Journal of Civil Structural Health Monitoring

https://doi.org/10.1007/s13349-018-0317-0

ORIGINAL PAPER

Sensitivity‑based damage detection algorithm for structures using

vibration data
C. G. Krishnanunni1   · R. Sethu Raj1 · Deepak Nandan1 · C. K. Midhun1 · A. S. Sajith1 · Mohammed Ameen1

Received: 30 March 2018 / Accepted: 12 November 2018

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract
Damage in a structure can lead to changes in the structural properties such as stiffness and natural frequencies. The ratio of
frequency changes in two modes is a function of the damage location. In this paper, vibration data and static displacement
measurements are used to detect and quantify structural damages. A sensitivity analysis is performed to study how natural
frequencies and static displacements change in the presence of a structural damage. An objective function representing
an error is defined using the sensitivity equation and minimized using Cuckoo Search algorithm. The effectiveness of the
technique is demonstrated with the help of cantilever beams and fixed–fixed beam in which different damage scenarios are
simulated using ANSYS and analyzed to obtain the modal parameters. In addition, a laboratory tested space frame model
has been used to demonstrate the proposed technique. Numerical results indicate that damages can be accurately detected
and quantified in a relatively shorter computational time using the Cuckoo Search algorithm.

Keywords  Damage · Sensitivity equation · Vibration · Objective function · Algorithm

1 Introduction be easily obtained from vibration response, damage detec-

tion by studying the change in natural frequency became
Mechanical, civil, and aerospace engineering communities a popular way to identifying and quantifying the damages
have already identified structural health monitoring as a [2]. Pandey, Biswas, and Samman [3] have investigated a
challenging problem. The ineffectiveness of visual exami- parameter called as the curvature mode shape as a possible
nation and destructive testing in damage detection has candidate for locating damage. Later, Artificial Neural Net-
been a major concern for the last few years. The aerospace work was used to solve the problem of damage detection [4,
industry is in need of new low-cost, nondestructive testing 5] and yielded excellent results. Hou, Noori, and Amand [6]
techniques for detecting damages in aerospace components. developed a wavelet-based approach for structural damage
Consequently, in the last few decades, many researchers have detection, wherein characteristics of the vibration signals
reported various nondestructive damage detection methods. under wavelet transformation were examined. Curadelli et al.
In particular, damage detection based on structural [7] have extended the use of wavelet transform to detect
response to vibration has gained popularity in recent years. structural damage by means of the instantaneous damp-
The basic principle involved in such damage detection meth- ing coefficient identification. Yet, another approach was to
ods is to compare the behaviour of structures to vibration in minimize an objective function, which is defined in terms of
the damaged and undamaged state [1]. the discrepancy between vibration data identified by modal
Structural damages lead to changes in dynamic charac- testing and those computed from analytical model [8–11].
teristics of the structure such as natural frequencies, mode Giuseppe Quaranta, Biagio Carboni, and Walter Lacarbon-
shapes, and modal damping. Since natural frequencies can ara [12] have investigated the numerical issues involved in
using modal curvature for damage detection and suggested
a more affordable way for detecting damage with modal cur-
* C. G. Krishnanunni
researchunni@gmail.com vature data.
In this paper, the change in natural frequency of the struc-
1
Structural Engineering Division, Department of Civil ture is used to locate and quantify the damage. A sensitivity
Engineering, National Institute of Technology, analysis [13] is performed to compute the change in natural
Calicut 673601, India

