Exact solution based finite element formulation of cracked beams for crack detection

Accepted Manuscript

EXACT SOLUTION BASED FINITE ELEMENT FORMULATION OF CRACKED BEAMS FOR CRACK DETECTION Ugurcan Eroglu , Ekrem Tufekci PII: DOI: Reference:

S0020-7683(16)30112-3 10.1016/j.ijsolstr.2016.06.005 SAS 9184

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

2 December 2015 3 May 2016 1 June 2016

Please cite this article as: Ugurcan Eroglu , Ekrem Tufekci , EXACT SOLUTION BASED FINITE ELEMENT FORMULATION OF CRACKED BEAMS FOR CRACK DETECTION, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.06.005

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EXACT SOLUTION BASED FINITE ELEMENT FORMULATION OF CRACKED

Ugurcan Eroglu

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BEAMS FOR CRACK DETECTION

Ekrem Tufekci*

[email protected]

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[email protected],

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Istanbul Technical University, Faculty of Mechanical Engineering,

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Gumussuyu 34437, Istanbul, TURKEY

Fax: + 90 212 245 07 95

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Phone: + 90 212 293 13 00 (2650 Extension)

This manuscript consists of 22 pages including title page and abstract, and 4672 words

*

Corresponding author

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ABSTRACT In this study, a new finite element formulation is presented for straight beams with an edge

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crack, including the effects of shear deformation, and rotatory inertia. The main purpose of the study is to present a more accurate formulation to improve the beam models used in crack detection problems. Stiffness matrix, consistent load vector, and mass matrix of a beam element is obtained using the exact solution of the governing equations. The formulation for frame structures is also presented. Crack is modelled utilizing from the concepts of linear

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elastic fracture mechanics. Several numerical examples existing in the literature related to the vibrations of such structures are solved to validate the proposed model. Additionally, an experimental modal analysis is performed to see the superiority of the present method for high modes of vibration, which are generally not taken into account in crack detection problems. The inverse problem is also solved using a well – known optimization technique called

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genetic algorithms. Effects of shear deformation, rotatory inertia, and number of natural frequencies considered, on the accuracy of the estimation of crack parameters are

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investigated. It is found that considering more number of frequencies yields better estimation of crack parameters, but require a better modelling of the dynamics of the beam. Therefore,

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the present formulation is found to be an essential tool in crack detection problems.

Keywords: cracked beam, crack detection, genetic algorithms, beam theory, finite element method, experimental modal analysis 2

ACCEPTED MANUSCRIPT 1. Introduction Beams are one of the simplest structural elements. The simplicity of the governing equations make the static, and dynamic behavior of beams easy to analyze and manipulate. Also, the manufacturing process is easy due to their simple geometries. Thus, beams have been used in many structures from different engineering disciplines such as civil, mechanical, ocean, aerospace to carry and/or transfer loads over the years. Since the application area of beams and beam-like structures is extensive, it is obvious that monitoring their health is a very

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important issue to maintain the functions of the entire structure, and to ensure the safety. Despite the advanced techniques to estimate the service life of structures, cracks or crack like defects may occur due to several effects which may not be taken into account, such as underestimated external loads, mechanical vibrations, corrosive environment, collisions, fatigue etc. That is why there is a great number of studies on non-destructive testing (NDT)

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methods of structures. For local crack detection, it is well-known that the NDT methods such as x- ray, lamb waves, acoustic emission, etc. can be used. Nevertheless, these methods require a very hard work to examine the entire structure, inherently. For that reason, different methods which measure, and/or are based on a feature that is related to the whole structure

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gained a considerable attention. The natural frequencies, and corresponding mode shapes are unique for a structure since they contain information related to the loading and boundary conditions as well as the material and geometric properties of the structure. There are many

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works devoted to crack detection in beam – like structures using the modal properties. A comprehensive review of the literature on this subject is not within the scope of this study, but

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information about an enough number of studies for an overview are summarized below.

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Existence of a crack on an elastic beam causes a local flexibility due to the strain energy concentration. Irwin (1957) came up with the idea of modeling the strain concentration by using an equivalent spring. This idea led to the development of a more general factor which is

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called the stress intensity factor (SIF) (Papaeconomou and Dimarogonas, 1989). More information about SIFs, and their expressions for different loading conditions and geometries can be found in the work of Tada et al. (1973). The effects of the crack on the global behavior of the structure are modelled by different approaches such as smaller cross – section (Kirshmer, 1944), or a link element (Harrison, 1973). In the following years, a more convenient way to model the compliance due to the crack is implemented. The additional flexibility is modelled by a spring, and stability, and vibrations of cracked beams are examined by many authors, such as Dimarogonas (1976), Chondors and Dimarogonas (1979), 3

ACCEPTED MANUSCRIPT Anyfantis and Dimarogonas (1983), Ostachowicz and Krawwczuk (1992), and this idea is still being used widely. Reviews of especially the early stages of this topic can be found in the articles of Dimarogonas (1996), and Salawu (1997). One of the first studies about crack detection is the one by Adams et al. (1978), where changes in the axial vibrations of a bar with uniform or tapered cross – section are examined to detect the damage in such structures. Since the coupling between axial and transverse vibrations are not taken into account, the formulation was more accurate for small, or symmetric cracks. Transverse vibrations of a

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simply supported beam are used for the same purpose by Chang and Petroski (1986). Qian et al. (1990) used a finite element model to determine the natural frequencies of a cantilevered beam, taking the effect of crack closure into account. Also, a method based on the relationship between the crack, and modal parameters to determine the crack position from known natural frequencies is developed. Similar studies can be found in refs. (Rizos et al., 1990; Pandey et

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al., 1991; Liang et al., 1991), and many more. A detailed review of this subject can be found in Salawu (1997), for example. With the development of the computer science, different methods to solve the dynamic problem of a cracked beam, such as finite element method (FEM), and to solve the inverse problem (i.e. detecting crack location and depth using modal

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properties) are implemented. He et al. (2001) implemented the genetic algorithms (GA) to solve the inverse problem. FEM is used to model the crack shaft, and the shaft is assumed to have one surface crack. They concluded that the numerical problems encountered in the

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solution of the inverse problem are eliminated with using GA.

Saridakis et al. (2008)

approximated the analytical model of a shaft with 2 cracks using neural networks. Only the

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bending modes are considered. GA is used to solve the inverse problem. It is concluded that their method reduces the computational time, and, therefore may be considered as an efficient

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tool for real – time crack detection. Xiang et al. (2008) used wavelet – based finite elements in modelling the cracked shaft. Rotatory inertia, and shear deformation is taken into account.

