Element stiffness index distribution method for multi-crack detection of a beam-like structure

Abstract

In this article, a method using “element stiffness index distribution” for multi-crack detection of a beam-like structure is presented. The element stiffness index distribution is defined as a vector of norms of sub-matrices corresponding to element stiffness matrices calculated from the reconstructed global stiffness matrix of the beam. When there is a crack at an element, the element stiffness index of that element will be changed. Therefore, by monitoring the change in the element stiffness index distribution, the crack can be detected. A significant peak in the element stiffness index distribution is the indicator of the crack existence. The crack location and the crack depth can be determined from the location of the peak and the height of the peak. In this study, the global stiffness matrix is reconstructed from the measured frequency response functions instead of mode shapes to avoid some limitations of the mode shape-based methods for crack detection. Numerical simulation results for the cases of single crack and double cracks are provided. The results showed that cracks with a depth as small as 10% of the beam height can be detected by the method. The proposed method can be applied with the noise level up to 10%.

Keywords

crack detection damage detection element stiffness element stiffness index distribution multi-crack detection stiffness method

Introduction

A crack may cause serious failure of a structure; therefore, early crack detection is extremely important. The subject of crack detection has concerned many researches in the last three decades as reviewed by Dimarogonas (1996). In practice, it is expensive to carry out a proper inspection of structures by visual methods. In contrast, cracks can be detected by non-destructive techniques which are based on changes in the dynamic properties such as frequency and mode shape which can be obtained from vibration signals since cracks may have a serious influence on them (Chondros and Dimarogonas, 1980). Since vibration signals of structures can be measured easily, non-destructive methods will provide considerable savings in both time and cost.

Many of the studies for crack detection were concentrated on the analysis of natural frequency in the case of open crack for simplification reasons. Liang et al. (1992) developed theoretical relationships between eigenfrequency changes and magnitudes, and the locations of crack-induced damage for beam structures with either simply supported or cantilever boundary conditions. Hu and Liang (1993) proposed an integrated approach to detect cracks using the changes in natural frequencies. The spring model and continuum damage model were combined to analyze crack positions and crack depths. Multi-crack detection method using finite element model of the structure and genetic algorithms was reported by Ruotolo and Surace (1997). Shen and Taylor (1991) proposed a crack detection approach based on the minimization of the difference between measured data and a computational model of a beam-type structure. Some authors (Lee, 2009; Loya et al., 2006; Rizos et al., 1990) modeled cracks as massless rotational springs, whose stiffness were calculated using fracture mechanics to investigate the natural frequencies and mode shapes of cracked beams. In practice, a crack may not be only open or remaining closed at all times, but it can open and close depending on the loading circumstance and the cracked structure. This type of crack is called a “breathing crack” and has been discussed in various papers (Chondros et al., 2001, 1998; Kisa and Brandon, 2000; Loutridis et al., 2005; Luzzato, 2003; Nguyen, 2013; Ruotolo et al., 1996; Sundermeyer and Weaver, 1995). The authors of these papers related the change in frequency to the breathing crack and presented techniques based on this phenomenon for crack detection.

Although the natural frequency can be obtained easily for detecting the existence of cracks, it is difficult for inspecting the location of cracks. Meanwhile, the mode shape is distorted locally at the crack location and it can be used for localizing the crack. Some authors focused on using mode shapes for crack detection. Pandey et al. (1991) proposed the damage detection method using mode shape curvature. The reduction in cross section caused by the damage leads to an increase in the curvature of the mode shapes in the vicinity of the damage. Kisa et al. (1998) presented a method for free vibration analysis of cracked beams using a combination of finite elements and component mode synthesis methods. The beam was divided into two components related by a flexibility matrix which incorporated the interaction forces. In other papers (Abdel Wahab, 1999; Kisa and Arif Gurel, 2006; Rizos et al., 1990; Zhong and Oyadiji, 2011), the change in curvature of mode shapes was used to determine the location of the crack. Some other authors were interested in using stiffness and flexibility matrices of structures obtained from mode shapes for crack detection purpose since these matrices are related directly to cracks in structures. Pandey and Biswas (1994) presented a method for crack detection based on the difference between the flexibility matrices of the un-damaged and damaged structures. The study showed that the method works best when the damage is located in the section where high bending moments occur. Jaishi and Ren (2006) used modal flexibility residual for damage detection. Recently, Caddemi and Caliò (2014) presented the method to determine the exact location of multiple cracks on using frequency and mode shape measurements. However, the above methods for crack detection using mode shapes have some limitations. The distortion of a mode shape caused by cracks is very small and it can only be inspected visually with a very large crack depth (Caddemi and Caliò, 2013; Kisa et al., 1998; Rizos et al., 1990). Mode shapes are usually obtained from measured frequency response functions (FRFs), so there will be an error in the calculation of mode shapes from measured vibration signals. There is also an additional error in the stiffness and flexibility matrices calculated based on mode shapes since only some fundamental modes can be measured with a high accuracy in practice.

To this end, this article aims to extent the state-of-art of structural damage detection by proposing a new technique using the structural stiffness matrix. In this approach, the “element stiffness index distribution” is defined and calculated from the measured FRFs, not from the mode shapes as the previous methods to avoid the above limitations of the mode shapes. In the proposed method, the existence, the location, and the depth of cracks can be detected and estimated. A significant change in the element stiffness index distribution will be an indicator of the existence of crack(s). The location of the crack is the element which exhibits a significant change in the stiffness index. The crack depth can be estimated from the relationship between the height of the peak and the crack depth. The proposed method can be applied for both single-crack and multi-crack detections. Numerical simulation is provided to investigate the efficiency of the proposed method. However, the proposed method has its limitations such as the approximation source of the Rayleigh assumption, which can influence the accuracy of the method.

