Membership-set identification method for structural damage based on measured natural frequencies and static displacements

Abstract

Based on measured natural frequencies and static displacements, an improved interval analysis technique is proposed for structural damage detection by adopting membership-set identification and two-step model updating procedures. Due to the scarcity of uncertain information, the uncertainties are considered as interval numbers in this article. Via the first-order Taylor series expansion, the interval bounds of the elemental stiffness parameters of undamaged and damaged structures are obtained. The structural damage is detected by the quantitative measure of the possibility of damage existence in elements, which is more reasonable than the probability of damage existence in the condition of less measurement data. In this study, the conversation of the interval analysis method is remarkably reduced by the membership-set identification technique. The present method is applied to a truss structure and a steel cantilever plate for damage identification, and the damage identification results obtained by the interval analysis method and probabilistic method are compared. This article also discusses the effects of damage level and uncertainty level on detection results. The numerical examples show that the wide intervals resulting from the interval operation can be narrowed by the proposed non-probabilistic approach, and the feasibility and applicability of the present method are validated.

Keywords

Damage identification interval analysis membership-set identification model updating uncertainty

Introduction

The structural deterioration and damage resulting from environmental erosion, overloading, fatigue, material aging, or other unexpected events exist in many engineering fields. These damages affect the efficiency of the structure, contributing to possible failures that could be catastrophic in terms of loss in economy and life. Thus, structural damage detection is becoming a worldwide research subject.

During the last two decades, extensive researches have been performed in the area of non-destructive structural damage identification, and these are divided into two major categories: dynamic¹ and static.² Both techniques are based on the fact that local damages usually cause a decrease in structural stiffness, which produces changes in the global characteristics of the structure. With the development of measurement equipment and signal processing techniques, the vibration properties of a structure can be measured more accurately and conveniently. This method, therefore, has been developing rapidly and has found application in civil, mechanical, and aerospace engineering. The dynamic testing data used for damage detection are: natural frequency, mode shape, frequency response function, mode shape curvature, modal flexibility, modal strain energy, and so on.^3–7 Among these data types, natural frequency is the most widely used because it can be measured most conveniently and accurately. Moreover, natural frequencies are global properties of the structure; thus, they can be measured at a few locations or even at one point.⁸ However, dynamic parameter identification requires the use of the mass, stiffness, and damping properties, which is more complicated than the static method. Static tests are easily executable and provide additional information to dynamic identification without any introduction of uncertainties due to masses and damping ratios.⁹ Sanayei and Scampoli¹⁰ developed a computer program for parameter identification of a one-third scale-reinforced concrete deck using incomplete static test data. Banan et al.¹¹ used the incomplete sets of applied static forces and displacements to estimate element stiffness.

The sensitivity-based finite element (FE) model updating approach is an important method for damage detection, in which a theoretical FE model of a structure is adjusted to an updated model consisting of measured data. The efficiency of this damage identification method relies on the accuracy of the analytical FE model and the measured frequencies. Most studies assume that the analytical FE model is precise enough to represent the vibration properties of the structure and that the measurements are also accurate. In practice, however, there are many uncertainties during the model updating procedure, such as FE modeling error and measurement noise. Uncertainties in the FE model exist due to inaccurate physical parameters, non-ideal boundary conditions, and structural non-linear properties. In order to address the problem of these uncertainties, extensive research has been performed. Collins et al.¹² first derived a statistical identification procedure by treating the initial structural parameters as normally distributed random variables with zero means and specified covariance. Xia and Hao¹³ developed a statistical damage identification algorithm based on the change of natural frequency to account for the effects of random noise in the vibration data and variations in the FE model. The statistics of the parameters are estimated by the perturbation method and verified with the Monte Carlo technique. Yeo et al.¹⁴ presented a damage assessment algorithm for framed structures based on the static response with a regularization technique. Statistical distributions of the system parameters with a set of noise-polluted measured data were obtained by the data perturbation method, and the damage was then assessed by a statistical hypothesis test approach. However, in practice, it is hard to determine the probability density function due to the complexity of the sources of uncertainty and the impossibility of obtaining sufficient experimental data. Hence, most literatures assume that the uncertain variables obey a certain probability distribution, but such assumptions may lead to questionable results.

Set-based theoretical convex methods, including convex models and interval analysis, are effective non-probabilistic methods to deal with the uncertain problems.¹⁵ Interval analysis method has been applied in various fields of engineering.^16,17 In this method, the uncertain quantities are considered as interval numbers rather than as random variables. Compared with the probabilistic analysis approach, the interval analysis method requires less information, namely, only the bounds of the uncertain parameters, which can be easily obtained in practical engineering problems. Gabriele et al.¹⁸ combined the model updating and interval analysis methods to propose an interval damage identification method, but this method is so computationally expensive that it cannot be applied in the damage detection of large structures. García et al.¹⁹ proposed a new methodology based on a combination of interval analysis and constraint propagation techniques and applied it to the structural assessment of Sorraia River Bridge.

In order to obtain a more accurate estimation of identified parameters, more measured data should be utilized. Xia et al.²⁰ proposed a statistical method with combined uncertain frequency and mode shape data for structural damage identification. Nazin and Polyak²¹ proposed a method for the interval estimation of parameters of linear multi-output models based on a large number of measurements. In this study, an interval method based on uncertain frequencies and static measurements is developed for structural damage identification. In the non-probabilistic damage identification method, the intervals of elemental stiffness parameters are obtained by utilizing measured frequencies and static displacements, respectively. Based on these intervals, the membership-set operation is performed to obtain tighter interval estimations of the elemental stiffness parameters. Furthermore, the two-step model updating procedure is adopted in this method to make identification results more accurate.

