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Abstract

This article illustrates an exact approach to investigate natural frequencies and mode shapes of cracked multiple I-section beams with different boundary conditions and presents a damage identification method based on the first-order natural frequency and the corresponding mode shape. First, the local flexibility due to a sectional crack in multiple I-section beams is obtained following the procedures in deriving the equations for rectangular section. Next, according to the local flexibility, the mode shape function of vibration for the multiple I-section beams is determined using the recurrence formulas and fundamental solutions. Then, based on the calculation method of natural frequency and mode shape, the damage identification algorithm for crack locations and sizes is presented using the sensitivity method and the singular value decomposition method. Finally, experimental study on a cantilever beam with single I cross section and finite element method simulation of cracked cantilever beam with double I cross section are conducted to validate the proposed approach. Numerical simulation of direct and inverse problems of multiple I-section beams with different boundary conditions is carried out. The effects of crack location, depth, and other parameters on natural frequencies of cracked beams are also investigated and discussed.

Keywords

Multiple I-section beam free vibration damage identification crack different boundary conditions

Introduction

Cracks present a serious threat to proper performance of structures. It is an important means to ensure the safety of structures by taking some remedial measures or evaluating structural residual load-carrying capacity as soon as possible whenever cracks appear. The structural dynamic characteristics such as natural frequencies and mode shapes will be changed due to the presence of cracks, which means that the dynamic characteristics have great potential for diagnosis of cracks. I-section and multiple I-section hollow structures (such as box-section structure) are widely used in the field of aerospace, mechanical, civil engineering, and so on. Thus, it is of great significance to study the dynamic characteristics and crack identification based on dynamic characteristics of cracked I-section and multiple I-section hollow structures.

The research on the dynamic behavior of cracked beams has received a great deal of attention, and there is a wealth of related articles published. The detailed review on the vibration of cracked structures was given by Dimarogonas.¹ And the various kinds of analytical, semi-analytical, and numerical methods have been employed to solve the problem of a cracked simple beam.^2–4 In some articles, the beam was divided into several sub-beams by cracks simulated by massless rotational springs whose stiffness is dependent on the extent of damage.^5,6 Thereby, the general solution for eigenfunctions of every sub-beam contains four unknown constants and a system of 4(n + 1) equations in the case of n cracks.⁷ In order to decrease the determinant order, Shifrin and Ruotolo⁸ presented a system of (n + 2) linear equations for a beam with n cracks. The finite element method (FEM) is also widely used for free vibration analysis of damaged beams.^9–11 As a major drawback of FEM, eigenfunctions with high-order determinant should be used to ensure the accuracy of the eigensolutions. The spectral finite element method (SFEM) based on the fast Fourier transform has been used in dynamic analysis of structures,^12,13 which represents a structural member with the fewer spectral finite elements. By comparison with the FEM, the SFEM can divide the member into more refined elements regardless of its dimensions and any need in order to improve the solution accuracy. The transfer matrix method (TMM) is prevalently and effectively applied to the free vibration analysis of cracked beams. This method is first introduced by Pestel et al.¹⁴ Based on TMM, Lin et al.¹⁵ studied the natural frequencies and mode shapes of the beams with an arbitrary number of cracks. Modified TMMs are also used to study the dynamic behavior of stepped beams with edge cracks and different boundary conditions.¹⁶ Using the differential quadrature method (DQM), Hsu¹⁷ studied the vibration of edge-cracked beam on elastic foundation with the axial loading and excitation force. To overcome the difficulties in the boundary conditions implementation of the DQM, the generalized differential quadrature method (GDQM) is applied to solve the natural frequencies of some structures under various boundary conditions by Du et al.^18,19 Li^20,21 exploited the fundamental solutions for each sub-beam and obtained the eigenvalue equation from a second-order determinant by the use of the basic solution. However, the objects of the above researches are the solid rectangular or circular section beams. The free vibration analysis of cracked I-section and multiple I-section beams presented in this article has been barely studied.

Structural damage identification has played an important role for ensuring safety, implementing rescues, and avoiding emergency action. Various identification methods have been developed, especially the crack detection based on the vibration information, which has been received great attention. In the studies of the inverse problem, the genetic algorithms, artificial neural network, frequency contour lines method, waveform fractal dimension, and so on have been adopted to identify cracks in the structures. Genetic algorithms recognized as promising intelligent search techniques for the difficult optimal problems are widely employed to identify the damage in structures.^22,23 Based on the combination of wavelet-based elements and genetic algorithm, Xiang et al.²⁴ used a genetic algorithm with small population size and a large mutation rate to detect the damage in a shaft. And in this improved method, the great exploration of the search space with the small number of cost function evaluation is achieved. Artificial neural networks can be used as an alternative effective tool for solving the inverse problems because of the pattern-matching capability, and many neural network–based approaches are used for detecting structural damage.^25,26 An experimentally verified crack damage detection algorithm has been proposed by Li et al.,²⁷ in which a vibration-based analysis data as input in artificial neural networks are used. Dong et al.²⁸ modeled the rotor system by wavelet FEM which adopted frequency contour lines methods to detect the crack. Fractal dimension is used to quantify the difference in the irregularity and has been applied to establish damage feature in damage identification.^29–31 An and Ou³² proposed an improved damage identification method based on the fractal theory and curvature method, which exhibits high-noise insusceptibility. Based on the fractal dimension characteristics of the time–frequency feature, Li et al.³³ proposed a new data-driven approach to detect and localize damages in shear-type building structures.

The aim of this article is to present an analytical method to investigate the free vibration of a cracked multiple I-section beam with any number of cracks and the general form of boundary conditions and provide an efficient approach to solve the inverse problem of detecting multiple open cracks in multiple I-section beams. The presented damage identification method can be used to detect the location and depth of cracks which only requires the first-order natural frequency and the corresponding mode shape. The calculated results of direct and inverse problems are in good agreement with the experimental results and FEM simulation results, which demonstrates the validity and reliability of the presented method.

Local flexibility due to a crack in multiple I-section beam

The local flexibility arose in the crack location due to the existence of the crack and has been widely applied in the free vibration analysis of cracked beam structures. The local flexibility can adopt a unified calculational formula for the solid rectangular section when the depth of cracks is any value. However, the calculational formula of the local flexibility for the multiple I-section beam is different according to the different depths of cracks. In this article, the calculational formulas of the shallow open crack (Figure 1) and deeper open crack (Figure 2) existed in the multiple I-section beam are proposed. As shown in Figures 1 and 2, the cross section is composed of n I-section, in which the symbols of the width and height of the multiple I-section beam are b and h, respectively, the thickness of flange is t, the interval between webs is d_i, and the depth of crack is a.

Figure 1.

Multiple I cross section (shallow open crack).

Figure 2.

Multiple I cross section (deeper open crack).

The relative angle of rotation $θ$ caused by the virtual twin couple moment M across the crack location can be obtained by the following expression³⁴

$θ = \frac{\partial}{\partial M} [\int_{Ac} GdA]$ (1)

where G is a measure of energy-release rate and Ac is the surface area of the open crack. According to the geometrical relationship between Figures 1 and 2, the right side of equation (1) can be expanded as

$θ = \frac{\partial}{\partial M} [\int_{0}^{a} \int_{0}^{b} Gd ξ d η]$ (2)

where ξ and η are location variables. ξ is measured along the depth direction of the sectional crack, and η is the distance from the left vertical axis.