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They implemented a GA procedure to solve the inverse problem, by considering the first 3 frequencies. Formulation is validated through numerical examples as well as an experimental study. Another study which uses GA for crack detection is the one by Vakil-Baghmisheh et al. (2008). An analytical model without shear deformation and rotatory inertia effects is used to represent the dynamic behavior of a cracked beam. A detailed explanation about the GA is also presented in their study. Additionally, an experimental study is performed in order to validate the model. Khaji et al. (2009) proposed an analytical method to analyze cracked Timoshenko beams for crack detection. Mehrjoo et al. (2013) presented a new cracked finite beam element formulation based on the conjugate beam concept, and Betti’s theorem. Shear 4

ACCEPTED MANUSCRIPT deformation or rotatory inertia effects are not considered in the beam model, while crack is modelled as a linear spring. The inverse problem is solved using GA. Numerical examples are presented in order to validate the method. Caddemi and Morassi (2013) presented the exact closed form solutions of static and dynamic problems of multi cracked Euler-Bernoulli beams. Dirac’s delta function, and rotational spring are used to model the flexibility due to a crack, and a justification of rotational spring model is proposed. Caddemi et.al (2013) presented the explicit expressions of stiffness matrix, mass matrix, and element shape functions of their FE

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model which is constructed to examine the dynamic behavior of damaged frame structures. Heaviside function is used to include the effect of discontinuity of the cross-section or the material, while discontinuities in transverse deflection, and slope are modelled by Dirac’s delta function. Using these functions leads to the differential equation system of cracked beams, which does not require any additional transition conditions. The advantages of such a

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method are presented in the corresponding study. Mehrjoo et al. (2014) constructed a cracked finite beam element analogous to their earlier work (Mehrjoo et al., 2013), but used Timoshenko beam theory. GA is utilized in crack detection problem. Caddemi and Calio (2014) pursued the exact solution of the crack detection problem using the closed form

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expressions of vibration mode shapes, which are derived in the same work. Hakim et al. (2015) applied artificial neural networks to detect possible cracks in I-beams. For the training of the neural networks, results of 3D finite element simulations obtained from a commercial

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package are used. It is concluded that the method provides good results for even extra – light damages, which is possibly due to the fact that the first 5 natural frequencies are used in the

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inverse problem, unlike the majority of studies. Moezi et al. (2015) considered cantilever an Euler-Bernoulli beam with a crack. To estimate the crack location and depth, they applied a

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modified cuckoo optimization algorithm. It is concluded that modified cuckoo algorithm yields better results than cuckoo, and GA – Nelder-Mead algorithms. For other studies dealing

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with crack detection in beam – like structures, see (Chinchalkar et al. 2001; Patil and Maiti 2003; Moradi et al. 2011; Mazanoglu and Sabuncu 2012; Vakil-Baghmisheh et al. 2012; Surace et al. 2013; Pokale and Gupta 2014; Cao et al. 2014; Rubio et al. 2015) for example. The literature survey shows that the majority of the studies are focused on different techniques to solve the inverse problem, rather than a better modelling of the structure itself. It is also seen that most of the studies consider only the first 3 natural frequencies in order to detect cracks or crack-like defects. Even if the optimization algorithm used in the inverse problem may affect the computational time, the accuracy of the results is mainly based on the 5

ACCEPTED MANUSCRIPT mathematical model of the structure. More realistic models yield better estimations of the natural frequencies, which gains more importance if one wants to consider higher modes in the solution of the inverse problem, and wants more precision. On the other hand, practicality of the mathematical formulation should be considered. Solution of the equations to get the results should not take too much time since hundreds of analysis might be needed to solve the inverse problem. In this respect, FEM looks to be a convenient method to model the beam.

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In this study, an FEM formulation obtained by the exact solution of the beam equations is used for a better modelling of the structure. Only the in – plane motion of the beam is considered, and shear deformations, axial deformations, and rotatory inertia are included in the formulation. Crack is modelled as a rotational, and two translational springs. By using the transition conditions in the analytical modelling, a cracked beam finite element formulation is

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obtained. The formulation is also adopted to the frame structures. GA is used to solve the inverse problem. Proposed method is validated through numerous examples existing in the literature, as well as an experimental study. 2. Analysis

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2.1 Exact solution of the beam equations

Assuming the cross – section of the beam shown in Figure 1 is rigid, and symmetric about y-z

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dw Fz  dz EA k F du  x  y y dz GA d x M x  dz EI x

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plane, differential equation system of the static behavior is known as,

(1)

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dFz   qz dz dFy  q y dz dM x   Fy  mx dz

where E and G are elasticity and shear modulus of the material, A is the area of the cross – section, Ix is the moment of inertia of the cross – section about the x-axis ky is the shear correction factor, w and u are the displacements in z and y directions, Ωx is the rotation of the 6

ACCEPTED MANUSCRIPT cross – section about the x-axis, Fz and Fy are the internal forces in z and y directions, and Mx is the internal moment about the x-axis, qz and qy are external distributed loads in z and y directions, and mx is the external distributed moment about the x axis. It should be noted that all of the forces, and force couples are assumed to be in y-z plane, which ensures the beam axis to remain in y-z plane, together with the assumption of symmetric cross – section.

dy  A( z ) y  f ( z ) dz

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If the equation system (1) is represented in matrix form as below, (2)

where the vector y  y( z ) includes the deflection and internal load variables seen in Equation (1), A( z ) is 6  6 coefficient matrix, and f ( z ) is the resultant vector of external distributed

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loads, the solution can be written in the following form. z

y ( z )  Y( z, zo ) y o  Y( z, zo )  Y 1 ( , zo ) f ( )d zo

where

(3)

Y( z, zo ) is the principal matrix about the point zo, and yo is the initial values vector (yo

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= y(zo)). Taking zo = 0 simplifies the solution procedure. A more detailed information about the solution of such an equation system, and the principal matrix can be found in any

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differential equations book, such as Hubbard and West (1995).

0 0

z EA

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 1  0   Y  z, 0    0  0  0 0

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The principal matrix for our problem is given as below.

ky z

0 

z3 6 EI x

1 z

0

0 1

0

0 0

1

0

0 0

0

1

0 0

0

z

GA 

z2 2 EI x

 0   z2  2 EI x   z  EI x  0   0  1 

2.2 Finite element formulation 2.2.1 Stiffness matrix and consistent load vector The solution given in Equation (3) can be re-written in the following form. 7

(4)

ACCEPTED MANUSCRIPT  X   A( z ) B( z )  Xo   p1  z         F   S( z ) D( z )   Fo  p 2  z 

(5)

where  Fz    F   Fy  ; M   x

 wo    X o   uo  ;    xo 

 Fzo    Fo   Fyo  M   xo 

Y14 Y15 Y16  B( z )  Y24 Y25 Y26  Y34 Y35 Y36 

Y41 Y42 Y43  S( z )  Y51 Y52 Y53  ; Y61 Y62 Y63 

Y44 Y45 Y46  D( z )  Y54 Y55 Y56  Y64 Y65 Y66 

Y22 Y32

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Y13  Y23  ; Y33 

Y12

and if, z

E( z )  Y( z, 0)  Y 1 ( , 0) f ( ) d

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 E4  p 2  z    E5   E6 

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 E1  p1  z    E2  ;  E3 

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A beam element with a length of

(6)

(7)

M

0

then,

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w   X   u ;    x Y11 A( z )  Y21 Y31

(8)

ze is given in Figure 2. Considering the difference between

the positive direction assumption in analytical and finite element approaches (see Figure 1,

z A  zo  0 and zB  ze , equation (5) can be written as follows.

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and Figure 2), and assuming

X B   A( ze ) B( ze )   X A   p1  ze      F   p z  S ( z ) D ( z ) F e   B  e A   2  e 

(9)

With the minus sign in front of FA, the force vectors FA and FB become the external nodal loads. After some algebra, Equation (9) can be arranged into the following form.