Vibration of a beam-like structure

Intact beam

Let us consider the bending vibration of a beam-like structure. The governing equation of the beam-like structure can be written in finite element model as follows

M \overset{\cdot\cdot}{y} (t) + C \overset{\cdot}{y} (t) + Ky (t) = N^{T} f (t) = \tilde{f} (t)

(1)

where M, C, and K are structural mass, damping, and stiffness matrices, respectively; f is the excitation force; N^T is the transposition of the shape functions at the position x of the interaction force; and y is the nodal displacement of the beam. The displacement of the beam u at the arbitrary position x can be obtained from the shape functions N and the nodal displacement y as (Lin and Trethewey, 1989)

u = Ny

(2)

The shape function of an element can be obtained as

N = [\begin{matrix} N_{1} & N_{2} & N_{3} & N_{4} \end{matrix}]

(3)

where

\begin{matrix} N_{1} = 1 - 3 {(\frac{x}{l})}^{2} + 2 {(\frac{x}{l})}^{3}; N_{2} = x {(\frac{x}{l} - 1)}^{2} \\ N_{3} = 3 {(\frac{x}{l})}^{2} - 2 {(\frac{x}{l})}^{3}; N_{4} = x [{(\frac{x}{l})}^{2} - \frac{x}{l}] \end{matrix}

(4)

with l being the length of the element.

Multi-cracked beam-like model

Figure 1 shows a uniform beam-like structure with the ith cracked element. Neglecting shear action, the strain energy of an element without a crack can be written as (Qian et al., 1990)

W^{(0)} = \frac{1}{2 EI} (M^{2} l + MP l^{2} + \frac{P^{2} l^{3}}{3})

(5)

where P and M are the shear and bending internal forces at the right node of the element. The additional energy due to the crack of a rectangular beam with thickness h and width b can be expressed as follows (Qian et al., 1990)

W^{(1)} = b \int_{0}^{a} (\frac{K_{I}^{2} + K_{II}^{2}}{E'} + \frac{(1 + ν) K_{III}^{2}}{E}) da

(6)

where $E' = E$ for plane stress; $E' = (E / (1 - ν^{2}))$ for plane strain; a is the crack depth; and K_I, K_II, and K_III are stress intensity factors for opening-type, sliding-type, and tearing-type cracks, respectively.

Figure 1.

Model of a cracked element.

Taking into account only bending, equation (6) leads to

W^{(1)} = b \int_{0}^{a} \frac{{(K_{I M} + K_{I P})}^{2} + K_{II P}^{2}}{E'} da

(7)

where

K_{I M} = \frac{6 M \sqrt{π a} F_{I} (s)}{b h^{2}}; K_{I P} = \frac{3 P l \sqrt{π a} F_{I} (s)}{b h^{2}}; K_{II P} = \frac{P \sqrt{π a} F_{II} (s)}{b h}

(8)

F_{I} (s) = \sqrt{\frac{2}{π s} tg (\frac{π s}{2})} \frac{0.923 + 0.199 {[1 - \sin ((π s / 2))]}^{4}}{\cos (π s / 2)}

(9)

F_{II} (s) = (3 s - 2 s^{2}) \frac{1.122 - 0.561 s + 0.085 s^{2} + 0.18 s^{3}}{\sqrt{1 - s}}

(10)

Components of the flexibility matrix $\tilde{c}$ of the intact element can be calculated as

{\tilde{c}}_{ij}^{(0)} = \frac{\partial^{2} W^{(0)}}{\partial P_{i} \partial P_{j}}; i, j = 1, 2; P_{1} = P; P_{2} = M

(11)

The additional flexibility coefficient is

{\tilde{c}}_{ij}^{(1)} = \frac{\partial^{2} W^{(1)}}{\partial P_{i} \partial P_{j}}; i, j = 1, 2; P_{1} = P; P_{2} = M

(12)

The total flexibility coefficient is then the sum of the flexibility coefficient of the intact element and the additional flexibility coefficient due to the crack

{\tilde{c}}_{ij} = {\tilde{c}}_{ij}^{(0)} + {\tilde{c}}_{ij}^{(1)}

(13)

From the equilibrium condition, the following equation can be obtained

(\begin{matrix} P_{i} & M_{i} & P_{i + 1} & M_{i + 1} \end{matrix})^{T} = T (\begin{matrix} P_{i + 1} & M_{i + 1} \end{matrix})^{T}

(14)

where

T = {(\begin{matrix} \begin{matrix} - 1 \\ 0 \end{matrix} & \begin{matrix} - L \\ - 1 \end{matrix} & \begin{matrix} 1 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 1 \end{matrix} \end{matrix})}^{T}

(15)

The stiffness matrix of the cracked element is derived by applying the principle of virtual work

K_{c} = T^{T} \tilde{c} T

(16)

Finally, the global stiffness matrix K of the cracked beam is assembled from the element stiffness matrix for intact elements defined in finite element method and matrix K_c for cracked elements. Rayleigh damping in the form of $C = α M + β K$ is used for the beam, where α and β are calculated as follows (Lin and Trethewey, 1989)

α = \frac{2 ω_{1} ω_{2} (ξ_{1} ω_{2} - ξ_{2} ω_{1})}{ω_{2}^{2} - ω_{1}^{2}}; β = \frac{2 (ξ_{2} ω_{2} - ξ_{1} ω_{1})}{ω_{2}^{2} - ω_{1}^{2}}

(17)

Substituting the global matrix K into equation (1), the dynamic response of the cracked beam can be determined.

Element stiffness index distribution

In order to present the concept of “element stiffness index distribution,” the ith element stiffness matrix is denoted as

K_{e}^{i} = [\begin{matrix} k_{11}^{i} & k_{12}^{i} & k_{13}^{i} & k_{14}^{i} \\ k_{21}^{i} & k_{22}^{i} & k_{23}^{i} & k_{24}^{i} \\ k_{31}^{i} & k_{32}^{i} & k_{33}^{i} & k_{34}^{i} \\ k_{41}^{i} & k_{42}^{i} & k_{43}^{i} & k_{44}^{i} \end{matrix}]