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frequencies due to the presence of damage. The sensitivity If 𝜹𝐊 is the small change in the stiffness matrix due to the
equation which defines the sensitivity of natural frequen- damage and 𝛿𝜆i0 is the change in the eigenvalue, then neglect-
cies of the system to small changes in stiffness is used in ing the change in mass matrix, Eq. (1) becomes:
defining the objective function. The optimization algorithm
used is the Cuckoo Search (CS) [14], which is a relatively
[𝐊𝟎 + 𝜹𝐊 − (𝜆i0 + 𝛿𝜆i0 )𝐌𝟎 ](𝝓𝐢𝟎 + 𝜹𝝓𝐢𝟎 ) = 0. (2)
new optimization algorithm and has shown great potential Expanding Eq. (2) and neglecting second-order terms yield:
and simplicity when compared to well-developed algo- 𝐊𝟎 𝝓𝐢𝟎 + 𝐊𝟎 𝜹𝝓𝐢𝟎 + 𝜹𝐊𝛟𝐢𝟎 − 𝜆i0 𝐌𝟎 𝝓𝐢𝟎
rithms like Genetic Algorithm. Cuckoo Search algorithm is (3)
− 𝛿𝜆i0 𝐌𝟎 𝝓𝐢𝟎 − 𝜆i0 𝐌𝟎 𝜹𝝓𝐢𝟎 = 0
known for its easy implementation and faster convergence
rate. The objective function compares the changes in vibra- Using Eq. (1) in Eq. (3), we get:
tion data measured before and after the damage with that
𝐊𝟎 𝜹𝝓𝐢𝟎 + 𝜹𝐊𝝓𝐢𝟎 − 𝛿𝜆i0 𝐌𝟎 𝝓𝐢𝟎 − 𝜆i0 𝐌𝟎 𝜹𝝓𝐢𝟎 = 0. (4)
of the analytical model. The analytical method used is the
Finite-Element Method (FEM). The results are obtained in Multiplying Eq. (4) throughout by 𝝓𝐓𝐢𝟎 gives:
the form of Stiffness Reduction Factors (SRFs) of different
𝝓𝐢𝟎 T 𝜹𝐊𝝓𝐢𝟎 − 𝛿𝜆i0 𝝓𝐢𝟎 T 𝐌𝟎 𝝓𝐢𝟎
elements of the structure. The results show that the dam- (5)
aged elements can be detected with high accuracy and with + (𝝓𝐢𝟎 T 𝐊𝟎 − 𝜆i0 𝝓𝐢𝟎 T 𝐌𝟎 )𝜹𝝓𝐢𝟎 = 0.
relatively shorter computational time.
However, in the case of symmetrical structures, two The transpose of Eq. (1) gives:
or more damage sites will be predicted, depending on the 𝝓𝐢𝟎 T (𝐊𝟎 − 𝜆i0 𝐌𝟎 ) = 0. (6)
degree of geometric and material symmetry [15]. The
situation is investigated with the help of a symmetrical Multiplying Eq. (6) by 𝛅𝛟𝟎𝐢
fixed–fixed beam. In such cases, to uniquely determine (𝝓𝐢𝟎 T 𝐊𝟎 − 𝝓𝐢𝟎 T 𝜆i0 𝐌𝟎 )𝜹𝝓𝐢𝟎 = 0. (7)
the damage location, displacement measurements or mode
Therefore, Eq. (5) reduces to:
shape measurements have to be made. In this paper, in addi-
tion to natural frequencies, the structural response (vertical 𝝓𝐢𝟎 T 𝛅𝐊𝝓𝐢𝟎 − 𝛿𝜆i0 𝝓𝐢𝟎 T 𝐌𝟎 𝝓𝐢𝟎 = 0 (8)
displacements) to a static load is considered to detect the Equation (8) can be rewritten as follows:
damage in the case of symmetric fixed–fixed beam. Such
measurements could be easily made in a real structure such 𝝓𝐢𝟎 T × 𝜹𝐊 × 𝝓𝐢𝟎
as a Truss bridge where vehicle loads could be considered 𝛿𝜆i0 = , (9)
𝝓𝐢𝟎 T × 𝐌𝟎 × 𝝓𝐢𝟎
as a typical static load and displacements could be meas-
ured at various points. However, for minor damages, the where:
stiffness reduction will be very low, and hence, change in
displacements would be negligible. In such cases, both fre- n
∑
quency measurements and displacement measurements have 𝜹𝐊 = − 𝛿ke × [𝐊𝐞 ]. (10)
to be used to detect the damage. Two criteria are consid- e=1

ered for damage detection, namely, the frequency changes Here, summation implies assembly. 𝛿ke is defined as the stiff-
and a combination of frequency changes and displacement ness reduction factor (SRF). [𝐊𝐞 ] is the element stiffness
changes (or changes in acceleration–time response). matrix. From Eq. (9), it is evident that the change in natural
frequency of the structure is a function of the change in stiff-
ness of the structure, which by itself is a function of stiffness
2 Dynamic analysis reduction factors (SRFs) of elements. Hence, the change in
natural frequency can be represented as function of SRFs of
2.1 Sensitivity equation elements of the structure.
Furthermore, the relative change in the ith eigenvalue as
The equation of motion for undamped free vibration of a sys- a measure of damage, known as damage index (DI), can be
tem can be written in the form of an eigenvalue problem as: expressed as follows:
(𝐊𝟎 − 𝜆i0 𝐌𝟎 )𝝓𝐢𝟎 = 0 i = 1, 2, 3, 4 … n. (1) 𝜆i − 𝜆i0 𝛿𝜆
Here, 𝐊𝟎 and 𝐌𝟎 are the stiffness and mass matrices for DI = = i0 . (11)
𝜆i0 𝜆i0
undamaged structure, respectively. 𝝓𝐢𝟎 represents the mode
shape vector for 𝜆i0 (eigenvalue) and 𝜆i0 = 𝜔2i0.