Sensitivity-based FE model updating for damage identification

FE model updating for structures with frequency changes

Consider the free vibration problem of an undamped structure with $N$ degrees of freedom (DOFs):

M \overset{\cdot\cdot}{x} (t) + Kx (t) = 0

(1)

where: $M$ is the $N \times N$ mass matrix; $K$ is the $N \times N$ stiffness matrix; and $x (t)$ and $\overset{\cdot\cdot}{x} (t)$ are the displacement and acceleration vectors, respectively. The eigenvalue problem corresponding to equation (1) can be expressed as:

K ϕ_{i} = λ_{i} M ϕ_{i}, i = 1, 2, \dots, N

(2)

where $λ_{i}$ and $ϕ_{i}$ are the ith eigenvalue and mass-normalized mode shape, respectively. Due to structural damage, we suppose that changes occur in the quantities $K$ , $λ_{i}$ , and $ϕ_{i}$ , whereas the mass, $M$ , remains unchanged. These can be expressed as:

\begin{matrix} K_{c} = K + d K, M_{c} = M \\ λ_{ci} = λ_{i} + d λ_{i}, ϕ_{ci} = ϕ_{i} + d ϕ_{i} \end{matrix}

(3)

where: $K_{c}$ , $M_{c}$ , $λ_{ci}$ , and $ϕ_{ci}$ are the stiffness matrix, mass matrix, ith eigenvalue, and mass-normalized mode shape, respectively, in the changed state; and $d K$ , $d λ_{i}$ , and $d ϕ_{i}$ denote the respective change in the stiffness matrix, the eigenvalue, and eigenvector.

Substituting equation (3) into equation (2) and left-multiplying $ϕ_{i}^{T}$ , the following expression can be obtained by neglecting the high-order terms:

ϕ_{i}^{T} \cdot d K \cdot ϕ_{i} = d λ_{i} = λ_{ci} - λ_{i}

(4)

For the FE model of a structure, the matrices $K$ and $K_{c}$ can be expressed in the form of non-positive parameter decomposition, as in the following:

K = \sum_{i = 1}^{m} α_{i} K_{i} = α_{1} K_{1} + α_{2} K_{2} + \dots + α_{m} K_{m}

(5)

K_{c} = \sum_{i = 1}^{m} α_{ci} K_{i} = α_{c 1} K_{1} + α_{c 2} K_{2} + \dots + α_{cm} K_{m}

(6)

where: $m$ is the number of elements in the structure; $α_{i}$ is the initial elemental stiffness parameter (bending stiffness in this article); $K_{i}$ is the ith elemental stiffness matrix divided by $α_{i}$ ; and $α_{ci}$ is the structural elemental stiffness parameter in the changed state.

According to equations (5) and (6), equation (4) can be rewritten as follows:

S_{f} \cdot d α = d λ

(7)

where: $d α$ and $d λ$ are the elemental stiffness change and eigenvalue change vectors, respectively; and $S_{f}$ is the eigensensitivity matrix, whose elements are:

S_{fij} = ϕ_{i}^{T} K_{j} ϕ_{i}, i = 1, 2, \dots, n_{1}, j = 1, 2, \dots, m

(8)

where $n_{1}$ is the number of available measured frequencies. If $n_{1}$ equals the number of elements in the structure, $m$ , then $S_{f}$ will be a square matrix. However, in practice, $n_{1}$ is usually less than $m$ . Normally, its solution is written as:

d α = S_{f}^{+} \cdot d λ

(9)

where $S_{f}^{+}$ is the Moore–Penrose generalized inverse of matrix $S_{f}$ .

FE model updating for structures with static displacement changes

Consider the static problem of a structure with $N$ DOFs:

Ku = P

(10)

where: $K$ is the $N \times N$ stiffness matrix; $u$ is the displacement vector; and $P$ is the force vector. The damage identification equation based on static displacement sensitivity is given as:

S_{s} \cdot d α = d u = u_{c} - u

(11)

where: $u_{c}$ is the measured static displacement vector at $n_{2}$ DOFs in the changed state; and $S_{s}$ is the sensitivity matrix of the static displacements at $n_{2}$ DOFs with respect to a change in structural stiffness parameters, which can be written as:

S_{s} = [\begin{matrix} \frac{\partial u_{1}}{\partial α_{1}} & \frac{\partial u_{1}}{\partial α_{2}} & \dots & \frac{\partial u_{1}}{\partial α_{m}} \\ \frac{\partial u_{2}}{\partial α_{1}} & \frac{\partial u_{2}}{\partial α_{2}} & \dots & \frac{\partial u_{2}}{\partial α_{m}} \\ ⋮ & ⋮ & ⋮ \\ \frac{\partial u_{n_{2}}}{\partial α_{1}} & \frac{\partial u_{n_{2}}}{\partial α_{2}} & \dots & \frac{\partial u_{n_{2}}}{\partial α_{m}} \end{matrix}] \in R^{n_{2} \times m}

(12)

Similarly, the solution for equation (11) can be written as:

d α = S_{s}^{+} d u

(13)

where $S_{s}^{+}$ is the Moore–Penrose generalized inverse of matrix $S_{s}$ .

