A Continuous Vibration Theory for Beams with a Vertical...

Transaction B: Mechanical EngineeringVol. 17, No. 3, pp. 194{204c Sharif University of Technology, June 2010

A Continuous Vibration Theory forBeams with a Vertical Edge Crack

M. Behzad1;�, A. Ebrahimi1 and A. Meghdari1

Abstract. In this paper, a continuous model for exural vibration of beams with an edge crackperpendicular to the neutral plane has been developed. The model assumes that the displacement �eldis a superposition of the classical Euler-Bernoulli beam's displacement and of a displacement due tothe crack. The additional displacement is assumed to be a product between a function of time and anexponential function of space. The unknown functions and parameters are determined based on the zerostress conditions at the crack faces and the concept of J-integral from fracture mechanics. The governingequation of motion for the beam has been obtained using the Hamilton principle and solved using a modi�edGalerkin method. The results have been compared with �nite element results and an excellent agreementis observed.

Keywords: Vibration; Cracked beam; Vertical crack; J-integral.

INTRODUCTION

Fatigue and crack initiation and propagation in struc-tures and machinery subjected to dynamic loading areone of the main concerns of designers and users. An un-controlled crack can lead to a catastrophic failure undercertain conditions. The importance of early detectionof cracks makes researchers study various aspects ofthe behavior of structures defected by cracks. Oneof these aspects is the vibration of cracked structures.Crack creation and development in a system changesthe dynamic and vibration behavior of that system.With measurement and analysis of these vibrations, thecracks can be identi�ed well in advance and appropriateactions can be taken to prevent more damage to thesystem.

The vibration behavior of cracked structures hasbeen investigated by many researchers. Dimaragonaspresented a review on the topic of the vibration ofcracked structures [1]. His review contains vibration ofcracked rotors, bars, beams, plates, pipes, blades andshells. Two more literature reviews are also availableon the dynamic behavior of cracked rotors by Wauerand Gasch [2,3].

Beams are important elements in structures and

1. School of Mechanical Engineering, Sharif University of Tech-nology, Tehran, P.O. Box 11155-9567, Iran.

*. Corresponding author. E-mail: m [email protected].

Received 18 May 2009; received in revised form 2 February 2010;accepted 6 April 2010

machinery, so the vibration behavior of cracked beamshas been studied by researchers widely. One mayde�ne a crack with its edge parallel or perpendicularto the neutral axis as horizontal or vertical cracks,respectively, as shown in Figure 1.

There exist three methods for the vibration mod-eling of beams with horizontal transverse cracks:

1. Discrete models with a local exibility model forcracks.

2. Continuous models with a local exibility model forcracks.

3. Continuous models with a continuous model for thecrack.

The local exibility model for the crack hasbeen suggested by Dimaragonas for the �rst time [4].He replaced the cracked beam with two undamagedhalf beams connected by a rotational spring. Thesti�ness of this spring is obtained from the conceptof the J-integral in fracture mechanics. Papadopoulospresented a complete literature review on the methodof using the J-integral for �nding the local exibility of

Figure 1. A beam with (a) horizontal edge crack and (b)vertical edge crack under bending.

A Vibration Theory for Beams with Vertical Edge Crack 195

cracks [5]. The local exibility idea has been followedby several researchers till now. Some researchersmodeled two undamaged half beams discretely andadded the exibility of the rotational spring to the exibility matrix of the system [6,7]. Others modeledtwo undamaged half beams continuously and usedappropriate boundary conditions for each part to linkthem through the rotational spring [8,9]. Some otherresearchers have tried to modify and improve the local exibility model of the crack by adding one or twolinear springs besides the rotational one [10]. Thesemethods have also been extended for beams with morethan one crack [11-13]. The local exibility model forthe crack is a simple approach and has a relatively goodresult in �nding the fundamental natural frequencyof a cracked beam. However, this method cannot beimplemented for �nding stress at the crack area underdynamic loads, mode shapes in free vibrations andoperational deformed shapes in forced vibrations.

Another approach to the vibration analysis ofcracked beams is continuous modeling of the crack.Christides and Barr developed a continuous theoryfor the vibration of a uniform Euler-Bernoulli beamcontaining one or more pairs of symmetric cracks [14].They suggested some modi�cations on the familiarstress �eld of an undamaged Euler-Bernoulli beam inorder to consider the crack e�ect. The di�erentialequation of motion and corresponding boundary condi-tions are given as the results. However, in their model,two di�erent and incompatible assumptions have beenmade for displacement and strain �elds. Althoughthe accuracy of the results in �nding the naturalfrequencies is acceptable for some applications, theirmodel is still not reliable for more accurate analysessuch as stress analysis near the crack tip under dynamicloading and mode shape analysis. In addition, theobtained partial di�erential equation is complicatedand dependent on some constants that are unknownand must be calculated by correlating the analyticallyobtained results with those calculated by �nite elementin each case. Several researchers followed the Christidesand Barr approach by modifying their method andgained some improvements [15-19]. However, there stillexists the inconsistency between strain and displace-ment �elds, which causes inaccuracy in the results,especially in mode shapes and stress analysis.