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3 Static analysis √
√nmodes ([ )2
√ ∑ 𝜆i − 𝜆i0 ]AN [ 𝜆i − 𝜆i0 ]EX
J= √ − , (16)
As discussed earlier, for symmetric structures, damage site 𝜆i0 𝜆i0
i=1
cannot be uniquely determined [15]. In this case, structural
response to a static load is considered to accurately detect dam- where suffixes ‘AN’ and ‘EX’ represent analytical and
age. This is based on the fact that, due to stiffness reduction, experimental values, respectively. The objective function is
the structural response to a given load vector would be more a global error function which defines the error between the
in the damaged state when compared to the undamaged state. experimentally and analytically obtained frequency changes.
In this paper, the vertical displacement measurements are con- The objective function based on relative change was first
sidered to detect the damages. proposed by Hong Hao and Yong Xia [10]. Here, ‘ 𝜆i ’ rep-
resents the eigenvalue in the damaged state. In our case,
3.1 Sensitivity equation least square error has been considered. The solution of this
optimization problem is unique for asymmetric structures
In dynamic analysis, a sensitivity equation which defines sen- [15]. At least two modes of vibration must be considered for
sitivity of natural frequency to small change in stiffness was formulating the objective function.
derived. On similar terms, for static analysis, the sensitivity Equation (16) can be rewritten as follows:
equation describes the sensitivity of displacement vector {x} √
to a small change in stiffness 𝜹𝐊 , under the action of a given √nmodes ([
√ ∑ 𝛿𝜆i0 ]AN [ 𝜆i − 𝜆i0 ]EX
)2
static load vector 𝐅. J= √ − . (17)
𝜆i0 𝜆i0
The equilibrium equation in the undamaged state can be i=1

written as follows:
𝛿𝜆i0 is obtained from Eq. (9). From Eqs. (9) and (17), it is
𝐅 = 𝐊𝟎 𝐱. (12) evident that J is a function of all the stiffness reduction fac-
In the damaged state, under the action of the same load vec- tors ( 𝛿 ke ). Minimizing J gives the values of all the SRFs.
tor 𝐅 , Eq. (12) becomes: The information contained in the mode shape vectors in the
predamaged state is incorporated in the objective function.
𝐅 = (𝐊𝟎 + 𝜹𝐊)(𝐱 + 𝜹𝐱). (13) The analytical values of eigenvalues and mode shape vec-
Expanding Eq. (13) and using Eq. (12) in Eq. (13), we get: tors in the predamaged condition are obtained by FEM. The
optimisation is carried out using Cuckoo Search algorithm
𝐊𝟎 𝜹𝐱 + 𝜹𝐊𝐱 + 𝜹𝐊𝜹𝐱 = 0. (14)
(CSA). The method does not require any accurate analytical
Rearranging Eq. (14), we get:
modelling to detect damage. The incorporation of sensitiv-
𝜹𝐱 = −(𝐊𝟎 + 𝜹𝐊)−1 𝜹𝐊𝐱. (15) ity equation in the objective function has also resulted in
𝛿x is a column vector containing the change in the displace- relatively lower computational time.
ment at various positions in the structure due to the occur- In the present study, numerical experiments are per-
rence of a damage. This expression is used in the formulation formed with the help of ANSYS. Two examples of cantilever
of objective function based on displacement measurements. beams with different damage scenario have been considered.

4.2 Objective function based on displacement

4 Objective function measurements

The objective function defines the error between vibration Objective function for the optimisation problem based on
data obtained from analytical model computations and experi- displacement measurements can be defined in a similar way
mental model testing. Objective function could be defined in as follows:
three different ways depending on the static and vibration data √
available. )2
√npoints ([ ]
√∑ 𝛿xi AN [ xi − xi0 ]EX
J=√ − , (18)
xi0 xi0
4.1 Objective function based on frequency i=1

measurements
where 𝛿xi represents the change in vertical displacement at
the ith position and can be obtained from Eq. (15). The num-
The objective function for the optimisation problem is defined
ber of points in which displacement measurements has to
as follows:
be made is decided by the investigator. More displacement
measurements must be made at those points where damage

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is suspected. However, the displacement changes tend to be FEM analysis to obtain