The expression of relationship between the energy-release rate G and stress intensity factor K_I has been given by Dimarogonas and Papadopoulos³⁵

$Gd ξ d η = \frac{K_{I}^{2}}{E} d ξ d η$ (3)

where E is Young’s modulus.

Tada et al.³⁶ provided the formula of stress intensity factor K_I

$K_{I} = \frac{Mh}{2 I_{0}} \sqrt{π ξ} F$ (4)

where I₀ is the moment of inertia that can be computed by the formula $I_{0} = [b h^{3} - \sum_{i = 0}^{n} d_{i} {(h - 2 t)}^{3}] / 12$ , which is same for both the shallow and deeper open cracks, and F is a function of the relative position z $(z = ξ / h)$ that can be given by

$F = \frac{\sqrt{\frac{2}{π z} \tan \frac{π z}{2}} [0.923 + 0.199 {(1 - \sin \frac{π z}{2})}^{4}]}{\cos \frac{π z}{2}}$ (5)

The crack depth is constant in the range of the cross-sectional width for the shallow open crack $(0 \leq a \leq t)$ . Substituting equations (3)–(5) into equation (2), we obtain

$θ = \frac{M π b h^{4}}{2 E {I_{0}}^{2}} \int_{0}^{a / h} z F^{2} dz$ (6)

The local flexibility coefficient can be defined according to the concept of the local flexibility as follows

$C = \frac{\partial θ}{\partial M}$ (7)

The computational formula of the local flexibility coefficient C for shallow open crack $(0 \leq a \leq t)$ can be given when substituting equation (6) and I₀ into equation (7)

$C = \frac{72 π b h^{4}}{E {[b h^{3} - \sum_{i = 0}^{n} d_{i} {(h - 2 t)}^{3}]}^{2}} \int_{0}^{a / h} z F^{2} dz$ (8)

For the deeper open crack $(t \leq a \leq h - t)$ , the relative angle of rotation $θ$ in the crack site by M can be expressed as

$θ = \frac{M π h^{4}}{2 {EI}_{0}^{2}} \int_{0}^{a / h} \int_{0}^{b} z F^{2} dzd η$ (9)

The computational formula of the local flexibility coefficient C for deeper open crack can be obtained based on the constant hypothesis of crack depth in the range of cross section when equations (7) and (9) are considered together

$C = \frac{72 π h^{4}}{E {[b h^{3} - \sum_{i = 0}^{n} d_{i} {(h - 2 t)}^{3}]}^{2}} [b \int_{0}^{t / h} z F^{2} d z + (b - \sum_{i = 0}^{n} d_{i}) \int_{t / h}^{a / h} z F^{2} d z]$ (10)

Direct problem: solution of natural frequency and mode shape

Considering an intact Euler–Bernoulli beam with uniform cross section, the differential equation that governs transverse motion can be expressed as

$EI \frac{\partial^{4} y (x, t)}{\partial x^{4}} + ρ A \frac{\partial^{2} y (x, t)}{\partial t^{2}} = 0$ (11)

where I is the moment of area, ρ is the material density, A is the cross-sectional area, and y(x, t) is the transverse deflection function of the beam at the axial coordinate x and time t.

Let

$y (x, t) = ϕ (x) q (t)$ (12)

Using equation (12) in equation (11), we obtain

$ϕ (x) = \sum_{i = 1}^{4} A_{i} S_{i} (x)$ (13)

where $ϕ (x)$ is the mode shape function of vibration, $S_{1} (x) = e^{β x}$ , $S_{2} (x) = e^{- β x}$ , $S_{3} (x) = \sin β x$ , $S_{4} (x) = \cos β x$ , $β^{4} = ω^{2} ρ A / EI$ , and A_i (i = 1, 2, 3, 4) is integral constants.

Using a similar approach as Li,²¹ we could construct a set of fundamental functions ${\bar{S}}_{i} (x)$ (i = 1, 2, 3, 4) as follows

$\begin{matrix} {\bar{S}}_{1} (x) = \frac{1}{2} [\cosh (β x) + \cos (β x)] \\ {\bar{S}}_{2} (x) = \frac{1}{2 β} [\sinh (β x) + \sin (β x)] \\ {\bar{S}}_{3} (x) = \frac{1}{2 β^{2}} [\cosh (β x) - \cos (β x)] \\ {\bar{S}}_{4} (x) = \frac{1}{2 β^{3}} [\sinh (β x) - \sin (β x)] \end{matrix}$ (14)

Now, a multiple I-section beam with an arbitrary number of open nonpropagating edge cracks is considered as shown in Figure 3. It consists of n cracks which are located at points x₁, x₂, …, x_n with sizes a₁, a₂, …, a_n. The multiple I-section beam is divided into n + 1 segments by n cracks. The symbols of 1, 2, …, i, …, n + 1 refer to the numbering of n + 1 segments. To model a general form of boundary conditions, the ends of the beam carry lumped masses (m_s with eccentricity e_s and rotary inertia J_s, s = 1, 2) supported by linear springs (k_t₁ and k_t₂) and rotational springs (k_R₁ and k_R₂) as shown in Figure 3. Thereby, all kinds of two-end supports can be obtained by choosing the appropriate value for lumped masses and stiffness of springs.

Figure 3.

A beam with n edge open cracks with lumped masses and linear and rotational springs at both ends.

A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in this article. The following expressions are required to build in order to satisfy the continuity of deformation and equilibriums of bending moment and shear force at the crack position for segments i and i + 1

$ϕ_{i + 1} (x_{i}) = ϕ_{i} (x_{i})$ (15a)

$ϕ_{i + 1}^{″} (x_{i}) = ϕ_{i}^{″} (x_{i})$ (15b)

$ϕ ″'_{i + 1} (x_{i}) = ϕ ″'_{i} (x_{i})$ (15c)

But the slope has a jump at the ith crack position

$ϕ'_{i + 1} (x_{i}) - ϕ'_{i} (x_{i}) = EI C_{i} ϕ_{i}^{″} (x_{i})$ (15d)

where C_i is the local flexibility coefficient of the ith crack location and can be computed by equations (8) and (10).

From equations (14) and (15), the vibration mode shape function $ϕ_{i + 1} (x)$ of the (i + 1)th segment can be obtained

$ϕ_{i + 1} (x) = ϕ_{i} (x) + EI C_{i} ″_{i}^{ϕ} (x_{i}) {\bar{S}}_{2} (x - x_{i}) H (x - x_{i})$ (16)

where $ϕ_{i} (x)$ is the vibration mode shape function of the ith segment, $x \in [x_{i - 1}, x_{i})$ , and H(x) is Heaviside function.