(18)

the global stiffness matrix will have the form of

K = [\begin{matrix} k_{11}^{1} & k_{12}^{1} & k_{13}^{1} & k_{14}^{1} & 0 & 0 & \dots \\ k_{21}^{1} & k_{22}^{1} & k_{23}^{1} & k_{24}^{1} & 0 & 0 & \dots \\ k_{31}^{1} & k_{32}^{1} & (k_{33}^{1} + k_{11}^{2}) & (k_{34}^{1} + k_{12}^{2}) & k_{13}^{2} & k_{14}^{2} & 0 & 0 & \dots \\ k_{41}^{1} & k_{42}^{1} & (k_{43}^{1} + k_{21}^{2}) & (k_{44}^{1} + k_{22}^{2}) & k_{23}^{2} & k_{24}^{2} & 0 & 0 & \dots \\ 0 & 0 & k_{31}^{2} & k_{32}^{2} & (k_{33}^{2} + k_{11}^{3}) & (k_{34}^{2} + k_{12}^{3}) & k_{13}^{3} & k_{14}^{3} & 0 & 0 & \dots \\ 0 & 0 & k_{41}^{2} & k_{42}^{2} & (k_{43}^{2} + k_{21}^{3}) & (k_{44}^{2} + k_{22}^{3}) & k_{23}^{3} & k_{24}^{3} & 0 & 0 & \dots \\ 0 & 0 & k_{31}^{3} & k_{32}^{3} & (k_{33}^{3} + k_{11}^{4}) & (k_{34}^{3} + k_{12}^{4}) & k_{13}^{4} & k_{14}^{4} & 0 & 0 & \dots \\ 0 & 0 & k_{41}^{3} & k_{42}^{3} & (k_{43}^{3} + k_{21}^{4}) & (k_{44}^{3} + k_{22}^{4}) & k_{23}^{4} & k_{24}^{4} & 0 & 0 & \dots \\ ⋮ \\ 0 & 0 & \dots & k_{31}^{i - 1} & k_{32}^{i - 1} & (k_{33}^{i - 1} + k_{11}^{i}) & (k_{14}^{i - 1} + k_{12}^{i}) & k_{13}^{i} & k_{14}^{i} & 0 \\ 0 & 0 & \dots & k_{41}^{i - 1} & k_{42}^{i - 1} & (k_{43}^{i - 1} + k_{21}^{i}) & (k_{44}^{i - 1} + k_{22}^{i}) & k_{23}^{i} & k_{24}^{i} & 0 \\ 0 & 0 & \dots & k_{31}^{i} & k_{32}^{i} & (k_{33}^{i} + k_{11}^{i + 1}) & (k_{34}^{i} + k_{12}^{i + 1}) \\ 0 & 0 & \dots & k_{41}^{i} & k_{42}^{i} & (k_{43}^{i} + k_{21}^{i + 1}) & (k_{44}^{i} + k_{22}^{i + 1}) \\ ⋮ \end{matrix}]

(19)

As can be seen from equation (19), any sub-matrix

{\tilde{K}}_{e}^{i} = [\begin{matrix} (k_{33}^{i - 1} + k_{11}^{i}) & (k_{34}^{i - 1} + k_{12}^{i}) & k_{13}^{i} & k_{14}^{i} \\ (k_{43}^{i - 1} + k_{21}^{i}) & (k_{44}^{i - 1} + k_{22}^{i}) & k_{23}^{i} & k_{24}^{i} \\ k_{31}^{i} & k_{32}^{i} & (k_{33}^{i} + k_{11}^{i + 1}) & (k_{34}^{i} + k_{12}^{i + 1}) \\ k_{41}^{i} & k_{42}^{i} & (k_{43}^{i} + k_{21}^{i + 1}) & (k_{44}^{i} + k_{22}^{i + 1}) \end{matrix}]

(20)

in the global stiffness matrix is constructed mainly from the ith element stiffness matrix with some additional components of the (i − 1)th and (i + 1)th element stiffness matrices. Therefore, the sub-matrix ${\tilde{K}}_{e}^{i}$ can be used to characterize the stiffness of the ith element for the crack detection problem. Indeed, when there is a crack at the ith element, only three sub-matrices ${\tilde{K}}_{e}^{i - 1}$ , ${\tilde{K}}_{e}^{i}$ , and ${\tilde{K}}_{e}^{i + 1}$ will be changed as can be seen from equation (20). Since the sub-matrix ${\tilde{K}}_{e}^{i}$ reflects the local stiffness, its norm should serve as a good local indication of its stiffness condition. From this point of view, the change in norm of the sub-matrix ${\tilde{K}}_{e}^{i}$ can be used as an indicator of the damage at the ith element. In order to detect the change in norm of the sub-matrix ${\tilde{K}}_{e}^{i}$ from the global stiffness matrix, we define an element stiffness index distribution as follows

Λ = [λ_{1} λ_{2} \dots λ_{Q}]

(21)

where $λ_{i} = norm ({\tilde{K}}_{e}^{i})$ is called ith element stiffness index; Q is the number of finite elements. When there is a crack at the ith element, the element stiffness index distribution is expected to have a significant change in the ith element.

If the global stiffness matrix can be reconstructed from measured data, the element stiffness index distribution (21) can be derived to detect the crack on the beam. To do that, equation (1) is rewritten as follows

Ky (t) = \tilde{f} (t) - M \overset{\cdot\cdot}{y} (t) - C \overset{\cdot}{y} (t)

(22)

Taking the Fourier transform on both sides of equation (22), we have

KY (ω) = \tilde{F} (ω) + ω^{2} MY (ω) - i ω CY (ω)

(23)

If $\tilde{F} (ω), Y (ω)$ , M, and C are known, matrix K can be reconstructed from equation (23). However, too much data require measuring simultaneously. In order to overcome this, we apply the FRFs. Indeed, assuming that there is only one external force with constant amplitude acting on the fixed kth degree of freedom (DOF) and displacement signals are measured successively at every DOF, the FRF corresponding to the ith DOF can be written as follows

H_{i} (ω) = \frac{Y_{i} (ω)}{F_{k} (ω)}

(24)

where Y_i(ω) is the ith component of vector Y and ${\tilde{F}}_{k} (ω)$ is the kth component of vector $\tilde{F} (ω)$ .

From equation (24), we have

Y_{i} (ω) = H_{i} (ω) {\tilde{F}}_{k} (ω)

(25)

Therefore, vector Y can be expressed as follows

Y (ω) = H_{k} (ω) {\tilde{F}}_{k} (ω)

(26)

where $H_{k} (ω) = [H_{1} H_{2} \dots H_{n}]^{T}$ is the vector of FRFs at frequency ω obtained by acting as an external force at the fixed kth DOF with the response signals being measured at all of the DOFs. The FRFs can be measured by single input–single output vibration instruments.