very small in the case of minor damages. In such cases, the Structural Model
natural frequency
accuracy of damage detection might get reduced. In addi-
Static Analysis
tion, huge static loads have to be applied in such cases. The Displacement measurement
Measurements:
analytical values of the displacement vector is obtained by Intact state Dynamic Analysis
Frequency measurement
FEM. In the previous case, ‘J’ is a function of all SRFs. ‘J’
could be minimized to obtain the SRFs. Static Analysis
Displacement measurement
Measurements:
4.3 Objective function based on both frequency damaged State Dynamic Analysis
Frequency measurement
and displacement measurements
Objective function using
Objective function for the optimization problem based on both frequency measurements
displacement and frequency measurements can be defined as Selection of
J  , where: Objective function using
√
objective
function displacement measurements
npoint ([ )2
∑ 𝛿xi ]AN [ 𝛿xi ]EX Objective function using
J = Wi −
xi0 xi0 frequency and displacement
i=1
nmode ([ )2 (19) Function optimization measurements
∑ 𝛿𝜆i0 ]AN [ 𝛿𝜆i0 ]EX using CSA
+ Wj − .
j=1
𝜆i0 𝜆i0
Damage detection
(SRF identification)
The relative accuracy in the measurement of natural fre-
quency and displacement will determine the weights Wi and
Wj . The value of Wi and Wj can be chosen between 0 and 1. Fig. 1  Procedure for damage detection
When the frequency data are more reliable, Wj is chosen
higher than Wi and closer to 1. In most cases, the relative acceleration–time plot could be used to define an objective
measurement error of frequency is about 1% [10], whereas function as J  , where:
√
the displacement or acceleration measurements are more
contaminated with noise and it is customary to add 5–10% npoint ntime ([
∑ ∑ 𝛿tij ]AN [ 𝛿tij ]EX )2
white noise to the analytically obtained displacement pro- J = W1
tij
−
tij
file to simulate field measurements [16]. For pursuing the i=1 j=1
(20)
damage detection procedure shown in Fig. 1, the parameters nmode ([
∑ 𝛿𝜆 ]AN i0
[ 𝛿𝜆 ]EX )2
i0
Wi and Wj have to be known before hand. This is achieved + W2 − ,
j=1
𝜆i0 𝜆i0
by developing the finite-element model of the structure to
be assessed and simulating a suitable damage scenario.
Adequate measurement noise is added to the frequencies where tij represents acceleration of ith location at jth time
and displacements obtained from the numerical model and step. npoint refers to the number of measurement sites. W1
the weights are determined, so that the damage is located and W2 are the weights as discussed previously. The general
accurately based on the procedure in Fig. 1. The computed damage detection procedure is shown in Fig. 1.
weights could be used for assessing the health of the struc-
ture based on field measurements. This objective function
should be used when dealing with symmetric structures,
where frequency measurements alone are not sufficient to 5 Overview of Cuckoo Search algorithm
guarantee a unique solution to the optimisation problem
[15]. Cuckoo Search is a relatively new metaheuristic optimization
algorithm [14, 17]. The algorithm is based on the aggressive
4.4 Objective function based on frequency data breed characteristics of cuckoo birds. This is incorporated
and acceleration–time history in combination with Lévy flight behaviour of certain birds
to get a simple and effective optimization algorithm. This is
If accelerometers are used to record the dynamic response of used in minimizing Eqs. (17), (18), (19), and (20).
a structure subjected to an arbitrary excitation, the obtained

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Journal of Civil Structural Health Monitoring

5.1 Cuckoo breed characteristics example, the inverse of the function value may be taken as the
fitness of the function.
Cuckoos are well known for their breeding strategy. They Each egg in a nest is taken to represent a solution. A cuckoo
lay their eggs in the nests of other host birds (particularly is taken to represent a new solution. The main aim of the pro-
crows). If the host bird identifies the cuckoo’s egg, it may gram is to discard the old solutions and accept potentially new
throw away the egg or abandon the nest. Some cuckoo spe- and improved solutions.
cies belonging to the ‘Tapera’ genus have evolved, such that When generating a new solution, xt+1 for a cuckoo i, Lévy
female parasitic cuckoos are often found to be extremely flight is performed as:
specialized in mimicking host eggs. Thus, it becomes very
xit+1 = xit + 𝛼 ⊕ Levy(𝜆), (21)
difficult for the host bird to detect the cuckoo’s egg.
where 𝛼 > 0 is the step size, which is related to the scales of
5.2 Lévy flight the new problem of interest. In most cases, we can use 𝛼 = 1 .
The product ⊕ means entrywise multiplication.
A Lévy flight is a random walk in which the step lengths have Based on these steps, the pseudocode for CS algorithm
a probability distribution. When defined as walks in space, the can be written as [14]:
steps made are in random directions. However, this direction
depends on the current position. begin
In nature, animals and birds search for food in a random
Objective function f (x), x = (x1 , ...., xd )T
or quasi-random manner. The path in which an animal moves
becomes a random walk, because its next step is based on the Generate initial population of n host nests xi (i = 1,
current location. Hence, the direction that it chooses depends 2, ...., n)
on a probability model which can be modelled mathematically.
Thus, it is a type of random walk in which each successive while (stop criterion)
move is chosen randomly and is uninfluenced by any previ-
Get a cuckoo randomly by Levy flights
ous moves. For example, consider a drunk person walking on
a street. The likelihood of the person taking a step to the right Evaluate it’s quality/fitness Fi Eq. (17),
is the same as that of the person taking a step to the left, with (18) or (19)
no memory of the previous routes. In a Lévy walk, most steps
are taken within a small area; however, longer routes may be Choose a nest among n (say, j) randomly
taken occasionally. A wide variety of animals, such as marine
if (Fi > Fj )
predators, birds, terrestrial mammals, and various insects
exhibit Lévy-like patterns. Evidence of Lévy walks has also replace j by new solution
been found in the ways that people wander freely.
end
5.3 Cuckoo Search algorithm A fraction (pa ) of worse nests are abandoned and
new ones are built
For simplicity in defining the CSA, the following idealised
rules are considered: Keep the best solutions
(or nests with best quality solutions)
1. Each cuckoo lays one egg at a time and dumps it in a
randomly chosen nest. Rank the solutions and find the current best.
2. The best nests with high-quality eggs (solutions) will
end while
carry over to the next generation.
3. The number of available host nests is fixed, and a Postprocess results and visualization
host can discover an alien egg with a probability of
pa ∈ [0, 1] . In this case, the host bird can either throw end
away the egg or build a completely new nest in a com-
pletely new location.

