A method is presented to detect and quantify structural damages from changes in modal parameters (such as natural frequencies and mode shapes). An inverse problem is formulated to minimize the objective function, defined in terms of discrepancy between the vibration data identified by modal testing and those computed from analytical model, which then solved to locate and assess the structural damage using continuous ant colony optimization algorithm. The damage is formulated as stiffness reduction factor. The study indicates potentiality of the developed code to solve a wide range of inverse identification problems.
Structural health monitoring (SHM) has become an important area of research within the civil, mechanical, aerospace engineering community in recent years. Damage to structure may be caused as a result of normal operations, accidents, deterioration, or severe natural events such as earthquake and storms. Sometimes the extent and location of damage can be determined through visual inspection. But visual inspection technique has a limited capability to detect the damage, especially when the damage lies inside the structure and is not visible. So an effective and reliable global damage assessment methodology is necessary for determination of damage state particularly for these inaccessible regions.
Modal parameters based damage detection method has several advantages over alternative techniques due to the fact that the modal parameters depend only on the mechanical characteristics of the structure and not on the excitation applied. Review of modal parameter based damage detection methods was carried out by Doebling et al. [
A damage detection problem using changes in natural frequencies and/or mode shapes is basically an inverse problem, where one objective function, defined in terms of discrepancies between the vibration data identified by modal testing and those computed from analytical model, is minimized or maximized. However, these relationships are very complex involving a large number of local optima, hence making the problem too difficult to be solved by conventional optimization algorithms such as conjugate gradient method. In comparison, recent computational intelligence methods, such as artificial neural network [
In the present study an extended form of ant colony optimization (ACO) [
Ant colony optimization [
In case ACO applied to combinational optimization problems, the set of available solution components are defined by the problem formulation and the ant samples a component to be added to the current solution set based upon the discrete probability distribution function associated with each element of the set. In ACO_{R} this idea has been shifted to use a continuous probability density function (PDF) instead of discrete probability distribution one. One of the most popular functions to be used as a PDF is the Gaussian function, where a Gaussian kernel
The algorithm is started with random solution set of cardinality
Mathematically, for constructing a solution, an ant chooses at each construction step
Exactly one of the Gaussian functions
For each selected Gaussian function, the values of the
One of the disadvantages associated with ACO algorithm is that it may converge into some local optimum thereby leading to wrong results. Again, it may require great amount of computation time for getting the results. To minimize the convergence time and to increase the accuracy there have been a number of works on algorithm refinements and hybridization [
These parameters for ACO_{R} algorithm are not independent of each other rather they depend on the dimensions of the problem. When the solution archive size
It may probable that the ants may converge to a local optima or it may true for some run that the number of iteration is not enough for convergence thus not achieving target accuracy. Hence, to ensure proper convergence, each problem is run multiple times and the run yielding lowest objective function value is considered as actual damage scenario.
For a properly modeled structure, the eigenvalue equation is given by
In practice for damage identification the natural frequencies and mode shapes are identified from modal testing, and it is assumed that the finite element model representing the structure will provide the same modal values as those identified from modal testing. However, it does not happen due to several errors associated with inaccurate modeling, erroneous measurement and environmental noises, and so forth. If these noises are greater than the actual changes of modal parameters due to structural damage, then the information of real structural damage cannot be accurately identified. One method to minimize these discrepancies is model updating. The basic assumption behind model updating is that for a linear and undamped system, the errors in modeling of boundary conditions and joints can be eliminated by adjusting the material properties of the elements [
Due to associated uncertainties in test results, there is always a discrepancy between modal predictions by mathematical model and test results. For the numerical simulation study, the simulated noisy natural frequencies and mode shapes are obtained by adding a random value as given by
Computer codes are developed based on formulations outlined in previous sections and applied to beam and frame type structural systems. The structures are modeled with EulerBernoulli beam element and damage is represented in terms of SRF. The modal parameters are calculated numerically from eigenvalue analysis. The result and discussion portion is broadly divided into three parts. First part deals with damage detection in beam type structures. In this part the natural frequency from experiment is used for damage detection in a beam, which is then extended to include a numerically simulated beam for detecting single and multiple damages using noisy frequency along with mode shape data. The second part deals with damage detection in frame type structures. A 2storey rigid frame structure as considered by Yu and Xu [
Modal testing is carried out to capture vibration properties of a steel cantilever beam of dimensions 530 mm × 24 mm × 6 mm in order to demonstrate the applicability of developed algorithm in damage detection of a real structure. These experiments are carried out in the Structural Engineering and Material Testing Laboratory of Indian Institute of Technology, Kharagur. Figure
Experimental setup.
Initially, the undamaged beam is tested and its response is recorded. Then single damage of depth 2 mm and width 1.8 mm is introduced at 100 mm distance from fixed support by making a fine saw cuts perpendicular to the longitudinal axis. Further another cut of same amount is done at a distance of 200 mm from the support in addition to previous cut. This represents double damage case. The response is calculated after each cut respectively. Young’s modulus and mass density of the beam is considered as 200 × 10^{9} N/m^{2} and 7800 kg/m^{3}, respectively. Figure
Experimental beam model (not to scale).
Single damage case
Double damage case
Damage detail at “
Damages in experimental beam.
For numerical simulation, the experimental beam divided into 20 equal EulerBernoulli beam elements. ABAQUS FEA software is used to numerically estimate the stiffness reduction factor for cracked beam element. 4noded shell elements (S4R) available in ABAQUS are used to analyze the beam and crack is simulated by removing the elements at the cracked location. A static load of 100 N is applied at the free end of this model and the resulting deflection is calculated. In next step, second beam model is selected with similar element and with similar loading conditions. Young’s modulus for the elements near the cracked location (i.e., those falls within element 4 and element 8 of the Figure
The first five natural frequencies are measured and used for the present study. Table
Natural frequency of the structure (in Hz).
Mode  Undamaged structure  Single damage case  Double damage case  

