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Abstract

A static algorithm is presented in this article for damage localization in beam structures by moving load. Central to the damage localization approach is the combination of the moving load technique and the pure bending theory. The first advantage of the presented algorithm is that only a few sensors are needed in the static test. The second advantage of the presented algorithm is that only the deflection parameters of the current structure are needed in damage localization process. The proposed method is applied to a simple supported beam and its feasibility is verified using damage simulations. It was found that the defect location can be identified by inspecting the sudden increase in the curvature of the deflection curve for the beam. The results showed that the presented algorithm is effective on the localization of single damage or multiple damages in the beam structure without baseline data.

Keywords

Damage localization moving load static test deflection curvature

Introduction

Many load-carrying structural systems have been in use for tens or even hundreds of years. It is unavoidable for these structures to continuously accumulate damage in their service environment. Thus, it is most desirable that this damage be detected at the earliest possible stage to assure safety. Many research studies have been conducted during the past years in the area of damage detection–based structural response characteristics with different algorithms and useful databases.^1–3 Generally, the existing damage detection approaches can be clarified into two major categories, that is, the dynamic detection methods and the static detection methods. Significant efforts have already been devoted to developing dynamic detection methods using various vibration characteristics, such as natural frequencies,^4–6 mode shapes,^7–11 modal flexibility,^12–15 curvature,^11,16 modal strain energy,^17–19 and moving load responses.^20–22 Radzieński et al.¹⁰ defined new damage indicators based on the change in natural frequencies and any mode shape (measured or modeled) for beam-like structures. Kopsaftopoulos and Fassois¹¹ proposed a vibration-based statistical time series method that is capable of effective damage detection, precise localization, and magnitude estimation within a unified stochastic framework. Sung et al.¹⁶ present a damage detection method for beam-like structure by the normalized uniform load surface (NULS) curvature obtained by modal flexibility. It has shown that changes in NULS curvature only occur at damaged elements in the structure. Xiang et al.¹⁷ used the changes in damage-induced curvature mode shape and natural frequency to form a hybrid damage detection approach to detect damages on the surface of conical shell. Xu et al.¹⁸ developed an energy damage detection strategy by disposing strain responses for a long-span cable-stayed bridge. Yi et al.¹⁹ proposed a multi-stage method based on “energy-damage” theory using the wavelet packet transform (WPT) and the artificial neural network (ANN) for structural damage diagnosis. Using modal strain energy, Cha and Buyukozturk²⁰ developed hybrid multi-objective optimization algorithms to detect damages in various three-dimensional steel structures. Khorram et al.²¹ compared the performances of two wavelet-based damage detection approaches to find the location and the size of a crack in a beam subjected to a moving load. Using Hilbert–Huang transform, Roveri and Carcaterra²² presented a novel method for damage detection of bridge structures under a traveling load. The technique uses a single-point measurement and is able to identify the presence and the location of the damage along the beam. Cavadas et al.²³ discussed the application of data-driven methods on moving load responses to detect the occurrence and the location of damage. However, these dynamic detection algorithms are faced with some difficulties in applications.²⁴ One is that the precision of dynamic data are affected by many factors such as stiffness, mass, and damping. Therefore, many dynamic approaches unavoidably suppose that there is no damping and the mass is unchanged before and after damage. The next issue is that modal data for some stiff civil structures are difficult to be measured accurately, especially for high-order modes. In view of this, the static detection algorithms^25–34 also attracted much attention in civil engineering area in recent years due to their simplicity for implementation. However, most of these methods need a detailed finite element model (FEM) or the baseline parameters of the intact system. It is sometimes impossible to obtain a detailed FEM or the baseline data for many existing structures. Thus, a simple and effective localization method that does not need an FEM or the baseline data of the intact structure will be welcomed.

For this reason, this article presents a static method for damage localization in beam structures by moving load. The key point of the damage location algorithm lies in the combination of the moving load technique and the pure bending theory. The first advantage of the presented algorithm is that only a few sensors are needed in the static test. The second advantage of the presented algorithm is that only the deflection parameters of the current structure are needed in damage localization process. The proposed method is verified by a numerical example to show its feasibility. It has been shown that the presented algorithm is effective on the localization of single damage or multiple damages in the beam structure.

Theory

Figure 1 presents a simple supported beam “A–B” under two symmetrical concentrated forces F and the corresponding bending moment diagram. Apparently, the bending moment at any cross section in the segment “1 − n” of this beam is a constant $(M = F \times Δ x)$ , as shown in Figure 1.

Figure 1.

Simple supported beam and corresponding bending moment diagram under two symmetrical concentrated forces.