The sensitivity matrix, $S_{s}$ , can be computed by taking the first differential of equation (10) with respect to the structural stiffness parameter $α_{i}$ as:

\frac{\partial K}{\partial α_{i}} u + K \frac{\partial u}{\partial α_{i}} = \frac{\partial P}{\partial α_{i}}

(14)

where:

\frac{\partial u}{\partial α_{i}} = {[\frac{\partial u_{1}}{\partial α_{i}}, \frac{\partial u_{2}}{\partial α_{i}}, \dots, \frac{\partial u_{n}}{\partial α_{i}}]}^{T}

which can be derived from equation (14):

\frac{\partial u}{\partial α_{i}} = K^{- 1} (\frac{\partial P}{\partial α_{i}} - \frac{\partial K}{\partial α_{i}} u)

(15)

where $\partial P / \partial α_{i} = 0$ because the force vector is independent of the elemental stiffness parameters. Furthermore, equation (15) can be written as:

\frac{\partial u}{\partial α_{i}} = - K^{- 1} \frac{\partial K}{\partial α_{i}} u

(16)

According to equation (5), we can obtain:

\frac{\partial K}{\partial α_{i}} = \frac{\partial (\sum_{i = 1}^{m} α_{i} K_{i})}{\partial α_{i}} = K_{i}

(17)

The substitution of equations (16) and (17) into equation (12) leads to the expression for $S_{s}$ :

S_{s} = - K^{- 1} [\begin{matrix} K_{1} u & K_{2} u & \dots & K_{m} u \end{matrix}] \in R^{n_{2} \times m}

(18)

where $K_{1}, K_{2}, \dots, K_{m}$ are known structural parameters. So far, $d α$ could be obtained from equation (13).

Interval damage detection for structures based on the membership-set identification technique

It is understandable that if the uncertainty level is larger than or close to the measured quality changes due to damages, the damaged members cannot be identified or the healthy members may be identified as damaged. In this study, the measurement uncertainties are considered as the main factors affecting the damage identification result. Based on interval mathematics, the measured frequency and displacement intervals in the changed state can be expressed as:

λ_{c}^{I} = [\underline{λ_{c}}, \bar{λ_{c}}] = {(λ_{c 1}^{I}, λ_{c 2}^{I}, \dots, λ_{c n_{1}}^{I})}^{T}, λ_{ci}^{I} = [\underline{λ_{ci}}, \bar{λ_{ci}}], i = 1, 2, \dots, n_{1}

(19)

u_{c}^{I} = [\underline{u_{c}}, \bar{u_{c}}] = {(u_{c 1}^{I}, u_{c 2}^{I}, \dots, u_{c n_{2}}^{I})}^{T}, u_{ci}^{I} = [\underline{u_{ci}}, \bar{u_{ci}}], i = 1, 2, \dots, n_{2}

(20)

where: the superscript “I” represents the interval number or vector; “ $\bar{s}$ ” denotes the upper bounds of the interval number or vector; and “ $\underline{s}$ ” denotes the lower bounds of the interval number or vector. For simplicity, the uncertain measured parameters (natural frequencies and static displacements) in the undamaged or damaged state of the structure can be written as a vector: $X = (X_{q})^{T}, q = 1, 2, \dots, (n_{1} + n_{2})$

\begin{matrix} X^{I} = X^{c} + Δ X^{I} = X^{c} + Δ X [- 1, 1] \\ X^{c} = {(X_{1}^{c}, X_{2}^{c}, \dots, X_{n_{1} + n_{2}}^{c})}^{T}, \\ Δ X = {(Δ X_{1}, Δ X_{2}, \dots, Δ X_{n_{1} + n_{2}})}^{T} \end{matrix}

(21)

where the superscript “c” and symbol “ $Δ$ ” represent the corresponding middle value and radius of the interval number or vector, respectively.

Based on the interval expression (21), vector $X$ can be put in the following form:

{\begin{matrix} X = X^{C} + d X, | d X | \leq Δ X \\ X_{q} = X_{q}^{C} + d X_{q}, | d X_{q} | \leq Δ X_{q} \end{matrix}

(22)

Interval damage detection utilizing measured frequencies

Consider the damage identification formula (9). The vectors $d α$ and $d λ$ and the matrix $S_{f}$ are functions of $X = (X_{1}, X_{2}, \dots, X_{n_{1}})^{T}$ , so they can be expanded as the first-order Taylor series in terms of $d X_{q}$ :

d α (X) = d α (X^{c} + d X) \approx d α (X^{c}) + \sum_{q = 1}^{n_{1}} \frac{\partial (d α (X^{c}))}{\partial X_{q}} d X_{q}

(23)

d λ (X) = d λ (X^{c} + d X) \approx d λ (X^{c}) + \sum_{q = 1}^{n_{1}} \frac{\partial (d λ (X^{c}))}{\partial X_{q}} d X_{q}

(24)

S_{f} (X) = S_{f} (X^{c} + d X) \approx S_{f} (X^{c}) + \sum_{q = 1}^{n_{1}} \frac{\partial S_{f} (X^{c})}{\partial X_{q}} d X_{q}

(25)

Substituting equations (23) to (25) into equation (9) and neglecting the high-order terms, the following equations can be obtained:

d α (X) = S_{f} {(X^{c})}^{+} \cdot d λ (X^{c}) + S_{f} {(X^{c})}^{+} \sum_{q = 1}^{n_{1}} (\frac{\partial (d λ (X^{c}))}{\partial X_{q}} - \frac{\partial S_{f} (X^{c})}{\partial X_{q}} \cdot d α (X^{c})) d X_{q}

(26)

Using the natural interval extension, we can obtain the interval of the elemental stiffness change as follows:

d α^{I} (X) = S_{f} {(X^{c})}^{+} \cdot d λ (X^{c}) + \sum_{q = 1}^{n_{1}} S_{f} {(X^{c})}^{+} (\frac{\partial (d λ (X^{c}))}{\partial X_{q}} - \frac{\partial S_{f} (X^{c})}{\partial X_{q}} \cdot d α (X^{c})) Δ X_{q}^{I}

(27)

Since $Δ X_{q}^{I} = Δ X_{q} [- 1, 1]$ , the lower and upper bounds of the interval vector $d α^{I}$ are:

\underline{d α (X)} = S_{f} {(X^{c})}^{+} \cdot d λ (X^{c}) - \sum_{q = 1}^{n_{1}} | S_{f} {(X^{c})}^{+} (\frac{\partial (d λ (X^{c}))}{\partial X_{q}} - \frac{\partial S_{f} (X^{c})}{\partial X_{q}} \cdot d α (X^{c})) | Δ X_{q}

(28)

