1. Introduction
The identification of bridge structure damage has attracted extensive attention recently [
1,
2,
3]. According to the type of data used, damage identification methods can be divided into dynamic methods and static methods. The fundamental concept behind these techniques is to determine the location and extent of structural damage by determining the damage characteristics from changes in static or dynamic factors. These damage detection methods can be subdivided into the static displacement-based method, the vibration frequency-based method, the vibration mode-based method, the dynamic flexibility-based method, the best achievable theory-based method, etc.
For the static displacement-based method, Lee and Eun [
4] proposed an analytical method for damage detection by using displacement curvature and deflection data expanded from the measured data. Jin et al. [
5] proposed a static sensitivity method for damage detection by measuring the static displacement of the structure. By solving the static damage identification equation, the damage parameters of each element can be obtained and used to determine the location of damage in the structure. Song et al. [
6] used the static strain data to obtain the corresponding static displacement for detecting structural damages. Ma et al. [
7] employed wavelet analysis to diagnose structural defects using the static displacements of the structures. The fault position can be determined from the wavelet maxima lines and the fault severity can be assessed by the wavelet coefficients along the corresponding maxima lines. Wang et al. [
8] proposed a technique for hanger fault diagnosis using static deflection change and the cable force of an arch bridge hanger. Numerical and experimental results indicated that their approach can successfully assess the location and reliability of faults in arch bridges.
For the vibration frequency-based method, Wang et al. [
9] presented a damage detection method based on electromechanical admittances of multiple PZT patches and the cross-correlation coefficient. The damage sites and intensities of the plain concrete beam may be identified by monitoring the electromechanical admittance signals of each PZT over a range of frequencies. Dahak et al. [
10] proposed a frequency contour method to detect the damage location and depth in a beam. The contour line was plotted using only the value of the changes in the measured natural frequencies and the vectors of the curvature mode shapes of the intact structure. Zhong et al. [
11] developed a new approach based on auxiliary mass spatial probing using the spectral center correction method for the damage detection of beam-like structures. They discovered that fracture information for beam-like structural damage detection may be obtained from the derivatives of the natural frequency curve. Gillich et al. [
12] found that relative frequency shifts are associated with structural defects and can be used to locate the cracks in a structure. Mousavi et al. [
13] discovered that the instantaneous frequency of the first intrinsic mode function is related to the fault location in the structure. Mostafa et al. [
14] employed the shape of the instantaneous frequency as a fault indicator to exclude the influence of the vehicle dynamics. They found that a high-resolution instantaneous frequency derived from the dynamic response could accurately identify structural defects.
For the vibration mode-based approach, Unger et al. [
15] detected structural damage by minimizing the differences between the experimental mode shape and the corresponding analytical mode shape. A prestressed concrete beam experiment revealed that modal curvatures are very sensitive to local changes in bending stiffness close to the sensor position, but are insensitive to local changes farther from the measurement point. Tran-Ngoc et al. [
16] proposed a damage detection method according to a combination of the artificial neural network (ANN) and cuckoo search (CS) algorithm. They discovered that, for localizing and quantifying structural damage, ANN-CS (ANN-CS) is more accurate than ANN and takes less time to compute. Altunışık et al. [
17] carried out a detailed investigation on the modal parameter identification and vibration-based damage detection of a multiple-crack cantilever beam with a hollow circular cross-section. The disparities between the analytical data and the measured data for damage detection were reduced using the modal sensitivity approach based on Bayesian parameter estimation. To solve the problem of incomplete measurement, Yang and Peng developed the model condensation method [
18] and proposed the modal condensation sensitivity [
19] for the damage identification of beam structures. Pooya and Massumi [
20] used the difference between mode shape curvature and the mode shape curvature estimation of damaged structures as an indicator to identify the damage location.
For the dynamic flexibility-based method, Ahmadi-Nedushan and Fathnejat [
21] presented a two-stage fault diagnosis approach based on the dynamic modal flexibility and the improved teaching–learning-based optimization technique. To find the defect location in the structure, He et al. [
22] employed the deflection computed by the modal flexibility matrix. Using modal flexibility, Bernagozzi et al. [
23] developed the data-driven criteria for structure-type classification to obtain the best approach for identifying structural defects. Liu et al. [
24] developed a generalized flexibility matrix algorithm to address the issue of defect identification using incomplete mode shape data. Peng and Yang [
25] carried out sensor placement and defect detection in steel truss bridges based on changes in generalized modal flexibility.
As the best achievable theory-based method, Lim and Thomas [
26] suggested the most viable eigenvector technique to ascertain the position and severity of damage in spatial truss structures. Despite the measurement inaccuracies unavoidably making the damaged location more challenging, their system works effectively. Zhao [
27] and Ricci [
28] further verified the best achievable eigenvector method with a shallow-arch structure and a 10-bay truss laboratory structure, respectively. Based on Bayesian inference, Prajapat and Ray-Chaudhuri [
29] improved the detection accuracy of the best achievable eigenvector method under the interference of data noise. According to an experimental study involving a laboratory scale shear building and different stiffness modification scenarios, their algorithm is efficient enough to localize the stories with stiffness modifications.
In this work, a novel damage diagnosis approach is proposed using the best achievable displacement variation. The improvements to the suggested technique concentrate on the following two features when compared to the best feasible eigenvector methods currently used [
26,
27,
28,
29]. The first innovation is that the proposed method is based on static test displacement, while the existing methods are based on dynamic test modal data. Instead of performing dynamic tests, static tests have the advantage of being more accurate and having a simpler measuring approach. Moreover, the static test is more commonly used in the field of bridge engineering. The second innovation is that a successive elimination approach is proposed to determine the true damaged elements and exclude the pseudo-damaged elements more reliably. The suggested technique, which employs this successive elimination procedure, exceeds the current static sensitivity method in terms of accuracy and dependability. The proposed method is demonstrated by a numerical example and two experimental examples. The results of the three examples revealed that the proposed method can be successfully used for structural damage assessments. The basic framework of this work is as follows:
Section 2 describes the theoretical basis, main formulas, and implementation steps of the best achievable displacement method for damage identification. In
Section 3 and
Section 4, the results of numerical verification and experimental verification are presented, respectively. Several conclusions are drawn in
Section 5.
2. Theoretical Development
For a structure before and after damage, the static response equations can be expressed as:
where
and
are the stiffness matrices of the undamaged and damaged structure,
and
are the corresponding displacement vectors under the static load vector
, and
and
are the changes in the stiffness matrix and the displacement owing to damage, respectively. Substituting Equations (3) and (4) into (1) gives
Substituting Equation (1) into (5), while excluding the high-order component, gives
Substituting Equation (1) into (6) gives
Equation (7) can be rewritten as
In Equation (8), the stiffness change
can be obtained by multiplying the sum of the elemental stiffness matrix by the damage coefficient as
where
is the
-th elementary stiffness matrix,
is the
-th elementary damage coefficient,
is the total number of elements.
denotes that the
-th element is undamaged.
equals 1, and less than 1 denotes that the damage to the
-th element is either complete or partial. Substituting Equation (9) into (8) yields
As mentioned before,
is valid for most undamaged elements. For convenience, the subsequent derivation process is carried out using three damaged elements as examples (other damage cases are also applicable). Without loss of generality, it is assumed that the
-th,
-th, and
-th elements are the potentially damaged elements. Thus, Equation (10) can be simplified by keeping only the damaged elements as
The link between the displacement variation and the damaged elements is established by Equation (12). The physical meaning of Equation (12) is crucial and useful for damage localization. It is observed that Equation (12) is valid merely when
is a linear combination of the vectors
,
, and
associated with the damaged elements. Equation (12) can be rewritten as
Based on Equation (16), the best feasible displacement variation can be defined to determine whether or not
is the linear combination of the column vectors of
as
where
is the best achievable displacement variation corresponding to the
-th,
-th, and
-th elements, and the Moore–Penrose inverse is denoted by the symbol “+”. If
is basically equal to
, it can be concluded that
is the linear combination of the column vectors of
, and vice versa. In the meantime, the difference between
and
can be calculated by
In which
denotes the 2-norm. For other element combinations, the corresponding
can also be calculated by the above process. The minimum value of all
will correspond to the damaged element combination. This means that the damage locations can be identified through searching the minimum value in all
. For convenience, the relative distance
can be defined for damage localization as
where
denotes the maximum value in all
. If
is equal to zero or extremely near to zero, it can be concluded that the elements related to
are the damaged elements. Notice that the procedure described above may be applied to exclude those pseudo-damaged components. For instance, the best achievable displacement variation and the distance for the combination of the
-th and
-th elements can be calculated by
If
and
are extremely close, it can be inferred that the
-th element is a pseudo-damaged element. In the end, the damage coefficients (i.e., damage extents)
,
and
can be calculated easily from the linear Equation (16) as
Note that a linear approximation that ignores the higher-order term is used in the derivation from Equation (5) to (6). Therefore, to obtain a more accurate damage coefficient, the acceleration formula derived from flexibility disassembly perturbation [
30,
31] should be employed as
To further assess the final damaged pieces, the calculated values of the damage coefficients can corroborate the findings of the damage localization based on the aforementioned best achievable hypothesis.
As a summary, a step-by-step description of the whole approach is illustrated as follows:
Step 1: Construct the finite element model of the structure to obtain the global stiffness matrix and the elementary stiffness matrices (), and construct the finite element model of the structure.
Step 2: Perform the static test on the structure before and after damage to obtain the displacement , and the corresponding change .
Step 3: For the combination of the possible damaged elements, use the best achievable theory described by Equations (16)–(23) to identify the genuine injured components and exclude the pseudo-damaged elements.
Step 4: Calculate the relevant damage coefficients (i.e., damage extents) for the detected damaged elements using Equations (24) and (25).
3. Verification by the Numerical Example
The proposed damage identification method is illustrated numerically using the beam structure in
Figure 1. To build the finite element model, the beam is evenly split into 26 segments using plane beam elements. Note that the finite element model is obtained by the MATLAB 2016b software, and the type of discretization element used is the Bernoulli–Euler plane beam element. The length of each segment is 0.1m. The physical and geometric parameters of this beam are as follows: Young’s modulus—30GPa; density—2500 kg/m
3; cross-sectional area—0.21 m
2; moment of inertia—8.575 × 10
−3 m
4. The decrease in the elastic modulus before and after damage simulates structural deterioration. It is assumed that the static vertical load of 4000 kN is applied at node 13, as shown in
Figure 1. The analytical displacement data computed by the finite element models before and after damage are assumed to be the measured displacement data of the static test. The measurement error in practice is simulated by adding uniformly distributed random numbers to the analytical displacement data.
Two damage scenarios are then simulated to verify the proposed method. The first damage scenario estimates a 20% reduction in the elastic modulus of element 13. Using the displacement data without noise,
Figure 2 presents all relative distances
(
) for each element based on the above best achievable technique. According to
Figure 2, it can be discovered that only
and the others are greater than zero. As a result, element 13 is the only one that can be determined to be damaged, and its damage extent can be calculated as
, which is exactly the value that was presumed. When 3% noise is added to the displacement data,
Figure 3 presents all the relative distances
(
) for each element based on the above best achievable technique.
Figure 3 reveals that
is the minimum value for all
. This means that element 13 can be recognized as the damaged element and its damage extent can be calculated as
, which is close to the assumed value 0.2. The second damage scenario implies that the elastic moduli of elements 7 and 13 are reduced by 15% and 20%, respectively.
Figure 4 displays the relative distances for a selection of element combinations using noise-free displacement data. From
Figure 4, it can be found that only the distance associated with the combination of components 7 and 13 is equal to zero. This indicates that elements 7 and 13 can be determined to be the damaged elements. Their damage extents can be calculated as
and
, which are exactly the assumed values.
Figure 5 illustrates the relative distances for several element combinations using displacement data with 3% noise and the above-mentioned best method. From
Figure 5, it can be found that the combination of elements 7 and 13 corresponds to the smallest of all calculated distances. Thus, elements 7 and 13 can be determined to be the damaged elements, and the damage extents can be calculated as
and
, which are close to the assumed values (0.15 and 0.2). The findings of this numerical example demonstrate that the suggested technique may successfully locate and assess damage in the beam structure.