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ACCEPTED MANUSCRIPT FA   B( ze ) O   p1  ze    B( ze ) O   A( ze ) -I  X A           FB   D( ze ) I  p 2  ze   D( ze ) I   S( ze ) O  X B  1

1

(10)

where I is 3  3 identity matrix, and O is 3  3 zero matrix. Equation (10) is nothing but the exact solution represented in with a different notation. This representation makes it easier to implement any transition condition, as encountered in cracked beam modelling, without losing the exactness of the solution. Using the analogy between the well – known basic finite

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element formulation, the stiffness matrix of a beam element Kel, and the consistent load vector fel is obtained as below. 1

 B( ze ) O   A( ze ) -I  K      D( ze ) I   S( ze ) O  el

 B ( ze ) O    p1  ze    f el       D ( z ) I p z  e     2  e 

(11)

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1

It should be noted that the exact solution of the homogeneous beam equations are used as shape functions. External distributed loads contribute to the nodal loads, as expected.

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Therefore, it can be said that even if the displacement values on the nodes are equal to the exact solution, their variations within the beam element differ from the exact solution if an

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2.2.2 Mass matrix

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external distributed load is applied.

The un – damped free vibration equation is known as, (12)

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Mu  K u  0

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where M and K are the mass matrix and the stiffness matrix of the structure, respectively, overdot stands for the time derivative, and u is the displacement vector. For harmonic motion,

u  ueit

where



(13) and t stand for the frequency and time, respectively. Then, the Equation (12)

becomes as below.

(K   2M)u  0

(14) 9

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det(K   2M)  0

(15)

gives the natural frequencies. Kinetic energy of the beam element, including the rotatory inertia, is known as, 1 B XT ( z, t ) X( z, t ) T  P dz 2 zA t t

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z

(16)

where  is the density of the material, and the matrix P is as below.

0 A 0  0 I x  0

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A P   0  0

(17)

X( z)   A( z) B( z) y o  H( z)y o

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The shape functions for the displacements and rotation are as follows. (18)

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If the displacement components of each end of the beam element shown in Figure 2 are denoted as a vector u el , the following relations are obtained.  X( z A )   H( z A )  uel     yo  G yo  X( zB )   H( zB ) 

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(19)

y o  G 1uel

(20)

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Then, the equation (18) becomes,

X( z)  H( z)G 1uel

(21)

Considering the displacement is in the form given in Equation (13), if one places the displacement vector given in (21) to Equation (16), z

B T 1 T   2  uel   (G 1 )T  HT ( z )P H( z )dz G 1u el 2 ZA

10

(22)

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1 2 el T el el  u  M u 2

(23)

the mass matrix of a beam element is obtained as below. M el   (G 1 )T

zB

H

T

( z )P H( z )dz G 1

(24)

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zA

It should be noted that, analogous to the stiffness matrix, the exact solution of the displacement components with no external distributed loads are used as the shape functions to obtain the mass matrix. Therefore, a single element is not expected to represent the mode shapes of free vibration. That is why the convergence of the natural frequencies should be

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investigated to find the minimum number of elements needed to obtain reliable results within a certain frequency range.

2.3 Application of the formulation to frame structures

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The presented formulation can also be adopted to frame structures. Consider a part of a frame, as shown in Figure 3.

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The transition condition is as follows.

 T O  T O y i 1  0   y A   y z    i e  O T  y A O T   

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(25)

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where ze denotes the element length, and the matrix T is defined as,

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 cos  0 sin   T   0 1 0    sin  0 cos  

(26)

Then, deflections and internal loads at point B is written as, ze

y B  Y  ze , 0  y A  Y  ze , 0   Y  , 0 f   d

(27)

0

Using Equation (25), 11

ACCEPTED MANUSCRIPT ze  T O y B  Y  ze , 0    y A  Y  ze , 0   Y  , 0 f   d O T  0

(28)

is obtained. Using the similar manipulations in Equations (5-11), stiffness matrix and consistent load vector for a rotated element are obtained as follows. 1

1

K

el rot .

1

 T O   B( ze ) O   A( ze ) -I   T O   T O   T O   K el           O I   D( ze ) I   S( ze ) O  O I  O I  O I 

 B ( ze ) O    p1  ze    el    f   D( ze ) I   p 2  ze   

el f rot .

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1

(29)

It should be noted that the transformation is not identical to those encountered especially in

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standard FEM textbooks since the Frenet coordinate system is used in this study. Similarly, mass matrix is transformed as follows. 1

2.4 Transversely cracked beams

(30)

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 T O  T O M elrot .   M el    O I  O I 

energy has the form U   JdAc

(31)

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Ac

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The local compliance due to a crack is modelled through an energy approach. The strain

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where Ac is the area of the crack, and J is the strain energy release rate (Tada et al. 1973) which is known as

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2 2 2  m  m   1  m J    K I j     K II j   k y   K III j   E '  j 1   j 1   j 1   

(32)

where E '  E 1  2  , or E '  E for plane strain, and plane stress, respectively, and Kij are the SIFs of ith mode for the load j (i = I, II, III; j = Fz, Fy, Mx). For in – plane problems, m = 3. By definition, the local compliance is written in the following form (Papaeconomou and Dimarogonas, 1989).

12

ACCEPTED MANUSCRIPT

cij 

2 Pi Pj

    JdAc  A   c 

(33)





where Pi is the ith load P  Fz

Fy



M x . T

SIFs are known as below (Tada et al., 1973).

K I1   Fz A   aF1  a h 

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K I2  0

K I3   M x h  2 I x    aF2  a h  K II1  0

K II2   k y Fy / A   aFII  a h 

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K II3  0 where,

2h tan

(34)

a

2h tan

a

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F1  a h  

M

3 2h  0.752  2.02 a  0.37 1  sin  a      a h 2h     a cos 2h

(35)

AC

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4 2h  0.923  0.199 1  sin  a      a 2h     F2  a h   a cos 2h 2 3  a a a  1.22  0.561  0.085  0.18       h h h   FII  a h   a 1 h

A beam element with a crack on its node B is shown in Figure 4. At the crack location, the following transition conditions are written.

 Fz  ze    Fz  ze             F z  F z  y e   y e         M x  ze    M x  ze 

(36)

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ACCEPTED MANUSCRIPT  w  ze      c11      u  ze    c21   c   31  z     x e  

c12 c22 c32 C

   c13   Ft  ze    w  ze       c23   Fn  ze     u  ze      c33    M x  ze    x  ze    

(37)

or in matrix form,  I C y  ze    y  ze   0 I  

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

(38)

where ze , and ze denote the coordinates just after, and just before the cracked cross – section, respectively. 2.5 Finite element formulation of cracked beams

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In the absence of external distributed loads, using Equation (38), the displacements, and internal loads for the right face of the cracked cross –section  z  ze  is written as follows.

(39)

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y B

ze   I C  1   Y  ze ,0  y A  Y( ze ,0)  Y ( ,0) f ( )d   0 I   0 

Analogous to Equation (9), Equation (38) can be written as follows.

(40)

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X B   I C  A( ze ) B( ze )   X A  p1  ze   Cp 2  ze        p 2  ze   FB  0 I   S( ze ) D( ze )  FA   

After some algebra, the stiffness matrix, and the consistent load vector of a cracked beam

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element is obtained as below.

1

B( ze )  CD( ze ) O   A( ze )  CS( ze ) -I  K  D( ze ) I   S ( ze ) O  

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el C

1

B( ze )  CD( ze ) O   p1 ( ze )  Cp 2 ( ze )  fCel     D( ze ) I   p 2 ( ze )  

(41)

It is seen from Equation (41) that the effects of a transverse crack is also observed in the consistent load vector. This is due to the coupling between axial, and bending deformations, e.g. a normal force creates also a bending moment at the cracked section. 2.6 Genetic algorithms

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ACCEPTED MANUSCRIPT As stated in the literature survey, it is known that the detection of crack location and depth can be considered as an optimization problem. GA is an optimization technique that simulates the evolution of a population for survival, based on the principles of genetics. It basically applies the rules of natural selection on a population, which consists of the information about the crack location and depth for our problem, to find the best individual. GA is found to be more advantageous than other classical optimization techniques since, for example, it does not require a derivative information, can deal with large number of variables, and can optimize a

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variable even if the cost surface is complex, thanks to its ability to jump out of local minimums (Haupt and Haupt, 2004). More detailed explanations can be found in the studies of Goldberg (1989), and Haupt and Haupt (2004), for example. However, a brief explanation of some of the important steps of GA related to the problem of this study is given below.

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Initial population: The starting point of the GA is to create a set of chromosomes, so – called the population. A chromosome consists of the variables in the optimization problem, which are the crack location and depth in this case. The number of chromosomes in the initial population is arbitrary, and the values of the variables are random. As the initial population gets more crowded, the chance of obtaining the best individual in less number of iterations

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increases, however, it also means more computational time. In this study, initial population size is selected as 15 times of the number of variables (Mehrjoo et al. 2014; Khaji and

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Mehrjoo 2014).

Cost function: Cost function refers to the function to be minimized. In this study, the cost

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function is defined similar to defined by Casciati (2008). Only the part of the cost function that is related to the natural frequencies is considered. That is, 2

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 iex  ic  f1   wi   ex i 1  i 

(42)

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n

where n is the number of frequencies used in calculations, ic and iex are calculated and measured frequencies, respectively, and wi is a weighting factor. This factor is defined as wi = 1/i

(43)

in order to reduce the effect of measurement noise which affects higher modes more than the lower ones (Casciati 2008). Additionally, a second cost function is considered in order to see the effectiveness of the presented method, which is defined as the absolute value of sum of the relative errors between the measured and calculated natural frequencies. This is given by 15

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n

f2   1  i 1

ic iex

(44)

The details, and more information about the steps of GA can be found in Haupt and Haupt (2004). The algorithm continues with natural selection, crossover, and mutation, until obtaining a new population (see Figure 5). The algorithm stops when the cost value of the best chromosome

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reaches a certain level 103  , or if the average change in the cost value of best chromosome is lower than 106 for 50 iterations, or if maximum number of iterations is achieved (100 times number of variables). 3. Numerical examples

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In this section, the proposed method is tested through some numerical examples, related to cracked, and un – cracked beams, existing in the literature.

Example 1: Vakil – Baghmisheh et al. (2008) represented the first 4 natural frequencies of an intact cantilever beam with 82 cm length, 2 cm width, 1 cm height, and 2700 kg/m3 density.

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Elastic modulus is taken to be 70 GPa, and Poisson’s ratio is 0.33. Variation of the first 4 natural frequencies with number of elements used is given in Figure 5. ref ,i stands for the ith

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frequency, obtained with only using 5 elements. Such a normalization is performed to be able to see the effects of number of elements on all 4 frequencies more clearly. It is seen from

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Figure 6 that results converge after about 20 elements. Table 1 shows the comparison of the results with the ones existing in the literature. It should be noted that 80 elements are used to

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obtain more accurate results. It is seen from Table 1 that the results are in a good comparison with the analytical solution. Since it is a slender beam, effects of shear deformation and

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rotatory inertia have limited effect on the frequencies. Example 2: Mehrjoo et al. (2013) considered a beam with 4 m length, 0.1 m width, and 0.2 m height. Material is assumed to be steel with an elasticity modulus of 200 GPa, and Poisson’s ratio of 0.3. The beam has a crack with 80 mm depth located 1.5 m from the left support (see Figure 7). Natural frequencies are calculated for 2 different boundary conditions: Clamped – Free (CF), and simply supported (SS). Results are given in Table 2, with comparison. The effects of shear deformation on the natural frequencies are seen more clearly for higher modes.

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ACCEPTED MANUSCRIPT Example 3: This example is to illustrate the effects of shear deformation and rotatory inertia on the natural frequencies. Experimental results of Ruotolo and Surace (1997), as well as the FEM results of Mehrjoo et al. (2014) are used for validation. The cantilever beam has following properties: 20  20 mm2 cross – section, 800 mm length, two transverse cracks located 254 mm and 545 mm from the clamped end with depths of 4 mm and 6 mm, respectively. Beam is made of C30 steel.

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3 cases are considered: (1) Shear deformation and rotatory inertia effects are taken into account. (2) Shear deformation is neglected. (3) Rotatory inertia is neglected.

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The results are given in Table 3, with comparison. It is seen that results of this study, including all effects, are very close to experimental results. Shear deformation has a significant effect on especially the high modes. Rotatory inertia does not have a major effect for the considered modes of this problem.

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Example 4: This example is to show the capability and the accuracy of the present method on modelling different boundary conditions, different crack locations and depth ratios (Figure 8). The beam is assumed to have 25 mm height (h = 25 mm), and 12.5 mm width. The material of

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(Khaji et al. 2009).

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the beam has elastic modulus of 210 GPa, Poisson’s ratio of 0.3, and density of 7860 kg/m3

Table 4 shows the comparison of the results of present study with exact solutions, and 2-D

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FEM solutions, performed by Khaji et al. (2009), for different crack depths, and boundary conditions. The length of the beam is assumed to be 100 mm, as in the corresponding reference. Even though the dimensions of the structure is not within the limits of a beam, the

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results are in comparison with the 2-D FEM solution. The average of errors for all boundary conditions, and crack depths are given in Table 5. 2DFEM solution is taken as the reference, and relative errors of exact results and present results are calculated accordingly. It is seen that relative errors are close to each other for both approaches, up to η = 0.5. For the deeper crack, error is lower for present approach if there is no pinned end. If at least one of the boundaries are pinned, exact solution yields better results. This is due to fact that the effect of crack is modelled only for bending in the corresponding reference. In the present approach, the effect of the crack on the shear behavior is also 17

ACCEPTED MANUSCRIPT considered. Since the structure should not be assumed as a beam, as mentioned before, the effects of the shear stresses on the cross – section cannot be captured well enough via beam theory approach. Therefore, the effect of the crack on shear stiffness, which becomes more important for deeper cracks, cannot be modelled accurately. The numerical experiments show that if the crack is modelled only with a coil spring, the results improve dramatically, for this specific example. But the results of that case are not presented here, to keep the integrity of

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the study. Table 6 shows the natural frequencies for different crack locations, and boundary conditions. The crack depth is the same for all problems (η = 0.5). The length of the beam is assumed to be 175 mm, as in Khaji et al. (2009). The results are in a better agreement with the 2D FEM solution this time, due to the fact that the length to the height ratio is 7 for this problem, and it

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can be said that the structure can be approximated as a very thick beam.

