Damage assessment of laminated composite beam structures using damage locating vector (DLV) method

Abstract

In this paper, the damage locating vector (DLV) method using normalized cumulative energy (nce) is employed to locate multiple damage sites in laminated composite beam structures. Numerical simulations of two laminated composite beams are employed to investigate several damage scenarios in which the degradation of elements is modeled by the reduction in the longitudinal Young’s modulus and transverse Young’s modulus of beam layers. The results show that the DLV method gives good performance for this kind of structure.

Keywords

damage locating vector method (DLV) / laminated composite beam structure / normalized cumulative energy / structural health monitoring (SHM)

Introduction

Many terrible accidents caused by unanticipated damages in structures have motivated the development of structural health monitoring (SHM) [ 1]. Besides predicting early the damages in structures, SHM also reduces significantly repair and maintenance cost. In general, when a structure is damaged its modal parameters (frequency, mode shape, damping ratio) are changed. Based on this property, many vibration-based structural health monitoring methods have been proposed to handle different kinds of materials. Some typical methods that can be named are the frequency change [ 2, 3], the mode shape change [ 4], the curvature mode shape change [ 5], the flexibility change [ 6− 9], the modal strain energy [ 10, 11]. In addition, many other methods can be referred from some meticulous literature reviews [ 12− 15].

Nowadays, along with materials such as graphite, ceramics, polymers, etc., composite with many excellent characteristics (strength and lightweight) has been widely used in civil aerospace, military aircraft and industrial manufactures. Due to the complexity in the forming of composite structures, the integrity assessment and the rehabilitation of this kind of structures are more difficult [ 16, 17] than the corresponding isotropic structures. In a few recent decades, the damage identification for composite structures is promoted by many researchers. Lee et al. [ 18] found the correlation between the change in damping coefficient and the severity of damaged composite beam under notches and artificial delamination. Minak et al. [ 17] used the difference between undamaged and damaged first natural frequencies and a pattern recognize method to identify the delamination in a laminate composite beam. Lestari et al. [ 19] applied successfully the curvature mode shape method to locate damages of three types (delamination, impact and saw-cut) in carbon/epoxy laminated composite beams. Moreno et al. [ 20] utilized the mode shape derivatives to localize damages in composite plate.

In SHM, a practicable damage identification method that uses efficiently measured data gains much interest from terminal users. The damage locating vector (DLV) method, a kind of flexibility change method, is one of applicable methods. This method is first proposed by Bernal [ 6] and then developed by many researchers [ 21− 25]. Gao et al. [ 22] implemented an experiment of a 5.6 m-long three-dimension truss that demonstrated the efficiency and reality of DLV method. Quek et al. [ 25] extended the DLV method by proposing a new indicator called normalized cumulative energy (nce) and an intersection scheme to carry out the issue of limited sensors in evaluating damage in truss and frame structures. Although, the DLV method has been examined on structures made of isotropic material like beam, truss and frame, it hasn’t been verified on composite structures. Hence, in this work, the damage locating vector (DLV) method using the normalized cumulative energy (nce) is applied to locate the damage (reduction of elemental stiffness matrix) in laminated composite beam. The numerical simulation is implemented on an asymmetric cross-fly (0°/90°) and a symmetric cross-fly (0°/90°/0°) cantilevered beams in which several damaged scenarios are considered.

Finite element formulation of a laminated composite beam

Consider a laminated composite beam consisting nL layers as depicted in Fig. 1. A global coordinate Oxyz is attached at center of the beam such that the x-axis is in longitudinal direction. The size of the beam is characterized by the length L, the width b and the thickness h. Based on the first-order shear deformation theory (FSDT) [ 26], the displacement field of the laminated composite beam is given by

(1) \begin{array}{l}u(x,y,z)=u_{\mathrm{0}}(x)+z{\beta }_x(x),\hfill \\ w(x,y,z)=w_{\mathrm{0}}(x),\hfill \end{array}

where u and w are x-direction and z-direction displacements of the beam respectively, u_{\mathrm{0}} and w₀ are the x-direction and z-direction displacement of the beam neutral axis respectively and {\beta }_x is the rotation of the normal to the middle plane around y-axis. Here the bending of the beam on the yz-plane is not considered.