Behzad et al. presented a new continuous theoryfor the bending analysis of a cracked beam [20]. Abilinear displacement �eld has been suggested for thebeam strain and stress calculations and the bendingdi�erential equation has been obtained using equilib-rium equations. The model can predict the load-de ection relation of the beam near or far from thecrack tip accurately and can be also used for stress-strain analysis in a cracked beam. This model is alsoused for the vibration analysis of a cracked beam and

showed an excellent performance in dynamic loadingtoo [21,22]. They used this method also for the forcevibration analysis of beams with a horizontal edgecrack [23].

In all the above approaches, the crack is assumedto be horizontal. This type of crack is more probable tobe created and other forms of crack tend to grow hor-izontally in bending. However, by using a pre-crackedelement with speci�c orientation in a structure, thecrack may become horizontal. The other applicationof this research is in the area of rotor dynamics, wherea cracked beam rotates. Figure 1 shows a beam withhorizontal and vertical cracks.

In this paper, a continuous approach for the ex-ural vibration analysis of a beam with a vertical edgecrack has been presented for the �rst time. The crackis assumed to be an open edge notch and the crackedge is perpendicular to the neutral plane. A quasi-linear displacement �eld has been suggested for thecracked beam and the strain and stress �elds have beencalculated. The di�erential equation of motion of thecracked beam has been obtained using the Hamiltonprinciple. This partial di�erential equation has beensolved with a special numerical algorithm based on theGalerkin projection method. The constants needed inthis model can be obtained using fracture mechanics.The results of this study are compared with the �niteelement results for veri�cation.

CRACK BEHAVIOR ANALYSIS INBENDING AND HYPOTHESES

The basic assumption in the Euler-Bernoulli vibrationtheory for beams is that the plane sections of a beamwhich are perpendicular to the neutral axis remainplane and perpendicular to the neutral axis afterdeformation. In the presence of an edge crack, thisassumption, especially near the vicinity of the crack, isno longer correct.

Behzad et al. suggested that for a horizontalcrack, the crack faces have an additional displacementdue to the absence of the normal stress [20,21]. Theydiscussed that this additional displacement is inheritedby the adjacent area with lesser magnitude, and dissi-pates away with an exponential regime along the beamlength. Consequently, for planes far from the crack tip,this warping will be negligible and the displacement�eld can be assumed linear.

This idea can be followed here for a verticalcrack with some modi�cations. In order to have abetter sense of the exural vibrations in a crackedbeam, a �nite element model has been produced inthis research, and the mid span vertical crack exuralbehavior can be seen in Figure 2. This �nite elementmodel is made using ANSYS software [24]. The crackis modeled as a vertical U-shape notch at the mid-span

196 M. Behzad, A. Ebrahimi and A. Meghdari

as a crack, and a �ne singular mesh is used. Note thatthe grid lines of Figure 2 are only some hypotheticallines that show the deformation �eld of the beam, andthese lines are not referring to the �nite element mesh.

Near the crack area, the plane sections will nolonger remain plane. In fact, the crack faces have anadditional rotation, in comparison with the remainingpart of the section, due to the absence of normal stress.This additional rotation dissipates gradually, while thedistance from the crack tip increases. With a goodapproximation, it can be supposed that each planesection turns into two straight planes after deformation.Each straight plane section turns into two planes withdi�erent slopes, one on the right side and the other onthe left side of the crack edge. The slope di�erencebetween these two planes decreases with distance fromthe crack tip. These two straight planes connectto each other through a nonlinear part near the xz-

Figure 2. Displacement �eld illustration in a beam witha vertical edge crack subject to bending.

plane. Figure 3 shows the coordinate system and theparameter de�nitions, graphically.

In order to �nd the stress, strain and deformationfunctions for a beam with a vertical crack in exuralvibration, a displacement �eld for the beam has beensuggested in this research. In fact, it is assumed thateach plane section turns into two straight planes and anonlinear connector after deformation. In this research,the beam is assumed to be a slender prismatic beamand the crack is considered as an open edge U-shapenotch. The cross section of the beam is assumed tobe symmetric about the y-axis, so the y-axis can beassumed to be the neutral axis in pure bending. Thedisplacements and stresses are supposed to be smalland the crack does not grow. Finally, the material isassumed to be linear elastic.