\bar{d α (X)} = S_{f} {(X^{c})}^{+} \cdot d λ (X^{c}) + \sum_{q = 1}^{n_{1}} | S_{f} {(X^{c})}^{+} (\frac{\partial (d λ (X^{c}))}{\partial X_{q}} - \frac{\partial S_{f} (X^{c})}{\partial X_{q}} \cdot d α (X^{c})) | Δ X_{q}

(29)

Thus, the lower and upper bounds of $d α^{I}$ can be obtained from equations (28) and (29), and the sensitivity coefficients $\partial (d λ (X^{c})) / \partial X_{q}$ and $\partial S_{f} (X^{c}) / \partial X_{q}$ can be obtained according to the following expressions:

\frac{\partial (d λ_{j} (X^{c}))}{\partial X_{q}} = \frac{\partial λ_{cj} (X^{c})}{\partial X_{q}} - \frac{\partial λ_{j} (X^{c})}{\partial X_{q}}, j = 1, 2, \dots, n_{1}, q = 1, 2, \dots, n_{1}

(30)

\frac{\partial S_{fjk} (X^{c})}{\partial X_{q}} = \frac{\partial (ϕ_{j} {(X^{c})}^{T} K_{k} ϕ_{j} (X^{c}))}{\partial X_{q}} = 2 ϕ_{j} {(X^{c})}^{T} K_{k} \frac{\partial ϕ_{j} (X^{c})}{\partial X_{q}}, j = 1, 2, \dots, n_{1}, k = 1, 2, \dots, m, q = 1, 2, \dots, n_{1}

(31)

For the initial analytical FE model, it is clear that $λ_{j}$ and $ϕ_{j}$ are independent of measured parameters. Hence:

\frac{\partial λ_{j} (X^{c})}{\partial X_{q}} = 0, \frac{\partial ϕ_{j} (X^{c})}{\partial X_{q}} = 0, \frac{\partial λ_{cj}}{\partial X_{q}} = 1

(32)

Substituting equations (30) to (32) into equations (28) and (29), the lower and upper bounds of interval vector $d α^{I}$ in the changed state are determined.

Interval damage detection utilizing measured static displacements

Consider the damage identification formula (11). The vectors $d α$ and $d u$ and the matrix $S_{s}$ are functions of $X = (X_{n_{1}}, X_{n_{1} + 1}, \dots, X_{n_{1} + n_{2}})^{T}$ , so they can be expanded as the first-order Taylor series in terms of $d X_{q}$ :

d α (X) = d α (X^{c} + d X) \approx d α (X^{c}) + \sum_{q = n_{1}}^{n_{1} + n_{2}} \frac{\partial (d α (X^{c}))}{\partial X_{q}} d X_{q}

(33)

d u (X) = d u (X^{c} + d X) \approx d u (X^{c}) + \sum_{q = n_{1}}^{n_{1} + n_{2}} \frac{\partial (Δ u (X^{c}))}{\partial X_{q}} d X_{q}

(34)

S_{s} (X) = S_{s} (X^{c} + d X) \approx S_{s} (X^{c}) + \sum_{q = n_{1}}^{n_{1} + n_{2}} \frac{\partial S_{s} (X^{c})}{\partial X_{q}} d X_{q}

(35)

Substituting equations (33) to (35) into equation (11) and neglecting high-order terms, the following equations can be obtained:

d α = S_{s} {(X^{c})}^{+} d u (X^{c}) + S_{s} {(X^{c})}^{+} \sum_{q = n_{1}}^{n_{1} + n_{2}} [\frac{\partial (d u (X^{c}))}{\partial X_{q}} - \frac{\partial S_{s} (X^{c})}{\partial X_{q}} d α (X^{c})] d X_{q}

(36)

Using the natural interval extension, we can obtain the interval of the elemental stiffness change as follows:

d α^{I} = S_{s} {(X^{c})}^{+} d u (X^{c}) + S_{s} {(X^{c})}^{+} \sum_{q = n_{1}}^{n_{1} + n_{2}} [\frac{\partial (d u (X^{c}))}{\partial X_{q}} - \frac{\partial S_{s} (X^{c})}{\partial X_{q}} d α (X^{c})] Δ X_{q}^{I}

(37)

Since $Δ X_{q}^{I} = Δ X_{q} [- 1, 1]$ , the lower and upper bounds of the interval vector $d α^{I}$ are:

\underline{d α} = S_{s} {(X^{c})}^{+} d u (X^{c}) - | \sum_{q = n_{1}}^{n_{1} + n_{2}} S_{s} {(X^{c})}^{+} [\frac{\partial (d u (X^{c}))}{\partial X_{q}} - \frac{\partial S_{s} (X^{c})}{\partial X_{q}} d α (X^{c})] | Δ X_{q}

(38)

\bar{d α} = S_{s} {(X^{c})}^{+} d u (X^{c}) + | \sum_{q = n_{1}}^{n_{1} + n_{2}} S_{s} {(X^{c})}^{+} [\frac{\partial (d u (X^{c}))}{\partial X_{q}} - \frac{\partial S_{s} (X^{c})}{\partial X_{q}} d α (X^{c})] | Δ X_{q}

(39)

The lower and upper bounds of $d α^{I}$ can be obtained from equations (38) and (39). The sensitivity coefficients $\partial (d u (X^{c})) / \partial X_{q}$ and $\partial S_{s} (X^{c}) / \partial X_{q}$ can be obtained according to the following expressions:

\frac{\partial (d u_{j} (X^{c}))}{\partial X_{q}} = \frac{\partial u_{cj} (X^{c})}{\partial X_{q}} - \frac{\partial u_{rj} (X^{c})}{\partial X_{q}}, j = 1, 2, \dots, n, q = n_{1}, (n_{1} + 1), \dots, (n_{1} + n_{2})

(40)