Equation (16) is a recurrence formula of mode shape functions, and the mode shape function of (n + 1)th segment of the beam can be expressed as

$ϕ_{n + 1} (x) = ϕ_{1} (x) + EI \sum_{i = 1}^{n} C_{i} ϕ_{i} ″ (x_{i}) {\bar{S}}_{2} (x - x_{i}) H (x - x_{i})$ (17)

where $ϕ (0), ϕ' (0), M (0)$ , and $Q (0)$ refer to the displacement, the slope, bending moment, and shear force at the left side of the beam, respectively. They are called the initial parameters in this article. The mode shape function $ϕ_{1} (x)$ for the first interval $x \in [0, x_{1})$ can be expressed as

$\begin{array}{l} ϕ_{1} (x) = ϕ (0) {\bar{S}}_{1} (x) + ϕ^{'} (0) {\bar{S}}_{2} (x) - \frac{M (0)}{E I} {\bar{S}}_{3} (x) \\ - \frac{Q (0)}{E I} {\bar{S}}_{4} (x), x \in [0, x_{1}) \end{array}$ (18)

Equation (18) can be applied to the beam at x = 0 of any boundary conditions.

Using an equilibrium approach, the boundary condition at x = 0 can be derived as

$\begin{array}{l} M (0) = - e_{1} m_{1} ω^{2} ϕ (0) + k_{R 1} ϕ^{'} (0) \\ - (J_{1} + m_{1} e_{1}^{2}) ω^{2} ϕ^{'} (0) \end{array}$ (19a)

$Q (0) = - k_{t 1} ϕ (0) + m_{1} ω^{2} ϕ (0) + e_{1} m_{1} ω^{2} ϕ' (0)$ (19b)

The boundary condition at x = L can also be derived as

$\begin{array}{l} M (L) = e_{2} m_{2} ω^{2} ϕ_{n + 1} (L) - k_{R 2} {ϕ^{'}}_{n + 1} (L) \\ + (J_{2} + m_{2} e_{2}^{2}) ω^{2} {ϕ^{'}}_{n + 1} (L) \end{array}$ (19c)

$\begin{array}{l} Q (L) = k_{t 2} ϕ_{n + 1} (L) - m_{2} ω^{2} ϕ_{n + 1} (L) \\ - e_{2} m_{2} ω^{2} {ϕ^{'}}_{n + 1} (L) \end{array}$ (19d)

Substituting equations (19a) and (19b) into equation (18), we obtain

$\begin{matrix} ϕ_{1} (x) & = ϕ (0) {\bar{S}}_{1} (x) + ϕ' (0) {\bar{S}}_{2} (x) \\ + \frac{[e_{1} m_{1} ω^{2} ϕ (0) - k_{R 1} ϕ' (0) + (J_{1} + m_{1} e_{1}^{2}) ω^{2} ϕ' (0)]}{EI} {\bar{S}}_{3} (x) \\ + \frac{[k_{t 1} ϕ (0) - m_{1} ω^{2} ϕ (0) - e_{1} m_{1} ω^{2} ϕ' (0)]}{EI} {\bar{S}}_{4} (x) \end{matrix}$ (20)

Substituting equations (17) and (20) into equations (19c) and (19d), the general equations of cracked beam (shown in Figure 3) can be gained in the following

$T_{11} ϕ (0) + T_{12} ϕ' (0) + C_{1} = 0$ (21a)

$T_{21} ϕ (0) + T_{22} ϕ' (0) + C_{2} = 0$ (21b)

where

$\begin{matrix} T_{11} = & EI {\bar{S}}_{1} ″ (L) + e_{1} m_{1} ω^{2} {\bar{S}}_{3} ″ (L) + (k_{t 1} - m_{1} ω^{2}) {\bar{S}}_{4} ″ (L) \\ - e_{2} m_{2} ω^{2} [{\bar{S}}_{1} (L) + \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{3} (L) + \frac{(k_{t 1} - m_{1} ω^{2})}{EI} {\bar{S}}_{4} (L)] \\ + [k_{R 2} - (J_{2} + m_{2} e_{2}^{2}) ω^{2}] . [{\bar{S}'}_{1} (L) + \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{3}' (L) + \frac{(k_{t 1} - m_{1} ω^{2})}{EI} {\bar{S}}_{4}' (L)] \end{matrix}$ (22a)

$\begin{matrix} T_{12} = & EI {\bar{S}}_{2}^{″} (L) - [k_{R 1} - (J_{1} + m_{1} e_{1}^{2}) ω^{2}] {\bar{S}}_{3} ″ (L) - e_{1} m_{1} ω^{2} {\bar{S}}_{4} ″ (L) \\ + [k_{R 2} - (J_{2} + m_{2} e_{2}^{2}) ω^{2}] [{\bar{S}}_{2}' (L) + \frac{[- k_{R 1} + (J_{1} + m_{1} e_{1}^{2}) ω^{2}]}{EI} {\bar{S}}_{3}' (L) - \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{4}' (L)] \\ - e_{2} m_{2} ω^{2} [{\bar{S}}_{2} (L) + \frac{[- k_{R 1} + (J_{1} + m_{1} e_{1}^{2}) ω^{2}]}{EI} {\bar{S}}_{3} (L) - \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{4} (L)] \end{matrix}$ (22b)

$\begin{matrix} T_{21} = & EI \bar{S} ″'_{1} (L) + e_{1} m_{1} ω^{2} S ″'_{3} (L) + (k_{t 1} - m_{1} ω^{2}) S ″'_{4} (L) \\ - (k_{t 2} - m_{2} ω^{2}) [{\bar{S}}_{1} (L) + \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{3} (L) + \frac{(k_{t 1} - m_{1} ω^{2})}{EI} {\bar{S}}_{4} (L)] \\ + e_{2} m_{2} ω^{2} [{\bar{S}}_{1}' (L) + \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{3}' (L) + \frac{(k_{t 1} - m_{1} ω^{2})}{EI} {\bar{S}}_{4}' (L)] \end{matrix}$ (22c)

$\begin{matrix} T_{22} = & EI S ″'_{2} (L) + [(J_{1} + m_{1} e_{1}^{2}) ω^{2} - k_{R 1}] S ″'_{3} (L) - e_{1} m_{1} ω^{2} S ″'_{4} (L) \\ - (k_{t 2} - m_{2} ω^{2}) [{\bar{S}}_{2} (L) + \frac{[- k_{R 1} + (J_{1} + m_{1} e_{1}^{2}) ω^{2}]}{EI} {\bar{S}}_{3} (L) - \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{4} (L)] \\ + e_{2} m_{2} ω^{2} [{\bar{S}}_{2}' (L) + \frac{[(J_{1} + m_{1} e_{1}^{2}) ω^{2} - k_{R 1}]}{EI} {\bar{S}}_{3}' (L) - \frac{e_{1} m_{1} ω^{2}}{EI} {\bar{S}}_{4}' (L)] \end{matrix}$ (22d)

$\begin{matrix} C_{1} = & EI \sum_{i = 1}^{n} C_{i} {ϕ ″}_{i} (x_{i}) [EI {\bar{S}}_{2}^{″} (L - x_{i}) H (L - x_{i}) - e_{2} m_{2} ω^{2} {\bar{S}}_{2} (L - x_{i}) H (L - x_{i}) \\ + [k_{R 2} - (J_{2} + m_{2} e_{2}^{2}) ω^{2}] {\bar{S}}_{2}' (L - x_{i}) H (L - x_{i})] \end{matrix}$ (22e)