DISPLACEMENT FIELD DEFINITION

With reference to Figures 2 and 3 and the aboveassumptions, the displacement �eld for a beam with avertical edge crack can be de�ned. It is well known thatthe displacement and stress �elds near the crack tip are3D functions. In this paper, at �rst, a 3-dimensionaldisplacement �eld has been introduced for the beambut, afterwards, the equations are integrated over thecross-section area of the beam and a 1-dimensionalrelation is obtained for the beam vibrations. This

Figure 3. Coordinate system and parameters de�nition.


equation is not an exact equation, but the results ofthis research show the good engineering approximationof this relation.

The crack section consists of two parts: Thecrack face, which is denoted in Figure 2 by Ac, andthe remaining part of the section, which is denotedby Ah in this research. Under pure bending, thehealthy part of the cross section (Ah) rotates aboutits neutral axis, which is coincident with the y-axis inthis research. This planar part remains plane afterrotation and perpendicular to the neutral axis. Thecrack face rotates about the y-axis too, but more thanthe remaining part of the section and, consequently,does not remain perpendicular to the neutral axisdue to the shear stress near the crack tip. Thecrack face can also be assumed to remain plane afterdeformation, except at a small area near the crack tip.The rotation di�erence between the crack face and theremaining part of the section inherits to the adjacentcross sections but, gradually, the magnitude of thisdi�erence decreases. As a side e�ect, deformation ofthe beam along the z-axis is a function of y. In fact,the parts of the beam sections which have more rotationcause more vertical displacement, too. The numericalsimulations con�rm this phenomenon.

Based on the above explanations, the followingdisplacement �eld is introduced for a beam with avertical edge crack in exural vibration:8>>>>>><>>>>>>:

u =

(�z�(x; t) y < 0�z(�(x; t) + (x; t)) y > 0

w =

(w0(x; t) y < 0w0(x; t) + �(x; t) y > 0

v = 0

(1)

In which u; v and w are the displacement componentsalong x; y and z axes. �(x; t) is the rotation of thatpart of the section with y < 0, as shown in Figure 3. (x; t) is the additional rotation of that part of thesection with y>0 and �(x; t) is the additional verticaldisplacement of the beam for y > 0. By assuming thatthe plane sections in y < 0 remain perpendicular to theneutral axis, one has:

�(x; t) =@w0(x; t)

@x: (2)

The additional rotation (x; t) of that part of the planesections with y > 0 has its maximum value at the crackface and decreases gradually with distance from thecrack tip. This additional rotation is a nonlinear andcomplicated variable with respect to x. Here, in thisresearch, an exponential regime has been assumed forfunction (x) along the x-axis as follows:

(x; t) = m(t)e�� jx�xcjb sgn(x� xc): (3)

In Equation 3, m(t) is the magnitude of additional rota-tion of the crack faces, � is a dimensionless exponentialdecay rate, which will be obtained later in this paper,xc is the crack position, b is the depth of the beam andsgn(x�xc) is the sign function, which is �1 for x < xcand +1 for x > xc. The application of a sign functionis due to the fact that the additional rotation functionhas a discontinuity at the crack position and the sign ofits value changes when passing through the crack tip.

In order to �nd the value of m(t), zero normalstress conditions at the crack faces can be used. Thenormal strain function can be found using Equation 1:

"x = u;x

=

8>>>>>><>>>>>>:�zw0;xx y < 0

�z w0;xx �m(t)

�be�� jx�xcjb

+2m(t)e�� jx�xcjb �(x�xc)!

y > 0

(4)

in which �(x� xc) is the Dirac delta function and thesubscript ; x denotes the partial derivative with respectto x. The normal stress at the crack faces where y > 0and x = x+

c or x�c should be zero, so one has:

m(t) =b�w0;xx(xc; t): (5)

The additional displacement, �(x; t), which also de-creases gradually with distance from the crack tip canbe assumed to be a function similar to (x; t) as follows:

�(x; t) = n(t)e�� jx�xcjb ; (6)

where n(t) is the magnitude of additional displacementat the crack face, which can be found using a zero shearstress condition at the crack faces. The shear strainfunction, xz, can be found using Equation 1:

xz =12

(u;z + w;x)

=

(0 y<0��m(t)+n(t)�b

�e�� jx�xcjb sgn(x�xc) y>0

(7)

The shear stress at the crack faces, where y > 0 andx = x+

c or x�c , should be zero so one has:

n(t) = � b�m(t) = � b2

�2w0;xx(xc; t): (8)

To avoid discontinuity and considering the nonlinearityat the crack tip, it is assumed that the displacement�eld at y > 0 transforms into the de�ned functionsin Equation 1 with an exponential regime from the


displacement �eld at y < 0. So, the displacement �eldis modi�ed in this paper as follows:

u =

8>><>>:�zw0;x y < 0

�z (w0;x +b�

�1� e�� yd�

w0;xx(xc)e��jx�xcjb sgn(x� xc)

�y > 0

w =

8>><>>:w0(x; t) y < 0

w0(x; t)� b2�2

�1� e�� yd�

w0;xx(xc)e��jx�xcjb y > 0

v =

(0 y < 0b2�2

�d ze�� yd e�� jx�xcjb y > 0

(9)

In Equation 9, � is a dimensionless parameter and willbe discussed later in this paper. The term (1 � e�� yd )prevents the discontinuity at the crack tip. Thedisplacement component, v, is modi�ed in order that yz becomes zero at the crack faces.