\frac{\partial S_{s} (X^{c})}{\partial X_{q}} = \frac{\partial (\begin{matrix} \frac{\partial u (X^{c})}{\partial α_{1}} \frac{\partial u (X^{c})}{\partial α_{2}} \dots \frac{\partial u (X^{c})}{\partial α_{m}} \end{matrix})}{\partial X_{q}}, q = n_{1}, (n_{1} + 1), \dots, (n_{1} + n_{2})

(41)

where $\partial^{2} u (X^{c}) / \partial α_{i} \partial X_{q}$ can be computed by taking the first differential of equation (14) with respect to the structural stiffness parameter $X_{q}$ as:

K \frac{\partial^{2} u}{\partial α_{i} \partial X_{q}} = - \frac{\partial^{2} K}{\partial α_{i} \partial X_{q}} u - \frac{\partial K}{\partial α_{i}} \frac{\partial u}{\partial X_{q}} - \frac{\partial K}{\partial X_{q}} \frac{\partial u}{\partial α_{i}}, i = 1, 2, \dots, m, q = n_{1}, (n_{1} + 1), \dots, (n_{1} + n_{2})

(42)

For the initial analytical FE model, it is clear that $u_{rj}$ and $K$ are independent of measured parameters. Hence:

\frac{\partial u_{j} (X^{c})}{\partial X_{q}} = 0, \frac{\partial K (X^{c})}{\partial X_{q}} = 0, \frac{\partial u_{cj} (X^{c})}{\partial X_{q}} = 1

(43)

Membership-set identification technique

In the previous sections, the interval estimations of elemental stiffness parameters are obtained, respectively, utilizing different measurement data (frequency and static displacement) by the natural interval extension. However, the natural interval extension may lead to conservative results. In order to reduce the conversation and make the interval damage identification results more accurate, a membership-set identification (MSI) technique was applied in this study. The damage information obtained using various measured data is investigated synthetically to assess the true structural damage. The interval stiffness parameter vectors, updated by different measured data, can be expressed as:

(α_{c}^{I})^{k} = [\underline{α_{c}^{k}}, \bar{α_{c}^{k}}], k = 1, 2, \dots, M

(44)

where $k$ denotes different measured data.

The tighter interval estimation is obtained by the membership-set operation on the interval vectors $(α_{c}^{I})^{k}$ :

α_{c}^{I *} = ⋂_{k = 1}^{M} {(α_{c}^{I})}^{k}

(45)

where $α_{c}^{I *}$ is the optimized interval estimation of the structural elemental stiffness vector in the changed (damaged) state. The planar schematic diagram is shown in Figure 1. In this study, only the natural frequency and static displacement are used as measured data for structural damage identification. Hence, $k$ = 2, and the optimized interval estimation can be rewritten as:

α_{c}^{I *} = {(α_{c}^{I})}_{f} \cap {(α_{c}^{I})}_{s}

(46)

where ${(α_{c}^{I})}_{f}$ and ${(α_{c}^{I})}_{s}$ are the interval vectors of elemental stiffness parameters obtained using measured frequency and measured static displacement, respectively. The schematic diagram is shown in Figure 2.

Possibility of damage existence

Usually, the initial analytical model is different from the real structure due to modeling errors. In this study, two procedures of FE model updating will be performed. The initial analytical FE model is updated to obtain the undamaged FE model and the damaged FE model, respectively, and then the damage is identified by comparing the differences of elemental stiffness parameters between the undamaged and the damaged FE models.

Applying the model updating method and the MSI technique developed in previous sections, the elemental stiffness parameter vectors within the undamaged and damaged FE models can be obtained as two interval vectors:

α_{r}^{I} = {(α_{r 1}^{I}, α_{r 2}^{I}, \dots, α_{ri}^{I}, \dots, α_{rm}^{I})}^{T}, i = 1, 2, \dots, m

(47)

α_{d}^{I} = {(α_{d 1}^{I}, α_{d 2}^{I}, \dots, α_{di}^{I}, \dots, α_{dm}^{I})}^{T}, i = 1, 2, \dots, m

(48)

The structural damage can easily be identified through the comparison of these two vectors, when $α_{ri}$ and $α_{di}$ are real numbers. Nevertheless, due to uncertainty, the real elemental stiffness parameters $α_{ri}$ and $α_{di}$ will belong to the intervals denoted here as $[\underline{α_{ri}}, \bar{α_{ri}}]$ and $[\underline{α_{di}}, \bar{α_{di}}]$ , respectively. For interval numbers $[\underline{α_{ri}}, \bar{α_{ri}}]$ and $[\underline{α_{di}}, \bar{α_{di}}]$ , we cannot directly identify the damage of structures. Generally, the middle value of $α_{di}^{I}$ is less than the middle value of $α_{ri}^{I}$ . It cannot be assured, however, that the value of $α_{ri}$ is always larger than the value of $α_{di}$ . In order to address this problem, the elemental stiffness parameters of the undamaged and damaged FE models ( $α_{di}^{I}$ and $α_{ri}^{I}$ ) are represented on the same axis, as represented in Figure 3.

Here, the quantitative measure of the possibility of damage existence (PoDE) can be introduced. The two rectangles in Figure 3 show the region of variation of $α_{ri}$ and $α_{di}$ , respectively. The interval of $α_{ri}^{I}$ obtained by the first model updating procedure represents the healthy interval for the ith element. Therefore, the hatched region denotes the possible damaged region. The PoDE in the elements denoted as $p_{i}$ can be defined as:

p_{i} = \frac{\underline{α_{ri}} - \underline{α_{di}}}{\bar{α_{di}} - \underline{α_{di}}} \times 100 %

(49)

where $p_{i}$ takes a value of between 0% and $100 %$ . If the damage in an element does not take place or the lower bound of $α_{ri}^{I}$ is equal to or less than the lower bound of $α_{di}^{I}$ , $p_{i}$ will vanish. On the other hand, if the $p_{i}$ of an element is $100 %$ , we conclude that the upper bound of $α_{di}^{I}$ is equal to or less than the lower bound of $α_{ri}^{I}$ ; in other words, damage exists in the element to a certain degree.