$\begin{matrix} C_{2} = & EI \sum_{i = 1}^{n} C_{i} ϕ_{i} ″ (x_{i}) [EI \bar{S} ″'_{2}' (L - x_{i}) H (L - x_{i}) \\ - (k_{t 2} - m_{2} ω^{2}) {\bar{S}}_{2} (L - x_{i}) H (L - x_{i}) \\ + e_{2} m_{2} ω^{2} {\bar{S}}_{2}' (L - x_{i}) H (L - x_{i})] \end{matrix}$ (22f)

From equations (21a) and (21b), it can be seen that only two initial parameters $ϕ (0)$ and $ϕ' (0)$ of any kind of support configuration (shown in Figure 3) are unknown because all $ϕ_{i}^{″} (x)$ (i = 1, 2, …, n) of C₁ and C₂ have the two same initial parameters $ϕ (0)$ and $ϕ' (0)$ . The frequency equation can be obtained by setting the determinants consisting of the coefficients of $ϕ (0)$ and $ϕ' (0)$ in equations (21a) and (21b) as zero. By solving frequency equation, the natural frequency ω, two initial parameters $ϕ (0)$ and $ϕ' (0)$ can be obtained. The mode shape function of each segment can be obtained by substituting ω, $ϕ (0)$ , and $ϕ' (0)$ into equations (16) and (17).

For clamped–free, pinned–pinned, clamped–pinned, clamped–clamped, and other common boundary conditions, there are only two of $ϕ (0)$ , $ϕ' (0)$ , M(0), and Q(0) unknown for any kind of support form at x = 0. And there are also only two of $ϕ (L)$ , $ϕ' (L)$ , M(L), and Q(L) unknown at x = L. Similar to equations (21a) and (21b), the coefficients of the two unknown initial parameters are used to establish the frequency equation. In this article, the frequency equations of clamped–free and pinned–pinned common boundary conditions in the civil engineering structures are listed.

Under clamped–free boundary condition, $ϕ (0) = 0$ , $ϕ' (0) = 0$ , $ϕ_{n + 1}^{″} (L) = 0$ , and $ϕ ″'_{n + 1} (L) = 0$ , that is

$ϕ_{1} (x) = - \frac{M (0)}{EI} {\bar{S}}_{3} (x) - \frac{Q (0)}{EI} {\bar{S}}_{4} (x)$ (23)

Using equations (23) and (17), the right boundary conditions of cracked beam (x = L) may be written as

$- \frac{M (0)}{EI} {\bar{S}}_{3}^{″} (L) - \frac{Q (0)}{EI} {\bar{S}}_{4}^{″} (L) + EI \sum_{i = 1}^{n} C_{i} ϕ_{i}^{″} (x_{i}) {\bar{S}}_{2}^{″} (L - x_{i}) = 0$ (24a)

$- \frac{M (0)}{EI} \bar{S} ″'_{3} (L) - \frac{Q (0)}{EI} \bar{S} ″'_{4} (L) + EI \sum_{i = 1}^{n} C_{i} ϕ_{i}^{″} (x_{i}) \bar{S} ″'_{2} (L - x_{i}) = 0$ (24b)

In equations (24a) and (24b), there are only two unknown parameters M(0) and Q(0) due to all the $ϕ_{i}^{″} (x)$ $(i = 1, 2, \dots, n)$ with the same two initial parameters M(0) and Q(0) as $ϕ_{1} (x)$ . The frequency equation of the clamped–free beam is obtained by setting the second-order determinants consisting of the coefficients of $M (0)$ and $Q (0)$ in equations (24a) and (24b).

Under pinned–pinned boundary condition, $ϕ (0) = 0$ , $ϕ ″ (0) = 0$ , $ϕ_{n + 1} (L) = 0$ , and $ϕ_{n + 1}^{″} (L) = 0$ , therefore

$ϕ_{1} (x) = ϕ' (0) {\bar{S}}_{2} (x) - \frac{Q (0)}{EI} {\bar{S}}_{4} (x)$ (25)

The right boundary conditions of cracked beam (x = L) may be obtained using equations (25) and (17)

$ϕ' (0) {\bar{S}}_{2} (L) - \frac{Q (0)}{EI} {\bar{S}}_{4} (L) + EI \sum_{i = 1}^{n} C_{i} ϕ_{i}^{″} (x_{i}) {\bar{S}}_{2} (L - x_{i}) = 0$ (26a)

$ϕ' (0) {\bar{S}}_{2}^{″} (L) - \frac{Q (0)}{EI} {\bar{S}}_{4}^{″} (L) + EI \sum_{i = 1}^{n} C_{i} ϕ_{i}^{″} (x_{i}) {\bar{S}}_{2}^{″} (L - x_{i}) = 0$ (26b)

Similar to equations (24a) and (24b), equations (26a) and (26b) have only two unknown parameters $ϕ' (0)$ and $Q (0)$ . The frequency equation of pinned–pinned beam is obtained by setting the second-order determinant consisting of the coefficients of $ϕ' (0)$ and $Q (0)$ in equations (26a) and (26b).

Inverse problem: crack damage identification

From section “Direct problem: solution of natural frequency and mode shape,” the presence of cracks changes the characteristic equation and has influenced on the mode shapes and natural frequencies of the beam, which provides the theory of crack identification. In an inverse problem, the crack positions and the corresponding depths are unknown parameters. In the modal experiment of structure, the first-order mode has the highest accuracy which is also the easiest to be tested. Therefore, the aims of this article are to form the crack location and depth identification algorithm based on the first-order mode. Two equations determined by boundary conditions at x = L of the cracked beam with arbitrary boundary conditions can be expressed as

$B_{11} f_{11} + B_{22} f_{12} = 0$ (27a)

$B_{11} f_{21} + B_{22} f_{22} = 0$ (27b)

$ω_{s}$ denotes the measured first natural frequency, and d_s is the measured first mode shapes corresponding to the m measure position, that is

$d_{s} = {\begin{matrix} d_{s 1} & d_{s 2} & \dots & d_{si} & \dots & d_{sm} \end{matrix}}^{T}$ (28)

where d_si is the vibration mode value of the ith measure position; m is the number of measure position; B₁₁ and B₂₂ are two unknown parameters in the four initial parameters; B₁₁ and B₂₂ are dependent on boundary conditions; f₁₁ and f₂₁ are the coefficients of B₁₁; f₁₂ and f₂₂ are the coefficients of B₂₂; and f₁₁, f₂₁, f₁₂, and f₂₂ are the functions of crack location, crack depth, and natural frequency, respectively.

The procedure of crack identification based on the first natural frequency and mode shape is applied as follows:

Let B₁₁ = C (C is an arbitrary real number);

Assume the initial crack location vector x^o and depth vector r^o

$x^{o} = {\begin{matrix} x_{1}^{o} & x_{2}^{o} & \dots & x_{i}^{o} & \dots & x_{n}^{o} \end{matrix}}^{T}$

$r^{o} = {\begin{matrix} r_{1}^{o} & r_{2}^{o} & \dots & r_{i}^{o} & \dots & r_{n}^{o} \end{matrix}}^{T}$

$x_{i}^{o}$ is the ith crack location, and $r_{i}^{o}$ is the ith crack depth. Substituting $x^{o}$ , $r^{o}$ , and the first natural frequency into equations (22a)–(22f), f₁₁, f₂₁, f₁₂, and f₂₂ can be obtained. And then substitute the determined value of C obtained in step 1, f₁₁, f₂₁, f₁₂, and f₂₂ together into equations 27(a) and 27(b), we obtain

$B_{22} = - \frac{C f_{11}}{f_{12}}$ (29a)

$B_{22} = - \frac{C f_{21}}{f_{22}}$ (29b)

The first mode shape function can be obtained by substituting B₁₁, B₂₂, x^o, r^o, and the first natural frequency into equation (17). The vibration mode vector d_c at test points can be obtained from the calculated first mode shape

$d_{c} = {\begin{matrix} d_{c 1} & d_{c 2} & \dots & d_{ci} & \dots & d_{cm} \end{matrix}}^{T}$ (30)

where d_ci is the calculated value of vibration mode at the ith test point.

In order to make d_s and d_c comparable, d_s and d_c are processed by modal regularization. There are two main regularization methods, that is, regularization with mass and the maximum number of 1. In this article, d_s and d_c are regularized with the maximum number of 1, respectively. The normalized mode shape vectors $\bar{d_{s}}$ and $\bar{d_{c}}$ can be obtained as follows

$\bar{d_{si}} = \frac{d_{si} - min (d_{s})}{max (d_{s}) - min (d_{s})}, \bar{d_{ci}} = \frac{d_{ci} - min (d_{c})}{max (d_{c}) - min (d_{c})}$ (31)

The changes in structural physical parameters lead to changes in dynamic characteristics. Using the mathematical expressions, we obtain

$[{\bar{d}}_{c} - {\bar{d}}_{s}] = [S] {\begin{matrix} Δ x \\ Δ r \end{matrix}}$ (32)

where [S] is the sensitivity matrix and $Δ x$ and $Δ r$ are the residuals of the crack location and crack depth, respectively.

The residuals $Δ x$ and $Δ r$ can be obtained from the following expressions

$\begin{array}{l} Δ x = {\begin{matrix} Δ x_{1} \\ Δ x_{2} \\ ⋮ \\ Δ x_{i} \\ ⋮ \\ Δ x_{n} \end{matrix}} = {\begin{matrix} x_{1} - x_{1}^{0} \\ x_{2} - x_{2}^{0} \\ ⋮ \\ x_{i} - x_{i}^{0} \\ ⋮ \\ x_{n} - x_{n}^{0} \end{matrix}}, \\ Δ r = {\begin{matrix} Δ r_{1} \\ Δ r_{2} \\ ⋮ \\ Δ r_{i} \\ ⋮ \\ Δ r_{n} \end{matrix}} = {\begin{matrix} r_{1} - r_{1}^{0} \\ r_{2} - r_{2}^{0} \\ ⋮ \\ r_{i} - r_{i}^{0} \\ ⋮ \\ r_{n} - r_{n}^{0} \end{matrix}} \end{array}$ (33)

The specific form of the sensitivity matrix [S] is as follows

$[S] = [\begin{matrix} \frac{\partial \bar{d_{c 1}}}{\partial x_{1}} & \frac{\partial \bar{d_{c 1}}}{\partial x_{2}} & \dots & \frac{\partial \bar{d_{c 1}}}{\partial x_{n}} & \frac{\partial \bar{d_{c 1}}}{\partial r_{1}} & \frac{\partial \bar{d_{c 1}}}{\partial r_{2}} & \dots & \frac{\partial \bar{d_{c 1}}}{\partial r_{n}} \\ \frac{\partial \bar{d_{c 2}}}{\partial x_{1}} & \frac{\partial \bar{d_{c 2}}}{\partial x_{2}} & \dots & \frac{\partial \bar{d_{c 2}}}{\partial x_{n}} & \frac{\partial \bar{d_{c 2}}}{\partial r_{1}} & \frac{\partial \bar{d_{c 2}}}{\partial r_{2}} & \dots & \frac{\partial \bar{d_{c 2}}}{\partial r_{n}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{\partial \bar{d_{cm}}}{\partial x_{1}} & \frac{\partial \bar{d_{cm}}}{\partial x_{2}} & \dots & \frac{\partial \bar{d_{cm}}}{\partial x_{n}} & \frac{\partial \bar{d_{cm}}}{\partial r_{1}} & \frac{\partial \bar{d_{cm}}}{\partial r_{2}} & \dots & \frac{\partial \bar{d_{cm}}}{\partial r_{n}} \end{matrix}]$ (34)

The elements in matrix [S] are theoretically calculated by partial differential method, while it is very difficult to solve the partial differential in practical engineering. In this article, the perturbation method³⁷ is employed to compute the elements of the sensitivity matrix [S] numerically. For the perturbation method, assume that a physical parameter has a very small change, which leads to a changed value of $\bar{d_{ci}}$ . Then, the difference between the changed value and the original value of $\bar{d_{ci}}$ (i = 1, 2, …, m) is divided by the small change of the parameter. That is, the following expression

$\frac{\partial {\bar{d}}_{ci}}{\partial x_{k}} = \frac{{\bar{d}}_{ci} (ω_{s}, x_{1}, x_{2}, \dots, x_{k} + ε, \dots, x_{n}, r_{1}, r_{2}, \dots, r_{n}) - {\bar{d}}_{ci} (ω_{s}, x_{1}, x_{2}, \dots, x_{k}, \dots, x_{n}, r_{1}, r_{2}, \dots, r_{n})}{ε}, | ε | < < 1$ (35a)

$\frac{\partial {\bar{d}}_{ci}}{\partial r_{k}} = \frac{{\bar{d}}_{ci} (ω_{s}, x_{1}, x_{2}, \dots, x_{n}, r_{1}, r_{2}, \dots, r_{k} + ε, \dots, r_{n}) - {\bar{d}}_{ci} (ω_{s}, x_{1}, x_{2}, \dots, x_{n}, r_{1}, r_{2}, \dots, r_{k}, \dots, r_{n})}{ε}, | ε | < < 1$ (35b)

According to equation (32), the residuals $Δ r$ and $Δ x$ can be solved by the following expression

${\begin{matrix} Δ x \\ Δ r \end{matrix}} = {[S]}^{+} [{\bar{d}}_{c} - {\bar{d}}_{s}]$ (36)

where $[S]^{+}$ is the generalized inverse matrix of [S]. Through singular value decomposition,³⁸ the decomposition form of matrix [S] could be expressed as $[S] = [U] [\begin{matrix} Σ & 0 \\ 0 & 0 \end{matrix}] [V]^{H}$ . [U] and [V] are the m-order unitary matrix and 2n-order unitary matrix, respectively. $\sum = diag (σ_{1}, σ_{2}, \dots, σ_{r})$ which has $σ_{1} \geq σ_{2} \geq \dots \geq σ_{r} > 0$ . $σ_{i}$ (i = 1, 2, …, r) are the singular values of [S]. $[V]^{H}$ is the conjugate transpose of [V] which has $[V] [V]^{H} = I$ and $[V]^{H} [V] = I$ . r is the rank of the matrix [S], $r = rank ([S])$ . Then, the generalized inverse matrix $[S]^{+}$ can be obtained from the following formula: $[S]^{+} = [V] [\begin{matrix} Σ^{- 1} & 0 \\ 0 & 0 \end{matrix}] [U]^{H}$ .

Update the crack parameters

$x_{new}^{o} = x_{old}^{o} + p \cdot Δ x, r_{new}^{o} = r_{old}^{o} + p \cdot Δ r$

where p is an underrelaxation parameter and can be used to avoid overshoots and increase reliability of the algorithm in early steps of iterations.

Iterate the procedures (2)–(7) until the residuals of $Δ r$ and $Δ x$ become sufficiently small and satisfactory.

Examples of direct problem

It should be noted that all the numerical results of this article are obtained based on Young’s modulus E = 206 GPa and mass density ρ = 7850 kg/m³.

Reliability of the proposed method

Comparison with experimental results

A cantilever beam with I cross section experienced the dynamic property test which is used to verify the validity and reliability of the present method for the direct problem. The geometry parameters of the cantilever beam shown in Figure 4 are given as follows: L = 1.8 m, b = 0.06 m, h = 0.102 m, t = 0.0064 m, and d = 0.028 m.

Figure 4.

A cracked cantilever beam with I cross section.

Two cracked cases are investigated:

Case 1. Only one crack occurs at x₁ = 0.3 m from the left end of the beam, and the depth of the crack changes from 0% to 30% at intervals of 5%.

Case 2. The same crack position and depth (30%) with case 1 except the added another crack at x₂ = 0.75 m from the left end of the beam also changes from 0% to 30% at intervals of 5%.

In the procedure of the dynamic property test, the cantilever is equally divided into 12 parts, that is, there are 13 test points measured. In total, five acceleration sensors of the 131E type produced by the Jiangsu Donghua Testing Technology Co., Ltd are used to measure the dynamic property. In detail, the dynamic test is carried out in three batches due to the lack of acceleration sensors, in which the seventh test point at 0.9 m from the left end of the beam is set as the fixed point, while another four sensors are located at different test points in every batch. A device similar to the exciting hammer is adopted to exert the pulsed excitation on the cantilevered beam. The acceleration time history response of the dynamic test point is collected by the vibration test system of the DH5920 type produced by the Jiangsu Donghua Testing Technology Co., Ltd, while the input response is not collected. The field test is shown in Figure 5.

Figure 5.

The field test.

The reduction coefficient η denotes the effect of cracks on the natural frequency which is calculated by equation (37)

$η_{j} = \frac{ω_{cj}}{ω_{j}}$ (37)

where $ω_{cj}$ is the jth natural frequency of the cracked beam, $ω_{j}$ is the jth natural frequency of the beam without crack, and $η_{j}$ is the reduction coefficient of the jth natural frequency. The reduction coefficients $η$ of the first two natural frequencies using the presented method and the experimental measurement are shown in Figure 6.

Figure 6.

Reduction coefficients of the first two natural frequencies: (a) the first-order natural frequency of two cases and (b) the second-order natural frequency of two cases.

It can be seen from Figure 6 that the reduction coefficients of the first two natural frequencies decrease with the increase in crack depth. The reduction coefficients obtained by both the presented method and the experiment show good agreement, which demonstrates the validity and reliability of this method. In addition, the calculated relation curves between the reduction coefficients of the natural frequency and the relative crack depth have a sudden change. It shows that the different local flexibility coefficients of the cracked beam with single I cross section are provided in the shallow open crack $(0 \leq a \leq t)$ and deeper open crack $(t \leq a \leq h - t)$ .

With the relative depths of two cracks are 30% in case 2, the first-order mode shapes obtained by the presented method and the experiment are shown in Figure 7.

Figure 7.

The first-order normalized mode shapes (case 2: a₁/h = a₂/h = 0.3).

Comparison with FEM results

The second example (shown in Figure 8) is a fixed–free beam with double I-section. The geometrical parameters of the double I-section beam are L = 5 m, b = 0.2 m, h = 0.2 m, t = 0.05 m, and d = 0.025 m. Two cases of cracked double I-section beam are investigated:

Case 1. Only single crack occurs at x₁ = 1.5 m from the left end of the beam, and the relative depth of the crack changes from 0% to 30% at intervals of 5%.

Case 2. Double cracks occur at x₁ = 1.5 m and x₂ = 2.5 m from the left end of the beam, respectively. The same relative depths of two cracks vary from 0% to 30% at intervals of 5%, simultaneously.

Figure 8.

A cracked fixed–free beam with double I cross section.

Finite element analysis is carried out by ANSYS software. A three-dimensional (3D) FEM model of the double I-section beam is built with the eight-node SOLID45 elements. Most elements of SOLID45 have the length of 0.02 m, the width of 0.02 m, and the height of 0.02 m. The number of SOLID45 elements and nodes in intact FEM model are 27,000 and 36,144, respectively. The 3D FEM model is shown in Figure 9(a), and the corresponding elevation view of the FEM model is shown in Figure 9(b).

Figure 9.

FEM model of a cracked fixed–free beam with double I cross section: (a) 3D FEM model and (b) elevation view of FEM model.

It is worth mentioning that the boundary condition shown in Figure 3 is used in the calculation process by the proposed method in order to verify the validity of the proposed method more fully. Let k_R₁ = 10¹⁴, k_t₁ = 10¹⁴, m₁ = 0, e₁ = 0, J₁ = 0; k_R₂ = 0, k_t₂ = 0, m₂ = 0, e₂ = 0, and J₂ = 0. The reduction coefficients of the first two natural frequencies under two cases are calculated by the proposed method and FEM, respectively. And the corresponding results are presented in Figure 10. While the relative depth of double cracks is 20% in case 2, the first two-order mode shapes obtained by the proposed method and FEM, respectively, are shown in Figure 11.

Figure 10.

Reduction coefficients of the first two natural frequencies for two cases: (a) reduction coefficient of case 1 and (b) reduction coefficient of case 2.

Figure 11.

The first two-order normalized mode shapes (case 2: a₁/h = a₂/h = 0.2): (a) the first-order normalized mode shapes and (b) the second-order normalized mode shapes.

As can be seen from Figure 10, the reduction coefficients for the first two natural frequencies of the fixed–free beam with double I-section decrease as the relative crack depth increases. The more the number of cracks, the more significant the changes. One sees that the good agreement between the results of the presented method and FEM is achieved. It also can be seen from Figure 11 that the first two mode shapes obtained by the presented method and FEM are very close. Thus, the validity and reliability of the proposed method are verified.

Numerical simulation for a cracked beam of double I-section with spring supports

In this example, a beam of double I-section supported by linear springs (k_t₁ = k_t₂ = 100 × EI/L³) and rotational springs (k_R₁ = k_R₂ = EI/L) without lumped masses is analyzed. The geometrical parameters of the double I-section beam shown in Figure 12 are given as follows: L = 1.8 m, b = 0.12 m, h = 0.102 m, t = 0.0064 m, and d = 0.028 m.

Figure 12.

A cracked beam of double I-section with linear spring and rotational spring supports.

