This research entails a theoretical and numerical study on a new damage detection method for bridges, using response sensitivity in time domain. This method, referred to as “adjoint variable method,” is a finite element model updating sensitivity based method. Governing equation of the bridge-vehicle system is established based on finite element formulation. In the inverse analysis, the new approach is presented to identify elemental flexural rigidity of the structure from acceleration responses of several measurement points. The computational cost of sensitivity matrix is the main concern associated with damage detection by these methods. The main advantage of the proposed method is the inclusion of an analytical method to augment the accuracy and speed of the solution. The reliable performance of the method to precisely identify the location and intensity of all types of predetermined single, multiple, and random damages over the whole domain of moving vehicle speed is shown. A comparison study is also carried out to demonstrate the relative effectiveness and upgraded performance of the proposed method in comparison to the similar ordinary sensitivity analysis methods. Moreover, various sources of errors including the effects of noise and primary errors on the numerical stability of the proposed method are discussed.
The main objective of developing the structural health monitoring (SHM) system for structures 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 reduce maintenance costs [
SHM algorithms are identified as static system identification (SI) and dynamic SI, according to the types of structural response used. Dynamics-based SI 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 interrogation and on the modality used to excite the component [
The frequency domain SI and time-domain SI are more practical than static SI, as the static response, for instance, displacements of a structure are very difficult to measure in most cases.
The developments in the field of SI using vibration data of civil engineering structures have been recently reviewed by several authors. Some recent studies are briefly described in the following.
Doebling et al. [
Alampalli and Fu [
The frequency-domain SI 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 damages 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 SI.
The time-domain SI may be an attractive one to overcome the drawbacks of the frequency-domain SI. For time-domain SI, 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. The most important difficulty in sensitivity based SI methods is calculation of sensitivity matrix. Calculation of this massive matrix is repeated in each iteration and it is so time-consuming and has a significant effect on the efficiency of method. Despite the high importance of calculation method of sensitivity matrix and the optimization of its performance in SI procedure, there is no literature on this regard. In this paper, computational methods for sensitivity matrix are discussed and a novel sensitivity based damage detection method in time-domain referred to as “Adjoint variable method,” is developed. Computational algorithm of proposed method is 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
Since many algorithms of damage detection are based on the difference between modified model before occurrence of damage and after that, problems such as parameter identification and damage detection are closely related to model updating. Discrepancy between two models is used for detection and quantification of damage.
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 dynamic responses of the structures can be obtained by direct numerical integration using Newmark method.
The approach minimizes the difference between response quantities (usually 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. The measured response is a nonlinear function of the parameters. So, minimizing the error between the measured and predicted response will produce a nonlinear optimization problem.
Penalty function method is generally used for modal sensitivity with a truncated Taylor series expansion in terms of the unknown parameters. In this paper, the truncated series of the dynamic responses in terms of the system parameter
Let
Three-dimensional sensitivity matrix.
One of the important difficulties in parameter estimation is ill-conditioning. In the worst case, this can mean that there is no unique solution to the estimation problem and many sets of parameters are able to fit the data. Many optimization procedures result in the solution of linear equations for the unknown parameters. The use of the singular value decomposition (SVD) [
From experiences gained in model updating with simulated structures, Li and Law [
Like many other inverse problems, (
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 the sensitivity analysis.
When the parameter variations are small, the traditional way to assess their effects on calculated responses is the employment of 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 sensitivity analysis are listed in Figure
Different approaches to sensitivity analysis.
In the approximation approach, sensitivity matrix is obtained by either the forward finite difference or by the central finite difference method.
If the design is perturbed to
In the discrete method, sensitivity matrix is obtained by design derivatives of the discrete governing equation. For this process, it is necessary to take the derivative of the stiffness matrix. If this derivative is obtained analytically using the explicit expression of the stiffness matrix with respect to the variable, it is an analytical method, since the analytical expressions of stiffness matrix are used. However, if the derivative is obtained using a finite difference method, the method is called a semianalytical method. 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 finite element 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
Sensitivity analysis can be performed very efficiently by using deterministic methods based on adjoint functions. The use of adjoint functions for analyzing the effects of small perturbations in a linear system was introduced by Wigner [
This method constructs an adjoint problem that solves the adjoint variable, which contains all implicit dependent terms.
For the dynamic response of structure, the following form of a general performance measure will be considered:
To obtain the design sensitivity of
In order to write
Since (
Since the terms involving a variation in the state variable in (
Since (
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. 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 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 the expressions in (
Major disadvantage of the finite difference method is the accuracy of its sensitivity results. Depending on perturbation size, sensitivity results are quite different. For a mildly nonlinear performance measure, relatively large perturbation provides a reasonable estimation of sensitivity results. However, for highly nonlinear performances, a large perturbation yields completely inaccurate results. Thus, the determination of perturbation size greatly affects the sensitivity result. And even though it may be necessary to choose a very small perturbation, numerical noise becomes dominant for a too-small perturbation size. That is, with a too-small perturbation, no reliable difference can be found in the analysis results.