Average relative errors of both approaches for different crack locations are given in Table 7. 2D FEM solution is taken as the reference again. It is seen that present approach yields very satisfactory results for different boundary conditions, and crack locations.

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Example 5: Two bay, single storey frame structure is considered in this example, to validate the applicability of the present formulation to frames. The structure is shown in Figure 9. Both intact, and damaged configurations are considered. The cross – section of the members are

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assumed to be 198 x 122 mm. The length of the members are L = 12 m, and the material of the structure is assumed to be steel with E = 206 GPa, and ν = 0.3. Mass per unit length is taken

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to be 185.4 kg/m. The same problem is also considered in the works of Labib et al. (2014), and Caddemi and Calio (2013). 3 different cases, as mentioned in Example 3, are considered.

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Natural frequencies of intact frame is given in Table 8, for all cases. For Case (3), which excludes the effects of shear deformation, and rotatory inertia, the results are almost identical

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to those of Labib et al. (2014), since those effects are not considered in their study. The results of Caddemi and Calio (2013) are slightly higher than present study, and Labib et al. (2014), due to fact that the axial deformation is also neglected in that work. For the damaged frame, the ratio of crack depth to the height of the section is taken as 0.9, for all cracks. Along with the presented crack formulation, the effect of the crack is also modelled as in Labib et al. (2014) (i.e. crack is modelled by using only a rotational spring, with a spring constant given in the corresponding reference). The results of both approaches are given in Table 9. It is seen that the present formulation gives almost identical results to those obtained 18

ACCEPTED MANUSCRIPT by Labib et al. (2014), when the beam, and the crack is modelled as in that study. If the structure is modelled as presented in this study, the results changes considerably. The difference between the results increase for higher modes. Example 6: In this example, the inverse problem of Example 3 is solved. Experimental results fo Ruotolo and Surace (1997) are used as the inputs, and the crack parameters are calculated according to the formulation presented in this study.

performs quite satisfactory, as the results are highly accurate. 4. Experiments 4.1 Experimental setup

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Estimated crack parameters are shown in Table 10. It is seen that the present formulation

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Even though there are many studies performing experiments of such structures, as mentioned earlier, most of them consider first 3 natural frequencies to solve the inverse problem. In this study, to see the effect of number of natural frequencies on the accuracy of crack detection, first 5 natural frequencies are determined experimentally.

The experimental setup is shown in Figure 10. The beam is excited by Brüel&Kjaer (BK)

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8206 Impact Hammer. Response of the structure is measured by CLV – 2534 Compact Laser Vibrometer. Both signals are acquired using the data acquisition hardware BK PULSE 3050-

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A-060, and the software BK – PULSE LabShop. To identify the modal parameters, the software ME’SCOPE 5.0™ is utilized.

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Experiments are performed on St32 steel beam with rectangular cross – section. Height, and

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width of the cross – section are 20 mm, and 8 mm, respectively. The beam has a length of 80 cm, and is clamped at one end using a milling vise. 2 specimens are used during the tests. One of them has no cracks or crack – like defects, and the other one has ac = 3 mm depth crack in a

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distance of Lc = 50 cm from the clamped end. Wire erosion with 0.1 mm diameter is used to obtain about 0.15-0.18 mm thick notch (Figure 11). Both of the beams are excited with the hammer from 8 equidistant points, and the response is measured at a fixed point in a distance of 30 cm from the clamped end. Frequency Response Functions (FRFs) of each excitation point are obtained, and the natural frequencies of the structure is determined by averaging the results of all 8 measurements. Frequency span is chosen as 1.6 kHz to capture the first five frequencies of the structure. FRFs of all points of the cracked beam can be seen in Figure 12. 19

ACCEPTED MANUSCRIPT 4.2 Results The first 5 natural frequencies of cracked, and un – cracked beam is given in Table 11. It is seen that relative error between the results is about %1 at most for both cracked, and un – cracked beams. The results for the cracked beam are used to solve the inverse problem. Effects of the cost function, number of natural frequencies taken into account n, shear deformation, and rotatory

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inertia on the accuracy of the crack detection is investigated. Aforementioned 3 cases, which includes or excludes the effects of shear deformation, and rotatory inertia, are considered. Also, the inverse problem is solved considering the first 3, 4, and 5 natural frequencies, for each case. Variations of the best cost value, calculated by using f1, through generations are given in Figure 13.

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The estimated crack location, and depth are given in Table 12. It is seen that the accuracy of the estimations on the location of the crack highly depend both on the number of frequencies taken into account, and on the beam model. If shear deformation and rotatory inertia are taken into account, the optimization process yields better results for the position of the crack, whereas if the shear deformation is not taken into account, only the frequencies that are

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bending – dominant must be considered. It is observed that the crack parameters are not

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estimated accurate enough for n = 3,4 in Cases 2, and 3, and for n = 3, for Case 1. This is thought to be due to measurement errors, and uncertainty of the material properties, considering the present formulation yielded satisfactory results in Example 6. It can be said

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that the best estimation is for the Case 1 with n =5, for both cost functions, which is the main contribution of the present study.

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5. Conclusion

This paper introduces a new FE model for cracked beam – like structures. The formulation

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includes shear deformation, and rotatory inertia. The formulation also includes the axial – bending coupling, arising due to the crack, which effects the dynamic behavior of the frame structures. The validity of the model is shown via some numerical examples existing in the literature, as well as an experimental study. It is seen that the formulation yields excellent results, even for higher modes of vibration than generally considered in crack detection problems. The problem of crack detection is also revisited. Experimental results of a previous study are used to estimate the crack parameters of a double – cracked beam. Additionally, a set of 20

ACCEPTED MANUSCRIPT experimental modal analysis is performed to obtain the natural frequencies of a single – cracked beam, to use as the input of the inverse problem. GA is utilized to solve the inverse problem. Effects of shear deformation, rotatory inertia, and number of natural frequencies considered, on the accuracy of the crack detection are investigated. Unlike the majority of the past studies on this subject, up to first five frequencies are used to solve the inverse problem. It can be concluded that considering more frequencies yields a better estimations of the crack

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parameters, only if the cracked beam model is capable of modelling the considered modes well enough. When the first five natural frequencies are considered, the crack location, and depth are estimated by errors of %2, and %2.66, respectively. This shows the superiority of the presented FE formulation, especially in modelling the modes which shear deformations and rotatory inertia cannot be neglected.

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For a higher precision in crack detection in beam – like structures, presented formulation can be considered as an excellent tool, since it enables us to examine beams with any boundary conditions accurately, via a standard FEM procedure. This formulation can be incorporated with different optimization techniques for faster convergence of the inverse problem. Also, more advanced crack models, such as breathing crack, can be implemented for a better

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modelling.