Experimental  Analytical  Updated  
1st  16.41  17.4722  16.5412  16.25  16.24 
2nd  103.63  109.4966  103.6623  103.99  103.32 
3rd  290.00  306.5980  290.2616  289.52  288.60 
4th  567.76  600.8375  568.8231  564.38  563.76 
5th  938.31  993.3336  940.4060  932.63  924.87 
Various parameters required for ACO_{R} algorithm are considered as follows. The size of solution archive is taken as 40. The value of
Damage detection results for the experimental beam.
Results of damage detection for single damage
Results of damage detection for double damage
It is seen from Figure
Numerical simulations are carried out to demonstrate the effectiveness of the proposed damage assessment algorithm. A steel cantilever beam as described in previous section is taken in this study. Single and multiple damaged conditions are simulated for the purpose. In total, three random cases, as shown in Table
Various damage cases.
Case  Description 


10% damage at element 5 




The algorithm is evaluated for its performance when both natural frequency and mode shape (in the form of MAC value) is used as damage indicator. Noise is added up to 1% to the theoretically calculated natural frequencies and up to 10% noise in theoretically calculated mode shape values. The parameters of ACO_{R} algorithm is kept similar to previous section. Like the previous section, three experiments are conducted considering different initial seeds for each case and the experiment providing minimum objective function is considered as the final damage scenario. The maximum iteration per experiment is fixed at 1000. The results are shown in Figure
Statistical results for damage identification in cantilever beam.
Damage case  Detected damage amounts  

10% @ 5  15% @ 9  10% @ 11  20% @ 15  
Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  
Noise free data  
Single element  10.00 


Twoelement  10.00 

15.00  0.000  
Fourelement  10.00 

15.00 

10.00 

20.00  0.00 


0.5% noise in natural frequency and 5.0% noise in mode shape data  
Single element  9.14 


Twoelement  9.73 

15.20 


Fourelement  8.46 

14.41 

9.40 

19.47 



1.0% noise in natural frequency and 10.0% noise in mode shape data  
Single element  11.10 


Twoelement  11.87 

14.07 


Fourelement  13.66  0.00  13.81 

10.30 

20.43 

Damage detection results considering variation in stiffness/mass matrix.
Damage case  Detected damage amounts  

10% @ 5  15% @ 9  10% @ 11  20% @ 15  
Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  
0.5% variation in stiffness/mass matrix  
Single element  8.34 