As for the effect of crack number, if cracks occur in turn, the relative depths for three cracks are the same as a₁/h = a₂/h = a₃/h = 0.2, and the locations of cracks are x_i = 0.45, 0.9, and 1.35 m (i = 1, 2, 3), respectively. The lowest two circular natural frequencies of different crack numbers are shown in Table 1. And the mode shapes of the intact beam are shown in Figure 13.

Table 1.

The lowest two circular natural frequencies.

Crack number	First (rad/s)	Second (rad/s)
0	577.5860	1870.1147
1	573.2324	1856.3127
2	567.8596	1849.6680
3	566.0136	1848.0242

Figure 13.

The first two-order mode shapes of the intact double I-section beam: (a) the first-order normalized vibration mode and (b) the second-order normalized vibration mode.

From the calculated circular natural frequencies in Table 1, one sees that the calculated circular natural frequencies decrease with the increase in crack number.

If the beam (shown in Figure 12) which has only one crack at x = 0.9 m and the relative depth of the crack is 30% only supported by linear springs with stiffness k_t₁ = k_t₂ = EI/L³, the lowest two circular natural frequencies are 122.9353 and 1461.1205 rad/s. For the cracked beam under pinned–pinned boundary condition, the lowest two circular natural frequencies are 647.0382 and 2628.0501 rad/s. Change the stiffness of linear springs and let k_t₁ = k_t₂ = nEI/L³, n is the magnification factor of stiffness. The corresponding lowest two circular natural frequencies are calculated and are presented in Figure 14.

Figure 14.

Natural frequencies with respect to linear springs with different stiffnesses: (a) the first natural frequency and (b) the second natural frequency.

As can be seen from Figure 14, for a cracked beam with double I-section which has only one crack, the first two circular natural frequencies increase nonlinearly with the increase in stiffness of linear springs. At first, the lowest two circular natural frequencies increase rapidly and then slowly and gradually approaching the natural frequencies of the cracked beam under pinned–pinned boundary condition.

Results and discussion for the parameters of crack

The beam with the length of 3 m and three I-section is used to explore the effect of the crack parameters (crack depth and location) on the natural frequency. The geometric parameters of the beam shown in Figure 15 are given as follows: b = 0.18 m, h = 0.2 m, t = 0.01 m, d₁ = 0.02 m, and d₂ = 0.06 m. In this article, the effect of crack parameters on the natural frequency of the beam with fixed–free and pinned–pinned boundary conditions is discussed.

Figure 15.

A three-I-section beam with single crack.

In order to demonstrate the effect of crack depth, the beam with only one crack at x = 1.5 m and relative crack depth a/h varying from 0% to 30% is adopted. The first two natural frequencies of the beam are calculated by the proposed method. Corresponding results are shown in Figures 16 and 17.

Figure 16.

Effect of the crack depth on the natural frequencies of the fixed–free beam.

Figure 17.

Effect of the crack depth on the natural frequencies of the pinned–pinned beam.

It can be seen from Figures 16 and 17 that the first two natural frequencies of the fixed–free beam and the first natural frequency of the pinned–pinned beam decrease as the depth of crack increases. In addition, the relation curves between the reduction coefficients of the natural frequency and the relative crack depth exhibit a sudden change. It reveals that shallow open crack $(0 \leq a \leq t)$ and deeper open crack $(t \leq a \leq h - t)$ have different local flexibility coefficients of the cracked beam with multiple I cross section. However, the second-order natural frequency of the pinned–pinned beam does not change with the increase in the depth of the crack, and the reason is that the bending moment generated by the second-order mode shape is zero at the midspan section. This phenomenon is demonstrated by equation (15).

In order to discuss the effect of the crack position on the natural frequency, the reduction coefficients of the natrual frequency in the cases of single carck with relative depth a₁/h = 0.2 located at x₁ = 0.3, 0.6, …, 2.7 m are calculated by the presented method, respectively. The results are shown in Figures 18 and 19.

Figure 18.

Effect of the crack position on the natural frequencies of the fixed–free beam.

Figure 19.

Effect of the crack position on the natural frequencies of the pinned–pinned beam.

It can be seen from Figures 18 and 19:

The effect of cracks on the natural frequency of the same order is different for the beam with different boundary conditions, even if the cracks are in the same position.

The effect of the cracks at the same position on the natural frequency of the different order is different under the same boundary conditions.

The effect of the cracks at the different locations on the natural frequency of the same order is also different under the same boundary conditions.

The reason is analyzed that the bending moment generated by the same order mode is different under the different boundary conditions, and the distribution of the bending moment generated by different order modes is different under the same boundary condition, that is, the bending moment at the crack position is different, which leads to the different effects of the crack on the natural frequency. The presented equation (15) verifies the above viewpoints.

For the fixed–free beam (shown in Figure 15) in the range of 0.3–2.7 m, the bending moment of the first-order mode at the 0.3 m from fixed end is maximum, while the one at the 2.7 m from fixed end is minimum. Thereby, in the range of 0.3–2.7 m, the effect of the crack on the first natural frequency of the fixed–free beam is largest when the crack is located at 0.3 m from the fixed end, while the crack at 2.7 m has little effect on the first natural frequency. Similarly, the value of the moment generated by the second-order mode reaches the maximum at about 1.5 m from the fixed end in the range of 0.3–2.7 m, thereby the effect of the crack on the second natural frequency of the fixed–free beam is largest when the crack is located at 1.5 m from the fixed end. While the bending moment at 0.6 m from the fixed end is close to zero, the crack located at 0.6 m from the fixed end has little effect on the second natural frequency, which leads to the effect of the crack position on the natural frequency of fixed–free beam shown in Figure 18.

For the pinned–pinned beam in the range of 0.3–2.7 m, the bending moment generated by the first-order mode shape is minimum at 0.3 and 2.7 m, while the moment is largest at the midspan section. And the bending moment generated by the second-order mode shape is zero at midspan section, while the bending moments at the 1/4 and 3/4 sections are maximum. Therefore, the effect of the cack position on the first two-order natural frequencies of the pinned–pinned beam is determined as shown in Figure 19.

Examples of inverse problem

Crack identification of experimental cantilever beam

In order to verify the validity and reliability of the present technique for prediction of crack parameters in multiple I-section beams, the inverse problem of various cases presented in section “Comparison with experimental results” is solved. The first-order natural frequency and normalized mode shapes obtained from modal testing of the cantilever I-section beam (Figure 4) are used as input data to identify the crack positions and depths. The actual and predicted crack parameters calculated by the proposed identification algorithm are listed in Tables 2 and 3.

Table 2.

Crack identification of the cantilever I-section beam with single open crack.

Case	Crack location (m)			Crack depth (a/h)
	Actual	Predicted	Error (%)	Actual	Predicted	Error (%)
1	0.30	0.3009	0.30	0.05	0.0507	1.40
2	0.30	0.3060	2.00	0.10	0.1020	2.00
3	0.30	0.2977	−0.77	0.15	0.1520	1.33
4	0.30	0.3043	1.43	0.20	0.2059	2.95
5	0.30	0.3035	1.17	0.25	0.2544	1.76
6	0.30	0.2987	−0.43	0.30	0.3004	0.13

Table 3.

