

[1] Armon, D., Benhaim, Y. and Braun, S. (1994). "Crack detection in beams by rank-ordering of eigenfrequency shifts",

Mechanical Systems and Signal Processing, Vol. 8, pp. 81-91.

[2] Cawley, P. and Adams, R. D. (1979). "Location of defects in structures from measurementsof natural frequencies", Journal of Strain Analysis for Engineering Design, Vol. 14, pp. 49-57.

[3] Cerri, M. N. and Ruta, G. C. (2004). "Detection of localised damage in plane circular arches by frequency data", Journal of Sound and Vibration, Vol. 270, pp. 39-59.

[4] Chondros, T. G. and Dimarogonas, A. D. (1980). "Identification of cracks in welded-joints of complex structures", Journal of Sound and Vibration, Vol. 69, pp. 531-538.

[5] Dimarogonas, A. and Massouros, G. (1981). "Torsional vibration of a shaft with a circumferential crack", Engineering Fracture Mechanics, Vol. 15, pp. 439-444.

[6] Gounaris, G. and Dimarogonas, A. (1988). "A finite-element of a cracked prismatic beam for structural-analysis", Computers & Structures, Vol. 28, pp.309-313.

[7] Gudmundson, P. (1982). "Eigenfrequency changes of structures due to cracks, notches or other geometrical changes", Journal of the Mechanics and Physics of Solids, Vol. 30, pp. 339-353.

[8] Hearn, G. and Testa, R. B. (1991). "Modal-analysis for damage detection in structures", Journal of Structural Engineering-Asce, Vol. 117, pp. 3042-3063.

[9] Ju, F. D. and Mimovich, M. E. (1988). "Experimental diagnosis of fracture damage in structures by the modal frequency method", Journal of Vibration Acoustics Stress and Reliability in Design-Transactions of the Asme, Vol. 110, pp. 456-463. [10] Lifshitz, J. M. and Rotem, A. (1969). "Determination of reinforcement unbonding of composites by a vibration

technique", Journal of Composite Materials, Vol. 3, pp. 412-423.

[11] Maia, N. M. M., Silva, J. M. M., Almas, E. A. M. and Sampaio, R. P. C. (2003)."Damage detection in structures: From mode shape to frequency response function methods",

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[12] Narkis, Y. (1994). "Identification of crack location in vibrating simply supported beams", Journal of Sound and Vibration, Vol. 172, pp. 549-558.

[13] Pandey, A. K. and Biswas, M. (1994). "Damage detection in structures using changes in flexibility", Journal of Sound and Vibration, Vol. 169, pp. 3-17.

[14] Pandey, A. K., Biswas, M. and Samman, M. M. (1991). "Damage detection from changes in curvature mode shapes", Journal of Sound and Vibration, Vol. 145, pp. 321-332.

[15] Park, K. C., Reich, G. W. and Alvin, K. F. (1998). "Structural damage detection using localized flexibilities", Journal of Intelligent Material Systems and Structures, Vol. 9, pp. 911-919.

[16] Parloo, E., Guillaume, P. and Van Overmeire, M. (2003). "Damage assessment using mode shape sensitivities", Mechanical Systems and Signal Processing, Vol. 17, pp. 499-518.

[17] Ratcliffe, C. P. and Bagaria, W. J. (1998). Vibration technique for locating delamination in a composite plates. AIAA Journal, 36, 1074-1077.

[18] Rizos, P. F., Aspragathos, N. and Dimarogonas, A. D. (1990). "Identification of crack location and magnitude in a cantilever beam from the vibration modes", Journal of Sound and Vibration, Vol. 138, pp. 381-388.

[19] Salawu, O. S. (1997). "Detection of structural damage through changes in frequency: A review", Engineering Structures, Vol. 19, pp. 718-723.

[20] Shang, D. G., Barkey, M. E., Wang, Y. and Lim, T. C. (2003). "Effect of fatigue damage on the dynamic response frequency of spot-welded joints", International Journal of Fatigue, Vol. 25, pp. 311-316.

[21] Stubb, N. and Kim, J.T. 1996. ‘‘Damage Localization in Structures without Baseline Modal Parameter,’’ AIAA Journal,34:1644–1649.

[22] Tracy, J. J. and Pardoen, G. C. (1989). "Effect of delamination on the natural frequencies of composite laminates", Journal of Composite Materials, Vol. 23, pp. 1200-1215.

[23] Vestroni, F. and Capecchi, D. (2000). "Damage detection in beam structures based on frequency measurements", Journal of Engineering Mechanics-Asce, Vol. 126, pp. 761-768.

[24] Xia, Y. and Hao, H. (2003). "Statistical damage identification of structures with frequency changes", Journal of Sound and Vibration, Vol. 263, pp. 853-870.

Referências

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