For a maximization problem, the fitness or quality of a solu-

tion can be taken to be directly proportional to the value of the
objective function. For other types of problems, other fitnesses
can be defined accordingly. For a minimization problem, for

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6 Example 1: cantilever beam (a) Table 1  Analytical and experimental natural frequencies (Yang et al.
1985)
To demonstrate the proposed damage detection technique, a Mode Analytical values Experimental values (Hz)
cantilever beam has been used as an example. The example (Hz)
Undamaged Damaged
problem is taken from an experimental study reported by
Yang et al. [18]. The geometric and material properties of 1 23.70 19.53 19.00
the beam are as follows: 2 148.53 122.05 115.85
3 415.88 339.26 332.36
Material: Aluminium 4 815.00 661.73 646.91
Length: 495.3 mm 5 1347.40 1085.22 1037.46
Width: 25.4 mm 6 2013.20 1594.59 1591.36
Depth: 6.35 mm
Young’s modulus: 7.1 × 1010 N/m2
Mass density: 2210 kg/m3. are sufficient to detect the damage and mode shape measure-
ments are not made.
The beam was damaged by a saw cut, as shown in Fig. 2b. The objective function was created using the sensitivity
The beam was modelled using 20 plane frame elements hav- equation and the error is minimized using CSA to obtain the
ing six degrees of freedom at each node. The natural fre- SRF’s. For the comparison of results, the objective function
quencies of the undamaged beam obtained by eigenvalue defined by Hong and Hao [10] is also minimized using CSA
analysis are given in Table 1. The damage was induced in and it was found that the results so obtained agreed well with
the ninth element. the reported results in the reference, which was obtained
The experimental values of natural frequency in the using genetic algorithm.
predamaged and damaged states as obtained by Yang et al. The results are shown in Fig. 3. The number of nests was
[18] are also provided in Table 1. Only the first six natural set to 25. The probability pa is set to 0.35. The process is
frequencies were used for damage detection. Since it is an iterated 65,000 times for good convergence.
unsymmetrical structure, natural frequency measurements The results show that the damage is correctly detected in
element 9. Small magnitude of SRF is detected in the 2nd
and 12th elements which may be due to noise in the fre-
quency measurements and nonlinearity caused due to severe
damage [10]. The use of sensitivity equation in the objective
function has improved the accuracy in damage detection.
Figure 3 shows that the computed SRF in ninth element is
three times the SRF of Element 12. Therefore, it can be
concluded with a reasonable level of confidence that the

80
Present result
Reference result
70

60

50
SRF

40

30

20

10

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
ELEMENT NUMBER

Fig. 2  Configuration of the cantilever beam [10] Fig. 3  SRF of the cantilever beam (in %)

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Journal of Civil Structural Health Monitoring

major damage is in element 9. The computational time is to the adjacent finite element. It is to be understood that,
also significantly lower compared to the other methods. This when the number of elements used is 15, 18, 20, 23, or 27,
is because the time required for eigenvalue analysis is saved the damage is located approximately in the middle of the
in each iteration. The method also has the advantage that it finite element which leads to a reasonably good estimate
does not require any accurate analytical modelling or finite- of the damage magnitude as observed in Fig. 4. However,
element model updating. Algorithm detects very minor dam- for other cases in Fig. 4, due to the proximity of damage
age in element 8 which is adjacent to the actual damage loca- to adjacent element, the damage magnitude gets distributed
tion. This is due to the involvement of mode shape vector in across elements leading to a lower detected SRF. Numerical
the objective function and very severe damage in element 9 experiments by varying the number of elements have to be
(the section was actually weakened by 93.75% [18]). done for accurately determining the damage magnitude or
an objective function defined by Eq. (19) could be used. In
6.1 Number of modes and number of finite the simulations presented here, the number of elements is
elements to be used for damage detection fixed at 20 to demonstrate that accurate analytical modelling
(finer mesh) is not required for efficient damage detection.
Theoretically, the measurement of frequency changes in one
pair of modes will yield a locus of possible damage sites
[15]. Only an optimum number of modes are needed to be 7 Example 2: cantilever beam (b)
considered so as to transform the problem into a well-posed
one, i.e., for unique determination of the damage location. In this example, a cantilever beam with damages at multiple
In the above analysis, it is essential to use the first six natural positions has been considered. The beam is simulated using
frequencies to achieve a unique solution to the optimisation ANSYS and the modal parameter obtained from ANSYS
problem. Numerical experiment by considering only the first is used in place of the experimental values. The beam is
five natural frequencies leads to an ill-posed problem, i.e., a shown in Fig. 5.
unique solution was not obtained. The noise in the frequency The geometric and material properties of the beam are
measurement is the major reason that caused the problem as follows:
to be ill-posed. Since there is practical limit to the range of
frequencies that a structure can be tested for, only the first Material: Steel
few natural frequencies are used in all the demonstrations. Length: 600.0 mm
The effect of number of finite elements used for damage Width: 25.0 mm
detection is demonstrated in Fig. 4. It can be observed from
Fig. 4 that, irrespective of the finite-element mesh used, the
damage is accurately detected. However, the magnitude of
the detected SRF depends on the proximity of the damage