According to Eq. (1), the strain field of the beam is expressed as follows:

(2) \mathbf{\epsilon }=\left\{\begin{array}{l}{\epsilon }_x\\ {\gamma }_{xz}\end{array}\right\}=\left\{\begin{array}{l}{u}_{\mathrm{0},x}+z{\beta }_{x,x}\\ w_{\mathrm{0},x}+{\beta }_x\end{array}\right\},

where the notation f_,x is referred to the derivative of a function f with respect to x. Then the Galerkin weak form of free vibration analysis of the laminated composite beam is given by

(3) {\displaystyle {\int }_L\delta {(\mathrm{\Lambda }\mathbf{u})}^T\mathbf{D}(\mathrm{\Lambda }\mathbf{u})dx}+{\displaystyle {\int }_L\delta {\mathbf{u}}^T\mathbf{m}\ddot{\mathbf{u}}dx}=\mathrm{0},

where

(4) \mathbf{\Lambda }=\left[\begin{array}{lll}{\displaystyle \frac{\partial }{\partial x}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\displaystyle \frac{\partial }{\partial x}}\\ \mathrm{0}& {\displaystyle \frac{\partial }{\partial x}}& \mathrm{1}\end{array}\right], \mathbf{u}=\left[\begin{array}{l}{u}_{\mathrm{0}}\\ w_{\mathrm{0}}\\ {\beta }_x\end{array}\right],

\delta is the notation of variation operator and \ddot{\mathbf{u}} is the second-order derivative with respect to time of u. The formulas for D and m are represented as follows:

(5) \mathbf{D}=\left[\begin{array}{lll}{A}_{\mathrm{11}}& B_{\mathrm{11}}& A_{\mathrm{12}}\\ B_{\mathrm{11}}& D_{\mathrm{11}}& B_{\mathrm{12}}\\ A_{\mathrm{21}}& B_{\mathrm{21}}& \kappa A_{\mathrm{22}}\end{array}\right], \mathbf{m}=\left[\begin{array}{lll}{I}_{\mathrm{0}}& \mathrm{0}& I_{\mathrm{1}}\\ \mathrm{0}& I_{\mathrm{0}}& \mathrm{0}\\ I_{\mathrm{1}}& \mathrm{0}& I_{\mathrm{2}}\end{array}\right],

where \kappa =\mathrm{5}/\mathrm{6} is the shear correction factor and A_{ij},B_{ij},D_{ij},I_{\mathrm{0}},I_{\mathrm{1}},I_{\mathrm{2}} are defined by

(6) \begin{array}{l}(A_{ij},B_{ij},D_{ij})=b{\displaystyle \sum _{k=\mathrm{1}}^{nL}{\displaystyle {\int }_{z_{k-\mathrm{1}}}^{z_k}{\bar{D}}_{ij}^k(\mathrm{1},z,z^{\mathrm{2}})dz}}\hspace{1em}(i,j=\mathrm{1},\mathrm{2}),\hfill \\ (I_{\mathrm{0}},I_{\mathrm{1}},I_{\mathrm{2}})=b{\displaystyle \sum _{k=\mathrm{1}}^{nL}{\displaystyle {\int }_{z_{k-\mathrm{1}}}^{z_k}{\rho }^k(\mathrm{1},z,z^{\mathrm{2}})dz}},\hfill \end{array}

where {\rho }^k is the mass density of the kth layer, \left[{\overline{D}}_{ij}^k\right]={\mathbf{a}}^k-{\mathbf{b}}^k{({\mathbf{c}}^k)}^{-\mathrm{1}}{({\mathbf{d}}^k)}^T with