EQUATION OF MOTION

Now, the strain �eld can be extracted from the dis-placement �eld. The normal strain component of thestress �eld can be written using Equation 9 as follows:

"x =

8>>>>>><>>>>>>:�zw0;xx y < 0�z �w0;xx �

�1� e�� yd��

1� 2b��(x� xc)

�w0;xx(xc)e��

jx�xcjb

�y > 0

(10)

The normal stress energy of the beam can be obtainedusing the following relation:

V =12

ZV

�x"xdV =12

EZV

"2xdV: (11)

In which V is the normal strain energy function, Vis the volume of the beam and E is the modulus ofelasticity.

In this research, the cracked beam is assumed tobe slender. So, the Euler-Bernoulli assumption canbe used and one can neglect the shear strain energyin comparison with the normal strain energy. Thedisplacement �eld is de�ned in order to force averageshearing strain components to be zero; similar to anundamaged Euler-Bernoulli beam.

The kinetic energy of the cracked beam can bealso calculated as follows:

T =12

ZV

�w2;tdV: (12)

In Equation 12, the rotational inertia has been ne-glected, similar to the undamaged Euler-Bernoullibeam theory.

Using the Hamilton principle, one has:

�Z t1

t0(T � V )dt = 0: (13)

Now, using Equations 10 to 13 and performing ap-propriate calculations, the following equations can beobtained:

E@2

@x2

�Iw0;xx� kw0;xx(xc; t)e��

jx�xcjb

�+�Aw;tt=0;

k =ZAc

z2�

1� e�� yd� dA = Ic �ZAc

z2e�� yd dA: (14)

Equation 14 is the governing equation of motion ofa beam with a vertical edge crack. In this equation,k is a geometrical factor which can be found forevery given cross-section and crack. I and Ic are themoment inertia of the cross-section and the crack face,respectively. In an undamaged beam, the geometricalparameter, k, is zero and, hence, Equation 14 turns intoa familiar form of the Euler-Bernoulli vibration equa-tion for slender beams. The dimensionless exponentialdecay rates (�; �) are the only factors that have notbeen discussed yet. In the next section, the parameters,� and �, are calculated and then, the solution for thepartial di�erential Equation 14 is presented.

EXPONENTIAL DECAY RATES � AND �CALCULATION

The exponential decay rates presented in this researchcan be obtained using the concepts of additionalremote point rotation and the J-integral, which aretwo familiar concepts in fracture mechanics. Severalresearchers used a similar method to evaluate the crackproperties [5,20].

When a pair of static bending moments, M , areapplied to the cracked beam, an additional relativerotation, ��, will exist between two ends of the beam,due to the crack, as shown in Figure 4.

For an Euler-Bernoulli simply supported beam,the slope function of the neutral axis is as follows:

�h =dwdx

=MEI

�x� l

2

�: (15)

Figure 4. Additional rotation of a beam with a verticalcrack under bending.


For a beam with a vertical crack under pure staticbending, the time derivatives vanish from Equation 14.Integrating two sides of the obtained equation and us-ing appropriate boundary conditions for pure bending,the following equation will be obtained:

d2w0

dx2 =MEI

�1 +

kI � k e��

jx�xcjb

�: (16)

Solving Equation 16 will result in the load-de ectionrelation of a beam with vertical crack under static purebending. The results are as follows [20]:

w0 =8>><>>:MEI

�x2

2 + c1x+ c2 + b2�2

kI�ke�

x�xcb

�x�xc

MEI

�x2

2 + c3x+ c4 + b2�2

kI�ke��

x�xcb

�x>xc (17)

in which constants c1, c2, c3 and c4 will be as fol-lows [20]:8>>><>>>:

c1 = c3 � 2 b�kI�k

c2 = � b2�2

kI�ke��

xcb

c3 = � l2 � c4

l � 1lb2�2

kI�ke��

l�xcb

c4 = c2 � 2xc b�kI�k

(18)

Using Equations 15 and 17, one can obtain the addi-tional remote point rotation, ��, as follows:

�� = (�c(0)� �h(0))� (�c(l)� �h(l))

=MEI

b�

kI � k

�2� e�� xcb � e�� l�xcb � ; (19)

where �c and �h are the rotation of a cracked beamand an undamaged or healthy beam under staticbending, respectively. In Equation 19, parameter k isa function of �. However, the �nite element results, incomparison with those obtained by this model, showthat parameter � is a large enough parameter and,accordingly, it can be assumed that parameter � tendsto in�nity. It must be noticed that, despite the factthat the exponential decay rate, �, is obtained hereby �nite element analysis and correlating analyticaland �nite element results, this value for � is a generalvalue and, in other cases, can be used without separatecalculations.