The continuum-continuum approach is so limited and is not applicable in complex engineering structures because very simple, classical problems can be solved analytically.
The discrete and continuum-discrete methods are equivalent under the conditions given below, using a beam as the structural component. It has also been argued that the discrete and continuum-discrete methods are equivalent under the conditions given below [
First, the same discretization (shape function) used in the FEA method must be used for continuum design sensitivity analysis. Second, an exact integration (instead of a numerical integration) must be used in the generation of the stiffness matrix and in the evaluation of continuum-based design sensitivity expressions. Third, the exact solution (and not a numerical solution) of the finite element matrix equation and the adjoint equation should be used to compare these two methods. Fourth, the movement of discrete grid points must be consistent with the design parameterization method used in the continuum method.
In this paper, two different analytical discrete methods, including direct differential method (DDM) and adjoint variable method (ADM) are presented and efficiency of proposed method is investigated when compared with DDM method.
While structural vibration responses are used for damage detection, assuming
Using (
In this equation,
The computational algorithm that leads to the determination of sensitivity matrix is as follows.
Calculate
Calculate
For the
For the
Consider
Calculate
Calculate
Calculate
If
If
If
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 the proposed algorithm or direct differentiate method (
Equation (
To illustrate the formulations presented in the previous sections, we consider the system shown in Figures
Multispan bridge model used in detection procedure.
The relative percentage error (RPE) in the identified results is calculated from (
A three-span bridge as shown in Figure
The transverse point load
For the forced vibration analysis, an implicit time integration method, called “the Newmark integration method” is used with the integration parameters
Speed parameter is defined as
Five damage scenarios of single, multiple, and random damages in the bridge without measurement noise are studied and they are 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 | Multi | 8, 13, and 29 | 11%, 4%, and 7% | Nil |
M1-3 | Multi | 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 |
Local damage is simulated with a reduction in the elastic modulus of material of an element. The sampling rate is 10000 Hz and 450 data of the acceleration response (degree of indeterminacy is 15) collected along the z-direction at nodes 5, 15, and 25 are used in the identification.
Scenario 1 studies the single damage scenario. The iterative solution converges in all speed parameter ranges with a maximum RPE of 0.088 in DDM method and 0.0354 in ADM method.
Scenarios 2 and 3 are on multiple damages with different amount of measured responses for the identification and Scenarios 4-5 are on random damages with different average for the identification. These scenarios also converge in all speed parameter ranges. One more scenario with model error is also included as in Scenario 6. This scenario consists of no simulated damage in the structure but with the initial elastic modulus of material of all the elements underestimated by 5% in the inverse identification.
Using both described methods, including DDM and proposed method, the damage locations and amount are identified correctly in all scenarios (Figure
RPE of DDM method for model 1.
Damage scenario | Speed parameter | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
M1-1 | 0.0465 | 0.0461 | 0.0457 | 0.0454 | 0.045 | 0.0743 | 0.0416 | 0.0471 | 0.088 |
M1-2 | 0.3135 | 0.317 | 0.3165 | 0.3157 | 0.315 | 0.2937 | 0.291 | 0.2967 | 0.0038 |
M1-3 | 0.0273 | 0.0268 | 0.0265 | 0.0262 | 0.0259 | 0.0281 | 0.0007 | 0.0007 | 0.0007 |
M1-4 | 0.052 | 0.0525 | 0.0516 | 0.0522 | 0.0531 | 0.0382 | 0.0576 | 0.0346 | 0.0155 |
M1-5 | 0.0411 | 0.0395 | 0.0408 | 0.0367 | 0.0403 | 0.06 | 0.0542 | 0.0207 | 0.0091 |
M1-6 | 0.0502 | 0.0546 | 0.0485 | 0.0471 | 0.0431 | 0.046 | 0.0422 | 0.041 | 0.0007 |
RPE of ADM method for model 1.