Substituting equation (26) into equation (23), the following equation is obtained

K H (ω) {\tilde{F}}_{k} (ω) = \tilde{F} (ω) + ω^{2} M H (ω) {\tilde{F}}_{k} (ω) - i ω C H (ω) {\tilde{F}}_{k} (ω)

(27)

Divide both sides of equation (27) by ${\tilde{F}}_{k} (ω)$ we have

KH (ω) = \frac{\tilde{F} (ω)}{{\tilde{F}}_{k} (ω)} + ω^{2} MH (ω) - i ω CH (ω)

(28)

Since only the kth component of the external force is non-zero, the first term of the right side of equation (28) will be a constant vector with the kth component equals to unity and the other components equal to zero. Denote this constant vector by L, equation (28) can then be rewritten as follows

KH (ω) = L + ω^{2} MH (ω) - i ω CH (ω)

(29)

In practice, the rotations are difficult to be measured. Nonetheless, since the rotation is the differential of the translation

y_{r} (t) = \frac{\partial y_{t} (t)}{\partial x}

(30)

where subscripts r and t stand for “rotation” and “translation,” then, in finite element analysis, the rotation at a node can be approximated from the translations at two successive nodes as follows

y_{r, i} (t) = \frac{y_{t, i + 1} (t) - y_{t, i} (t)}{Δ x}

(31)

where y_r,i(t) is the rotation at the ith node and y_t,i(t) and y_t,i + 1(t) are translations at the ith and (i + 1)th nodes, respectively. Taking Fourier transform on both sides of equation (31), one can obtain the following

Y_{r, i} (ω) = \frac{Y_{t, i + 1} (ω) - Y_{t, i} (ω)}{Δ x}

(32)

Substituting equation (25) into equation (32), one obtains the following

H_{r, i} (ω) {\tilde{F}}_{k} = \frac{H_{t, i + 1} (ω) - H_{t, i} (ω)}{Δ x} {\tilde{F}}_{k}

(33)

H_{r, i} (ω) = \frac{H_{t, i + 1} (ω) - H_{t, i} (ω)}{Δ x}

(34)

Therefore, only the FRFs corresponding to translations need to be measured, the FRFs corresponding to rotations can be calculated from equation (34) to fully obtain vector H. Vector H does not depend on the force and the displacements, it depends only on properties of the beam.

Now, if H, M, and C are known, K can be calculated from equation (29) by dividing both sides of this equation by H. It is assumed that the influence of the crack on the mass matrix can be neglected; thus, the finite element matrix M of the intact beam can be considered as accurate data. However, in practice, matrix C cannot be measured, it can only be calculated approximately from M and K of the intact beam by applying equation (17). Therefore, when there is a crack and the crack location is unknown, K calculated from equation (29) is not entirely accurate. To overcome this, components of the stiffness matrix from each row are calculated separately from equation (29). The components in each row of the stiffness matrix corresponding to intact elements will be accurate since they are all calculated from components of M and C obtained from intact elements. Only components of the stiffness matrix in rows corresponding to the cracked elements are not accurate. This will lead to an additional change in the element stiffness index distribution. The components of the stiffness matrix from each row are calculated step by step from equation (29) as follows.

First, components in the first row of K are analyzed. As can be seen from equation (19), only the first four components of the row are non-zero; therefore, four independent equations are needed to calculate these components. In order to calculate these four independent equations, dynamic responses at four frequencies $ω_{1}, ω_{2}, ω_{3}, ω_{4}$ are substituted into equation (29) as follows

\begin{matrix} k_{11} H_{1} (ω_{1}) + k_{12} H_{2} (ω_{1}) + k_{13} H_{3} (ω_{1}) + k_{14} H_{4} (ω_{1}) \\ = L_{1} (ω_{1}) + ω_{1}^{2} m_{11} H_{1} (ω_{1}) + ω_{1}^{2} m_{12} H_{2} (ω_{1}) \\ + ω_{1}^{2} m_{13} H_{3} (ω_{1}) + ω_{1}^{2} m_{14} H_{4} (ω_{1}) - i ω_{1} c_{11} H_{1} (ω_{1}) \\ - i ω_{1} c_{12} H_{2} (ω_{1}) - i ω_{1} c_{13} H_{3} (ω_{1}) - i ω_{1} c_{14} H_{4} (ω_{1}) \\ k_{11} H_{1} (ω_{2}) + k_{12} H_{2} (ω_{2}) + k_{13} H_{3} (ω_{2}) + k_{14} H_{4} (ω_{2}) \\ = L_{1} (ω_{2}) + ω_{2}^{2} m_{11} H_{1} (ω_{2}) + ω_{2}^{2} m_{12} H_{2} (ω_{2}) \\ + ω_{2}^{2} m_{13} H_{3} (ω_{2}) + ω_{2}^{2} m_{14} H_{4} (ω_{2}) - i ω_{2} c_{11} H_{1} (ω_{2}) \\ - i ω_{2} c_{12} H_{2} (ω_{2}) - i ω_{2} c_{13} H_{3} (ω_{2}) - i ω_{2} c_{14} H_{4} (ω_{2}) \\ k_{11} H_{1} (ω_{3}) + k_{12} H_{2} (ω_{3}) + k_{13} H_{3} (ω_{3}) + k_{14} H_{4} (ω_{3}) \\ = L_{1} (ω_{3}) + ω_{3}^{2} m_{11} H_{1} (ω_{3}) + ω_{3}^{2} m_{12} H_{2} (ω_{3}) \\ + ω_{3}^{2} m_{13} H_{3} (ω_{3}) + ω_{3}^{2} m_{14} H_{4} (ω_{3}) - i ω_{3} c_{11} H_{1} (ω_{3}) \\ - i ω_{3} c_{12} H_{2} (ω_{3}) - i ω_{3} c_{13} H_{3} (ω_{3}) - i ω_{3} c_{14} H_{4} (ω_{3}) \\ k_{11} H_{1} (ω_{4}) + k_{12} H_{2} (ω_{4}) + k_{13} H_{3} (ω_{4}) + k_{14} H_{4} (ω_{4}) \\ = L_{1} (ω_{4}) + ω_{4}^{2} m_{11} H_{1} (ω_{4}) + ω_{4}^{2} m_{12} H_{2} (ω_{4}) \\ + ω_{4}^{2} m_{13} H_{3} (ω_{4}) + ω_{4}^{2} m_{14} H_{4} (ω_{4}) - i ω_{4} c_{11} H_{1} (ω_{4}) \\ - i ω_{4} c_{12} H_{2} (ω_{4}) - i ω_{4} c_{13} H_{3} (ω_{4}) - i ω_{4} c_{14} H_{4} (ω_{4}) \end{matrix}

(35)

where $k_{ij}, m_{ij}, c_{ij}, L_{i} (ω_{j}), H_{i} (ω_{j})$ are the components of K, M, C, L, and H, respectively.