PoDE in one-dimensional description with multiple load cases (MLCs)

In the previous two sections, the MSI technique and PoDE were introduced. MSI in MLCs has many advantages when compared with the single load case (SLC). In this article, SLCs can be classified into two cases based on frequency and static displacement identification, and we write them as LC-F and LC-S, respectively.

Now we give the figure in the form of a one-dimensional description for the PoDE of MLCs. As we can see in Figure 4, $[\underline{α_{df}}, \bar{α_{df}}]$ and $[\underline{α_{ds}}, \bar{α_{ds}}]$ are the damaged intervals of the elemental stiffness parameters based on frequency identification and static displacement identification, respectively; $[\underline{α_{rf}}, \bar{α_{rf}}]$ and $[\underline{α_{rs}}, \bar{α_{rs}}]$ are the undamaged intervals based on frequency identification and static displacement identification, respectively; and $[\underline{α_{d}}, \bar{α_{d}}]$ and $[\underline{α_{r}}, \bar{α_{r}}]$ are the damaged interval and undamaged interval after intersection, respectively.

The center value and radius are two key elements to an interval number. If we make these two parameters certain, we can describe an interval easily. In this article, the center values of the undamaged intervals are always the same; the center values of the damaged intervals are determined by the damage level; and the radii of all undamaged and damaged intervals are determined by uncertainty (noise). Ignoring the computation error, the center values of the undamaged interval and the damaged interval under the same model are the same, respectively.

In Figure 4, the advantage of MSI has been shown. The undamaged and damaged intervals in MLCs are the intersection of the LC-F and LC-S. Because of the advantage in the interval intersection, the PoDE in MLCs is higher than both the load cases.

Study on damage level

As we all know, the center values of the damaged intervals are determined by damage level, so different damage models with different damage levels may lead to different center values. This is represented in Figure 5 as follows.

In Figure 5, two damaged model samples are set: Sample A and Sample B – both with the same undamaged model but with the degree of damage in Sample A set higher than that in Sample B. Therefore, the center value of Sample A’s undamaged elemental stiffness parameters is higher than that of Sample B’s. The uncertainties (noise) are the same in both the samples, so the interval radii are also the same.

In Figure 5: LC-FA and LC-FB represent Sample A and Sample B based on the frequency identification load case, respectively; LC-SA and LC-SB represent Sample A and Sample B based on the static displacement identification load case, respectively; MLC-A and MLC-B represent Sample A and Sample B based on MSI MLC, respectively; $[\underline{α_{dfA}}, \bar{α_{dfA}}]$ and $[\underline{α_{dfB}}, \bar{α_{dfB}}]$ are the damaged intervals of Sample A and Sample B based on frequency identification, respectively; $[\underline{α_{dsA}}, \bar{α_{dsA}}]$ and $[\underline{α_{dsB}}, \bar{α_{dsB}}]$ are the damaged intervals of Sample A and Sample B based on static displacement identification, respectively; and $[\underline{α_{dA}}, \bar{α_{dA}}]$ and $[\underline{α_{dB}}, \bar{α_{dB}}]$ are the damaged intervals of Sample A and Sample B after intersection.

It can be seen from the interval figure that the PoDE of Sample A is less than that of Sample B, and this is because the damage information for low damage levels is more easily interfered with by noise than that for high damage levels. In other words, in the interval figure, since a part of the damaged interval of MLC-A is more easily covered by the undamaged interval than in MLC-B, we can conclude that higher damage levels can easily be identified. We will give two numerical examples in the next section.

Study on noise

As we all know, the radii of the intervals are determined by uncertainty (noise), so different noise may lead to different radii, as represented in Figure 6.

In Figure 6, two damage model samples are set (Sample A and Sample B) with the same damage level, so undamaged and damaged center values are the same. The noise of damaged and undamaged cases in Sample A is less than that in Sample B. Therefore, the interval radii of Sample A’s elemental stiffness parameters are less than that of Sample B’s.

From Figure 6: $[\underline{α_{rfA}}, \bar{α_{rfA}}]$ and $[\underline{α_{rfB}}, \bar{α_{rfB}}]$ are the undamaged intervals of Sample A and Sample B based on frequency identification, respectively; $[\underline{α_{rsA}}, \bar{α_{rsA}}]$ and $[\underline{α_{rsB}}, \bar{α_{rsB}}]$ are the undamaged intervals of Sample A and Sample B based on static displacement identification, respectively; and $[\underline{α_{rA}}, \bar{α_{rA}}]$ and $[\underline{α_{rB}}, \bar{α_{rB}}]$ are the undamaged intervals of Sample A and Sample B after intersection. Other nomenclature are similar to that presented in the earlier sections.

From the interval figure, we can see that the PoDE of Sample A is higher than that of Sample B, and this is because the undamaged interval radii become too large as the noise goes up, with the real damaged information being submerged by noise. In other words, in the interval figure, as the undamaged radius of MLC-B is too large, a part of the damaged interval of MLC-B is more easily submerged by the undamaged interval than in MLC-A. Thus, we can conclude that lower uncertainty (noise) can easily be identified. We will give two numerical examples for this in the next section.

Two numerical examples

Examples

In order to illustrate the validity of the present interval method for damage identification, a 10-bar truss structure and a cantilever plate structure are utilized, as shown in Figures 7 and 8, respectively.