Damage scenario | Speed parameter | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
M1-1 | 0.0354 | 0.0346 | 0.0342 | 0.0338 | 0.0337 | 0.0003 | 0.0214 | 0.0107 | 0.0079 |
M1-2 | 0.0496 | 0.0338 | 0.0493 | 0.0585 | 0.0575 | 0.0294 | 0.024 | 0.0396 | 0.0214 |
M1-3 | 0.0008 | 0.0005 | 0.0005 | 0.0005 | 0.0005 | 0.0007 | 0.0007 | 0.0007 | 0.0006 |
M1-4 | 0.0271 | 0.0247 | 0.0222 | 0.0077 | 0.0071 | 0.0161 | 0.0006 | 0.0287 | 0.0007 |
M1-5 | 0.0051 | 0.0047 | 0.0028 | 0.0035 | 0.0031 | 0.1971 | 0.0171 | 0.001 | 0.0134 |
M1-6 | 0.0526 | 0.0237 | 0.0156 | 0.0009 | 0.0008 | 0.065 | 0.001 | 0.0008 | 0.0007 |
Detection of damage location and amount in elements 3, 7, 19, 25, and 28 and distribution of error in different elements with ADM scheme.
Further studies on Scenario 6 shows that both methods are sensitive to the initial model error and for the maximum 20% initial error can be converged and a relatively good finite element model is therefore needed for the damage detection procedure.
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 (r.m.s.) value is equal to a certain percentage of the r.m.s. 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 the effect of noise on stability of sensitivity methods, Scenario 2 (speed ratio of moving load is considered to be constant and equal to 0.5) is considered and different levels of noise pollution are investigated, and RPE changes with increasing number of loops for the iterative procedure have been studied.
Results are illustrated in Figure
RPE contours with respect to noise level and loops.
These contours show that both ADM and DDM methods are sensitive to the noise and if the noise level becomes greater than 1.3%, these methods lose their effectiveness and are not able to detect damage. So, in cases with noise level greater than 1.3%, a denoising tool alongside sensitivity methods should be used.
In order to compare and quantify the performance of different methods and evaluate the proposed method, relative efficiency parameter (REP) is defined as follows:
Figure
REP changes in different scenarios with respect to speed parameter.
Plane grid bridge model used in detection procedure.
Table
REP ranges in different scenarios.
Damage scenario | Max REP | Min REP | Average |
---|---|---|---|
M1-1 | 12.3739 | 4.9093 | 7.6744 |
M1-2 | 3.5953 | 2.2271 | 2.7166 |
M1-3 | 5.4912 | 4.5801 | 4.9990 |
M1-4 | 6.0214 | 2.287 | 4.6553 |
M1-5 | 3.8383 | 2.1599 | 3.1221 |
M1-6 | 7.6027 | 3.2449 | 4.7804 |
|
|||
Total | 12.3739 | 2.1599 | 4.6580 |
A plane grid model of bridge is studied as another numerical example to illustrate the effectiveness of the proposed method. The finite element model of the structure is shown in Figure
Five damage scenarios of single, multiple, and random damages in the bridge without measurement of the noise are studied and they are shown in Table
Damage scenarios for grid model.
Damage scenario | Damage type | Damage location | Reduction in elastic modulus | Noise |
---|---|---|---|---|
M2-1 | Single | 41 | 7% | Nil |
M2-2 | Multi | 3, 26, 35, and 40 | 9%, 14%, 3%, and 8% | Nil |
M2-3 | Multi | 5, 7, 12, 15, 24, and 37 | 4%, 11%, 6%, 2%, 10%, and 16% | Nil |
M2-4 | Random | All elements | Random damage in all elements with an average of 5% | Nil |
M2-5 | Random | All elements | Random damage in all elements with an average of 15% | Nil |
M2-6 | 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 10) collected along the
Similar to the previous model, Scenario 1 studies the single damage scenario. The iterative solution converges in all speed parameter ranges with a maximum RPE of 0.0006 in DDM method and 0.0011 in ADM method.
Scenarios 2 and 3 are on multiple damages with different amount of measured responses for the identification and Scenarios 4-5 are on random damages with different average for the identification. These scenarios also converge in all speed parameter ranges. One more scenario with model error is also included as Scenario 6. This scenario consists of no simulated damage in the structure, but with the initial elastic modulus of material of all the elements under-estimated by 5% in the inverse identification.
Using both described methods, including DDM and proposed method, the damage locations and amount are identified correctly in all the scenarios (Figure
RPE of DDM method for model 2.
Damage scenario | Speed parameter | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
M2-1 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0004 | 0.0004 | 0.0003 | 0.0006 | 0.0006 |
M2-2 | 0.0005 | 0.0006 | 0.0005 | 0.0004 | 0.0003 | 0.0004 | 0.0004 | 0.0005 | 0.0006 |
M2-3 | 0.0004 | 0.0004 | 0.0006 | 0.0003 | 0.0006 | 0.0005 | 0.0003 | 0.0005 | 0.0003 |
M2-4 | 0.0006 | 0.0006 | 0.0004 | 0.0005 | 0.0005 | 0.0005 | 0.0004 | 0.0002 | 0.0004 |
M2-5 | 0.0005 | 0.0006 | 0.0006 | 0.0004 | 0.0004 | 0.0003 | 0.0005 | 0.0004 | 0.0003 |
M2-6 | 0.0004 | 0.0004 | 0.0003 | 0.0005 | 0.0004 | 0.0004 | 0.0006 | 0.0003 | 0.0004 |
RPE of ADM method for model 2.