(7) \begin{array}{ll}{\mathbf{a}}^k=\left[\begin{array}{ll}{\overline{Q}}_{\mathrm{11}}^k& {\overline{Q}}_{\mathrm{15}}^k\\ {\overline{Q}}_{\mathrm{51}}^k& {\overline{Q}}_{\mathrm{55}}^k\end{array}\right],& {\mathbf{b}}^k=\left[\begin{array}{lll}{\overline{Q}}_{\mathrm{12}}^k& {\overline{Q}}_{\mathrm{14}}^k& {\overline{Q}}_{\mathrm{16}}^k\\ {\overline{Q}}_{\mathrm{52}}^k& {\overline{Q}}_{\mathrm{54}}^k& {\overline{Q}}_{\mathrm{56}}^k\end{array}\right],\\ {\mathbf{c}}^k=\left[\begin{array}{lll}{\overline{Q}}_{\mathrm{22}}^k& {\overline{Q}}_{\mathrm{24}}^k& {\overline{Q}}_{\mathrm{26}}^k\\ {\overline{Q}}_{\mathrm{42}}^k& {\overline{Q}}_{\mathrm{44}}^k& {\overline{Q}}_{\mathrm{46}}^k\\ {\overline{Q}}_{\mathrm{62}}^k& {\overline{Q}}_{\mathrm{64}}^k& {\overline{Q}}_{\mathrm{66}}^k\end{array}\right],& {\mathbf{d}}^k=\left[\begin{array}{lll}{\overline{Q}}_{\mathrm{21}}^k& {\overline{Q}}_{\mathrm{41}}^k& {\overline{Q}}_{\mathrm{61}}^k\\ {\overline{Q}}_{\mathrm{25}}^k& {\overline{Q}}_{\mathrm{45}}^k& {\overline{Q}}_{\mathrm{65}}^k\end{array}\right],\end{array},

where {\overline{\mathbf{Q}}}^k={\mathbf{T}}^k{\mathrm{Q}}^k{({\mathbf{T}}^k)}^T in which T^k is the transformation matrix for kth layer and Q^k is the material matrix [ 26] given by

(8) {\mathbf{Q}}^k=\left[\begin{array}{llllll}{Q}_{\mathrm{11}}^k& Q_{\mathrm{12}}^k& & & & \\ Q_{\mathrm{21}}^k& Q_{\mathrm{22}}^k& & & \mathrm{0}& \\ & & Q_{\mathrm{33}}^k& & & \\ & & & Q_{\mathrm{44}}^k& & \\ & \mathrm{0}& & & Q_{\mathrm{55}}^k& \\ & & & & & Q_{\mathrm{66}}^k\end{array}\right],

where

(9) Q_{\mathrm{11}}^k={\displaystyle \frac{E_{\mathrm{1}}^k}{\mathrm{1}-{\nu }_{\mathrm{12}}^k{\nu }_{\mathrm{21}}^k}},Q_{\mathrm{12}}^k={\displaystyle \frac{{\nu }_{\mathrm{12}}^k{E}_{\mathrm{2}}^k}{\mathrm{1}-{\nu }_{\mathrm{12}}^k{\nu }_{\mathrm{21}}^k}},Q_{\mathrm{22}}^k={\displaystyle \frac{E_{\mathrm{2}}^k}{\mathrm{1}-{\nu }_{\mathrm{12}}^k{\nu }_{\mathrm{21}}^k}},

(10) Q_{\mathrm{21}}^k=Q_{\mathrm{12}}^k,Q_{\mathrm{33}}^k=\mathrm{0},Q_{\mathrm{66}}^k=G_{\mathrm{12}}^k,Q_{\mathrm{44}}^k=G_{\mathrm{23}}^k,Q_{\mathrm{55}}^k=G_{\mathrm{13}}^k.

By using finite element method, the beam is divided into Ne elements and each element has two nodes. On each element, the displacement field of the beam, u^e, is approximated by linear shape functions, N_{\mathrm{1}}^e and N_{\mathrm{2}}^e , as follows:

(11) {\mathbf{u}}^e={\mathbf{N}}^e{\mathbf{d}}^e,

in which {\mathbf{d}}^e={\left[\begin{array}{llllll}{u}_{\mathrm{1}}\hfill & w_{\mathrm{1}}\hfill & {\beta }_{x\mathrm{1}}\hfill & u_{\mathrm{2}}\hfill & w_{\mathrm{2}}\hfill & {\beta }_{x\mathrm{2}}\hfill \end{array}\right]}^{\mathrm{T}} is the nodal displacements of element e and

(12) {\mathbf{N}}^e=\left[\begin{array}{llllll}{N}_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& N_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& N_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e\end{array}\right].

