2019 16th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)Mexico City, Mexico. September 11-13, 2019

Structural analysis of superficial cracks on structuralelements

Adriana Jimenez-Sanchez, Gerardo Silva-NavarroCentro de Investigacion y de Estudios Avanzados del I.P.N.

Depto. Ingenierıa Electrica, Seccion de Mecatronica,Mexico City, Mexico

adriana.jimenez.s@protonmail.com, gsilva@cinvestav.mx

Francisco Beltran-CarbajalUniversidad Autonoma Metropolitana, Unidad Azcapotzalco

Departamento de EnergıaMexico City, Mexicofbeltran@azc.uam.mx

Abstract—This paper addresses the structural dynamics ofmodels of cracked Euler-Bernoulli beams by using finite elementmethods and modal analysis techniques. The natural frequency isalso characterized and analyzed in order to present the decreasein natural frequencies due to the presence of a superficialcrack. Some numerical and experimental results are providedto illustrate the effects of a crack on aluminum beams.

Index Terms—Cantilever beam, Crack, Finite element, Identi-fication, Modal analysis.

I. INTRODUCTION

The presence of a crack in structural elements leads todynamic effects such as the introduction of local flexibilitiesinto the element, modification of the vibration response, adecrease of local stiffness and, consequently, a decrease ofthe associated natural frequencies and mode-shapes [1].

The development of methods for crack detection, moni-toring and failure modeling in mechanical elements startedin the early 1970s [1]. The continuous failures presented inturbine blades, shafts and fixed-wing aircrafts were among themain causes that drove the development of these theories andexperimental studies. Some of these mechanical structures canbe treated as cantilever beams.

In order to study the effects of cracks in mechanicaland structural elements, like added local flexibility and de-crease/variations on natural frequencies, commonly a dynamicapproach is used. In [2] is shown that through a spring locatedin the fixed-end of the beam is possible to simulate theflexibility and changes on the natural frequencies due to thepresence of a crack; it also shows that, for small cracks, thechanges on the natural frequencies also depend on the strainenergy. In [3] is shown the first-order method to predict suchchanges on the resonance frequencies of a cracked element.

In the mechanical behavior of cracks, two cases are ob-served: the opening crack, and the closing crack. The closingcrack dynamic response is similar to the uncracked elementresponse. Because of the opening and closing cracks aredepending on time, then it is generated a nonlinear dynamicbehavior. In [4] the eigenvectors and eigenfrequencies areevaluated from the characteristic equation solution when thecrack is open. It is also important to perform a vibratoryanalysis on damaged elements to know their behavior underforced vibrations and the effects of the crack location and

size on the overall structure. In [5] the beam is modeledby triangular disk finite elements, which have two degreesof freedom at each node. The method described in thispaper presents the possibility of modeling the elastic-dampingproperties on specific (point) finite elements. Finally, expertssystems utilizing predicate logic, or fuzzy logic and artificialneural networks have been developed for empirical crackidentification as shown in [6], [7].

The vibratory analysis in structures allows the estimation ofthe location and depth of the crack, only if one have a propermodel that relates the natural frequencies with the variables tobe estimated. In this work, the model of a cantilever beam isobtained using finite element formulation. To characterize thecrack, the effects on the strain energy are also considered. Thenatural frequencies for each mode are obtained by means ofthe characteristic equation of the system to then analyze anddiscuss the variation of the frequencies due to the presence ofa crack. Some numerical and experimental results are providedto illustrate the effects of a crack on aluminum beams.

II. EULER-BERNOULLI THEORY

The fundamental Euler-Bernoulli beam theory is an effectivemodel to study beams under axial forces and bending. Theirthree fundamental assumptions are the following [1], [11]:

1) Vertical displacements of the cross-section are smalland equal to the axis of the beam.

2) Lateral displacements do not exist.3) Cross-sections normal to the deformation axis remain

flat and orthogonal to the axis after deformation.

A. Finite element formulation

The beam element is obtained by discretization of the beamin a series of finite elements. Each element has two nodes and,therefore, it has four degrees of freedom, two at each node (onetransverse displacement and one for rotation).

Fig. 1. Schematic diagram of a beam element

978-1-7281-4840-3/19/$31.00 ©2019 IEEE

It is well-known that the stiffness matrix of the element oflength L between the i and j nodes, is given by

Ke =EI

L3

12 6L −12 6L. . . 4L2 −6L 2L2

. . . 12 −6Lsim

. . . 4L2

(1)

where E is the Young’s modulus and I is the moment ofinertia of the cross section.