According to the theory of material mechanics, the curvature of the deflection curve for this beam can be calculated by

$c_{v} = \frac{M}{EI}$ (1)

where c_v is the curvature, M is the bending moment, E is the elastic modulus, and I is the moment of inertia of the beam’s cross section with respect to the transversal axis. From equation (1), one can conclude that the curvature c_v of the deflection curve is a constant if EI is constant for the beam. When EI is reduced due to structural damage, the curvature c_v of the deflection curve at the damaged location will increase accordingly. This means that the defect location for this beam can be identified by inspecting the sudden increase in curvature of the deflection curve.

On the other hand, the curvature can be calculated by the central difference technique of the deflection vector d for the beam as follows

$c_{v}^{i} = \frac{d^{i + 1} - 2 d^{i} + d^{i - 1}}{{(Δ x)}^{2}}$ (2)

where dⁱ is the ith coefficient of d, $c_{v}^{i}$ is the corresponding curvature, and $Δ x$ is the length of structural element. The deflection vector consists of the vertical displacement data of the nodes in the beam, as shown in Figure 1. Equation (2) shows that damage identification can be carried out if the deflection parameters of the beam were measured through static testing. However, it is unrealistic and uneconomic to place many sensors in the static test of the beam. According to reciprocal theorem of displacement in structural mechanics, this difficulty can be avoided by increasing the loading conditions instead of increasing the number of measurement sensors. For example, as shown in Figure 2, the deflection of node “i” $(Δ i)$ in Figure 2(a) equals the deflection of node “1” $(Δ 1)$ in Figure 2(a′). This means that only a sensor is needed in measurement of the deflection of a beam under a concentrated force by this moving load technique.

Figure 2.

Reciprocal theorem of displacement: (a) the load F is at node 1 and (a') the load F is at node i.

On the other hand, according to superposition principle in linear elastic theory, the deflection of the beam in Figure 1 under two symmetrical concentrated forces will be equal to the summation of the deflections obtained by the two load cases as shown in Figure 3(a) and (b). Using the above moving load technique, the deflection vectors in Figure 3(a) and (b) can obtained as d₁ and d₂, respectively. And then the deflection vector of the beam as shown in Figure 1 or 3(c) can be calculated as

$d = d_{1} + d_{2}$ (3)

Figure 3.

Superposition principle of load: (a) the load F is at node 1, (b) the load F is at node n, and (c) the superposition of load cases (a) and (b).

In the end, a summary of the overall process for this algorithm is given as follows:

Step 1. Measure the deflection vectors (d₁ and d₂) for the two load cases as shown in Figure 3(a) and (b) using the moving load technique.

Step 2. Compute the deflection vector (d) of the beam as shown in Figure 1 or 3(c) by equation (3).

Step 3. Compute the curvatures $(c_{v}^{i})$ of the deflection curve for the beam by equation (2).

Step 4. Evaluate the possible damage locations according to the sudden increases in the curvatures $(c_{v}^{i})$ .

Numerical example

Figure 4 presents a simple supported beam used to demonstrate the feasibility of the moving load method for damage localization. The beam is divided into 36 segments and the length of each segment is $Δ x = 0.04 m$ . The physical parameters of this beam are as follows: Young’s modulus $E = 200 GPa$ , density $ρ = 7800 kg / m^{3}$ , moment of inertia $I = 2.133 \times 10^{- 10} m^{4}$ , and cross-sectional area $A = 1.6 \times 10^{- 4} m^{2}$ . Six damage cases as shown in Table 1 are studied in the example. Case 1: element 18 is damaged with a stiffness loss of 10%. Case 2: element 18 is damaged with a stiffness loss of 20%. Case 3: element 18 is damaged with a stiffness loss of 30%. Case 4: element 18 is damaged with a stiffness loss of 40%. Case 5: elements 9 and 18 are damaged with stiffness losses of 10% and 20%, respectively. Case 6: elements 9 and 18 are damaged with stiffness losses of 20% and 30%, respectively. For each damage case, suppose that the deflections at all nodes as shown in Figure 4 are measured by moving load $(F = 5 N)$ . Note that only two sensors are needed in the static test for this example. Certainly, the reliability of the damage localization result will increase as the number of sensors increases. From steps 1 to 3 of the proposed method, the values of curvatures $c_{v}^{i}$ for nodes 1–35 in damage cases 1–4 are shown in Table 2 and Figure 5. One can see from Table 2 and Figure 5 that nodes 17 and 18 have larger curvatures than the other nodes. The sudden increases in curvatures indicate that element 18 is damaged since nodes 17 and 18 are exactly associated with the 18th element as shown in Figure 4. Figure 6 presents the damage localization results for damage cases 1–4 with 1% noise on simulated data. One can see that the curvature indexes are obviously affected by the noise, and the damaged location (element 18) cannot be clearly identified by inspecting the corresponding curvatures for small damage cases 1 and 2. For large damage cases 3 and 4, the proposed method can still determine the damaged location even if the measurement error is considered.