90

80

70

60
SRF (%)

50

40

30

20

10

0
16 18 20 22 24 26 28 30
Number of elements

Fig. 4  Effect of number of finite elements on the identified SRF of

element containing damage (in %) Fig. 5  Configuration of the cantilever beam (b)

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Table 2  Analytical and ANSYS natural frequencies 70

Mode Analytical value ANSYS value (Hz)

60
(Hz)
Undamaged Damaged
50
1 15.85 15.89 15.63
2 99.36 99.55 98.82
40

SRF
3 278.21 278.55 272.19
4 545.21 545.40 522.85 30
5 901.37 900.53 862.26
6 1346.80 1343.47 1305.11 20
7 1881.70 1874.00 1873.95
10

Depth: 7.00 mm 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Young’s modulus: 2.0 × 1011 N/m2
ELEMENT NUMBER
Mass density: 7850 kg/m3.

Damage was simulated in the form of a crack of width 5 Fig. 6  SRF of the cantilever beam with multiple damage location (in
%)
mm, as shown in Fig. 5. The damage was induced in the 4th
and 17th element. 40% and 60% reduction in cross-sectional
area was provided in the 4th and 17th elements, respectively. Material: Aluminium
The values of natural frequency in the predamaged Length: 600.0 mm
and damaged states as obtained by ANSYS analysis and Width: 25.0 mm
obtained by analytical modelling are provided in Table 2. Depth: 6.00 mm
The beam was modelled using 20 plane frame elements Young’s modulus: 7.1 × 1010 N/m2
having six degrees of freedom at each node. The first seven Mass density: 2770 kg/m3.
natural frequencies are used for damage detection.
The damage detection procedure was carried out and
the results are shown in Fig. 6. The CSA parameters cho-
sen are the same as before. The process is iterated 65,000
times for good convergence of results.
The results show that the damage is correctly detected
in element 4 and element 17. Small magnitude of SRF
is detected in the 14th element which may be due to the
nonlinearity caused due to severe damage. The algorithm
detects SRF of 24.59% and 58.38% in the 4th and 17th
elements, respectively. The results show that the algorithm
gives a good insight into the relative magnitude of dam-
ages at different position.

8 Example 3: fixed–fixed beam

The special case of a symmetrical fixed–fixed beam is con-

sidered in this example. In this case, displacement meas-
urements have to be made for effective damage detection.
The beam was simulated and analyzed in ANSYS. Both
modal analysis and static analysis are done to obtain the
natural frequencies and the static displacement values.
A uniformly distributed load of magnitude 500 N/m is
applied for displacement values. The geometric and mate-
rial properties of the beam are as follows: Fig. 7  Configuration of the fixed–fixed beam

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Journal of Civil Structural Health Monitoring

Table 3  Analytical and ANSYS natural frequencies by Eq. (17) is minimized in this case. The results are shown
Mode Analytical value ANSYS value (Hz)
in Fig. 8.
(Hz) The results show that damage is detected in 12th element
Undamaged Damaged
and 9th element, whereas the actual damage is in 12th ele-
1 86.70 87.02 86.38 ment only. Perfect convergence in solution was not observed
2 239.09 239.70 238.80 in this case. This is because, with symmetric structures, the
3 468.74 469.48 465.98 damage cannot be uniquely located using frequency values
4 774.94 775.22 767.73 [15]. From the observed result, it can be concluded that fre-
5 1157.90 1156.50 1155.00 quency changes alone are not sufficient to detect the damage.
6 1617.80 1612.90 1593.30 Damage detection considering only displacement changes
7 2155.00 2143.80 2137.00 is not demonstrated in this example. The procedure requires
a large number of displacement measurements, since the
displacement changes are negligible for small damages.
Damage was simulated in the form of a saw cut, as shown in
Fig. 7. It was induced in the 12th element, the beam being 8.2 Damage detection with both frequency
modelled using 20 elements. and displacement measurements
The natural frequency values are provided in Table 3.
As before, the beam was modelled using plane frame ele- A uniformly distributed load of magnitude 500 N/m is
ments. The first six natural frequencies along with displace- applied on the beam and the vertical displacement at five
ment values at five points on the beam are used for damage different positions on the beam (at 90 mm, 210 mm, 330
detection. mm, 450 mm, and 570 mm from left end of the beam) is
For damage detection, the objective function defined by obtained before and after the damage. The load is chosen so
Eqs. (17) or (19) can be used. The relative effectiveness of as to obtain a maximum displacement of 5.3 mm, which is
the two objective functions is examined. The CSA param- of measurable order. The analysis could even be carried out
eters are already defined in the previous section and the pro- using a suitable concentrated load.
cess is iterated 65,000 times as before. The objective function defined by Eq. (19) is minimized
in this case. The weights Wi and Wj are set to unity, since
8.1 Damage detection with frequency both frequencies and displacements are obtained with the
measurements same degrees of accuracy. The results are shown in Fig. 9.
The figure pinpoints the damage to the 12th element.
In this case, only the first seven natural frequency values Thus, by combining frequency and displacement measure-
are used for damage detection. Objective function defined ments in the analysis, the damage could be detected accu-
rately in the case of symmetric structures. A small damage