The mass matrix is given by

Me =ρAL

420

156 22L 54 −13L. . . 4L2 13L −3L2

. . . 156 −22Lsim

. . . 4L2

(2)

where ρ is the material density and A is the cross section area.

Now, with the matrices of the beam element, we can obtainthe stiffness and mass matrices of a beam divided into Nelements (see Fig. 2), called global matrices. Such that, thedynamic model of the beam is described as

MX(t) +KX(t) = F (t), X ∈ Rn, F ∈ Rn (3)

where M and K are the stiffness and mass global matricesrespectively with 2n×2n dimension, n is the number of nodesin a beam, X is the displacement vector and F (t) is the forcevector.

Fig. 2. Finite element model of a cantilever beam

III. CRACK MODEL

Consider a cantilever beam with a cracked element, thebeam is divided into n elements. The presence of a crack inan element cause effects that are related to the stress intensityfactors of the crack. Now, using the strain energy of a beam,the stiffness matrix due to the presence of a crack can beobtained.

Fig. 3. Diagram of a cracked beam

The strain energy of the undamaged beam, subject to a totalbending moment Mt, is given by

U0 =1

2

∫ L

0

(Tx+M)2

EIdx

U0 =T 2L3

6EI+MTL2

2EI+M2L

2EI(4)

where T is the shear force and M is the bending moment onthe beam.

The flexibility coefficient for an uncracked element is ob-tained as

C(0)ij =

∂2U0

∂Ti∂Tj, T1 = T, T2 =M, i, j = 1, 2 (5)

Then, the flexibility matrix of an element without crack is

Cn =

L3

3EI

L2

2EIL2

2EI

L

EI

(6)

The additional strain energy due to the presence of a crackis given by [10]

U1 =

∫A

1

AE′

[(K2I +K2

II

)+

1

1− νK2III

]dA (7)

where E′ = E for plane stress and E′ = E1−ν2 for plane

strain, ν is the Poisson’s ratio and KI , KII and KIII arestress intensity factors for opening, sliding and tearing-typecracks, respectively.

To apply the linear fracture mechanics theory is necessaryto consider a plane strain state. Thus, neglecting the effects ofaxial forces, for a beam with cross-section b× h and a crackdepth a, the last equation may be written as

U1 = b

∫ a

0

[(KIM +KIT )

2+K2

IIT

]E′

da (8)

where KIM , KIT and KIIT are the stress intensity factorsand E′ = E

1−ν2 . These factors are given as [10]

KIM =6M

bh2√πaFI(r)

KIT =3TL

bh2√πaFI(r) (9)

KIIT =T

bh

√πaFII(r)

where r = a/h, and FI and FII the correction factors for arectangular cross section are given by

FI(r) =

√2 tan

(πr2

)πr

0.923 + 0.199[1− sin(πr2 )

]4cos(πr2

) (10)

FII(r) =(3r − 2r2

) 1.122− 0.561r + 0.085r2 + 0.18r3√1− r

Replacing (9) in (8), we have that

U1 = −

(9 (TL+ 2M)

2C1 + T 2h2C2

) (1− ν2

)π

bh2E(11)

where C1 =∫ r0rF 2

I (r)dr and C2 =∫ r0rF 2

II(r)dr (seeexpressions in the Appendix).

The flexibility coefficient introduced due to a crack is givenas

C(1)ij =

∂2U1

∂Ti∂Tj, T1 = T, T2 =M, i, j = 1, 2 (12)

And the flexibility matrix due to the presence of a crackedelement is the following

Cc =

[−

2π(9L2C1+h2C2)(ν2−1)bh2E

−36LπC1(ν2−1)

bh2E

−36LπC1(ν2−1)

bh2E−

72πC1(ν2−1)bh2E

](13)

A. Case of a damaged beam element

The element is assumed to have a transverse crack underbending and shearing forces as shown in Fig. 4. The equilib-