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ACCEPTED MANUSCRIPT REFERENCES Adams RD, Cawley, P Pye CJ, Stone BJ, 1978. A vibration technique for non - destructively assessing the integrity of structures. Journal of Mechanical Engineering Science. 20, 93-100. Anyfantis N, Dimarogonas AD, 1983. Stability of columns with a single crack subjected to follower and vertical loads. International Journal of Solids and Structures. 19, 281-291. Caddemi S, Calio I, 2013. The exact explicit dynamic stiffness matrix of multi-cracked Euler-

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Bernoulli beam and applications to damaged frame structures. Journal of Sound and Vibration. 332, 3049-3063.

Caddemi S, Calio I, 2014. Exact reconstruction of multiple concentrated damages on beams. Acta Mechanica. 225, 3137-3156.

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Caddemi S, Calio I, Cannizzaro F, Rapicavoli D, 2013. A novel beam finite element with singularities for the dynamic analysis of discontinuous frames. Archive of Applied Mechanics. 83(10), 1451-1468.

Caddemi S, Morassi A, 2013. Multi-cracked euler-bernoulli beams: mathematical modeling

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and exact solution. International Journal of Solids and Structures. 50, 944-956. Cao M, Radzienski M, Xu W, Ostachowicz W, 2014. Identification of multiple damage in

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beams based on robust curvature mode shapes. Mechanical Systems and Signal Processing. 46, 468-480.

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Casciati S, 2008. Stiffness identification and damage localization via differential evolution algorithms. Structural Control and Health Monitoring. 15, 436-449.

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Chang HY, Petroski HY, 1986. On detecting a crack by tapping a beam. International Journal of Pressure Vessels and Piping. 22, 41-55.

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Chinchalkar S, 2001. Determination of crack location in beams using natural frequencies. Journal of Sound and Vibration. 247(3), 417-429. Chondros TG, Dimarogonas AD, 1979. Identification of cracks in circular plates welded at the contour. Design Eng. Tech. Conf., ASME, St. Louis/MO, paper 79-DET-106. Dimarogonas AD, 1976. Vibration engineering. St. Paul/MN: West. Dimarogonas AD, 1996. Vibration of cracked structures: state of the art review. Engineering Fracture Mechanis. 55, 831-857. 22

ACCEPTED MANUSCRIPT Goldberg DE, 1989. Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading, MA. Hakim SJS, Abdul Razak H, Ravanfar SA, 2015. Fault diagnosis on beam-like structures from modal parameters using artificial neural networks. Measurement. 76. 45-61. Harrison HB, 1973. Computer methods in structural analysis. Englewood Cliffs, NJ: Prentince Hall.

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Haupt RL, Haupt SE, 2004. Practical genetic algorithms. 2nd ed., John Wiley and Sons, New Jersey.

He Y, Guo D, Chu F, 2001. Using genetic algorithms and finite element methods to detect shaft crack for rotor bearing system. Mathematics and Computers in Simulation. 57, 95-108.

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Hubbard JH, West BH, 1995. Differential equations. In: A dynamical systems approach: higher dimensional systems (with Beverly west). Text in applied mathematics, No:18. NY: Springer-Verlag.

Irwin, GR, 1957 Analysis of stresses and strains near the end of a crack traversing a plate.

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Journal of Applied Mechanics. 24, 361-364.

Khaji N, Shafiei M, Jalalpour M, 2009. Closed-form solutions for crack detection problem of

Sciences. 51, 667-681.

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Timoshenko beams with various boundary conditions. International Journal of Mechanical

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Kirshmer PG, 1944. The effect of discontinuities on the natural frequency of beams. Proc. ASTM 44, 897-904.

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Labib A, Kennedy D, Featherson C, 2014. Free vibration analysis of beams and frames with multiple cracks for damage detection. Journal of Sound and Vibration. 333, 4991-5003.

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Liang RY, Choy FK, Hu J, 1991. Detection of cracks in beam structures using measurements of natural frequencies. Journal of Franklin Institute. 328, 505-518. Mazanoglu K, Sabuncu M, 2012. A frequency based algorithm for identification of single and double cracked beams via a statistical approach used in experiment. Mechanical Systems and Signal Processing. 30, 168-185.

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ACCEPTED MANUSCRIPT Mehrjoo M, Khaji N, Ghafory-Ashtiany M, 2013. Application of genetic algorithm in crack detection of beam-like structures using a new cracked euler-bernoulli beam element. Applied Soft Computing. 13,867-880. Mehrjoo M, Khaji N, Ghafory-Ashtiany M, 2014. New timoshenko-cracked beam element and crack detection in beam-like structures using genetic algorithm. Inverse Problems in Science and Engineering. 22(3), 359-382.

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Moezi SA, Zakeri E, Zare A, Nedaei M, 2015. On the application of modified cuckoo optimization algorithm to the crack detection problem of cantilever euler-bernoulli beam. Computers and Structures. 157, 42-50.

Moradi S, Razi P, Fatahi L, 2011. On the application of bees algorithm to the problem of

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crack detection of beam-type structures. Computers and Structures. 89, 2169-2175.

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Pandey AK, Biswas M, Samman MM, 1991. Damage detection from changes in curvature

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mode shapes. Journal of Sound and Vibration. 145(2), 321-332.

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Mechanics. 5, 88-94.

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Engineering Fracture Mechanics. 70, 1553-1572. Pokale B, Gupta S, 2014. Damage estimation in vibrating beams from time domain

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measurements. Archive of Applied Mechanics. 84, 1715-1737. Qian GL, Gu SN, Jiang JS, 1990. The dynamic behaviour and crack detection of a beam with

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a crack. Journal of Sound and Vibration. 138(2), 233-243. Rizos PF, Aspragathos N, Dimarogonas AD, 1990. Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration. 138(3). 381-388. Rubio L, Fernandez-saez J, Morassi A, 2015. Crack identification in non-uniform rods by two frequency data. International Journal of Solids and Structures. 75-76, 61-80.

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ACCEPTED MANUSCRIPT Salawu OS, 1997. Detection of structural damage through changes in frequency: a review. Engineering Structures. 19(9), 718-723. Saridakis KM, Chasalevris AC, Papadopoulos CA, Dentsoras AJ, 2008. Applying neural networks, genetic algorithms and fuzzy logic for the identification of cracks in shafts by using coupled response measurements. Computers and Structures. 86, 1318-1338. Surace C, Archibald R, Saxena R, 2013. On the use of polynomial annihilation edge detection

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for locating cracks in beam-like structures. Computers and Structures. 114-115, 72-83.

Tada H, Paris P, Irwin G, 1973. The stress analysis of cracks hanbook. Del. Research Corp., Hellertown/PA.

Vakil-Baghmisheh MT, Peimani M, Sadeghi MH, Ettefagh MM, 2008. Crack detection in

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beam-like structures using genetic algorithms. Applied Soft Computing. 8, 1150-1160.

Vakil-Baghmisheh MT, Peimani M, Sadeghi MH, Ettefagh MM, Tabrizi, AF, 2012. A hybrid particle swarm-nelder-mead optimization method for crack detection in cantilever beams. Applied Soft Computing. 12, 2217-2226.

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Xiang J, Zhong Y, Chen X, He Z, 2008. Crack detection in a shaft by combination of waveletbased elements and genetic algorithm. International Journal of Solids and Structures. 45,

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4782-4795.