By substituting Eq. (11) into Eq. (3), the discrete equation for free vibration analysis of the laminated composite beam is represented by

(13) \mathbf{M}\ddot{\mathbf{d}}+\mathbf{K}\mathbf{d}=\mathrm{0},

where d is the displacement vector; \ddot{\mathbf{d}} is the second-order derivative with respect to time of the displacement; M and K are the global mass matrix and stiffness matrix which are assembled from elemental ones and given by

(14) \begin{array}{l}\mathbf{K}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\mathbf{K}}^e}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\displaystyle {\int }_{L_e}{({\mathbf{B}}^e)}^T\mathbf{D}{\mathbf{B}}^{e}dx}},\hfill \\ \mathbf{M}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\mathbf{M}}^e}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\displaystyle {\int }_{L_e}{({\mathbf{N}}^e)}^T\mathbf{m}{\mathbf{N}}^{e}dx}},\hfill \end{array}

where L_e is the domain of the eth element and B^e is defined by

(15) {\mathbf{B}}^e=\left[\begin{array}{llllll}{N}_{\mathrm{1},x}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2},x}^e& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& N_{\mathrm{1},x}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2},x}^e\\ \mathrm{0}& N_{\mathrm{1},x}^e& N_{\mathrm{1}}^e& \mathrm{0}& N_{\mathrm{2},x}^e& N_{\mathrm{2}}^e\end{array}\right].

In the literature, a damaged structure is usually presented as a cracked model which can be simulated by different approaches [ 27− 34]. So far, various damage identification methods [ 35− 38] for estimating the crack size and location in structures were reported. For composite material, the crack and delamination are considered as two of the most popular failure forms which lead to the reduction of stiffness of the structures [ 8, 39]. In this paper, the damage of the laminated composite beam is simulated by the reduction in the elemental stiffness matrix while the mass matrix is assumed unchanged. Specifically, the damage in a layer of an element is simulated by the reduction of Young’s moduli in x- and y-direction (i.e., E₁ and E₂) with the same ratio as

(16) \begin{array}{l}{\tilde{E}}_{\mathrm{1}}^k=(\mathrm{1}-{\alpha }^{e,k})E_{\mathrm{1}}^k,\hfill \\ {\tilde{E}}_{\mathrm{2}}^k=(\mathrm{1}-{\alpha }^{e,k})E_{\mathrm{2}}^k,\hfill \end{array}

where {\alpha }^{e,k}\in [\mathrm{0},\mathrm{1}] is denoted as the damage ratio of the kth layer of the eth element; and {\tilde{E}}_i, i=\mathrm{1}\mathrm{,}\mathrm{2} are the Young’s moduli of the damaged layers.

Damage locating vector method

In DLV method, load vectors named damage locating vectors (DLVs) were extracted from the null space of flexibility change matrix. The loads conduce zero stress at damaged elements and this property can be used to locate damage in structure. Next, the construction of the normalized cumulative energy (nce) as an efficient indicator for DLV method is briefly presented.

From modal properties of structure, the flexibility matrix is computed as follows [ 8]:

\mathbf{F}={\displaystyle \sum _{i=\mathrm{1}}^{sdof}{\displaystyle \frac{\mathrm{1}}{{\omega }_i^{\mathrm{2}}}}}{\mathbf{\phi }}_i{\mathbf{\phi }}_i^{\mathrm{T}},

(17)where F is the flexibility matrix, {\omega }_i is the ith frequency, {\mathbf{\phi }}_i is the ith mass-normalized mode shape [ 8] and sdof is the number of degrees of freedom. In above equation, the flexibility matrix can be well approximated by using a few low modes as

(18) \tilde{\mathbf{F}}={\displaystyle \sum _{i=\mathrm{1}}^{nmod}{\displaystyle \frac{\mathrm{1}}{{\omega }_i^{\mathrm{2}}}}}{\mathbf{\phi }}_i{\mathbf{\phi }}_i^{\mathrm{T}},

where nmod is the number of considered low modes. When a structure is damaged, the mode shapes, the frequencies and then the flexibility matrix are changed. Therefore, the change in flexibility matrix can be used as an indicator to detect damage locations. This change is expressed as

(19) {\tilde{\mathbf{F}}}_{\mathit{\Delta }}={\tilde{\mathbf{F}}}_{UD}-{\tilde{\mathbf{F}}}_D,

where the indexes UD and D mean respectively undamaged and damaged states.