Fig. 4. Schematic diagram of a cracked beam element

rium conditions are obtained as

Ti = −Ti+1

Mi = −LPi+1 −Mi+1

Pi+1 = Pi+1 (14)Mi+1 =Mi+1

Rewriting (14) in matrix form results inTiMi

Ti+1

Mi+1

=

−1 0−L −11 00 1

[ Ti+1

Mi+1

](15)

where

P =[−1 −L 1 0

0 −1 0 1

]T(16)

For a damaged beam element the stiffness matrix is givenas

Kc = [P ] [C]−1

[P ]T (17)

where C = Cn + Cc is the flexibility matrix

C =

[L3

3EI−

2π (9L2C1+h2C2)(ν2−1)bh2E

L2

2EI−

36LπC1(ν2−1)bh2E

L2

2EI−

36LπC1(ν2−1)bh2E

LEI

−72πC1(ν2−1)

bh2E

](18)

Adding the stiffness matrix due to crack in (3) is obtainedthe dynamic model of a cracked beam

MX(t) +KTX(t) = F (t), X ∈ Rn, F ∈ Rn (19)

where M is the mass matrix and KT is the stiffness matrixincluding those effects due to the presence of a crack. Notethat, the presence of damage into the beam does not have ahigh impact on the elements of the mass matrix and, therefore,these are neglected. Considering the boundary conditions fora cantilever beam, the naturals frequencies can be computedin both cases (undamaged and damaged beam) as follows

det(−ω2

iM +KT

)= 0, i = 1, 2, 3, . . . (20)

fni =ωi2π

(21)

where fni is the natural frequency associated to the i-th mode-shape in Hz.

IV. CRACKED BEAM BEHAVIOR

Consider a cantilever beam, as shown in Fig. 5. The modelof both beams (damaged and undamaged beam) is computedas shown in Sections II-A and III-A, then the modal analysisis performed for each position and depth of the crack to obtainand analyze the natural frequencies of the beam.

Fig. 5. Beam with a crack at the middle of its span

TABLE IGEOMETRICAL AND MATERIAL PROPERTIES OF THE BEAM

Cross section(m2) 0.0254× 0.0381

Density(kg/m3) 2700

Poisson’s ratio 0.33

Young’s modulus(GPa) 68.9

A. Natural frequencies

When there is a crack into a structural element the frequen-cies decrease as the depth ratio of the crack is increased. Thisrelationship is shown in Fig. 6. Also, it is observed how thelocation and depth of the crack affect each mode-shape, whenthe crack location goes away from the fixed end the frequencyapproximates the frequency of a healthy beam.

In Fig. 6a the effects of the crack on the first mode aremore significant when the crack is closer to the fixed end, butin the second mode (Fig. 6b) these effects are not important.

(a)

(b)

(c)

Fig. 6. Normalized frequencies vs depth ratio of a cracked cantilever beamfor the first three mode-shapes

However, the crack located at half of the length has a greatinfluence. In Fig. 7 the mode-shapes are shown for both cases,damaged and undamaged beam. The presence of a crack in astructural element also generates shifts on the mode-shapes,and each mode is affected by a different position of the crack.

B. Modal analysis

The numerical modal analysis of the beam in Fig. 5 isdescribed below. The first two mode-shapes of a beam withouta crack and a beam with a crack are observed, in additionto its frequency response function. In the frequency responsefunction in Fig. 8, the decrease in the natural frequency dueto the presence of a crack can be confirmed, for this case,at half of the length of the beam, such that the secondnatural frequency is the most affected value. Note the shifts(reductions) on these natural frequencies.

0 10 20 30 40 50 60 70 80 90 100

Discretized elements

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

No

de

sh

ap

es

Mode 3

Mode 2Mode 1

Crack position

- -Uncracked beam

-Cracked beam

(a)

0 10 20 30 40 50 60 70 80 90 100

Discretized elements

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

No

de

sh

ap

es

Mode 2Mode 1

Mode 3

Crack position

- -Uncracked beam

-Crakced beam

(b)

Fig. 7. Mode shapes of a cracked and uncracked beam

C. Experimental modal analysis

Some experimental measurements of natural frequencieswere made on a cracked and uncracked beam by using ex-perimental modal analysis techniques (modal impact hammertesting), with an accelerometer and excitation via an impacthammer, as it is shown in Figs. 9 and 10. Note how, in Fig.10b, one can verify the crack breathing phenomenon presentinto the second mode-shape due to the presence of a crack atthe middle of the length of the beam.