Crack identification of the cantilever I-section beam with double open cracks.

Case	Crack location (m)			Crack depth (a/h)
	Actual	Predicted	Error (%)	Actual	Predicted	Error (%)
1	0.30	0.3012	0.40	0.30	0.3076	2.53
1	0.75	0.7495	−0.07	0.05	0.0512	2.40
2	0.30	0.2987	−0.43	0.30	0.3096	3.20
2	0.75	0.7515	0.20	0.10	0.1048	4.80
3	0.30	0.2998	−0.07	0.30	0.3072	2.40
3	0.75	0.7499	−0.01	0.15	0.1453	−3.13
4	0.30	0.2987	−0.43	0.30	0.3096	3.20
4	0.75	0.7511	0.15	0.20	0.2036	1.80
5	0.30	0.3012	0.40	0.30	0.3019	0.63
5	0.75	0.7498	−0.03	0.25	0.2508	0.32
6	0.30	0.3012	0.40	0.30	0.3072	2.40
6	0.75	0.7499	−0.01	0.30	0.3047	1.57

From the reported results in Tables 2 and 3, the proposed method successfully identified the position and depth of the cracks in all case studies. It is evident that the presented method is powerful in crack iditification. It can be seen that the absolute value of maximum relative error for crack position is 2.00%, and the absolute value of maximum relative error for crack depth is 4.80%.

Crack identification of cracked beam supported by linear springs and rotational springs

A double I-section beam supported by linear springs (k_t₁ = k_t₂ = 100 × EI/L³) and rotational springs (k_R₁ = k_R₂ = EI/L) shown in Figure 12 is used to simulate the crack diagnose. There are nine kinds of crack cases, in which there are three cases for single crack, double cracks, and triple cracks, respectively. The first natural frequency and mode shape are adopted as the input data in all cases of the crack diagnose. Furthermore, in order to investigate the anti-noise ability of the crack identification algorithm in this article, different levels of noise is added into the first-order mode shape calculated by the present method. The expression of the first-order mode shape with noise $D_{input}$ is written as

$D_{input} = D_{1} (1 + α σ \times Randn)$ (38)

Randn is the randon generator function in MATLAB with a zero mean and a standard deviation of $σ$ , and $α$ is error level. The nosie level is controlled by the error level $α$ , and the inverse problem is solved under three different values of $α$ = 3%, 6%, and 9%, respectively. The results of crack identification of the double I-section beam are presented in Tables 4 –6.

Table 4.

Crack identification of the multiple I-section beam with single open crack.

Case	Error level $α$ (%)	Crack location (m)			Crack depth (a/h)
		Actual	Predicted	Error (%)	Actual	Predicted	Error (%)
1	3%	0.45	0.4483	−0.38	0.20	0.2051	2.55
2	6%	0.45	0.4466	−0.76	0.20	0.2038	1.90
3	9%	0.45	0.4450	−1.11	0.20	0.2033	1.65

Table 5.

Crack identification of the multiple I-section beam with double open cracks.

Case	Error level $α$ (%)	Crack location (m)			Crack depth (a/h)
		Actual	Predicted	Error (%)	Actual	Predicted	Error (%)
1	3%	0.45	0.4464	−0.80	0.20	0.1977	−1.15
1	3%	0.90	0.9031	0.34	0.20	0.2035	1.75
2	6%	0.45	0.4412	−1.96	0.20	0.1988	−0.60
2	6%	0.90	0.9025	0.28	0.20	0.2009	0.45
3	9%	0.45	0.4362	−3.07	0.20	0.2007	0.35
3	9%	0.90	0.9017	0.19	0.20	0.1975	−1.25

Table 6.

Crack identification of the multiple I-section beam with triple open cracks.

Case	Error level $α$ (%)	Crack location (m)			Crack depth (a/h)
		Actual	Predicted	Error (%)	Actual	Predicted	Error (%)
1	3%	0.45	0.4498	−0.04	0.20	0.1940	−3.00
		0.90	0.9001	0.01	0.20	0.2075	3.75
		1.35	1.3498	−0.01	0.20	0.2072	3.60
2	6%	0.45	0.4496	−0.09	0.20	0.1938	−3.10
		0.90	0.9002	0.02	0.20	0.2080	4.00
		1.35	1.3496	−0.03	0.20	0.2075	3.75
3	9%	0.45	0.4494	−0.13	0.20	0.1930	−3.50
		0.90	0.9004	0.04	0.20	0.2096	4.80
		1.35	1.3493	0.05	0.20	0.2076	3.80

According to the calculated results in Tables 4 –6, crack parameters of multiple I-section beam with an arbitrary number of cracks and different boundary conditions can be determined effectively and accurately by the presented method in this article. It is also observed that the reliability of the present method is influenced by the level of noise pollution. The absolute value of maximum relative error for crack position is 3.07% as shown in Table 5 with error level $α = 9 %$ , and the absolute value of maximum relative error for crack depth is 4.80% as shown in Table 6 with error level $α = 9 %$ .

Conclusion

It is an interesting topic for the crack diagnose based on the vibration test, which is the basis of free vibration analysis of crack. Multiple I-section hollow structures are the most widely used types in engineering fields, so it is significant to carry out analysis both in direct and inverse problems of a cracked multiple I-section beam. First, this article derives the local flexibility caused by the cracks of the multiple I-section beam and then presents an approach to the free vibration analysis of the beam with an arbitrary number of cracks under any general form of boundary conditions. And the inverse problem of multiple I-section beam is also solved by a powerful method. Experimental study and numerical simulation on I-section and multiple I-section beams are carried out in this article, and some conclusions are obtained as follows:

Whether the natural frequency and mode shape calculated or the crack parameters (location and depth) identified by the present method, all have good agreement with the experimental results and FEM simulation results, which fully demonstrates the validity and reliability of this approach.

For the beam only supported by linear springs, natural frequencies of the beam increase slowly with the increase in spring stiffness and gradually approaching the natural frequencies of the beam under pinned–pinned boundary condition.

Whether the fixed–free beam or the pinned–pinned beam, the natural frequency of the multiple I-section beam decreases with the increase in crack depth. And the relation curves between the reduction coefficients of the natural frequency and the relative crack depth exhibit a sudden change. It reveals that shallow open crack and deeper open crack have different local flexibility coefficients of the cracked beam with multiple I-section.

For some specific modes, the effects of crack position on this mode depend on the distribution of the bending moment along the beam length. That is to say, the different bending moments at crack location lead to the different effects on natural frequency.

The numerical example in section “Crack identification of cracked beam supported by linear springs and rotational springs” shows that the absolute value of maximum relative error for crack position is 3.07%, and the absolute value of maximum relative error for crack depth is 4.80% in the case of random error level α = 9%. This further illustrates the reliability and validity of the crack diagnose algorithm in this article, and it can be thought that the algorithm has strong anti-noise capability.

Footnotes

Handling Editor: Jose Antonio Tenreiro Machado

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 work was supported by National Natural Science Foundation of China (grant number 51478203) and Training Program for Outstanding Young Teachers of Jilin University.

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Direct and inverse problems on free vibration of cracked multiple I-section beam with different boundary conditions