The DLVs defined as a basis for the null space of the change in flexibility can be calculated from a singular value decomposition (SVD) of {\tilde{\mathbf{F}}}_{\mathrm{\Delta }} as follows [ 6]:

(20) {\tilde{\mathbf{F}}}_{\mathit{\Delta }}\stackrel{SVD}{=}\left[{\mathbf{U}}_{\mathrm{1}} {\mathbf{U}}_{\mathrm{0}}\right]\left[\begin{array}{cc}{\sum }_{\mathrm{1}}\hfill & \mathrm{0}\hfill \\ \mathrm{0}\hfill & \mathrm{0}\hfill \end{array}\right]{\left[{\mathbf{V}}_{\mathrm{1}} {\mathbf{V}}_{\mathrm{0}}\right]}^{\mathrm{T}}, \mathrm{with} \mathrm{D}\mathrm{L}\mathrm{V}s={\mathbf{V}}_{\mathrm{0}},

where {\mathbf{\sum }}_{\mathrm{1}} is a diagonal matrix including nonzero singular values of {\tilde{\mathbf{F}}}_{\mathrm{\Delta }} ; \left[\begin{array}{ll}{\mathbf{U}}_{\mathrm{1}}& {\mathbf{U}}_{\mathrm{0}}\end{array}\right] and \left[\begin{array}{ll}{\mathbf{V}}_{\mathrm{1}}& {\mathbf{V}}_{\mathrm{0}}\end{array}\right] are orthogonal matrices.

The strain energy for each element when structure is subjected to a DLV load is calculated by

(21) {\mathrm{\Xi }}_i^e={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{d}_i^{e\mathrm{T}}{\mathbf{K}}^e{d}_i^e,

where {\mathbf{d}}_i^e is the displacement of the eth element when the undamaged structure is subjected to the ith DLV load, K^e is the eth elemental stiffness matrix. The normalized cumulative energy (nce) is defined for each element by the following equation:

(22) {\bar{\mathrm{\Xi }}}^e={\displaystyle \frac{{\mathrm{\Xi }}^e}{\underset{k}{\max }\{{\mathrm{\Xi }}^k\}}},

where

(23) {\mathrm{\Xi }}^e={\displaystyle \sum _{i=\mathrm{1}}^{ndlv}{\displaystyle \frac{{\mathrm{\Xi }}_i^e}{\underset{k}{\max }\{{\mathrm{\Xi }}_i^k\}}}},

where ndlv is the number of DLVs.

It should be noted that due to the feature of DLVs that they cause zero stress at damaged elements when applied as static load to a reference structure, nce of damaged elements equal zero for all DLVs. If only one DLV load is employed, nce criteria may lead to misidentify some undamaged elements in the set of identified damaged elements. To tackle this issue all DLVs are supposed to be used.

Note also that the DLV method is one of the inverse analysis methods which help analyze the measurement data (such as frequencies and mode shapes, etc.) to identify damage sites of structures. It is different from surrogate model-based methods which locate and quantify damage in specific structures via the approximate models of the relation between the input data (such as structures with specific damage location and extent) and the output data.

Numerical examples

In this section, two examples of an asymmetric cross-fly (0°/90°) and a symmetric cross-fly (0°/90°/0°) cantilevered beams with several damage scenarios are considered. The simulation of laminated composite beam is conducted by using Matlab software.

Asymmetric cross-fly (0°/90°) cantilevered beam

The laminated composite cantilevered beam (See Fig. 2(a)) is referred to Ref. [ 40] in which the material properties are given particularly in Table 1. This beam is divided equally into 16 elements. In this example, to carry out multiple damage scenarios of the beam, three interested damage cases are summarized in Table 2. In the first case, only the layer 1 of the 4th element is damaged while in the second case, only the layer 2 of the 10th element is weaken. For the last case, damage appears at both layers of element 4 and 10. To obtain DLV loads, the flexibility matrices of both intact and damaged beams are approximated by all frequencies and mode shapes (48 modes). The first 15 non-dimensional angular frequencies of the beams are listed in Table 3. The nce values of all elements for the three damage cases are represented in Fig. 3. To investigate the consequence of the quality of the approximated flexibility matrix to nce values, in the third case, different numbers of modes are employed as illustrated in Fig. 4.