Twoelement  9.23 

14.31 


Fourelement  9.38 

14.48 

9.49 





1.0% variation in stiffness/mass matrix  
Single element  7.71 


Twoelement  7.91 

13.15 


Fourelement  8.57 

13.79 

8.82 



Results of damage detection in numerical cantilever beam.
Damage detection results for single element damage (E1) case
Damage detection results for double element damage (E2) case
Damage detection results for fourelement damage (E3) case
From Figures
A 2storey rigid frame structure as presented by Yu and Xu [
Twostorey plane frame as considered by Yu and Xu [
Four damage cases are considered as shown in Table
Various damage conditions of the frame.
Case  Description 

F1  0.5% at element 17 
F2  20% damage at element 8 and element 17, respectively 
F3  10% at element 8, 20%, at element 11, and 3% at element 17, respectively 
F4  15% at element 5, 15% at 8, 20% at 11, and 3% at element 17, respectively 
Comparison of damage scenario between the present algorithm and [
Damage case  Element  Actual damage  Detected damage  

As per Yu and Xu [ 
Proposed algorithm  Improvement in accuracy (%)  
Percentage of noise included*  0  5  10  0  5  10  0  5  10  
F1  17  0.5  0.42  0.42  0.40  0.5  0.5  0.4  16.0  16.0  0.0 


F2  8  20  18.8  17.5  16.5  20.0  20.8  19.6  6.0  8.5  15.5 
17  20  18.5  17.2  16  20.0  20.0  14.7  7.5  14.0  −6.5  


F3  8  10  11.8  10.8  14.5  10.0  10.2  16.8  18.0  6.0  −23.0 
11  20  17.2  18.5  14.3  20.0  20.0  20.9  14.0  7.5  24.0  
17  3  4.7  3.5  5.5  3.0  3.6  1.2  56.7  −3.3  23.3  


F4^{1}  5  15  13.2  14.5  14.7  14.9  13.4  14.4  11.3  −7.3  −2.0 
8  15  12.1  12.6  13.2  12.0  11.1  10.4  −0.7  −10.0  −18.7  
11  20  18.2  17.9  15.5  19.8  19.3  18.0  8.0  7.0  12.5  
17  3  4.1  4.4  4.7  2.7  2.7  0.0  26.7  36.7  −43.3  


F4^{2}  5  15  13.2  14.5  14.7  15.0  15.5  13.8  12.0  0.2  −6.1 
8  15  12.1  12.6  13.2  15.0  14.3  15.0  19.3  11.1  11.9  
11  20  18.2  17.9  15.5  20.0  20.3  18.6  9.0  9.1  15.5  
17  3  4.1  4.4  4.7  3.0  1.6  1.9  36.7  1.0  19.0 
^{ 1}Maximum number of iterations considered as 1000.
^{ 2}Maximum number of iterations considered as 2000.
The maximum number of iterations is kept as 1000 in all cases. However, for case F4, program is run for second time with 2000 iterations in order to get an improved result. The performance of proposed algorithm is measured in terms of improvement in accuracy which is expressed as follows:
Here the terms
It is seen from Table
Further, to demonstrate the effectiveness of this method a 4storey, 3bay steel space frame as shown in Figure
Damage conditions to study damage detection in plane frame.
Case  Description 

G1  10% damage at element 1 
G2  15% damage at element 1 and 10% damage at element 3 
G3  G2 + 15% damage at element 5 and 20% damage at element 21 
3Bay, 4storey frame model.
First six natural frequencies (Table
First six natural frequencies of undamaged frame.
Mode  1st  2nd  3rd  4th  5th  6th 


Frequency (Hz)  27.86  75.34  84.76  133.49  174.00  180.66 
Statistical results for damage detection in 3bay and 4storey frame structure.
Damage case  Detected damage amounts  

15% @ 1  10% @ 2  15% @ 5  20% @ 21  
Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  Mean  Std. Dev.  
Noise free data  
Single element  15.00 


Two element  15.00 

10.00 


Four element  15.00 

10.00 

15.00 

20.00 



0.5% noise in natural frequency and 5.0% noise in mode shape data  
Single element  9.14 