Fy

A Mx

B

Fz

Fz

Mx Fy

Figure 1. Straight beam, and positive internal load directions in analytical approach

A

25 Mx

B

Fz Fy

Mx

Fz

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Figure 2. Positive direction assumption in finite element approach

A-

C

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A+ ϴ

zi

zi+1=0

zi+1

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zi+1=ze

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Figure 3. Rotated element

B

A

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a

z = ze

z = zA = 0

Figure 4. Cracked beam

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B

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Start

Create Inital Population

Update Population

No

Natural Selection

Yes

End

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Best Individual

Crossover

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Stopping Criteria

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Evaluate Cost Function for Each Individual

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Figure 5. GA Flowchart

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Mutation

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1

ωi/ωref,i

0.995 i=1 i=2

i=3

0.985 10

15

20

25

30

35 40 45 50 No. of Elements

55

60

65

70

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5

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0.99

75

80

Figure 6. Convergence of natural frequencies.

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1.5 m

0.2 m

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4m

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Figure 7. Cracked beam considered in Example 2

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Figure 8. Cracked beam considered in Example 4

L

L

28 L

i=4

a2

L/2

L/2

a3

L/2

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L/2

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L/2 a1

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L/2

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(b)

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Figure 9. Two bay, single storey frame structure. (a) Intact; (b) Damaged.

sadasdasd

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L/2 a4 L/2

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Figure 10. Experimental setup. (a): Cantilevered beam and laser vibrometer. (b) Data acquisition hardware and impact hammer. (c) Vibrometer hardware

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Figure 11. Close – up of the crack

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Figure 12. FRFs of cracked beam

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ACCEPTED MANUSCRIPT Figure 13. Variation of the best cost function values with generation

This Study 12.2312 76.5972 214.2298 419.1086

PT

ED

1 2 3 4

Vakil – Baghmisheh et.al (2008) 12.2326 76.6606 214.6518 420.6318

M

Mode No.

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Table 1.

Mehrhoo et.al (2013) Boundary Mode This Study Conditions No. Beam element 2D FEM 1 9.78 9.71 9.79 CF 2 61.58 60.08 60.47 3 174.97 165.67 166.75 1 26.75 26.29 26.66 SS 2 111.64 108.11 108.89 3 261.19 244.81 246.36 Comparison of natural frequencies for intact beam (in Hz.)

AC

CE

Table 2. Natural frequencies of a cracked beam with different boundary conditions (in Hz.)

33

ACCEPTED MANUSCRIPT Table 3: Natural frequencies of a double cracked beam (in Hz.)

Ruotolo and Surace (1997)* 24.044 149.268 409.287 818.150

Mehrjoo et.al (2014) 23.962 148.660 412.464 -

(1) 24.044 148.731 408.729 817.353

*

(3) 24.045 148.759 409.091 818.834

AN US

Experimental results

This Study (2) 24.053 149.108 411.134 826.651

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Mode No. 1 2 3 4

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Table 4: Comparison of natural frequencies with exact solution, and 2D FEM method for different crack depths, and boundary conditions (in Hz.)

η

Exact* f

pinned - clamped

clamped - clamped

2-D Presen 2-D Presen 2-D Presen 2-D Presen Exact* Exact* Exact* FEM* t FEM* t FEM* t FEM* t

1948.2 1955.4 1949.0 4911.8 4821.5 4933.2 7139.0 7103.7 7152.0 9408.6 9525.4 9426.5

1

17603. 17487. 19600. 19167. 19491. 21397. 16417 8 3 2 2 4 5 0.2 f 22962. 23222. 30952. 33026. 31028. 32236. 30470. 32310. 33378. 22807 3 5 5 8 4 1 4 1 1 3 f 35888. 35800. 47851. 48771. 47266. 48519. 43563. 47887. 49141. 35711 7 1 1 8 4 2 5 9 0 4 f 1856.8 1876.4 1859.6 4221.3 4302.9 4270.4 6590.5 6690.7 6614.1 8859.9

PT

f

pinned - pinned

ED

clamped- free

9393.7 9416.0 9394.5

AC

1

CE

2

f 8350.7 8369.0 0.3 2 5 f 22803. 22978. 8 4 3 f 34219. 33570. 7 7 4 f 1704.9 1682.5

21740. 21285. 7 8 33792. 33462. 1 5 49654. 48458. 1 3 9120.6 8897.2

17603. 16176. 17229. 19553. 18860. 19198. 21397. 21642. 21028. 8 9 0 4 1 0 5 6 9 29030. 29165. 30280. 28714. 30409. 31313. 31784. 31461. 22315 31576 1 6 1 1 5 1 1 3 47851. 47272. 45741. 48491. 54350. 46233. 49141. 49442. 46726. 33559 1 9 2 9 1 2 0 2 4

8249.5

1705.7 3424.1 3618.0 3473.4 6011.1 6203.8 6014.5 8296.6 8703.2 8329.1

1

f

17603. 15980. 16782. 19501. 18751. 18710. 21397. 21299. 20574. 8 9 4 3 7 6 5 5 3 0.5 f 22646. 22390. 21637. 27363. 30635. 27472. 28609. 27080. 28707. 29560. 29523. 29680. 5 5 3 5 5 8 7 2 5 3 0 2 3 7268.6 6747.7 6944.0

2

f 32826. 31050. 31727. 47851. 47206. 42831. 48465. 52885. 43177. 49141. 46675. 43570.

34

ACCEPTED MANUSCRIPT 3

4

f

1

2

1

5

7

1

9

1

0

2

5

1446.2 1230.3 1371.4 2535.8 2977.2 2426.2 5457.5 5786.3 5325.6 7777.3 8523.6 7722.5

1

f

17603. 18080. 15708. 19449. 17483. 17585. 21397. 20313. 19430. 8 6 7 1 9 5 5 2 6 0.7 f 22507. 21024. 19737. 26078. 27025. 25979. 27345. 25122. 27227. 28237. 27793. 28143. 6 1 0 8 1 4 5 1 0 8 3 4 3 f 31801. 28882. 29923. 47851. 49439. 37023. 48441. 47405. 37298. 49141. 40391. 37660. 1 0 1 1 0 7 1 7 8 0 8 1 4 6261.2 5041.3 5341.9

2

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*Results are taken from the study of Khaji et al. (2009)

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Table 5: Average relative errors of exact solution (Khaji et al. 2009) and present approach for different crack depths and boundary conditions, taking the 2D FEM solution as the reference.