It can be observed that for all damage cases, the nce values are dropped dramatically to zero at damaged elements (case 1 at element 4, case 2 at element 10 and case 3 at elements 4 and 10) as depicted in Fig. 3. This helps us distinguish them as damaged elements clearly. Fig. 4 shows that the more numbers of modes used (i.e. the more exact the flexibility matrix is), the smaller the nce values at damaged elements are. In addition, when only a few first modes (like 5 modes) are exploited to estimate the flexibility matrix, the nce values still give a pretty good identification of damage locations.

Symmetric cross-fly (0°/90°/0°) cantilevered beam

The material properties of the symmetric cross-fly (0°/90°/0°) cantilevered beam, depicted in Fig. 2(b), are the same as the previous example. Similarly, the beam is divided into 16 equal elements. In Table 4, three damage scenarios are set up to reflect different damage kinds of the beam. The first case models a damage that occurs at the mid-layer of an element while the second case covers the situation that damage is exist at face-layers only. For the final case, the damage appears at two different elements and both mentioned damage kinds are considered mutually. In DLV implementation, the intact and damaged flexibility matrices for all cases are approximated using all modes (48 modes). The first 15 non-dimensional frequencies are listed in Table 5. Figure 5 depicts the nce values of all elements corresponding to three cases of damage. In case 3, the change of nce values when the number of approximating modes is increased is illustrated in Fig. 6.

As shown in Fig. 5, regardless of damage kinds, the nce values identify the position of damaged elements correctly. Similar to the previous example, it can be seen that, although using a small number of modes, DLV method gives fairly good judgement about damage location and the exactness is improved when the number of modes is increased (see Fig. 6).

Conclusions

In this paper, the damage locating vector (DLV) method using normalized cumulative energy (nce) is applied to locate the damage in laminated composite beam structures. Two numerical examples of different damage scenarios are implemented on an asymmetric cross-fly (0°/90°) and a symmetric cross-fly (0°/90°/0°) cantilevered beam. The damage in the structures is simulated by the reduction of longitudinal and transverse Young’s moduli of a specific layer of an element. The results show that regardless of damage scenarios, DLV method can diagnose successfully the damage sites in these structures. For future work, the DLV method should be combined with an optimization algorithm to locate damage as well as estimate the damage extent of structures. In addition, the effect of uncertain parameters of composite materials [ 41, 42] as well as measurement noise on the method should also be investigated.

References

[1] Naderi M, Khonsari M. Real-time fatigue life monitoring based on thermodynamic entropy. Wiley Online Library, 2011, vol. 10

[2] Messina A, Williams E J, Contursi T. Structural damage detection by a sensitivity and statistical-based method. Journal of Sound and Vibration, 1998, 216(5): 791–808

[3] Cawley P, Adams R D. The location of defects in structures from measurements of natural frequencies. Journal of Strain Analysis for Engineering Design, 1979, 14(2): 49–57

[4] Salawu O, Williams C. Bridge assessment using forced-vibration testing. Journal of Structural Engineering, 1995, 121(2): 161–173

[5] Pandey A K, Biswas M, Samman M M. Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration, 1991, 145(2): 321–332

[6] Bernal D. Load vectors for damage localization. Journal of Engineering Mechanics, 2002, 128(1): 7–14

[7] Chen X, Yu L. Flexibility-based objective functions for constrained optimization problems on structural damage detection. Advanced Materials Research, 2011, 186: 383–387

[8] Pandey A K, Biswas M. Damage detection in structures using changes in flexibility. Journal of Sound and Vibration, 1994, 169(1): 3–17

[9] Nobahari M, Seyedpoor S M. An efficient method for structural damage localization based on the concepts of flexibility matrix and strain energy of a structure. Structural Engineering and Mechanics, 2013, 46(2): 231–244

[10] Stubbs N, Kim J T, Topole K. An efficient and robust algorithm for damage localization in offshore platforms. Proc ASCE Tenth Struct Congr, 1992, 543–546

[11] Seyedpoor S M. A two stage method for structural damage detection using a modal strain energy based index and particle swarm optimization. International Journal of Non-linear Mechanics, 2012, 47(1): 1–8

[12] Doebling S W, Farrar C R, Prime M B, Shevitz D W. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review. Los Alamos National Laboratory, 1996