Figure 4.

The simple supported beam and sensor layout.

Table 1.

Damage cases in the example.

Damage case	Element number	Stiffness reduction (%)
Case 1	18	10
Case 2	18	20
Case 3	18	30
Case 4	18	40
Case 5	9, 18	10, 20
Case 6	9, 18	20, 30

Table 2.

Curvature values when element 18 is damaged with 10%, 20%, and 40% stiffness reductions (no noise).

Node number	Curvatures for case 1	Curvatures for case 2	Curvatures for case 3	Curvatures for case 4
1	0.0039	0.0039	0.0039	0.0039
2	0.0047	0.0047	0.0047	0.0047
3	0.0047	0.0047	0.0047	0.0047
4	0.0047	0.0047	0.0047	0.0047
5	0.0047	0.0047	0.0047	0.0047
6	0.0047	0.0047	0.0047	0.0047
7	0.0047	0.0047	0.0047	0.0047
8	0.0047	0.0047	0.0047	0.0047
9	0.0047	0.0047	0.0047	0.0047
10	0.0047	0.0047	0.0047	0.0047
11	0.0047	0.0047	0.0047	0.0047
12	0.0047	0.0047	0.0047	0.0047
13	0.0047	0.0047	0.0047	0.0047
14	0.0047	0.0047	0.0047	0.0047
15	0.0047	0.0047	0.0047	0.0047
16	0.0047	0.0047	0.0047	0.0047
17	0.0049	0.0057	0.0053	0.0063
18	0.0049	0.0057	0.0053	0.0063
19	0.0047	0.0047	0.0047	0.0047
20	0.0047	0.0047	0.0047	0.0047
21	0.0047	0.0047	0.0047	0.0047
22	0.0047	0.0047	0.0047	0.0047
23	0.0047	0.0047	0.0047	0.0047
24	0.0047	0.0047	0.0047	0.0047
25	0.0047	0.0047	0.0047	0.0047
26	0.0047	0.0047	0.0047	0.0047
27	0.0047	0.0047	0.0047	0.0047
28	0.0047	0.0047	0.0047	0.0047
29	0.0047	0.0047	0.0047	0.0047
30	0.0047	0.0047	0.0047	0.0047
31	0.0047	0.0047	0.0047	0.0047
32	0.0047	0.0047	0.0047	0.0047
33	0.0047	0.0047	0.0047	0.0047
34	0.0047	0.0047	0.0047	0.0047
35	0.0039	0.0039	0.0039	0.0039

The bold values denote the curvature values corresponding to the damaged element.

Figure 5.

Curvature values when element 18 is damaged (no noise).

Figure 6.

Curvature values when element 18 is damaged (1% noise).

For damage cases 5 and 6, elements 9 and 18 can be determined from Figure 7 without noise to be damaged since their corresponding nodes 8, 9, 17, and 18 have larger curvatures than the other nodes. When 1% noise is considered, the damage localization result for small damage case 5 is unreliable from Figure 8. For large damage case 6, the curvature indexes in Figure 8 can still indicate the damage locations by the notable increases in the curvatures. In addition, one can conclude that the curvatures will increase as the damage extents increase from Figures 5 –8. It has been shown that the proposed method is effective on the localization of single damage or multiple damages in the beam structure.

Figure 7.

Curvature values when elements 9 and 18 are damaged (no noise).

Figure 8.

Curvature values when elements 9 and 18 are damaged (1% noise).

It must be noted that the proposed method has some drawbacks as follows. The first is that the proposed approach can detect only damages that are between the two loading locations as shown in Figure 1. It is suggested to put the loads as close as possible to the support. This, in turn, complicates the things because deflections close to the support are extremely small and difficult to be measured with good accuracy. The second drawback is that the proposed method can detect only notable stiffness reduction in the structure. If the beam is a reinforced concrete one and rebars are cut at a certain location, the safety is dramatically reduced, while the stiffness is only slightly affected. This type of engineering disease cannot be detected by the proposed method. In addition, the support conditions of the beam structure have some impact on the proposed method. If the support conditions are non-ideal, the dynamic flexibility matrix obtained by the experimental modal parameters can be used to construct the virtual pure bending state for the beam structure. This issue will be studied in the further research.

Conclusion

A new deflection-based method has been proposed for beam damage localization by moving load. This method has two advantages: (1) only a few sensors are needed in static test and (2) only the deflection parameters of the current structure are needed in damage localization. To confirm the feasibility of the proposed method, a simple supported beam was investigated for several damage scenarios. The results showed that the presented algorithm is effective to identify structural damage locations. The presented algorithm may have good prospects in defect localization for beam structures without baseline data.

Footnotes

Academic Editor: Simon Laflamme

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 (41272345,11202138 and 41572305).

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