The properties of the truss are as follows: the cross-section of the bars is rectangular, with the area $A = 4 \times 10^{- 4} m^{2}$ ; the length $l$ is $0.6 m$ ; the mass density of the material is $7.67 \times 10^{3} kg / m^{3}$ ; and the actual Young’s modulus in the intact state is $E = 2.0 \times 10^{11} N / m^{2}$ . The properties of the plate are as follows: the plate’s side is $a = 0.1 m$ ; its thickness is $t = 0.01 m$ ; its density is $ρ = 7500 kg / m^{3}$ ; and its Poisson’s ratio is $υ = 0.3$ . Suppose that the elemental stiffness parameters of the initial analytical model are 1.1 times larger that of the undamaged model in both the structures and that the damaged model is similar to the undamaged model except that the stiffness parameter is reduced by 20% in Element Number 5 of the truss and by 30% in Element Numbers 6, 7, and 8 of the plate, respectively.

Uncertain parameters are independent, and we assume that: their uncertainty is 2%; the two methods are based on the uncertain frequencies and static measurements identification method in single mode; and the interval radius obtained by the interval method includes the results of the probability method. The probability method is based on some distribution assumptions, and, hence, it requires a lot of data validation. However, the interval method without any assumptions can obtain the reliable bounds of $α_{i}$ . On the other hand, because of the special properties of the interval, we can obtain integrated operations from the results of the two methods. The MLC damage identification results are listed in Tables 1 and 2.

Table 1.

Schematic diagram of truss for membership-set identification

Element number	LC		MLC (%)
	Frequency (%)	Static displacement (%)
1	2.6	11.7	2.6
2	0	0	0
3	0	15.4	15.4
4	2.6	15.5	40.1
5	72.2	71.4	91.2
6	31.8	2.2	31.8
7	17.4	1.8	17.4
8	17.4	2.1	17.4
9	31.8	0	31.8
10	20.1	3.5	20.1

LC: load case; MLC: multiple load case.

Table 2.

Schematic diagram of plate for membership-set identification

Element number	LC		MLC (%)
	Frequency (%)	Static displacement (%)
1	7.9	0	0
2	20.1	0.5	0.6
3	20.1	0.4	0.5
4	7.9	3.6	3.6
5	40.6	0	0
6	90.7	93.7	96.1
7	90.7	58.7	90.7
8	40.6	100	100
9	0	2.7	2.7
10	14.5	0.3	14.5
11	14.5	4.9	14.5
12	0	8.7	3.2
13	0	1.6	0
14	0	1.0	0
15	0	2.3	0
16	0	3.4	0
17	0	3.4	0
18	39.9	3.6	39.9
19	39.9	2.5	39.9
20	0	3.6	0

LC: load case; MLC: multiple load case.

Study on damage level

Different degrees of damage-to-damage identification results will have an important influence. The more serious the damage, the easier it is to identify. Therefore, if the degree of damage is minute, due to the influence of uncertain parameters, the real information will be submerged in the noise, and we will get wrong results. Thus, the advantage of working with integrated identification is then obvious.

As in Tables 3 and 4, suppose that the degrees of damage of Element Number 5 are 20%, 25%, and 30% in the truss and also assume that the degrees of damage are the same in the plate for Element Numbers 6, 7, and 8, respectively. The PoDE of Element Number 5 of the truss under the different degrees of damage is much larger than that in the other elements, as well as in the plate. As the degree of damage reduces, the PoDE of all bars decreases. This is because the results for the first-order Taylor expansion for small damages are more accurate, whereas the low degree of damage information in the damaged element is more easily interfered with by noise and, hence, may not be accurate. In the case of a high degree of damage, owing to the Taylor expansion of linearization, the undamaged elements will have a high PoDE. Compared with the identification results, under the SLC and the MLC methods, we can see that, for the damaged bar, the PoDE of MLCs are much higher than that of SLCs. This is because the interval radii of the undamaged and damaged elements and the stiffness parameters reduce more after the integrated operations, but the center values change only slightly. This decreases the denominator in equation (49) more, but the numerator is almost unchanged; the center values of the undamaged element-related and damaged element-related stiffness parameters are almost equal, and, hence, the denominator and the numerator in equation (49) are always low. After integrated operations, the denominator will increase, and, hence, the PoDE will decrease. The PoDE of MLCs is lower than that of SLCs for undamaged bars, whereas the PoDE of MLCs is higher than that of SLCs for damaged bars. The extent of change in the undamaged element is much smaller than that in the damaged element, and the identification results are both improved.

Table 3.

Schematic diagram of truss for PoDE under different damage levels

Damage level (%)	Element number	1	3	4	5	6	7	8	9	10
20	Frequency (%)	2.6	0	2.6	72.2	31.8	17.4	17.4	31.8	20.1
	Static displacement (%)	11.7	15.4	15.5	71.4	2.2	1.8	2.1	0	3.5
	MLC (%)	2.6	15.4	40.1	91.2	31.8	17.4	17.4	31.8	20.1
25	Frequency (%)	3.3	0	3.3	84.0	37.1	20.6	20.6	37.1	23.1
	Static displacement (%)	12.8	17.3	17.5	82.6	2.4	2.3	2.7	0	4.2
	MLC (%)	3.3	17.3	44.3	100	35.4	20.6	20.6	35.5	23.1
30	Frequency (%)	4.8	0	4.8	100	46.7	26.6	26.6	46.7	28.0
	Static displacement (%)	14.5	20.5	20.7	100	2.7	3.5	3.9	0	5.5
	MLC (%)	4.8	15.6	49.3	100	39.7	26.6	26.6	40.5	28.0

MLC: multiple load case; PoDE: possibility of damage existence.

Table 4.