On the other hand, additional rotation �� meansthat the cracked beam accumulates more strain energycompared with an undamaged beam. This extra strainenergy which is called UT here is stored at the vicinityof the crack. The additional rotation of a beamsubjected to a pair of bending moments at two ends, as

shown in Figure 4, can be obtained using Castigliano'stheorem as follows:

�� =@UT@M

: (20)

This additional strain energy is due to the crack andcan also be written in the following form [1,5]:

UT =ZAcJs(a)dA: (21)

In Equation 21, Ac is the crack face area. Equation 21is called the Paris equation, and Js in this equationis the strain energy release rate. There are severalexperimental, analytical and numerical formulas tocalculate the value of the J-integral based on thegeometry, loading and type of crack [25,26]. In thiscase, the value of the J-integral could be obtained fromthe following equation [1,5]:

Js =1E0

24 6Xi=1

KIi

!2

+

6Xi=1

KIIi

!2

+m

6Xi=1

KIIIi

!235 ; (22)

where KIi , KIIi and KIIIi are the Stress IntensityFactors (SIF), corresponding to three modes of frac-ture, which result for every individual loading mode, i.In pure bending, SIF is nonzero only for mode I. InEquation 22, if the plane stress assumption is used,then, E0 = E and, if the plane strain assumption isused, then E0 = E=(1 � �2). In this article the planestrain assumption is used.

The beam with a vertical edge crack can beassumed to consist of a set of thin plates along a z-axis, and each plane to contain an edge crack and besubjected to axial tension or compression. This StressIntensity Factor (SIF) for such plates is [26]:

KI = �0p�aF

�ab

�;

�0 =MzI;

F�ab

�=�

1 + 0:122 cos4 �a2b

�r 2b�a

tan�a2b: (23)

Equation 23 has an accuracy of 0.5% for any a=b. FromEquations 22 and 23, the energy release rate is:

Js =K2I

E0 =1� �2

E

�MzI

�2

�aF 2�ab

�: (24)


Now, substituting Equation 24 into 21 and, then, usingEquation 20, the additional rotation of the crackedbeam, ��, can be obtained in terms of bending momentand geometrical parameters. For a rectangular crosssection, this additional rotation is:

�� =1� �2

E2MI2 �';

' =Z d

2

� d2

Z a

0z2sF 2

�sb

�dsdz: (25)

The additional rotation can be evaluated using theobtained relation in Equation 19 too. Comparing thetwo sides of Equations 25 and 19, one has:

b�

kI � k

�2� e�� xcb � e�� l�xcb � = (1� �2)

2I�':

(26)

The numerical solution of Equation 26 will lead to�nding the value of exponential decay rate � for anyvalues of geometrical parameters and simply supportedends. From Equation 26, it can be shown thatexponential decay rate � is a function of a=d, l=dand xc=l. However, Behzad et al. discussed that, forslender beams (l=d � 10), slenderness factor (l=d) andcrack position ratio xc=l have a minor e�ect on � forhorizontal cracks [20]. It can be shown that exponentialdecay rate � is only a function of crack depth ratio (a=b)for slender beams with vertical cracks too. Figure 5shows � versus a=b for slender beams (l=d � 10).

EIGEN SOLUTION FOR SIMPLYSUPPORTED BEAM WITH VERTICALCRACK

In order to �nd the natural frequencies and modeshapes of a beam with vertical crack, the equation ofmotion presented in Equation 14 must be solved. How-ever, this equation cannot be solved analytically, and a

Figure 5. Exponential decay rate (�) versus crack depthratio (a=b) for a slender simply supported beam with avertical edge crack.

numerical method must be used. The especial form ofEquation 14 in which the solution at the crack positionis appeared in the governing equation prevents onefrom using the ordinary Galerkin projection method.Behzad et al. presented a modi�ed Galerkin projectionalgorithm for solving this type of equation [21]. Inthis paper, a similar approach has been used. Thebeam is assumed to be simply supported in this section.However, for every desired boundary condition, thepresented solution can be used.

It can be assumed that the solution is a harmonicfunction, so one has:

w(x; t) = X(x)ei!t; (27)

in which ! is the natural frequency of the beam. Substi-tuting Equation 27 into Equation 14 and assuming EIto be constant along the beam, the following eigenvalueproblem will be obtained:8>><>>:

d2

dx2

�X 00 � k

IX00(xc)e��

jx�xcjb

�� AEI!