Pinned – clamped Exact Present 4.599 4.584 5.58 5.944 5.379 6.909 6.599 9.56

Clamped – clamped Exact Present 1.282 1.629 1.551 2.948 2.626 3.722 8.51 5.442

PT

ED

M

Average relative errors (%) Clamped – free Pinned – pinned η Exact Present Exact Present 0.2 0.497 0.648 4.301 4.493 0.35 0.984 1.311 5.004 4.533 0.5 3.755 2.458 7.047 7.151 0.7 12.546 6.79 6.766 15.152

AC

CE

Table 6: Comparison of natural frequencies with exact solution, and 2D FEM method for different crack locations, and boundary conditions (in Hz.)

clamped- free

α

pinned - clamped

clamped - clamped

Exact*

2-D FEM*

Present

Exact*

2-D FEM*

Present

Exact*

2-D FEM*

Present

Exact*

2-D FEM*

Present

480.5

470.6

485.3

1610.8

1635.3

1617.5

2351.9

2415.7

2364.2

3819.1

3823.8

3779.1

f2 3856.5

3846.9

3808.2

5401.0

5409.7

5439.1

6541.8

6647.4

6579.9

8743.6

8783.2

8651.7

f3 9043.9

9374.9

9395.5 12108.1 12172.6 12085.1 13545.7 13792.0 13491.9 14691.4 15815.5 14707.7

f1 0.2

pinned - pinned

f4 15300.9 15667.8 15622.4 20953.3 21415.9 20579.2 22184.6 21863.4 21742.9 22849.1 23226.0 22674.5 0.4 f1

569.1

562.6

572.1

1376.5

1423.8

1391.1

35

2189.9

2264.2

2206.3

3395.3

3481.7

3401.4

ACCEPTED MANUSCRIPT

f2 3224.6

3157.4

3183.8

6328.9

6245.5

6294.8

7855.4

7821.9

7778.3

8679.5

8782.9

8644.7

f3 9176.2

9381.2

9377.5 12895.4 12355.3 12726.3 13665.9 13239.5 13557.2 15474.0 15702.0 15233.3

f4 15880.3 16019.1 15803.4 19535.3 20986.5 19505.9 20984.8 19809.8 20866.7 21518.7 21353.3 21503.3 f1 0.6

640.1

638.7

640.9

1376.5

1423.8

1391.1

2541.1

2569.8

2538.4

3395.3

3481.7

3401.4

f2 2938.5

2860.6

2924.2

6328.9

6245.5

6294.8

7186.7

7163.8

7187.2

8679.5

8782.9

8644.7

f3 8674.8

8921.7

8916.6 12895.4 12355.3 12726.3 14692.1 14522.8 14442.8 15474.0 15702.0 15233.3

f4 16605.3 16367.0 16124.7 19535.3 20986.5 19505.9 20234.2 19689.0 20254.6 21518.7 21353.3 21503.3 670.2

668.2

1610.8

1635.3

1617.5

2718.1

2708.7

f2 3561.8

3524.7

3548.7

5401.0

5409.7

5439.1

7732.5

7640.7

f3 7683.8

8037.2

8111.2 12108.1 12172.6 12085.1 13478.4 14041.5 13513.2 14691.4 15815.5 14707.7

2696

3819.1

3823.8

7640.8

8743.6

8783.2

3779.1 8651.7

f4 14117.7 13664.5 13677.6 20953.3 21415.9 20579.2 21672.9 21260.0 21.502.8 22849.1 23226.0 22674.5

*Results are taken from the study of Khaji et al. (2009)

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Table 7: Average relative errors of exact solution (Khaji et al. 2009) and present approach for different crack locations and boundary conditions, taking the 2D FEM solution as the reference.

ED

0.2 0.4 0.6 0.8

PT

α

Average relative errors (%) Clamped – free Pinned – pinned Exact Present Exact Present 2.092 1.16 1.106 1.564 1.584 0.978 4.093 3.286 1.788 1.026 4.093 3.286 2.284 0.499 1.106 1.564

Pinned – clamped Exact Present 1.898 1.469 3.135 2.712 1.324 1.243 1.904 1.344

Clamped – clamped Exact Present 2.469 3.011 1.495 1.892 1.495 1.892 2.469 3.011

CE

Natural frequencies of intact frame

AC

0.8

668.3

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f1

Frequency No. 1 2 3 4 5 6 7 8

Caddemi and Calio (2013) 0.5987 2.4667 3.1095 4.1894 -

Labib et al. (2014) 0.5987 2.4662 3.1080 4.1885 4.5085 4.5110 8.9315 10.0628 36

Case (1) 0.5986 2.4655 3.1067 4.1861 4.5052 4.5077 8.9233 10.0515

Present Case (2) 0.5987 2.4661 3.1079 4.1882 4.5082 4.5107 8.9210 10.0609

Case (3) 0.5987 2.4662 3.1080 4.1885 4.5084 4.5110 8.9315 10.0628

Ta ble 8:

ACCEPTED MANUSCRIPT -

11.3283 12.4102 12.7854

11.3128 12.3899 12.7645

11.3261 12.4079 12.7828

11.3283 12.4103 12.7855

CR IP T

9 10 11

*

Labib et al. (2014) 0.5919 1.7167 2.2836 3.2057 3.3609 4.2796 8.7921 9.5945 10.8836 12.4101 12.7785

ED

M

Caddemi and Calio (2013) 0.5919 1.7167 2.2845 3.2062 -

PT

Frequency No. 1 2 3 4 5 6 7 8 9 10 11

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Table 9: Natural frequencies of damaged frame

Present Case (1)* Case(3)** 0.5767 0.5919 1.2308 1.7165 1.8244 2.2833 2.6758 3.2051 2.8278 3.3602 4.2332 4.2795 8.7163 8.7918 9.3568 9.5935 10.6044 10.8823 12.3606 12.4100 12.7392 12.7784

AC

CE

Crack is modelled as presented in this study Crack is modelled only a rotational spring with a spring constant calculated as in Labib et al. (2014) **

Present Ruotolo and

Table 10: Estimation of the crack parameters in Example 6

Lc1 251 218

%error 1.18 14.2

Estimation Results (mm) ac1 %error Lc2 %error 3.79 5.25 548 0.55 2.60 35 581 6.6 37

ac2 5.56 7.40

%error 7.33 23.3

ACCEPTED MANUSCRIPT Surace (1997) Actual Results

254

4.0

545

6.0

un - cracked

Experimental 25.46 160.55 442.82 863.21 1404.9 25.432 158.96 439.84 863.93 1397.5

Present F.E. Method 25.43 158.89 442.80 861.81 1412.3 25.42 158.15 440.97 861.64 1405.2

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Lc = 50 cm ac = 3 mm

Mode No 1 2 3 4 5 1 2 3 4 5

AN US

Crack Parameters

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Table 11. Experimental and numerical natural frequencies (in Hz.)

ED

Table 12. Estimated crack parameters of the beam used in experimental study

AC

(1)

(2)

(3) Actual values (mm)

f1 Estimation Results (mm) Lc %error ac %error 695 39 6.56 118 545 9 2.69 10.3 510 2 3.08 2.66 630 26 5.09 69.7 595 19 4.36 45.3 700 40 6.70 123 665 33 6.71 124 595 19 4.60 33.3 700 40 6.98 133

f2 Estimation Results (mm) Lc %error ac %error 695 39 6.21 107 542 8.4 3.12 4.00 504 0.8 3.50 16.7 689 37.8 7.81 160 591 18.2 4.12 37.3 714 42.8 7.63 154 665 33.0 4.81 60.3 542 8.4 3.32 10.6 708 41.6 4.91 63.6

500

500

PT n 3 4 5 3 4 5 3 4 5

CE

Case

Cost function

3.00

38

3.00

AC

CE

PT

ED

M

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ACCEPTED MANUSCRIPT

39