Two element  9.73 

15.20 


Four element  8.46 

14.41 

9.40 





1.0% noise in natural frequency and 10.0% noise in mode shape data  
Single element  11.10 


Two element  11.87 

14.07 


Four element  13.66 

13.81 

10.30 

20.43 

Damage detection results for 3bay and 4storey frame structure.
Single element damage detection case
Double element damage detection case
Fourelement damage detection case
From the graphical results, it is clear that the proposed methodology can detect and quantify the damaged member with quite impressive accuracy in all cases. Though in fourelement damage detection case it has detected a false damage location at element 2, but it may not be harmful as it has detected and fairly quantified all true damage locations. Further, from the tabulated results it is observed that the standard deviation values are significantly less in comparison to actual damages in all the simulated runs which indicates the robustness of the present algorithm.
The same algorithm is used to detect damages in a 5storey steel space frame structure as shown in Figure
Space frame model.
Modal analysis is conducted using finite element model to generate natural frequency and mode shape. Nine random damage cases as shown in Table
Damage conditions to study the frame.
Case  Description 

H1  20% damage at element 3 
H2  10% damage at element 18 
H3  15% damage at element 27 
H4  H1 + 30% damage at element 22 
H5  H1 + H3 
H6  H2 + 25% damage at element 23 
H7  H2 + H5 
H8  H4 + 20% damage at element 32 
H9  H3 + H6 
Multiple damage detection in space frame structure.
Damage case  Element number  Actual damage  Detected damage  Percentage error  

Percentage of noise included*  0  5  10  0  5  10  
H1  3  20  20.00  18.67  17.96  0.00  6.65  10.20 


H2  18  10  10.00  9.28  11.92  0.00  7.20  19.20 


H3  27  15  15.00  14.26  13.22  0.00  4.93  11.87 


H4  3  20  20.00  18.41  18.62  0.00  7.95  6.90 
22  30  30.00  28.86  27.50  0.00  3.80  8.33  


H5  3  20  20.00  21.56  17.86  0.00  7.80  10.70 
27  15  15.00  14.69  12.84  0.00  2.07  14.40  


H6  18  10  10.00  9.50  9.03  0.00  5.00  9.70 
23  25  25.00  23.46  29.53  0.00  6.16  18.12  


H7  3  20  20.00  19.80  23.20  0.00  1.00  16.00 
18  10  10.00  8.93  9.34  0.00  10.70  6.60  
27  15  15.00  14.86  17.77  0.00  0.93  18.47  


H8  3  20  20.00  18.40  18.50  0.00  8.00  7.50 
22  30  30.00  29.68  26.80  0.00  1.07  10.67  
32  20  20.00  17.94  22.72  0.00  10.30  13.60  


H9  18  10  10.00  10.80  10.80  0.00  8.00  8.00 
23  25  25.00  26.70  23.30  0.00  6.80  6.80  
27  15  15.00  14.60  12.10  0.00  2.67  19.33 
From the Table
Damage assessment results for H2 and H4 conditions in space frame structure.
Single element damage case (H2)
Double element damage case (H4)
The number of false predictions is more in double element damage case (H4) than that in single element damage case (H2). Results are plotted only for two cases, that is, for actual damage and for a higher percentage of noise (1% and 10% noise in frequency and mode shape data, resp., in this case) in order to avoid clumsiness in graphical results. Similar observations are noticed for other cases also which has not been graphically represented to avoid repetitions.
A simple but robust damage detection methodology is presented to determine the locations and amount of damages in structures using continuous ant colony optimization algorithm. The algorithm is tested with a laboratory tested data. Further, the effectiveness of the algorithms is studied with cantilever beam model, a 3bay 4storey plane frame model, and a fivestorey space frame model. The parameters used in ACO_{R} algorithm are tuned to optimum values and care is taken to avoid local minima and initial imperfect pheromone depositions leading to preferential building of concentration to suboptimal results. The proposed damage detection method is found to be equally successful regardless of the damage location and extent of damage.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is financially supported by Aeronautical Research and Development Board (Structures Panel), Ministry of defense (R&D), Government of India.