2X = 0

X(0) = X(l) = 0X 00(0) = X 00(l) = 0 (28)

In Equation 28, simply supported boundary conditionshave been used. In a normal Strum-Louiville problem,one can easily consider function X to be in the formofPciSi(x) in which Si(x) are shape functions that

satisfy the physical boundary conditions. However, inthis research, the results show that such an approachwill lead to a divergence of the results. Since thefunction e�� jx�xcjd in Equation 28 is not a smoothfunction, it seems that the solution, especially for largercrack depth ratios, tends to have large derivativesnear the crack tip. Accordingly, extracting the valueof X 00(xc) from X by derivation can lead to large uctuations in the results and divergence. In orderto avoid the divergence problem, function X 00 and thevalue of X 00(xc) are not extracted from X by directderivation. Instead, X 00 is discretized independentlyfrom X and, then, a constraint equation is provided tolink X 00 to X.

Considering the above discussion, the followingrelations can be written:8>><>>:

X 00 � �IX00(xc)e��

jx�xcjd =

NPi=1

ciSi(x)

X =NPi=1

c0iSi(x)(29)

in which ci and c0i are two independent sets of constants,functions Si(x) are shape functions, which must satisfyphysical boundary conditions, and N is the numberof shape functions. Substituting Equation 29 intoEquation 28, multiplying two sides of the equation by


Sj(x), then, integrating along the length of the beam,one has:

NXi=1

ciZ l

0S00i (x)Sj(x)dx

� �AEI

!2NXi=1

c0iZ l

0Si(x)Sj(x)dx = 0;

j = 1; 2; � � � ; N: (30)

Or in the matrix form:

Kc� �AEI�

!2Pc0 = 0;

Kij =Z l

0S00i (x)Sj(x)dx;

Pij =Z l

0Si(x)Sj(x)dx: (31)

On the other hand, if one substitutes the secondequation of Equation 29 into the �rst one, the followingrelation will be obtained:

NXi=1

c0i�S00i (x)� �

IS00i (xc)e��

jx�xcjb

�=

NXi=1

ciSi(x):(32)

Multiplying two sides of Equation 32 by Sj(x) and,then, integrating along the length of the beam, onehas:

NXi=1

c0i

Z l

0S00i (x)Sj(x)dx

��IS00i (xc)

Z l

0Sj(x)e�� jx�xcjb dx

!=

NXi=1

ciZ l

0Si(x)Sj(x)dx; j=1; � � � ; N: (33)

Rearranging Equation 33 into matrix form, the follow-ing equation can be written:

Qc0 = Rc;

Qij =Z l

0S00i (x)Sj(x)dx

� �IS00i (xc)

Z l

0Sj(x)e�� jx�xcjb dx;

Rij =Z l

0Si(x)Sj(x)dx: (34)

Now, from Equation 34, coe�cients c0i can be relatedto ci, as follows:

c0 = Q�1Rc: (35)

Substituting Equation 35 into 31, the following equa-tion will be obtained:�

K� �AEI

!2M�

c = 0;

M = PQ�1R: (36)

The natural frequencies and corresponding modeshapes for the cracked beam can be calculated solv-ing the matrix eigenvalue problem of Equation 36.In the next section, the results are presented fora simply supported beam with a rectangular cross-section.

RESULTS FOR A SIMPLY SUPPORTEDBEAM WITH RECTANGULARCROSS-SECTION

In this section, the eigenvalue problem of Equation 36has been solved for free vibration analysis of a simplysupported slender prismatic beam with a vertical edgecrack and rectangular cross-section. In such a beam,the exponential decay rate, �, can be assumed to bein�nite and the exponential decay rate, �, can becalculated from Equation 26. The values of � versuscrack depth ratio (a=b) have been shown in Figure 5.

In a simply supported cracked beam, shape func-tions Si(x) can be assumed to be in the form ofsin(i�x=l), which satisfy physical boundary conditions.The natural frequency and mode shapes can be calcu-lated using the eigenvalue problem of Equation 36. Inthis research, the number of shape functions, N , is setto be 100. In order to generalize the results, the naturalfrequencies of the cracked beam have been divided intothe corresponding values for an undamaged beam (!0).Figures 6, 7 and 8 show the fundamental, second andthird natural frequency ratios of the cracked beam,respectively. In Figures 6 to 8, the natural frequencyratios have been plotted versus the crack depth ratio(a=b) for several crack positions.

In Figures 6 to 8, the results of Finite Element(FE) analysis are also presented for veri�cation. The�nite element results have been obtained using ANSYSsoftware. In order to have an accurate and reliablemodel, the PLANE183 singular element has been usedin the cracked area [22]. This element is an 8-nodequadratic solid singular element, especially designed forcrack analysis. In this research, a �ne mesh has beenused at the vicinity of the crack, and the dependency ofthe results on mesh size has been checked. In all results,


Figure 6. Fundamental natural frequency ratio for abeam with a vertical crack versus crack depth ratio.

Figure 7. Second natural frequency ratio for a beam witha vertical crack versus crack depth ratio.

Figure 8. Third natural frequency ratio for a beam witha vertical crack versus crack depth ratio.

there is good agreement between analytical results andthose obtained by FE analysis.

As can be seen in Figure 6, the reduction rate ofthe fundamental natural frequency has a direct relationwith the position of the crack. This rate reduces forcracks that have more distance from the mid span of thebeam. For the cracks at xc=l = 0:1, the fundamentalnatural frequency drops less than 1 percent when thecrack reaches half of the beam depth, while for thecracks at the mid span, this value is about 4 percent.

The dependency of the reduction of the naturalfrequency on the crack position is also seen in the �rstfew natural frequencies. For cracks at the mid span,the second natural frequency remains nearly constantwith the crack depth, because this point coincides withthe node of the second vibration mode of the beam.

Figures 9 and 10 compare the natural frequency

Figure 9. Fundamental natural frequency ratio versuscrack depth ratio for vertical crack and horizontalcrack [21] at xc=l = 0:5.

Figure 10. Second natural frequency ratio versus crackdepth ratio for vertical crack and horizontal crack [21] atxc=l = 0:25.


drop for horizontal and vertical cracks. In Figure 9, thefundamental frequency ratio for the mid span verticalcrack has been compared with a horizontal one inthe same position. It can be seen that a horizontalcrack has much more e�ect than a vertical one on thenatural frequency. This result was predictable, becausea horizontal crack reduces bending sti�ness more thana vertical crack. Figure 10 shows a similar comparisonfor the second natural frequency at xc=l = 0:25. It canbe seen that a horizontal crack has more e�ect on thesecond natural frequency, too.

Figures 11 to 13 show the �rst three normalizedmode shapes for a cracked beam with a=d = 0:5 andxc=l = 0:1, 0.3 and 0.5. Comparison of the analyticand �nite element results in this set of �gures showsthe e�ciency of the model presented in this research.

CONCLUSIONS

A continuous model for exural vibration analysis of abeam with a vertical edge crack has been developed inthis paper. It is assumed that the crack face rotates

Figure 11. First three normalized mode shapes of acracked beam with xc=l = 0:1 and a=b = 0:5. (|):Analytical results; (� � ��): Finite element results.



more than other parts of the section, as well as itsadjacent area. The additional rotation decays with anexponential regime along the beam length. On the baseof this assumption, a displacement �eld for the beamhas been suggested and modi�ed for compatibility,continuity and consistency. The strain and stress �eldsare calculated by direct derivation of the displacement�eld and by using the linear elastic material model.Then, the partial di�erential equation of motion hasbeen obtained using the Hamilton principle. Thisequation has been evaluated for static conditions andthe exponential decay rate has been obtained with theaid of the J-integral concept in fracture mechanics.

The obtained governing equation of motion fora simply supported beam with a rectangular cross-section and vertical edge crack has been solved witha modi�ed Galerkin projection method. The obtainedresults have been compared with �nite element resultsfor a few �rst natural frequencies and mode shapes,and an excellent agreement has been observed.

The obtained results have also been used forstudying the e�ect of crack parameters on naturalfrequencies and mode shapes. The calculated naturalfrequencies for a beam with vertical crack have beencompared with those obtained for horizontal crack, andit is observed that the natural frequencies are moresensitive to horizontal cracks. Finally, it must benoticed that the developed theory in this research isonly correct for open edge cracks without extension.

ACKNOWLEDGMENT

The authors would like to acknowledge the �nancialassistance of the \Iranian Gas Transmission Co."throughout this research.

REFERENCES

1. Dimarogonas, A.D. \Vibration of cracked structures-A state of the art review", Eng. Fract. Mech., 5, pp.


831-857 (1996).2. Wauer, J. \On the dynamics of cracked rotors: A

literature survey", Appl. Mech. Rev., 43(1), pp. 13-17(1990).

3. Gasch, R. \A survey of the dynamic behavior of asimple rotating shaft with a transverse crack", J.Sound. Vib., 160(2), pp. 313-332 (1993).

4. Dimarogonas, A.D. and Paipetis, S.A., AnalyticalMethods in Rotor Dynamics, London, Applied sciencepublisher (1983).

5. Papadopoulos, C.A. \The strain energy release ap-proach for modeling cracks in rotors: A state of theart review", Mech. Syst. Signal Pr., 22, pp. 763-789(2008).

6. Zheng, D.Y. and Fan, S.C. \Vibration and stability ofcracked hollow-sectional beams", J. Sound. Vib., 267,pp. 933-954 (2003).