V. DESCRIPTION IN MODAL COORDINATES

In terms of modal analysis, it is known that there is a lineartransformation decoupling the equations of motion, so that, amechanical system with n degrees of freedom can be obtained[11]. Then, the equation of motion describing the dynamics ofthe beam in (19), assuming F (t) ≡ 0, can be transformed tomodal coordinates as follows

qi(t) + ω2i qi(t) = fi(t), i = 1, 2, 3, . . . , n (22)

where qi denotes the i-th modal coordinate associated to thei-th mode-shape, ωi the corresponding natural frequency andfi the modal force.

Because the presence of a crack affects the frequencyresponse function and the location of the natural frequenciesωi, then by using real-time information on the displacementsor accelerations, it is possible to estimate such variationsto monitor and evaluate possible damages on a mechanical

(a)

(b)

Fig. 8. Modal analysis of a cantilever beam, first and second mode-shapes,(a) uncracked beam, and (b) cracked beam

Impact hamer

Accelerometer

Uncracked beam

(a)

(b)

Fig. 9. Experimental measurements, (a) Uncracked beam, and (b) Frequencyresponse function

Impact hammer

Accelerometer

Cracked beam

3mm

13.5mm

(a)

(b)

Fig. 10. Experimental measurements, (a) Cracked beam, and (b) Frequencyresponse function

element. To do this, it is possible the application of on-linealgebraic identification techniques presented in [12].

To illustrate the effect of a crack on the time responseof a cantilever beam, consider the time response for modalcoordinates in free vibrations of a damaged and undamagedbeam, for the first three mode-shapes, as shown in Fig. 11. Inthis case, the crack position is at the middle of the length ofthe beam and one can observe a notable shift in the secondmode-shape because this mode is the most affected due tothe crack, and, finally, in the third mode-shape there is not asignificant change. This change in the time response is becausethe presence of the crack causes reductions on specific lateralnatural frequencies. Thus, with a modal description and fastidentification techniques, the deflections of the beam can beused as information to compute the necessary parameters toget estimations of the location and dimensions of a crack.

0 0.05 0.1 0.15 0.2 0.25-0.01

-0.005

0

0.005

0.01

q(t

), 1

st

Mo

de Undamaged beam

Damaged beam

0 0.05 0.1 0.15 0.2 0.25-0.01

-0.005

0

0.005

0.01

q(t

), 2

nd

Mo

de

0 0.05 0.1 0.15 0.2 0.25

t [s]

-0.01

-0.005

0

0.005

0.01

q(t

), 3

rd M

od

e

Fig. 11. Time response of the beam

VI. DISCUSSION

It is possible to use the finite element method to obtain amathematical model of the beam in order to identify the effectson its mode-shapes due to the presence of a superficial crack.With this relationships, the location and dimensions of thecrack can be estimated, because as noticed in Fig. 7, differentlocations of the crack generate significant displacements inthe mode-shapes. The natural frequencies can be measuredmore easily than the mode-shapes and to identify the depthof the crack is necessary to consider their percentage ofreduction. The effects caused by a crack on a cantilever beamare summarized in Table II.

TABLE IINATURAL FREQUENCIES OF AN UNDAMAGED AND DAMAGED BEAM

Undamaged beam

Method First mode Second mode Third mode

This approach(Hz) 124.36 779.37 2182.3

ANSYS(Hz) 125.6 766.22 2062

Experimental(Hz) 113.15 697.25 2061.35

Variation (%) 9.9 11.77 5.86

Damaged beam

First mode Second mode Third mode

This approach(Hz) 121.04 702.07 2180.6

ANSYS(Hz) 121.56 679.75 2059.8

Variation (%) 0.46 3.28 5.86

Experimetal(Hz) 110.69 652.74 2045.82

Variation (%) 9.35 7.55 6.58

Finally, the model obtained for a cracked cantilever beamcan be used to analyze the dynamic behavior of the beamwith a superficial crack and, therefore, the model describedin modal coordinates (22) can be employed to algebraicallyobtain estimations for location and depth of a crack.

VII. CONCLUSIONS

This work presents the effects on the frequency responsefuntion and the natural frequencies caused by the presence ofa crack. There is a close relationship between the position anddepth of the crack and the shift on the natural frequencies.It is possible to compare the frequency response obtainedusing the dynamic model presented in this work with thefrequency response obtained by finite element methods andexperimental modal analysis. With this comparison, it isverified that the model obtained from the Euler-Bernoulli beamusing this approach is considered to approximate the dynamicbehavior observed in cracked elements. However, a high erroris observed between the frequencies of the physical model andthose obtained with this approach, because it is still necessaryto adjust the material properties. Besides, this model can beeasily represented in generalized coordinates, so that, in futurework the identification of cracks on the beam can be evaluated.

APPENDIXSOLUTION OF THE INTEGRALS C1 AND C2

C1 =1

7500000π2 cos2(

12πr

){594015 sin8(πr

2

)cos2

(πr2

)+ 594015 sin10

(πr2

)− 4752120 sin7

(πr2

)cos2

(πr2

)− 4752120 sin9

(πr2

)+ 17424440 sin6

(πr2

)cos2

(πr2

)+ 16632420 sin8

(πr2

)− 39917808 sin5

(πr2

)cos2

(πr2

)− 33264840 sin7

(πr2

)+ 73228020 sin4

(πr2

)cos2

(πr2

)+ 47091360 sin6

(πr2

)− 121835760 sin3

(πr2

)cos2

(πr2

)− 55306080 sin5

(πr2

)+ 49694280 tan2

(πr2

)cos2

(πr2

)+ 146456040 sin2

(πr2

)cos2

(πr2

)− 392300640 sin

(πr2

)cos2

(πr2

)+ 392300640 ln

[sec(πr

2

)+ tan

(πr2

)]cos2

(πr2

)+ 392300640 ln

[cos(πr

2

)]cos2

(πr2

)− 26793360 sin3

(πr2

)− 18883260 cos

(πr2

)+ 18883260}

C2 = −0.01178181818r11 + 0.01368r10 + 0.1101488889r9

− 0.374525r8 + 0.113413r7 + 1.573372167r6

− 3.5277072r5 + 3.135456r4 − 0.284592r3

− 0.426888r2 − 0.853776r − 0.853776 ln (1− 1r)

REFERENCES

[1] A.D. Dimarogonas, ”Vibration of cracked structures: a state of the artreview”, Engineering Fracture Mechanics, Vol. 55, No. 5, 1996, pp.831-857.

[2] R.C.M. Indersdorf and M.H. Munchen, ”Eigenfrequenzen eines angeris-senen Kragtragers”, Ingenieur-Archiv 55, 1985, pp. 237-241.

[3] P. Gudmundson, ”Eigefrequency changes of structures due to cracks,notches or other geometrical changes”, J. Mech. Phys. Solids., Vol. 30,No. 5, 1982, pp. 339-353.

[4] G.L. Qian, S.N. Gu and J.S. Jiang, ”The dynamic behavior and crackdetection of a beam with a crack”, Journal of Sound and Vibration, Vol.138, No. 2, 1990, pp. 233-243.

[5] W.M. Ostachowicz, and M. Krawczuk, ”Vibration analysis of a crackedbeam”, Computers and Structures, Vol. 36, No. 2, 1990, pp. 245-250.

[6] S. Moradi, P. Razi, and L. Fatahi, ”On the application of bees algorithmto the problem of crack detection of beam-type structures”, Computersand Structures, 2011, pp. 2169-2175.

[7] M. Pedroza, E. Mariotte, J. E. Quiroga, and Y. A. Munoz, ”Comparativeanalysis between SOM networks and Bayesian networks applied tostructural failure detection”, TECCIENCIA, Vol. 11, No. 20, 2016, pp.47-55.

[8] P.I. Kattan, ”MATLAB Guide to finite elements: an interactive ap-proach”, Springer, Second Edition, 2008, Berlin.

[9] H. Nahvi, and M. Jabbari, ”Crack detection in beams using experimentalmodal data and finite element model”, Journal of Sound and Vibration,402, 2017, pp. 70-84.

[10] H. Tada, P. Paris, and G. Irwin, ”The stress analysis of cracks handbook”,Hellertown, PA: Del Research Corp, 1973.

[11] S.S. Rao, ”Mechanical vibrations”, Second Edition, Addison-Wesley,1990.

[12] F. Beltran-Carbajal and G. Silva-Navarro, ”Adaptive-like vibration con-trol in mechanical systems with unknown paramenters and signals”’,Asian Journal of Control, 2013, Vol. 15, No. 6, pp. 1613-1626.