Denote that

A_{1} = [\begin{matrix} H_{1} (ω_{1}) H_{2} (ω_{1}) H_{3} (ω_{1}) H_{4} (ω_{1}) \\ H_{1} (ω_{2}) H_{2} (ω_{2}) H_{3} (ω_{2}) H_{4} (ω_{2}) \\ H_{1} (ω_{3}) H_{2} (ω_{3}) H_{3} (ω_{3}) H_{4} (ω_{3}) \\ H_{1} (ω_{4}) H_{2} (ω_{4}) H_{3} (ω_{4}) H_{4} (ω_{4}) \end{matrix}]

(36)

{\hat{K}}_{1} = {(k_{11} k_{12} k_{13} k_{14})}^{T}

(37)

B_{1} = [\begin{matrix} L_{1} (ω_{1}) + \sum_{j = 1}^{4} ω_{1}^{2} m_{1 j} H_{j} (ω_{1}) - \sum_{j = 1}^{4} i ω_{1} c_{1 j} H_{j} (ω_{1}) \\ L_{1} (ω_{2}) + \sum_{j = 1}^{4} ω_{2}^{2} m_{1 j} H_{j} (ω_{2}) - \sum_{j = 1}^{4} i ω_{2} c_{1 j} H_{j} (ω_{2}) \\ L_{1} (ω_{3}) + \sum_{j = 1}^{4} ω_{3}^{2} m_{1 j} H_{j} (ω_{3}) - \sum_{j = 1}^{4} i ω_{3} c_{1 j} H_{j} (ω_{3}) \\ L_{1} (ω_{4}) + \sum_{j = 1}^{4} ω_{4}^{2} m_{1 j} H_{j} (ω_{4}) - \sum_{j = 1}^{4} i ω_{4} c_{1 j} H_{j} (ω_{4}) \end{matrix}]

(38)

Equation (35) becomes

A_{1} {\hat{K}}_{1} = B_{1}

(39)

{\hat{K}}_{1} = A_{1}^{- 1} B_{1}

(40)

Four components in the first row of K are now obtained by taking the real part of A₁ and B₁ in equation (40)

{\hat{K}}_{1} = [real (A_{1}^{- 1})] [real (B_{1})]

(41)

Next, components in the second row of K can be derived by the same manner as calculated for the first row.

However, in the next rows, there are six components; thus, the general equations for determining components of the next ith row will have the form of

{\hat{K}}_{i} = [real (A_{i}^{- 1})] [real (B_{i})]

(42)

where

A_{i} = [\begin{array}{l} H_{n} (ω_{1}) H_{n + 1} (ω_{1}) H_{n + 2} (ω_{1}) H_{n + 3} (ω_{1}) H_{n + 4} (ω_{1}) H_{n + 5} (ω_{1}) \\ H_{n} (ω_{2}) H_{n + 1} (ω_{2}) H_{n + 2} (ω_{2}) H_{n + 3} (ω_{2}) H_{n + 4} (ω_{2}) H_{n + 5} (ω_{2}) \\ H_{n} (ω_{3}) H_{n + 1} (ω_{3}) H_{n + 2} (ω_{3}) H_{n + 3} (ω_{3}) H_{n + 4} (ω_{3}) H_{n + 5} (ω_{3}) \\ H_{n} (ω_{4}) H_{n + 1} (ω_{4}) H_{n + 2} (ω_{4}) H_{n + 3} (ω_{4}) H_{n + 4} (ω_{4}) H_{n + 5} (ω_{4}) \\ H_{n} (ω_{5}) H_{n + 1} (ω_{5}) H_{n + 2} (ω_{5}) H_{n + 3} (ω_{5}) H_{n + 4} (ω_{5}) H_{n + 5} (ω_{5}) \\ H_{n} (ω_{6}) H_{n + 1} (ω_{6}) H_{n + 2} (ω_{6}) H_{n + 3} (ω_{6}) H_{n + 4} (ω_{6}) H_{n + 5} (ω_{6}) \end{array}]

(43)

{\hat{K}}_{i} = (k_{i, n} k_{i, n + 1} k_{i, n + 2} k_{i, n + 3} k_{i, n + 4} k_{i, n + 5})^{T}

(44)

B_{i} = [\begin{matrix} I_{i} (t_{1}) + \sum_{j = n}^{n + 6} ω_{1}^{2} m_{ij} H_{j} (ω_{1}) - \sum_{j = n}^{n + 6} i ω_{1} c_{ij} H_{j} (ω_{1}) \\ I_{i} (t_{2}) + \sum_{j = n}^{k + 6} ω_{2}^{2} m_{ij} H_{j} (ω_{2}) - \sum_{j = n}^{n + 6} i ω_{2} c_{ij} H_{j} (ω_{2}) \\ I_{i} (t_{3}) + \sum_{j = n}^{k + 6} ω_{3}^{2} m_{ij} H_{j} (ω_{3}) - \sum_{j = n}^{n + 6} i ω_{3} c_{ij} H_{j} (ω_{3}) \\ I_{i} (t_{4}) + \sum_{j = n}^{k + 6} ω_{4}^{2} m_{ij} H_{j} (ω_{4}) - \sum_{j = n}^{n + 6} i ω_{4} c_{ij} H_{j} (ω_{4}) \\ I_{i} (t_{5}) + \sum_{j = n}^{k + 6} ω_{5}^{2} m_{ij} H_{j} (ω_{5}) - \sum_{j = n}^{n + 6} i ω_{5} c_{ij} H_{j} (ω_{5}) \\ I_{i} (t_{6}) + \sum_{j = n}^{k + 6} ω_{6}^{2} m_{ij} H_{j} (ω_{6}) - \sum_{j = n}^{n + 6} i ω_{6} c_{ij} H_{j} (ω_{6}) \end{matrix}]

(45)

where n is the column index of the first non-zeros component of the ith row.

In the last two rows of K, only four components are non-zero. Therefore, these components can be calculated similar to the case of first two rows.

Note that, the number of non-zero components of K in the first two rows and the last two rows depends on the boundary condition of the beam. Therefore, the number of equations for solving the components of K in the first two rows and the last two rows will vary depending on the boundary condition.