7. Yang, J., Chen, Y., Xiang, Y. and Jia, X.L. \Freeand forced vibration of cracked inhomogeneous beamsunder an axial force and a moving load", J. Sound.Vib., 312, pp. 166-181 (2008).

8. Lin, H.P. \Direct and inverse methods on free vibrationanalysis of simply supported beams with a crack", Eng.Struct., 26(4), pp. 427-436 (2004).

9. Zheng, D.Y. and Fan, S.C. \Vibration and stability ofcracked hollow-sectional beams", J. Sound. Vib., 267,pp. 933-954 (2003).

10. Loya, J.A., Rubio, L. and Fernandez-Saez, J. \Natu-ral frequencies for bending vibrations of Timoshenkocracked beams", J. Sound. Vib., 290, pp. 640-653(2006).

11. Orhan, S. \Analysis of free and forced vibration of acracked cantilever beam", NDT&E Int., 40, pp. 443-450 (2007).

12. Yang, X.F., Swamidas, A.S.J. and Seshadri, R. \Crackidenti�cation in vibrating beams using the energymethod", J. Sound. Vib., 244(2), pp. 339-357 (2001).

13. Wang, J. and Qiao, P. \Vibration of beams witharbitrary discontinuities and boundary conditions", J.Sound. Vib., 308, pp. 12-27 (2007).

14. Christides, S. and Barr, A.D.S. \One-dimensionaltheory of cracked Bernoulli-Euler beams", J. of Mech.Sci., 26(11/12), pp. 639-648 (1984).

15. Shen, M.H.H. and Pierre, C. \Natural modes ofBernoulli-Euler beams with symmetric cracks", J.Sound. Vib., 138(1), pp. 115-134 (1990).

16. Shen, M.H.H. and Pierre, C. \Free vibrations of beamswith a single-edge crack", J. Sound. Vib., 170(2), pp.237-259 (1994).

17. Carneiro, S.H.S. and Inman, D.J. \Comments on thefree vibration of beams with a single-edge crack", J.Sound. Vib., 244(4), pp. 729-737 (2001).

18. Chondros, T.G., Dimarogonas, A.D. and Yao, J. \Acontinuous cracked beam vibration theory", J. Sound.Vib., 215(1), pp. 17-34 (1998).

19. Chondros, T.G., Dimarogonas, A.D. and Yao, J.\Vibration of a beam with breathing crack", J. Sound.Vib., 239(1), pp. 57-67 (2001).

20. Behzad, M., Meghdari, A. and Ebrahimi, A. \A lineartheory for bending stress-strain analysis of a beam withan edge crack", Eng. Fract. Mech., 75(16), pp. 4695-4705 (2008).

21. Behzad, M., Meghdari, A. and Ebrahimi, A. \A newcontinuous model for exural vibration analysis ofa cracked beam", Pol. Mar. Res., 15(2), pp. 32-39(2008).

22. Behzad, M., Meghdari, A. and Ebrahimi, A. \A newapproach for vibration analysis of a cracked beam",Int. J. of Eng., 18(4), pp. 319-330 (2005).

23. Behzad, M., Meghdari, A. and Ebrahimi, A. \Acontinuous model for forced vibration analysis of acracked beam", ASME Int. Mech. Eng. Cong. andExp., Orlando, Florida USA (2005).

24. ANSYS User's Manual for Rev., 8, ANSYS Inc.(2004).

25. Barani, A. and Rahimi, G.H. \Approximate methodfor evaluation of the J-integral for circumferentiallysemi-elliptical-cracked pipes subjected to combinedbending and tension", Scientia Iranica, 14(5), pp. 435-441 (2007).

26. Tada, H., Paris, P.C. and Irvin, G.R., The Stress Anal-ysis of Cracks Handbook, Hellertown, Pennsylvania,Del Research Corp. (1973).

BIOGRAPHIES

Mehdi Behzad has a PhD in Mechanical Engineeringfrom the University of New South Wales, Sydney,Australia and is a faculty member of the mechanicalengineering department of Sharif University of Technol-ogy in Tehran, Iran. Professor Behzad is also chairmanof the Iran Maintenance Association.

Alireza Ebrahimi has a PhD in Mechanical Engi-neering from Sharif University of Technology in Tehran,Iran and is also a researcher of the condition monitoringcenter at that university.

Ali Meghdari has a PhD in Mechanical Engineeringfrom the University of New Mexico, Albuquerque,U.S.A. and is a faculty member of the Mechanical Engi-neering Department of Sharif University of Technologyin Tehran, Iran.

He is also a Distinguished Professor of Mechani-cal Engineering (MSRT), Vice-President of AcademicA�airs and a Fellow of the American Society of Me-chanical Engineers (ASME).

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