Schematic diagram of plate for PoDE under different damage levels

Damage level (%)	Element number	1	2	3	4	5	6	7	8	9	10
20	Frequency (%)	4.2	10.9	10.9	4.2	25.6	53.2	53.2	25.6	0	7.3
	Static displacement	0	0.4	0.3	2.2	0	58.8	36.2	63.1	1.6	0.2
	MLC (%)	0	0.5	0.3	2.2	0	59.3	53.2	67.0	1.6	7.3
25	Frequency (%)	5.8	15.1	15.1	5.8	32.9	70.7	70.7	32.9	0.0	10.5
	Static displacement (%)	0	0.4	0.3	2.8	0	75.7	47.0	72.7	2.1	0.3
	MLC (%)	0	0.5	0.4	2.8	0	77.7	70.7	81.4	2.1	10.5
30	Frequency (%)	7.9	20.1	20.1	7.9	40.6	90.7	90.7	40.6	0.0	14.5
	Static displacement (%)	0	0.5	0.4	3.6	0	93.7	58.7	100	2.7	0.3
	MLC (%)	0	0.6	0.5	3.6	0	96.1	90.7	100	2.7	14.5

Damage level (%)	Element number	11	12	13	14	15	16	17	18	19	20

20	Frequency (%)	7.3	0	0	0	0	0	0	23.4	23.4	0
	Static displacement (%)	3.0	7.0	1.0	0.7	1.5	2.2	2.1	2.3	1.5	2.2
	MLC (%)	7.3	5.5	0	0	0	0	0	23.4	23.4	0
25	Frequency (%)	10.5	0	0	0	0	0	0	31.1	31.1	0
	Static displacement (%)	3.9	1.3	1.3	0.9	1.9	2.8	2.7	2.9	2.0	2.9
	MLC (%)	10.5	0	0	0	0	0	0	31.1	31.1	0
30	Frequency (%)	14.5	0	0	0	0	0	0	39.9	39.9	0
	Static displacement (%)	4.9	8.7	1.6	1.0	2.3	3.4	3.4	3.6	2.5	3.6
	MLC (%)	14.5	3.2	0	0	0	0.0	0	39.9	39.9	0

MLC: multiple load case; PoDE: possibility of damage existence.

Study on noise

Another important factor influencing identification results is uncertainty (noise). A low degree of noise does not influence identification results much and is sometimes neglected; however, the identification results may be submerged by noise. Table 5 and Table 6, respectively, give the identification results for the truss and the plate with uncertainty of 1.5%, 2%, 2.5%, 3% and 4% when the degree of damage is 20%. From the results, it can be seen that the change of PoDE for the undamaged elements is slightly lower in the case of different noise. However, the PoDEs for the damaged elements decrease as the increasing noise in large degree of noise, the identification parameter interval radius may be too large. If the identification parameter intervals obtained from the damaged data are included in those obtained from the undamaged data, then the real information will be submerged by noise. Therefore, the identification results will be incredible. The PoDEs of the plate’s elements at different uncertainty levels are listed in Figure 9.

Table 5.

Schematic diagram of truss for PoDE under different noise

Element number	Uncertainty
	1.5%	2%	2.5%	3%	4%
1	3.9%	2.6%	1.9%	1.4%	0.8%
2	0%	0%	0%	0%	0%
3	17.6%	15.4%	14.0%	13.1%	12.0%
4	65.7%	40.1%	27.9%	15.3%	0.8%
5	100%	91.2%	69.6%	56.1%	40.1%
6	43.1%	31.8%	25.1%	20.6%	15.0%
7	23.6%	17.4%	13.6%	11.1%	8.0%
8	23.6%	17.4%	13.6%	11.1%	8.0%
9	29.8%	31.8%	25.1%	20.6%	15.0%
10	27.7%	20.1%	15.6%	12.6%	8.8%

PoDE: possibility of damage existence.

Table 6.

Schematic diagram of plate for PoDE under different noise

Element number	Uncertainty
	1%	1.5%	2%	3%	4%
1	0%	0%	0%	0%	0%
2	0.6%	0.6%	0.5%	0.5%	0.5%
3	0.5%	0.5%	0.5%	0.4%	0.4%
4	3.6%	3.6%	3.6%	3.6%	3.6%
5	0%	0%	0%	0%	0%
6	96.1%	61.1%	43.6%	26.1%	17.3%
7	90.7%	59.3%	43.7%	28.0%	20.2%
8	100%	51.0%	33.7%	20.4%	14.8%
9	2.7%	2.7%	2.7%	2.7%	2.7%
10	14.5%	8.6%	5.7%	2.8%	1.3%
11	14.5%	8.6%	5.7%	2.8%	1.3%
12	3.2%	4.2%	4.7%	5.2%	5.4%
13	0%	0%	0%	0%	0%
14	0%	0%	0%	0%	0%
15	0%	0%	0%	0%	0%
16	0%	0%	0%	0%	0%
17	0%	0%	0%	0%	0%
18	39.9%	26.0%	19.1%	12.1%	8.7%
19	39.9%	26.0%	19.1%	12.1%	8.7%
20	0%	0%	0%	0%	0%

PoDE: possibility of damage existence.

Conclusions

In this study, the interval analysis method for structural damage identification has been proposed. The measured static displacements in undamaged and damaged states were considered as uncertainties and described as interval numbers. Through first-order Taylor series expansion, the lower and upper bounds of the elemental stiffness parameters of the undamaged and damaged structures were obtained by the model updating method. The PoDE of each element was evaluated based on the intervals of elemental stiffness parameters in undamaged and damaged FE models. A higher value of PoDE implied more possibility of damage occurrence. Moreover, the MSI technique was utilized to reduce the conversation of the interval analysis method. The proposed method was applied to identify the damage to an uncertain truss structure. The comparison with the probabilistic approach demonstrated the validity of the present method. Using numerical examples, the interval damage identification method is illustrated to be an effective supplementary to the probabilistic damage identification method when the parameter information is limited.

Footnotes

Funding

We thank the National Nature Science Foundation of the P. R. China (No.11002013), 111 Project (No.B07009) and Defense Industrial Technology Development Program (No.A2120110001, No.B2120110011) for support.

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