Damage scenario | Speed parameter | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
M2-1 | 0.0002 | 0.0002 | 0.0001 | 0.0009 | 0.0011 | 0.0034 | 0.0014 | 0.0007 | 0.0007 |
M2-2 | 0.0006 | 0.0008 | 0.0004 | 0.0011 | 0.001 | 0.0014 | 0.0015 | 0.0012 | 0.0007 |
M2-3 | 0.0005 | 0.0007 | 0.0097 | 0.001 | 0.0011 | 0.0013 | 0.0018 | 0.001 | 0.0007 |
M2-4 | 0.0003 | 0.0003 | 0.0007 | 0.0013 | 0.0007 | 0.001 | 0.0012 | 0.0008 | 0.0011 |
M2-5 | 0.001 | 0.001 | 0.0008 | 0.0009 | 0.001 | 0.001 | 0.0014 | 0.0007 | 0.0012 |
M2-6 | 0.0007 | 0.0007 | 0.0007 | 0.0009 | 0.0011 | 0.0011 | 0.0011 | 0.0011 | 0.0011 |
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 effect of noise on stability of sensitivity methods, scenario 3 (speed ratio of moving load is considered to be constant and equal to 0.5) is considered and different levels of noise pollution are investigated, and RPE changes with increasing number of loops for the iterative procedure has been studied.
Figure
RPE contours with respect to noise level and loops.
Figure
REP ranges in different scenarios for model 2.
Damage scenario | Max REP | Min REP | Average |
---|---|---|---|
M2-1 | 2.423 | 1.4998 | 1.9089 |
M2-2 | 3.0713 | 1.8519 | 2.442633 |
M2-3 | 3.137 | 1.7166 | 2.443778 |
M2-4 | 2.8168 | 1.748 | 2.153533 |
M2-5 | 2.5382 | 1.6374 | 2.0865 |
M2-6 | 2.2976 | 1.4389 | 1.859456 |
|
|||
Total | 3.137 | 1.4389 | 2.117258 |
REP changes in different scenarios with respect to speed parameter for model 2.
A new damage detection method based on finite element model updating and sensitivity technique using acceleration time history data of a bridge deck affected by a moving vehicle with specified load, named “ADM” method, is presented. The updating procedure can be regarded as a parameter identification technique which aims to fit the unknown parameters of an analytical model such that the model behaviour corresponds as closely as possible to the measured behaviour.
Newmark method is used to calculate the structural dynamic response and its dynamic response sensitivity matrix is calculated by adjoint variable method. In order to solve ill-posed inverse problem Tikhonov regularization method is used and L-curve method is implemented to find optimum value of the regularization parameter.
In proposed method, an incremental solution for adjoint variable equation developed that calculates each element of sensitivity matrix separately. The main advantage is inclusion of an analytical method to augment the accuracy and speed of the solution.
Numerical simulations demonstrate the efficiency and accuracy of the method to identify location and intensity of single, multiple, and random damages in different bridge models.
Comparison studies confirmed that computational cost for this method is much lower than other traditional sensitivity methods. For modern, practical engineering applications, the cost of damage detection analysis is expensive. So, this method is feasible for large-scale problems.
Similar to other sensitivity methods, the drawback of proposed method is its low stability against input measurement noise, which can be easily improved by using low-pass denoising tools such as wavelets.
The structural mass, damping, and stiffness matrices of the bridge
Nodal displacement, velocity, and acceleration vectors, respectively
Vector of applied forces
The
The stiffness reduction of the element
Mapping force matrix to the associated Dof of the structure
Rayleigh damping coefficients
The measured and computed response vectors
Response residual vector
Matrix with elements of zeros or ones, matching the Dof corresponding to the measured response components
Sensitivity matrix
Vector of all unknown parameters
Regularization parameter
Loss in the element stiffness
General performance measure
Final time
Design parameter
Perturbation of design parameter in the direction of
Adjoint variable
Sensitivity of performance with respect to design parameter
Damping ratio
Modal matrix
Relative percentage of error
Relative efficiency parameter
Identified and the true elastic modulus
Total length of the bridge
Velocity of traveling load
Speed parameter
Critical speed
Mass per unit length
Root-mean-square
Noise-free acceleration and noise-polluted acceleration
Solution time of system identification method.
The authors declare that there is no conflict of interests regarding the publication of this paper.