30
18

16
25

14
20
12
SRF

10 15
SRF

8
10
6

4 5

2
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ELEMENT NUMBER
ELEMENT NUMBER

Fig. 9  Identified SRF with frequency and displacement changes (in

Fig. 8  Identified SRF with frequency changes (in %) %)

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 Journal of Civil Structural Health Monitoring

detected in the fifth element may be due to nonlinear- Focus is laid on detecting the storey in which the dam-
ity caused due to the damage. The computational time is age is present rather than identifying the structural element.
reduced by invoking gauss elimination procedure on banded Lumped mass modelling (LMM) is adopted as the analyti-
matrix to solve Eq. (15). cal model to compute the natural frequencies and dynamic
In most cases, the displacement changes tend to be very response. The geometric and material properties of the space
small, and hence, very sophisticated instruments are required frame are as follows:
for very accurate measurements. In such cases, the investiga- Column properties
tor is free to choose different weights Wi and Wj depending
upon the relative accuracy in the measurements of frequen- Material: Aluminium
cies and displacements. Shear modulus: 26 × 109 N/m2
Young’s modulus: 7.1 × 1010 N/m2
Mass density: 2770 kg/m3
9 Example 4: space frame model Length of column: 310.0 mm
Column-cross section: Trapezoid with dimensions:
The damage detection algorithm is applied to a three storey Longer side: 25 mm
frame model shown in Fig. 10a. Shorter side: 20 mm
Thickness: 2.56 mm.

Plate properties

Material: Steel
Young’s modulus: 2.1 × 1011 N/m2
Shear modulus: 77 × 109 N/m2
Mass density: 7850 kg/m3
Dimension of Plate: 250 × 200 × 10 mm.

The Lumped mass model is shown in Fig. 10b. The model

has three degrees of freedom and the shear building model
is used to compute the global stiffness and mass matrix.
The natural frequencies obtained by eigenvalue analy-
sis are given in Table 4. After constructing the damping
matrix, Newmark-beta method is used to obtain the dynamic
response to base excitation.
Uni-axial shake table experiment was conducted on the
specimen with accelerometers attached at individual storey
levels as shown in Fig. 10a. The recorded acceleration–time
history and the natural frequency obtained through a sine
(a) Configuration of the frame specimen sweep procedure (Table 4) are used as input to the damage
detection algorithm.
For this particular problem, a two-stage damage detection
procedure is carried out. In the first stage, the finite-element
model is updated, so that the acceleration–time plot obtained
experimentally and analytically has the same character in the
undamaged state. The model is updated by applying suitable
Stiffness reduction factor (SRF) for each storey. Stiffness
Table 4  Analytical and experimental natural frequencies
Mode Analytical value (Hz) Experimental (Hz)
Original Updated Undamaged Damaged
(b) Lumped mass model with mass lumped at each storey level 1 2.08 1.63 1.6 1.55
2 5.82 5.32 5.4 5.35
Fig. 10  Three storey frame specimen 3 8.39 6.94 – –

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Journal of Civil Structural Health Monitoring

Fig. 11  Acceleration–time plot for the top storey Fig. 13  Acceleration–time plot for the bottom storey

Fig. 14  Acceleration–time plot obtained analytically (undamaged

state)
Fig. 12  Acceleration–time plot for the middle storey

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 Journal of Civil Structural Health Monitoring

reduction of 5%, 62%, and 2% has been obtained for the top, 9.2 Damage identification with both frequency
middle, and bottom storey, respectively, by minimizing an changes and changes in acceleration time response
objective function based on direct comparison of frequencies
in the undamaged state. The flexibility provided by the bolted The frequency changes in the first two modes are also con-
connection is thus incorporated in the updated model. The sidered in addition to acceleration changes.
8
updated natural frequency values are provided in Table 4.
Shake table experiment was conducted for sinusoidal 7
wave of frequency 1.6 Hz and amplitude 1 mm. The accel-
6
eration–time plot for the damaged and undamaged state is
also shown in Figs. 11, 12, and 13. From the plots, it can 5
be inferred that damage has caused a shift in the system’s

SRF
4
natural frequency, thereby lagging the response.
The acceleration–time plot obtained after model updating 3

is shown in Fig. 14. The base excitation has a frequency of

2
1.6 Hz and amplitude 1 mm. 2% damping is considered in
the analytical model for computing the response. 1