[13] Gunes B, Gunes O. Structural health monitoring and damage assessment Part I: A critical review of approaches and methods. International Journal of Physical Sciences, 2013, 8(34): 1694–1702

[14] Markou M, Singh S. Novelty detection: a review—part 2: Neural network based approaches. Signal Processing, 2003, 83(12): 2499–2521

[15] Salawu O S. Detection of structural damage through changes in frequency: A review. Engineering Structures, 1997, 19(9): 718–723

[16] Trendafilova I, Palazzetti R, Zucchelli A. Damage assessment based on general signal correlation. Application for delamination diagnosis in composite structures. European Journal of Mechanics. A, Solids, 2015, 49(0): 197–204

[17] Minak G, Palazzetti R, Trendafilova I, Zucchelli A. Localization of a delamination and estimation of its length in a composite laminate beam by the VSHM and pattern recognition methods. Mechanics of Composite Materials, 2010, 46(4): 387–394

[18] Lee B T, Sun C T, Liu D. An assessment of damping measurement in the evaluation of integrity of composite beams. Journal of Reinforced Plastics and Composites, 1987, 6(2): 114–125

[19] Dixit A, Hanagud S. Comments on: Curvature mode shape-based damage assessment of carbon/epoxy composite beams. Journal of Intelligent Material Systems and Structures, 2010, 21(6): 659–663

[20] Moreno-García P, Araújo dos Santos J V, Lopes H. A new technique to optimize the use of mode shape derivatives to localize damage in laminated composite plates. Composite Structures, 2014, 108(1): 548–554

[21] Bernal D, Gunes B. Flexibility based approach for damage characterization: Benchmark application. Journal of Engineering Mechanics, 2003, 130(1): 61–70

[22] Gao Y, Spencer B Jr, Bernal D. Experimental verification of the flexibility-based damage locating vector method. Journal of Engineering Mechanics, 2007, 133(10): 1043–1049

[23] Qian J, Ji X, Xu Y. Two-stage damage diagnosis approach for steel braced space frame structures. Engineering Structures, 2007, 29(12): 3277–3292

[24] Sim S H, Jang S A, Spencer B F Jr, Song J. Reliability-based evaluation of the performance of the damage locating vector method. Probabilistic Engineering Mechanics, 2008, 23(4): 489–495

[25] Quek S T, Tran V A, Hou X Y, Duan W H. Structural damage detection using enhanced damage locating vector method with limited wireless sensors. Journal of Sound and Vibration, 2009, 328(4): 411–427

[26] Reddy J N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, 2004

[27] Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109

[28] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[29] Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[30] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29−30): 2777–2799

[31] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[32] Nguyen-Thoi T, Rabczuk T, Lam-Phat T, Ho-Huu V, Phung-Van P. Free vibration analysis of cracked Mindlin plate using an extended cell-based smoothed discrete shear gap method (XCS-DSG3). Theoretical and Applied Fracture Mechanics, 2014, 72(0): 150–163

[33] Liu G R, Chen L, Nguyen-Thoi T, Zeng K Y, Zhang G Y. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems. International Journal for Numerical Methods in Engineering, 2010, 83(11): 1466–1497

[34] Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: Framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998

[35] Friswell M I, Penny J E T. Crack modeling for structural health monitoring. Structural Health Monitoring, 2002, 1(2): 139–148

[36] Sinha J, Friswell M, Edwards S. Simplified models for the location of cracks in beam structures using measured vibration data. Journal of Sound and Vibration, 2002, 251(1): 13–38

[37] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of multiple flaws in piezoelectric structures using XFEM and level sets. Computer Methods in Applied Mechanics and Engineering, 2014, 275: 98–112

[38] Wu W, Zhu H, Zhuang X, Ma G, Cai Y. A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis. Theoretical and Applied Fracture Mechanics, 2014, 72: 136–149

[39] Della C N, Shu D. Vibration of delaminated composite laminates: A review. Applied Mechanics Reviews, 2007, 60(1): 1–20

[40] Khdeir A A, Reddy J N. Free vibration of cross-ply laminated beams with arbitrary boundary conditions. International Journal of Engineering Science, 1994, 32(12): 1971–1980

[41] Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464

[42] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96, Part B: 520–535

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg