This paper provides a comparative study on four different sensitivity-based damage detection methods for bridges. The methods investigated in this study are approximation approach, semianalytical discrete approach, and analytical discrete approach, which includes direct differential and adjoint variable methods. These sensitivity-based methods utilize finite element model updating procedure and allow a wide choice of physically meaningful parameters leading to vast range of applications in damage detection. The most important difficulty in these methods is calculation of sensitivity matrix. Calculation of this massive matrix is repeated in each iteration and has a significant effect on the efficiency of method. In this study, the acceleration measurements are simulated from the solution to the forward problem using finite element method under moving load with various speeds, along with the addition of artificially produced measurement noise. Various damaged structures with different damage patterns including single, multiple, and random damage are considered and efficiency of four sensitivity methods is compared. Moreover, various possible sources of error such as the effects of measurement noise as well as initial assumption error in stability of the methods are also discussed.
Structural health monitoring is the implementation of a damage identification strategy to different types of structures. SHM is vital to evaluate the fitness of a structure, in various disciplines including aerospace, mechanical and civil engineering, to perform its prescribed tasks properly. The necessity of SHM is highlighted when it is recognized that the performance of structures may change due to a gradual or sudden change in states, load conditions, or response mechanisms.
Bridges are truly the flagships of civil engineering, which attract the greatest attention within the engineering community. This is due to their small safety margins and their great exposure to the public [
The main objective of developing the SHM system for bridges is to enhance structural safety. However in bridges, SHM serves other economic benefits such as increased mission reliability, extended life of life-limited components, reduced tests, reduction in “down time,” increased equipment reliability, customization of maintenance actions, and greater awareness of operating personnel, resulting in fewer accidents. SHM also promises to help in reducing maintenance costs [
SHM is an inverse problem wherein the flaws in the structure are characterized using the measured data for some known inputs [ level 1: confirming the presence of damage; level 2: determination of location and orientation of the damage; level 3: evaluation of the severity of the damage; level 4: possibility of controlling or delaying the growth of damage; level 5: determining the remaining life in the structure (prognosis).
Dynamics-based SHM techniques assess the state of health of a structural component on the basis of the detection and analysis of its dynamic response. Such techniques can be classified on the basis of the type of response being considered for the investigations, on the frequency or time domain of investigation, and on the modality used to excite the component [
The developments in the field of structural damage detection (DD) using vibration data of civil engineering structures have been recently studied by several authors; some of them are briefly described herein.
Doebling et al. [
Alampalli and Fu [
Damage detection usually requires a mathematical model on the structure in conjunction with experimental modal parameters of the structure. The identification approaches are mainly based on the change in the natural frequencies [
The frequency-domain DD algorithms have been more widely developed and applied as the amount of measured data is reduced dramatically after the transform; thus they can be handled easily. Unfortunately, the effects of local damage on the natural frequencies and mode shapes of higher modes are greater than lower ones, but they are usually difficult to measure from experiments. In addition, structural damping properties cannot be identified in frequency domain DD.
The time-domain DD may be an attractive one to overcome the drawbacks of the frequency-domain DD. For time-domain DD, the forced vibration responses of the structure are needed in the identification. However, in some cases it is either impractical or impossible to use artificial inputs to excite the civil engineering structures, so natural excitation must be measured along with the structural responses to assess the dynamic characteristics [
Sensitivity-based methods allow a wide choice of physically meaningful parameters and this advantage has led to their widespread use in damage detection. Calculation of sensitivity matrix has a significant effect on the efficiency of these methods. Despite the high importance of calculation method of sensitivity matrix and optimizing its performance in DD procedure, there is not literature on this regard. In this paper, computational methods for sensitivity matrix are discussed and a novel sensitivity base damage detection method in time-domain referred to as “adjoint variable method” is introduced. Fundamental principles of proposed method are presented and its performance is compared with the conventional methods and it is shown that the numerical cost is considerably reduced by using the concept of adjoint variable.
The outline of the work is as follows: inverse problems along with model updating are briefly introduced in Section
A key step in model-based damage identification is the updating of the finite element model of the structure in such a way that the measured responses can be reproduced by the FE model. A general flowchart of this operation is given in Figure
General flowchart of a FEM-updating [
For a general finite element model of a linear elastic time-invariant structure, the equation of motion is given by
The approach minimizes the difference between response quantities (acceleration response) of the measured data and model predictions. This problem may be expressed as the minimization of
When the parameters of a model are unknown, they must be estimated using measured data. Since the relationship between the acceleration responses
The acceleration response vector
The iteration is terminated when a preselected criterion is met. The final identified damaged vector becomes [
Like many other inverse problems, the solution of (
The regularized solution from minimizing the function in (
The regularized solution in (
In the inverse problem of damage identification, it is assumed that the stiffness matrix of the whole element decreases uniformly with damage, and the flexural rigidity
The stiffness matrix of the damaged structure is the assemblage of the entire element stiffness matrix
The objective of sensitivity analysis is to quantify the effects of parameter variations on calculated results. Terms such as influence, importance, ranking by importance, and dominance are all related to sensitivity analysis.
The simplest and most common procedure for assessing the effects of parameter variations on a model is to vary selected input parameters, rerun the code, and record the corresponding changes in the results or responses, calculated by the code. The model parameters responsible for the largest relative changes in a response are then considered to be the most important for the respective response. For complex models, though, the large amount of computing time needed by such recalculations severely restricts the scope of this sensitivity analysis method; in practice, therefore, the modeler who uses this method can investigate only a few parameters that he judges a priori to be important [
When the parameter variations are small, the traditional way to assess their effects on calculated responses is by using perturbation theory, either directly or indirectly, via variational principles. The basic aim of perturbation theory is to predict the effects of small parameter variations without actually calculating the perturbed configuration but rather by using solely unperturbed quantities.
Various methods employed in design sensitivity analysis are listed in Figure
Approaches to design sensitivity analysis.
In the continuum approach, the design derivative of the variational equation is taken before it is discretized. If the structural problem and sensitivity equations are solved as a continuum problem, then it is called the continuum-continuum method. However, only very simple, classical problems can be solved analytically. Thus, the continuum sensitivity equation is solved by discretization in the same way that structural problems are solved. Since differentiation is taken at the continuum domain and is then followed by discretization, this method is called the continuum-discrete method.
The easiest way to compute sensitivity information of the performance measure is by using the finite difference method. Different designs yield different analysis results and, thus, different performance values. The finite difference method actually computes design sensitivity of performance by evaluating performance measures at different stages in the design process. If
A structural problem is often discretized in finite dimensional space in order to solve complex problems. The discrete method computes the performance design sensitivity of the discretized problem.
The design represents a structural parameter that can affect the results of the analysis.
The design sensitivity information of a general performance measure can be computed either with the direct differentiation method or with the adjoint variable method.
The direct differentiation method (DDM) is a general, accurate, and efficient method to compute FE response sensitivities to the model parameters. This method directly solves for the design dependency of a state variable and then computes performance sensitivity using the chain rule of differentiation. This method clearly shows the implicit dependence on the design, and a very simple sensitivity expression can be obtained.
Consider a structure in which the generalized stiffness and mass matrices have been reduced by accounting for boundary conditions. Let the damping force be represented in the form of
Mirzaee et al. [
The adjoint variable method yields a terminal-value problem, compared with the initial-value problem of response analysis.
For the dynamic structure, the following form of a general performance measure can be considered:
Mirzaee et el. [
Mirzaee et al. [
In this paper four different methods are presented to compute sensitivity information of the performance measure for damage detection procedure. First method is the finite Difference method (FD) which is classified as an approximation approach. Second, it is semianalytical discrete method which uses a finite difference scheme to calculate stiffness matrix in direct differential method (DDMFD) and third and fourth methods are two different analytical discrete methods: direct differential method (DDM) and adjoint variable method (ADM).
The first three methods including FD, DDMFD, and DDM are widely used by other authors [
The initial analytical model of a structure deviates from the true model and measurement from the initial intact structure is used to update the analytical model. The improved model is then treated as a reference model, and measurement from the damaged structure will be used to update the reference model.
When response measurement from the intact state of the structure is obtained, the sensitivities are computed from presented methods based on the analytical model of the structure and the well knowing input force and velocity. The vector of parameter increments is then obtained using the computed and experimentally obtained responses. The analytical model is then updated and the corresponding response and its sensitivity are again computed for the next iteration. When measurement from the damaged state is obtained, the updated analytical model is used in the iteration in the same way as that using measurement from the intact state. Convergence is considered to be achieved when the following criteria are met:
In order to evaluate the efficiency and competency of introduced methods, analysis of two FE models with extensive available numerical studies has been carried out. Representative examples are presented to demonstrate the effects of speed of loads, measurement noise level, and initial error on the accuracy and effectiveness of the methods.
The relative percentage error (
A single-span bridge is studied to compare different methods. This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFs. Total length of bridge is 10 m and height and width of the frame section are 200 mm. Rayleigh damping model is adopted with the damping ratios of the first two modes taken equal to 0.05.
Structural properties are summarized in Table
Properties of different models.
Model name | Number of elements | Number of nodes | Mass density | Modulus of elasticity | Rayleigh coefficients | Undamaged natural frequency of intact model | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
kg/mm3 | N/mm2 |
|
|
First mode | Second mode | Third mode | Fourth mode | Fifth mode | |||
Multispan | 10 | 11 | 7.8 × 10−9 | 2.1 × 10−5 | 0.1 | 2.860 × 10−5 | 29.6 | 118.3 | 266.2 | 473.8 | 742.1 |
Grid | 46 | 32 | 7.8 × 10−9 | 2.1 × 10−5 | 0.1 | 2.364 × 10−5 | 45.6 | 92.8 | 181.7 | 259.7 | 399.1 |
The transverse point load
For the forced vibration analysis an implicit time integration method called as “the Newmark integration method” is used with the integration parameters
Speed parameter is defined as
Six damage scenarios of single, multiple, and random damage along with initial error in the bridge without measurement noise are studied and shown in Table
Damage scenarios for single-span bridge.
Damage scenario | Damage type | Damage location | Reduction in elastic modulus | Noise |
---|---|---|---|---|
M1-1 | Single | 4 | 17% | Nil |
M1-2 | Multiple | 2, 7 | 4%, 21% | Nil |
M1-3 | Multiple | 3, 5, 6, and 8 | 12%, 6%, 5%, and 2% | Nil |
M1-4 | Random | All elements | Random damage in all elements with an average of 5% | Nil |
M1-5 | Random | All elements | Random damage in all elements with an average of 15% | Nil |
M1-6 | Estimation of undamaged state | All elements | 5% reduction in all elements | Nil |
Local damage is simulated with a reduction in the elastic modulus of material of an element. The sampling rate is 15000 Hz and 250 data of the acceleration response (degree of indeterminacy is 25) collected along the
Tables
Solution time, number of loops, and RPE of ADM method for model 1.
Damage scenario | Speed parameter | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |||||||||||
ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | |
M1-1 |
|
|
7.41 |
|
|
5.58 |
|
|
7.80 |
|
|
1.06 |
|
|
7.84 |
M1-2 |
|
|
5.92 |
|
|
5.35 |
|
|
6.91 |
|
|
1.02 |
|
|
7.00 |
M1-3 |
|
|
8.18 |
|
|
8.93 |
|
|
5.71 |
|
|
9.32 |
|
|
5.39 |
M1-4 |
|
|
9.06 |
|
|
1.03 |
|
|
6.14 |
|
|
1.16 |
|
|
5.19 |
M1-5 |
|
|
1.02 |
|
|
8.31 |
|
|
9.07 |
|
|
5.16 |
|
|
6.61 |
M1-6 |
|
|
5.83 |
|
|
5.12 |
|
|
7.64 |
|
|
1.70 |
|
|
6.25 |
Solution time, loops, and RPE of DDM method for model 1.
Damage scenario | Speed parameter | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |||||||||||
ST | NL | RPE | ST | NL | RPE | ST | NL | RPE | ST | NL | RPE | ST | NL | RPE | |
M1-1 |
|
|
2.25 |
|
|
1.59 |
|
|
2.17 |
|
|
2.89 |
|
|
2.00 |
M1-2 |
|
|
2.44 |
|
|
1.61 |
|
|
1.59 |
|
|
2.35 |
|
|
1.68 |
M1-3 |
|
|
2.37 |
|
|
1.85 |
|
|
1.42 |
|
|
2.16 |
|
|
3.03 |
M1-4 |
|
|
1.98 |
|
|
1.66 |
|
|
3.13 |
|
|
1.89 |
|
|
1.66 |
M1-5 |
|
|
1.73 |
|
|
2.18 |
|
|
1.98 |
|
|
2.05 |
|
|
2.88 |
M1-6 |
|
|
2.15 |
|
|
1.21 |
|
|
2.80 |
|
|
8.26 |
|
|
1.58 |
Solution time, loops, and RPE of FD method for model 1.
Damage scenario | Speed parameter | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |||||||||||
ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | |
M1-1 |
|
|
0.027 |
|
|
0.071 |
|
|
0.101 |
|
|
0.418 |
|
|
0.923 |
M1-2 |
|
|
0.028 |
|
|
0.048 |
|
|
0.189 |
|
|
1.322 |
|
|
0.627 |
M1-3 |
|
|
0.199 |
|
|
0.022 |
|
|
0.081 |
|
|
0.134 |
|
|
0.570 |
M1-4 |
|
|
0.127 |
|
|
0.057 |
|
|
0.072 |
|
|
0.391 |
|
|
0.627 |
M1-5 |
|
|
0.030 |
|
|
0.031 |
|
|
0.086 |
|
|
0.367 |
|
|
0.399 |
M1-6 |
|
|
0.019 |
|
|
0.029 |
|
|
0.064 |
|
|
0.536 |
|
|
0.732 |
*Max number of iterations reached and not converged.
Solution time, loops, and RPE of DDMFD method for model 1.
Damage scenario | Speed parameter | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |||||||||||
ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | ST (s) | NL | RPE | |
M1-1 |
|
|
0.065 |
|
|
0.067 |
|
|
0.118 |
|
|
0.243 |
|
|
0.830 |
M1-2 |
|
|
0.079 |
|
|
0.088 |
|
|
0.773 |
|
|
0.609 |
|
|
0.573 |
M1-3 |
|
|
0.227 |
|
|
0.096 |
|
|
0.143 |
|
|
0.492 |
|
|
0.449 |
M1-4 |
|
|
0.165 |
|
|
0.078 |
|
|
0.125 |
|
|
0.192 |
|
|
0.242 |
M1-5 |
|
|
0.107 |
|
|
0.049 |
|
|
0.161 |
|
|
0.254 |
|
|
0.670 |
M1-6 |
|
|
0.253 |
|
|
0.084 |
|
|
0.136 |
|
|
0.275 |
|
|
0.257 |
*Max number of iterations reached and not converged.
Noise is the random fluctuation in the value of measured or input that causes random fluctuation in the output value. Noise at the sensor output is due to either internal noise sources, such as resistors at finite temperatures, or externally generated mechanical and electromagnetic fluctuations [
To evaluate the sensitivity of results to such measurement noise, noise-polluted measurements are simulated by adding to the noise-free acceleration vector a corresponding noise vector whose root-mean-square (RMS) value is equal to a certain percentage of the RMS. value of the noise-free data vector. The components of all the noise vectors are of Gaussian distribution, uncorrelated and with a zero mean and unit standard deviation. Then on the basis of the noise-free acceleration
In order to study effects of noise in stability of different sensitivity methods, scenario 3 (speed ratio of moving load is considered to be fixed and equal to 0.5) is considered and different levels of noise pollution are investigated, and also RPE change with increasing the number of loops in iterative procedure has been studied. The results are illustrated in Figure
RPE contours with respect to noise level and loops in model 1.
ADM method
DDM method
FD method
DDMFD method
In order to study effects of initial error in stability of sensitivity methods, scenario 6 (speed ratio of moving load is considered to be fixed and equal to 0.5) is considered. This scenario consists of no simulated damage in the structure, but the initial elastic modulus of material of all the elements is underestimated and different levels of initial error are investigated; RPE changes with increasing the number of loops in iterative procedure have been studied. The results are illustrated in Figure
RPE contours with respect to initial error and loops in model 1.
ADM method
DDM method
FD method
DDMFD method
A plane grid model of bridge is studied as another numerical example to investigate the efficiency and accuracy of discrete methods. It is notable that according to complexity of this model, approximation and semidiscrete methods are not useable and diverged in all scenarios in all speed ranges.
The structure is modeled by 46 frame elements and 32 nodes with three DOFs at each node for the translation and rotational deformations. Rayleigh damping model is adopted with the damping ratios of the first two modes taken equal to 0.05. The finite element model of the structure is shown in Figure
Plane grid bridge model used in detection procedure.
Six damage scenarios of single, multiple, and random damage in the bridge without measurement noise are studied and shown in Table
Damage scenarios for multispan bridge.
Damage scenario | Damage type | Damage location | Reduction in elastic modulus | Noise |
---|---|---|---|---|
M1-1 | Single | 23 | 5% | Nil |
M1-2 | Multiple | 8, 13, and 29 | 11%, 4%, and 7% | Nil |
M1-3 | Multiple | 3, 7, 19, 25, and 28 | 12%, 6%, 5%, 2%, and 18% | Nil |
M1-4 | Random | All elements | Random damage in all elements with an average of 5% | Nil |
M1-5 | Random | All elements | Random damage in all elements with an average of 15% | Nil |
M1-6 | Estimation of undamaged state | All elements | 5% reduction in all elements | Nil |
The sampling rate is 14000 Hz and 460 data of the acceleration response (degree of indeterminacy is 20) collected along the
Tables
Solution time, number of loops, and RPE of ADM method for model 2.
Damage scenario | Speed parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | ||||||
ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | |
M1-1 |
|
1.26 |
|
2.03 |
|
1.09 |
|
2.30 |
|
1.74 |
M1-2 |
|
2.15 |
|
2.40 |
|
1.52 |
|
2.05 |
|
1.77 |
M1-3 |
|
2.62 |
|
1.86 |
|
1.26 |
|
2.12 |
|
1.71 |
M1-4 |
|
2.43 |
|
2.32 |
|
1.57 |
|
1.26 |
|
1.87 |
M1-5 |
|
2.59 |
|
3.76 |
|
2.00 |
|
3.35 |
|
1.48 |
M1-6 |
|
2.31 |
|
9.64 |
|
2.08 |
|
1.21 |
|
2.20 |
Solution time, loops, and RPE of DDM method for model 2.
Damage scenario | Speed parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.3 | 0.5 | 0.7 | 0.9 | ||||||
ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | ST (s) | RPE (%) | |
M1-1 |
|
2.15 |
|
3.68 |
|
2.64 |
|
3.42 |
|
2.91 |
M1-2 |
|
4.16 |
|
6.14 |
|
3.34 |
|
5.16 |
|
6.05 |
M1-3 |
|
5.89 |
|
4.91 |
|
2.87 |
|
4.57 |
|
4.70 |
M1-4 |
|
5.73 |
|
5.69 |
|
3.22 |
|
2.82 |
|
4.48 |
M1-5 |
|
3.55 |
|
5.58 |
|
5.02 |
|
6.32 |
|
3.28 |
M1-6 |
|
4.24 |
|
2.69 |
|
3.36 |
|
3.24 |
|
6.60 |
Detection of damage location and amount in elements 5, 7, 12, 15, 24, and 37 and distribution of error in different elements with ADM scheme.
In order to study effects of noise, scenario 3 is considered and different levels of noise pollution are investigated, and also RPE changes with increasing the number of loops for iterative procedure have been studied. The results are presented in Figure
RPE contours with respect to noise level and loops in model 2.
ADM method
DDM method
Scenario 6 (speed ratio of moving load is considered to be fixed and equal to 0.5) is considered to study effects of initial error in stability of sensitivity methods. RPE changes with increasing the number of loops for iterative procedure have been studied. The results are illustrated in Figure
RPE contours with respect to initial error and loops in model 2.
ADM method
DDM method
According to the results obtained in the previous section, different sensitivity methods considered in this study are compared in this section. Iteration number, relative percentage of error (RPE), solution time, noise level, and initial error effect in stability of damage detection procedure are effective parameters to compare efficiency and accuracy of the methods.
Average number of iterations for different scenarios with respect to the speed parameter is illustrated in Figure
Comparison of iteration number between different sensitivity methods.
All methods
Analytical discrete methods
As seen in Figure
Figure
Using the same convergence tolerance equal to 1 × 10−5 relative percentage of error in different methods is illustrated in Figure
Comparison of RPE between different sensitivity methods.
All methods
Analytical discrete methods
As seen in this figure, with a maximum number of iterations equal to 300, RPE in FD and DDMFD methods are considerably larger than DDM and ADM methods. DDMFD method has lower RPE than FD method in most cases. Average of RPE is 0.26329 for DDMFD compared to 0.27766 in FD method. Again, the larger the speed ratio of moving vehicle is, the larger the relative error percentage in approximation methods is.
Figure
Average of RPE is 0.000731 and 0.002037 for ADM and DDM methods, respectively. Therefore ADM is nearly 2.8 times more accurate than DDM.
In order to investigate computational cost of sensitivity methods, using the same amounts of RAM and CPU resources for the presented numerical simulations, solution time of different damaged models is evaluated and shown in Figure
Comparison of solution time between different sensitivity methods.
All methods
Analytical discrete methods
As seen in Figure
Figure
Average of solution time is 2.956 (s) and 7.794 (s) for ADM and DDM methods, respectively; consequently ADM is nearly 2.6 times faster than DDM.
In order to compare and quantify the performance of different methods, relative efficiency parameters of methods “
Figure
Model 1
Model 2
Table
Damage scenario | Max REP | Min REP | Average | |||
---|---|---|---|---|---|---|
Model 1 | Model 2 | Model 1 | Model 2 | Model 1 | Model 2 | |
M1-1 | 4.2291 | 1.9262 | 2.3373 | 1.5659 | 2.8785 | 1.7420 |
M1-2 | 3.8674 | 2.9893 | 2.3880 | 2.3178 | 3.1318 | 2.6008 |
M1-3 | 3.5062 | 2.8052 | 2.6852 | 2.0291 | 3.0292 | 2.4267 |
M1-4 | 3.6137 | 2.2448 | 1.9713 | 2.0117 | 2.5682 | 2.0893 |
M1-5 | 3.5047 | 2.9353 | 2.1168 | 1.6647 | 2.5290 | 2.2304 |
M1-6 | 5.5478 | 2.6344 | 2.2784 | 2.0035 | 3.2318 | 2.2587 |
|
||||||
Total | 5.5478 | 2.9893 | 1.9713 | 1.5659 | 2.8948 | 2.2248 |
Changes in the average of
Average of
Figure
Table
Damage scenario | Max REP | Min REP | Average | |||
---|---|---|---|---|---|---|
FD | DDMFD | FD | DDMFD | FD | DDMFD | |
M1-1 | 295.79 | 268.06 | 40.073 | 51.54 | 156.94 | 138.31 |
M1-2 | 478.80 | 307.33 | 53.87 | 76.39 | 200.82 | 183.35 |
M1-3 | 291.43 | 241.65 | 33.28 | 63.32 | 142.48 | 148.26 |
M1-4 | 298.07 | 174.74 | 43.55 | 49.16 | 146.99 | 102.53 |
M1-5 | 226.85 | 276.31 | 41.57 | 56.07 | 105.03 | 114.62 |
M1-6 | 783.07 | 505.42 | 31.93 | 65.47 | 250.56 | 187.09 |
|
||||||
Total | 783.07 | 505.42 | 31.93 | 49.16 | 167.14 | 145.69 |
Changes in the average of
Average of
Figure
Table
Damage scenario | Max REP | Min REP | Average |
---|---|---|---|
M1-1 | 1.23321 | 0.777484 | 1.095359 |
M1-2 | 1.557902 | 0.705181 | 0.964746 |
M1-3 | 1.226306 | 0.52565 | 0.931323 |
M1-4 | 1.726018 | 0.796804 | 1.277344 |
M1-5 | 1.349058 | 0.706861 | 0.949126 |
M1-6 | 1.686819 | 0.343253 | 1.04332 |
|
|||
Total | 1.726018 | 0.343253 | 1.043536 |
Changes in the average of
Average of
Figures
Stability of different methods against noise and initial error.
Method | Noise level | Initial error | ||
---|---|---|---|---|
Model 1 | Model 2 | Model 1 | Model 2 | |
ADM | 2.5% | 2.4% | 20% | 20% |
DDM | 2.5% | 2.3% | 20% | 20% |
FD | 2.7% | — | 30% | — |
DDMFD | 2.4% | — | 30% | — |
Stability comparison of different methods shows that FD is the most stable method and stability of analytical methods is slightly lower than that.
Figures
Comparing Figures
Different sensitivity-based damage detection methods are presented and acceleration time history data affected by a moving vehicle with specified load is used for damage detection procedure. Newmark method is used to calculate the structural dynamic response and its dynamic response sensitivities are calculated by four different sensitivity methods (finite difference method (FD), semianalytical discrete method (DDMFD), direct differential method (DDM), and adjoint variable method (ADM)).
Different damaged structures including single, multiple, and random damage are considered and efficiency of four aforementioned sensitivity methods is compared and following remarks are made. The advantage of the finite difference method is obvious. If structural analysis can be performed and the performance measure can be obtained as a result of structural analysis, then FD method is independent of the problem types considered. However, sensitivity computation costs become the main concern in the damage detection process for the large problems. Comparison of sensitivity methods shows that FD method is the most expensive method among other procedures. Semianalytical discrete method (DDMFD) alleviates some disadvantages of FD method and its computational cost is rather lower than FD method. The major disadvantage of the finite difference based methods (FD and DDMFD) is the accuracy of their sensitivity results. Depending on perturbation size, sensitivity results are quite different. As a result, it is very difficult to determine design perturbation sizes that work for all problems. The computational cost of damage detection procedure using these methods is too expensive and these methods are infeasible for large-scale problems containing many variables, for example, the second case study (model 2). The DDM is an accurate and efficient method to compute sensitivity matrix. This method directly solves for the design dependency of a state variable and then computes performance sensitivity using the chain rule of differentiation. The comparative study shows this has a very low computational cost method in all cases and is more accurate than finite difference methods. ADM calculates each element of sensitivity matrix separately by defining an adjoint variable parameter. The main advantage is in evaluating the dynamic response; an analytical solution exists which significantly increases the speed and accuracy of the solution. The comparative study shows that efficiency parameter of ADM is 2.89, 167.14, and 145.69 compared to DDM, FD, and DDMFD methods, respectively. This result indicates that ADM is extremely successful and can be applied as a powerful tool in SHM. Investigations of initial assumption error in stability of methods show that finite difference based methods have more enhanced stability than analytical discrete methods and all sensitivity-based methods have moderate stability against initial assumption error. The drawback of all sensitivity-based methods is their low stability against input measurement noise (about 2.5%), which can be improved by using low-pass denoising tools.
The authors declare that there is no conflict of interests regarding the publication of this paper.