In the second stage, the updated model is used for dam- 0

age detection. The objective function defined by Eq. (20) is TOP MIDDLE BOTTOM
minimized to obtain the SRF of the three storeys. To con- STOREY NUMBER
trol the growth of error, only those time steps for which the (a) Weight W2 = 1
8
acceleration in the intact and damaged state is either both
positive or both negative is considered. 7

6
9.1 Damage identification with changes
in acceleration–time response 5
SRF

4
The acceleration of the three storeys between 2 s and 8 s
with a time step of 0.3 s is considered for defining the objec- 3

tive function (npoint = 3). The response becomes steady 2

after 9 s. In this case, W2 is zero.
The identified SRF are shown in Fig. 15. The figure shows 1

that all the elements are detected as having damages. This may 0
be attributed to the errors in measurement and noise. However, TOP MIDDLE BOTTOM
STOREY NUMBER
the middle storey is detected as having the most severe damage.
(b) Weight W 2 = 0.1
8
8
7
7
6
6
5

5
SRF

4
SRF

4 3

3 2

2 1

0
1 TOP MIDDLE BOTTOM
STOREY NUMBER
0
TOP MIDDLE BOTTOM (c) Weight W 2 = 0.01
STOREY NUMBER
Fig. 16  Identified SRF considering both frequency changes and
Fig. 15  Identified SRF with acceleration changes acceleration changes with different weights

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Journal of Civil Structural Health Monitoring

The relative contribution of frequency and acceleration 25

to the objective function is considered as unity ( W1 = 1 ) and
( W2 = 1, 0.1, 0.001), respectively.
20
The obtained SRF are shown in Fig. 16. It can be inferred
that the damage is accurately detected in the middle storey
when W2 = 0.01 . Therefore, the relative weights W1 and W2 15
play a very major role in damage detection. In addition, since

SRF
the error in frequency measurement are relatively less, it is
more appropriate to make large number of frequency meas- 10
urements. The weights have to be chosen based on the rela-
tive accuracy in the measurements made. Thus, the proposed
5
algorithm is capable of detecting a minor damage.

9.3 Effect of damping ratio on damage detection 0

TOP MIDDLE BOTTOM
In the above detection, 2% damping ratio was considered STOREY NUMBER
for computing dynamic response. The procedure is repeated (a) ∆ t=0.1 s (ntime=61)
for 1.5% and 2.5% damping. Similar results were obtained,
16
indicating that change in damping ratio has negligible effect
on damage detection. This may be attributed to the peculiar 14
nature of the objective function defined based on relative
error. 12

10
9.4 Effect of number of time steps on damage
detection
SRF

In the above detection procedure, a total of 21 time steps 6

were considered [ntime = 21 in Eq. (20)]. The procedure

4
was repeated for different number of time steps between 2 s
and 8s. The results obtained are shown in Fig. 17. 2
Figure 17 shows that the damaged storey could be accu-
rately located and is independent of the number of time steps 0
TOP MIDDLE BOTTOM
considered. However the magnitude of the detected damage
STOREY NUMBER
depends on the number of time steps considered. This is
due to the error involved in instrumentation and faint noise (b) ∆ t=0.5 s (ntime=13)
in the data which causes the problem to be ill-conditioned.
This problem may be tackled by including a regularization Fig. 17  Effect of number of time steps ( ntime ) on damage detection
parameter in the objective function, which stabilizes the ill- ( W2 = 0.01)
conditioned problem [19]. The proposed damage detection
technique may find application in the health monitoring of readings on each of the columns could be used for dam-
buildings subjected to earthquake. The ground acceleration age detection. However, the tremendous computational
data could be obtained from an earthquake recording station time poses a serious problem for damage detection of
and the sensors embedded in different structural members large structures. In such cases, frequency and mode shape
of the building could record the acceleration–time response. measurements may be combined to solve the problem. It
The initial portion of the response represents the healthy is theoretically possible to identify the mode shape from
building and the occurrences of damage may be indicated by acceleration response. It is then possible to define an
the jumps in the acceleration–time response. With the base objective function based on the identified frequencies and
excitation data and acceleration–time response available, mode shapes of the structure. With the damaged storey
one could detect the possible damage sites in a multi-storey already identified, the number of parameters to be esti-
building, after the earthquake has subsided. mated (SRF) in the objective function and, consequently,
To detect the exact column containing the damage, a the computational time could be significantly reduced. Yet,
more rigorous analysis is required. The structure may be another way would be to use the obtained mode shape
modelled using space frame element [20]. Accelerometer

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 Journal of Civil Structural Health Monitoring

changes to detect the damage based on Modal Strain Funding  This research did not receive any specific grant from funding
energy change [21]. agencies in the public, commercial, or not-for-profit sectors.
The proposed damage identification technique with accel-
eration response may also find application in the damage
detection of bridges. In this case, the vehicle force acts as References
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