By this method, stiffness components in rows corresponding to intact elements will be reconstructed accurately because components of the matrix C are calculated from the intact beam. When the stiffness components in rows corresponding to the cracked element are reconstructed, the correlative components of matrix C do not correspond to the cracked element, leading to an error in the reconstructed components of the stiffness matrix of the cracked element. This leads to an additional change in the element stiffness index distribution. However, if the ith element is the cracked element, this additional change will influence only three sub-matrices ${\tilde{K}}_{e}^{i - 1}$ , ${\tilde{K}}_{e}^{i}$ , and ${\tilde{K}}_{e}^{i + 1}$ , in which the sub-matrix ${\tilde{K}}_{e}^{i}$ will be the most influenced. Therefore, it is expected that there will be a sharp change in the element stiffness index distribution at the cracked element.

Regularization

When the measured signals are contaminated by noise, both A and B in equations (41) and (42) will contain errors and these equations subsequently become ill-conditioned problem. This problem is very sensitive to changes in the data; even a small vector error may distort the computed solution considerably. The computed solution may be very different from the desired solution and ultimately meaningless. Therefore, we need to apply regularization methods which are available in the published literature. One of the most popular methods is Tikhonov regularization (Hansen, 2007).

It is noted that the noise in equations (41) and (42) comes mainly from FRF H. Therefore, it is important to filter out the noise of H before applying the regularization. Calculating matrix H at every frequency, vector H becomes a matrix so that each row corresponds to one FRF at one DOF. In this scenario, the truncated singularity value decomposition (TSVD) method can be applied to reduce the noise in H. The matrices U, S, and V are first decomposed from matrix H so that

H = US V^{T}

(46)

The most important aspects of the singularity value decomposition (SVD) are the singular values themselves. In discrete ill-posed problems, the singular values of the matrix H decay gradually to zero. The largest singularity values correspond to the smooth components of H, while the smaller values correspond to the noise in H. Therefore, in order to filter out the noise, the tiny positive singular values are set to zero (Hansen, 1987)

σ_{i} = {\begin{matrix} 1 & i ⩽ λ \\ 0 & otherwise \end{matrix}

(47)

where σ_i is the singularity of matrix S. The matrix H is then reconstructed from equation (46).

After filtering out the noise from H, the rotational components of H are calculated from equation (34), and Tikhonov regularization method is applied to solve equations (41) and (42). Tikhonov regularization method used to solve the problem is denoted as

\min_{x \in R^{n}} {| | Ax - B | |^{2} + | | L_{μ} x | |^{2}}

(48)

where $L_{μ} \in R^{m \times n}$ , m ≤ n is called the regularization matrix. In this study, m and n are the dimensions of A in equations (41) and (42) and m = n. The scalar µ > 0 is known as the regularization parameter. The standard form of Tikhonov regularization takes $L_{μ} = μ I$ , where I is the unit matrix. This choice leads to the minimization problem

\min_{x \in R^{n}} {| | Ax - B | |^{2} + μ^{2} | | x | |^{2}}

(49)

The positive scalar µ > 0 guarantees that $A^{T} A + μ^{2} I$ is invertible. The explicit solution of equation (49) is given by

x = (A^{T} A + μ^{2} I)^{- 1} A^{T} B

(50)

When SVD of A is available, that is, $A = US V^{T}$ , the solution vector x can be expressed as

x = V (S^{T} S + μ^{2} I)^{- 1} S^{T} B

(51)

The procedure for crack detection using element stiffness index distribution

The procedure for crack detection using element stiffness index distribution is performed in the following steps:

Calculate global matrices M, K, and C of the intact beam by finite element method.

Solving equation (1) when the excitation force is an impact roving along the beam to obtain the assumed “measured components of FRFs” corresponding to translations of the beam.

Filter out the noise from components of measured FRFs corresponding to translations using equations (46) and (47).

Calculate components of FRFs corresponding to rotations from measured FRFs by applying equation (34).

Construct matrix H from translation and rotation components of FRFs from steps 2 and 3.

Applying Tikhonov regularization method to solve equations (41) and (42) using equation (51), the global stiffness matrix K is reconstructed.

Construct the element stiffness index distribution using equations (20) and (21).

Inspecting the significant peaks in the element stiffness index distribution to determine the crack location and the crack depth.

Numerical simulation and discussions

Single-crack detection of a beam

A numerical example of a simply supported beam-like bridge divided by 30 elements which has a crack at the 15th element is analyzed. Parameters of the beam are as follows: mass density ρ = 7855 kg/m³; modulus of elasticity E = 2.1 × 10¹¹ N/m²; L = 40 m; b = 1 m; and h = 2 m. Modal damping ratios for the first two modes are 0.02. Six levels of the crack depth ranging from 0% to 50% have been applied. These six cases are numbered in Table 1. The excitation force, an impact with the amplitude of 20,000 N, is applied as follows

f (t) = {\begin{matrix} 20, 000 N & when t = 0 \\ 0 & when t \neq 0 \end{matrix}

(52)

Table 1.

Five cases with cracks of varying depths.

Case	Crack depth (%)
1	10
2	20
3	30
4	40
5	50

In order to check the method, the analytical element stiffness index distribution is calculated directly from the finite element matrix K for six levels of crack and is then illustrated as shown in Figure 2. It can be seen from this figure that there are peaks in the element stiffness index distributions at the 15th element as expected. It can be seen that, when the crack depth increases, the peak magnitude increases.

Figure 2.

Analytical element stiffness index distribution of five different crack depths.

Now, the matrix K is reconstructed from the measured response and finite element matrices M and C when the crack position and the crack depth are unknown. In this case, matrices M and C are both obtained from an intact beam.

Noise-free measurement

The FRFs H without noise is applied to reconstruct the stiffness matrix K from equations (41) and (42). Figure 3 presents the normalized element stiffness index distribution calculated from the reconstructed matrix K with five levels of the crack depth. As can be seen from this figure, since the rotations in vector H are approximated from the translations, not the exact value, then the element stiffness index distribution in Figure 3 is not exactly the same with that of Figure 2 where the rotations are measured data. However, there is still a clear peak in the element stiffness index distribution at the 15th element for every level of the crack as expected.

Figure 3.

Reconstructed element stiffness index distribution, noise 0%: (a) crack depth = 10%, (b) crack depth = 20%, (c) crack depth = 30%, (d) crack depth = 40%, and (e) crack depth = 50%.

It can be observed from Figure 3 that, when the crack depth increases, the peak in the element stiffness index distribution is sharper. One can define the height of the peak as the difference between the values at the top and the bottom of the peak. It is clear that when the crack depth increases, the stiffness sub-matrix of the cracked element decreases. This causes the change in the element stiffness index distribution at the cracked element to increase, that is, the height of the peak dh increases, as can be seen from Figure 3. Therefore, the height of the peak in the element stiffness index distribution can be considered as an intensity factor which relates the change in the element stiffness index distribution at the cracked element to the crack depth. Establishing a graph of the height of the peak versus crack depth from Figure 3, a relationship between the height of the peak (or intensity factor) and the crack depth is obtained as shown in Figure 4. It can be seen that this relationship is a straight line. This relationship is useful for crack depth estimation.

Figure 4.

The height of the peak versus crack depth, without noise.

Noisy measurement

The noise from measurements will influence the effectiveness of the method. Thus, in order to simulate polluted measurements, white noise is added to the simulated responses as shown in the following formula (Zhu and Law, 2006)

y_{noisy} = y + E_{p} N σ (y)

(54)

where y is the dynamic response of the beam; E_p is the noise level; N is the standard normal distribution vector with zero mean value and unit standard deviation; $y_{noisy}$ is the noisy response, and $σ (y)$ is its standard deviation.

Figure 5 shows the element stiffness index distributions for five levels of crack depth. As can be seen from these figures, there are significant peaks at the 15th element where the crack was located. When the crack depth is as small as 10% of the beam height, the proposed method is effective with a minor noise level of 1%. However, when the crack depth increases, the noise level in which the proposed method can be applied also increases. As can be seen from graphs (b), (c), (d), and (e) in Figure 5, when the environmental noise level ranges from 2% to 10%, the peak in the element stiffness index distribution is very stable at the cracked element as the crack depth increases from 20% to 50%.

Figure 5.

Reconstructed element stiffness index distribution: (a) crack depth 10%, noise 1%; (b) crack depth 20%, noise 2%; (c) crack depth 30%, noise 4%; (d) crack depth 40%, noise 6%; and (e) crack depth 50%, noise 10%.

What may also be observed from Figure 5 is that when the crack depth increases, the height of the peak also increases. The relationship between the height of the peak and the crack depth is established from this figure and presented in Figure 6. To enable comparison with and without the noise signal, the relationship between the height of the peak and the crack depth without noise is also presented in this figure. It is interesting that, after applying many different noisy signals, the relationship between the height of the peak and the crack depth is very stable and identical to the case of the noise-free signal.

Figure 6.

The height of the peak versus crack depth, with and without noise.

Multi-crack detection of a beam

Noise-free signals

In this section, two cracks are made at the locations of L/3 and 2L/3 corresponding to the 10th and 20th elements. The measurement is first assumed to be noise free. Figure 7 presents the element stiffness index distributions with five levels of crack depth. As can be seen from this figure, there are two peaks in the distribution at the cracked elements as expected. When the crack depth increases, the heights of the peaks also increase.

Figure 7.

Reconstructed element stiffness index distribution, noise 0%: (a) crack depth 10%, (b) crack depth 20%, (c) crack depth 30%, (d) crack depth 40%, and (e) crack depth 50%.

The relationships between the height of the two peaks and the crack depth are established from this figure and presented in Figure 8. It can be observed from Figure 8 that the heights of the two peaks are almost the same for different crack levels.

Figure 8.

The height of the two peaks versus crack depth, without noise.

Noisy signals

When the measurement is contaminated with noise, the effectiveness of the proposed method is influenced. Figure 9 presents the element stiffness index distribution with five levels of crack depth. As can be seen from this figure, similar to the case of a single crack, when the crack depth is small, the proposed method is effective with a small noise level of 1%. When the crack depth increases, the noise level in which the proposed method can be applied increases. As can be seen from Figure 9, when the crack depth is equal to or larger than 20%, the proposed method can be applied with noise levels ranging from 2% to 10%.

Figure 9.

In order to investigate the influence of noise on the height of the peaks, the relationship between the height of each peak and the crack depth with and without noise is established from Figure 9 and presented in Figures 10 and 11. It can be observed from Figures 10 and 11 that the relationship between the height of each peak and the crack depth with and without noise is close together. Thus, these relationships are useful for estimating the crack depth.

Figure 10.

The height of the first peak versus crack depth, with and without measurement noise.

Figure 11.

The height of the second peak versus crack depth, with and without measurement noise.

Multi-crack detection of a frame

In order to give a more solid evidence of the proposed method, the frame consisting of two vertical columns with the height of 3 m and one horizontal bar with length of 1 m in the X–Z plane as depicted in Figure 12 is analyzed. The frame is divided by 70 frame elements with the cross section of 0.4 m × 0.4 m in finite element model. Two cracks are made at 10th and 20th elements in the left column. Since the frame element has 12 DOFs, the stiffness matrix of a cracked element is adopted from Nguyen (2014). The dynamic response of the frame is calculated by finite element method. Because the excitation force acts in the X-direction, the frame will have bending vibration in the X–Z plane, thus only translations and rotations in the X–Z plane are non-zero while displacements of other DOFs are zero. Obtaining the translation displacements in the X-direction of the left column and applying the proposed method, the element stiffness index distribution will be obtained.

Figure 12.

The frame model in the X–Z plane.

The simulation results show that there are two clear peaks in the element stiffness index distribution of the left column at the cracked elements as presented in Figures 13 and 14 in the cases without and with measurement noise. However, as can be seen from these figures, the first peak which is close to the fixed end of the column is more significant than the second peak which is close to the free end of the column. This implies that the crack close to the fixed end can be detected more efficiently. In this case study, the relationships between the height of the first peak and the crack depth in the cases without and with measurement noise are established from Figures 13 and 14 and presented in Figure 15. As can be observed from Figure 15, the relationship between the height of the first peak and the crack depth with and without noise is close together. This means that these relationships can be used for estimating the crack depth.

Figure 13.

Reconstructed element stiffness index distribution of the left column, noise 0%.

Figure 14.

Reconstructed element stiffness index distribution of the left column: (a) crack depth 10%, noise 1%; (b) crack depth 20%, noise 2%; (c) crack depth 30%, noise 4%; (d) crack depth 40%, noise 6%; and (e) crack depth 50%, noise 10%.

Figure 15.

The height of the first peak versus crack depth, with and without measurement noise.

Conclusion

In this article, a method using “element stiffness index distribution” for single-crack and multi-crack detections of a beam-like structure is presented. The element stiffness index distribution is defined as a vector of norms of sub-matrices in the global stiffness matrix which serve as a local indication of stiffness condition of the beam. In this study, the global stiffness matrix is reconstructed directly from FRFs, not from the mode shapes to avoid some limitation of the measured mode shapes. Since the rotation is difficult to be measured in practice, it is calculated from the translation in this study. Therefore, only the translation of the translation needs to be measured for the proposed technique. Any change in the stiffness of an element will lead to a change in the element stiffness index of that element. By monitoring the change in the element stiffness index distribution, the location of the crack is detected. Some concluding remarks can be listed as follows:

The existence of cracks is detected by significant peaks in the element stiffness index distribution and the crack positions are determined by the locations of these peaks. The cracks with depths as small as 10% of the beam height can be detected by the proposed method.

The depth of cracks can be estimated from the relationship between the height of the peaks in the element stiffness index distribution and the crack depth.

The proposed method gives very good results for both single-crack and multi-crack detections. The proposed method can not only be applied for beam-like structures, but it can also be applied for more complex structures. The cracks can be detected by the proposed method with a level of random noise up to 10%.

However, the proposed method has some limitations. The crack location cannot be detected exactly since only the cracked elements are determined. The accuracy of the method for detecting the crack location depends on the number of measurement points. Also, the approximation of the rotations by equation (31) depends on the measurement points. In order to increase the accuracy of the method, more measurement points are needed.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2014.01, and Vietnam Academy of Science and Technology under grant number VAST.DLT.10/14-15.

References

Abdel Wahab

(1999) Damage detection in bridges using modal curvature: application to a real damage scenario. Journal of Sound and Vibration 226(2): 217–235.

Caddemi

Caliò

(2013) The exact explicit dynamic stiffness matrix of multi-cracked Euler–Bernoulli beam and applications to damaged frame structures. Journal of Sound and Vibration 332: 3049–3063.

Caddemi

Caliò

(2014) Exact reconstruction of multiple concentrated damages on beams. Acta Mechanica 225(11): 3137–3156.

Chondros

Dimarogonas

(1980) Identification of cracks in welded joints of complex structures. Journal of Sound and Vibration 69: 531–538.

Chondros

Dimarogonas

Yao

(1998) Longitudinal vibration of a bar with a breathing crack. Engineering Fracture Mechanics 61: 503–518.

Chondros

Dimarogonas

Yao

(2001) Vibration of a beam with a breathing crack. Journal of Sound and Vibration 239: 57–67.

Dimarogonas

(1996) Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics 5: 831–857.

Hansen

(1987) The truncated SVD as a method for regularization. BIT 27: 534–553.

Hansen

(2007) Regularization tools version 4.0 for Matlab 7.3. SIAM Numerical Algorithms 46: 189–194.

10.

Liang

(1993) An integrated approach to detection of cracks using vibration characteristics. Journal of the Franklin Institute 330(5): 841–853.

11.

Jaishi

Ren

(2006) Damage detection by finite element model updating using modal flexibility residual. Journal of Sound and Vibration 290: 369–387.

12.

Kisa

Arif Gurel

(2006) Modal analysis of multi-cracked beams with circular cross section. Engineering Fracture Mechanics 73: 963–977.

13.

Kisa

Brandon

(2000) The effects of closure of cracks on the dynamics of a cracked cantilever beam. Journal of Sound and Vibration 238(1): 1–18.

14.

Kisa

Brandon

Topcu

(1998) Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods. Computers & Structures 67: 215–223.

15.

Lee

(2009) Identification of multiple cracks in a beam using natural frequencies. Journal of Sound and Vibration 320: 482–490.

16.

Liang

Choi

(1992) Theoretical study of crack-induced eigenfrequency changes on beam structures. Journal of Engineering Mechanics: ASCE 118: 384–396.

17.

Lin

Trethewey

(1989) Finite element analysis of elastic beams subjected to moving dynamic loads. Journal of Sound and Vibration 136(2): 323–342.

18.

Loutridis

Douka

Hadjileontiadis

(2005) Forced vibration behaviour and crack detection of cracked beams using instantaneous frequency. NDT&E International 38(5): 411–419.

19.

Loya

Rubio

Fernandez-Saez

(2006) Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration 290: 640–653.

20.

Luzzato

(2003) Approximate computation of nonlinear effects in a vibrating cracked beam. Journal of Sound and Vibration 265: 745–763.

21.

Nguyen

(2013) Comparison studies of open and breathing crack detections of a beam-like bridge subjected to a moving vehicle. Engineering Structures 51: 306–314.

22.

Nguyen

(2014) Mode shapes analysis of a cracked beam and its application for crack detection. Journal of Sound and Vibration 333: 848–872.

23.

Pandey

Biswas

(1994) Damage detection in structures using changes in flexibility. Journal of Sound and Vibration 169(1): 3–17.

24.

Pandey

Biswas

Samman

(1991) Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration 145(2): 321–332.

25.

Qian

Jiang

(1990) The dynamic behaviour and crack detection of a beam with a crack. Journal of Sound and Vibration 138(2): 233–243.

26.

Rizos

Aspragathos

Dimarogonas

(1990) Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound Vibration 138: 381–388.

27.

Ruotolo

Surace

(1997) Damage assessment of multiple cracked beams: numerical results and experimental validation. Journal of Sound and Vibration 206(4): 567–588.

28.

Ruotolo

Surace

Crespo

. (1996) Harmonic analysis of the vibrations of a cantilevered beam with a closing crack. Computers & Structures 61: 1057–1074.

29.

Shen

M-HH

Taylor

(1991) An identification problem for vibrating cracked beams. Journal of Sound Vibration 150: 457–484.

30.

Sundermeyer

Weaver

(1995) On crack identification and characterization in a beam by non-linear vibration analysis. Journal of Sound and Vibration 183(5): 857–871.

31.

Zhong

Oyadiji

(2011) Detection of cracks in simply-supported beams by continuous wavelet transform of reconstructed modal data. Computers & Structures 89: 127–148.

32.

Zhu

Law

(2006) Wavelet-based crack identification of bridge beam from operational deflection time history. International Journal of Solids and Structures 43: 2299–2317.