Engineering, 2009, 1, 167-176
doi:10.4236/eng.2009.13020 Published Online November 2009 (http://www.scirp.org/journal/eng).
Detection and Quantification of Structural Damage of a
Beam-Like Structure Using Natural Frequencies
Saptarshi SASMAL, K. RAMANJANEYULU
Structural Engineering Research Centre, Council of Scientific and Industrial Research (CSIR),
Chennai, India
Received January 10, 2009; revised February 21, 2009; accepted February 23, 2009
Abstract
Need for developing efficient non-destructive damage detection procedures for civil engineering structures is
growing rapidly. This paper presents a methodology for detection and quantification of structural damage
using modal information obtained from transfer matrix technique. Vibration characteristics of beam-like
structure have been determined using the computer program developed based on the formulations presented
in the paper. It has been noted from reported literature that detection and quantification of damage using
mode shape information is difficult and further, extraction of mode shape information has practical
difficulties and limitations. Hence, a methodology for detection and quantification of damage in structure
using tranfer matrix technique based on the changes in the natural frequencies has been developed. With an
assumption of damage at a particular segment of the beam-like structure, an iterative procedure has been
formulated to converge the calculated and measured frequencies by adjusting flexural rigidity of elements
and then, the intersections are used for detection and quantification of damage. Eventhough the developed
methodology is iterative, computational effort is reduced considerably by using transfer matrix technique. It
is observed that the methodology is capable of predicting the location and magnitude of damage quite
accurately.
Keywords: Frequency, Mode Shape, Transfer Matrix, Damage Detection, Quantification
1. Introduction
The need for development of an efficient procedure for
non-destructive structural damage detection is increasing
in order to assess the integrity and serviceability of ex-
isting structures. This has led to continued research to
develop methods that could identify changes in vibration
characteristics of a structure. These methods are based on
the fact that modal parameters (notably frequencies and
mode shapes, and modal damping) are functions of the
physical properties of the structure (mass, damping, and
stiffness). Any change in the physical properties, such as
reduction in stiffness resulting from cracking or loosen-
ing of a connection, will cause detectable change in the
modal properties. Various methods have been employed
by researchers all over the world for damage detection of
structural systems, in frequency domain.
Perhaps, the first research article on damage detection us-
ing vibration measurements was by Lifshitz and Rotem
[1] where the change in the dynamic moduli was related
to the frequency shift and proposed as indicator of dam-
age in particle-filled elastomers. Cawley and Adams [2]
are the first researchers to give a formulation for damage
detection based on change in frequency of an undamaged
and damaged state of a structure. The systematic use of
mode shape information was proposed in [3] for localiz-
ing of structural damage without the use of a prior finite
element model (FEM) by using the modal assurance cri-
teria (MAC) to determine the level of correlation be-
tween modes from the test of an undamaged space shut-
tle orbiter body flap. Yuen [4] examined changes in the
mode shape and mode-shape-slope parameters to simu-
late the reduction of stiffness in each structural element
and compared predicted changes with the measured
changes to determine the damage location. Ismail et al.
[5] demonstrated that the frequency drop caused by an

S. SASMAL AND K. RAMANJANEYULU 168
opening and closing crack is less than that caused by an
open crack. This property is a potentially large source of
error that is considered by few of the researchers using
frequency changes. A simple and easy method for one-
dimensional structures by representing crack using a
spring that connects the two half components was pre-
sented by [6]. The natural frequencies were expressed as
functions of the crack depth and location. Hearn and
Testa [7] developed a damage detection method using
frequency shift of a structure due to damage. The fre-
quency sensitivity method combined with inter-
nal-state-variable theory to detect damage in composites
was used in [8]. They presented a damage indicator
which is capable of detecting damage due to 1) exten-
sional stiffness changes caused by matrix micro-cracking
and 2) changes in bending stiffness caused by transverse
cracks in the 90-degree plies. An experimental study on
the sensitivity of the measured modal parameters of a
shell structure was conducted in [9] to damage in the
form of a notch. A method for the detection of the exis-
tence and location of structural damage using the identi-
fied eigen solution together with properties of the eigen-
value problem was proposed in [10].
Slater and Shelley [11] presented a method based on
frequency-shift measurements to detect damage in a
smart structure by using the theory of modal filters to
track the frequency changes over time. Narkis [12] de-
duced a closed-form solution for the crack position, as
function of the frequency shift of two modes of the same
mechanical model and located the crack from measuring
either bending or axial frequencies of two modes only. A
transfer matrix technique was used in [13] to detect
damage for beam like structures. Ratcliffe [14] devel-
oped a technique for identifying the location of structural
damage in a beam using modified Laplacian Operator on
mode shape data. A sensitivity- and statistical- based
method to localize structural damage by direct use of
incomplete mode shapes was presented in [15]. and [16].
A numerical study of damage detection using the rela-
tionship between damage characteristics and the changes
in the dynamic properties was presented by [17]. It was
found that the rotation of mode shape is a sensitive indi-
cator of damage localisation. Another damage localisa-
tion method based on changes in uniform load surface
(ULS) curvature was developed by Wu and Law [18]. A
procedure using gap smoothing method was proposed in
[19] wherein local features in vibration curvature shapes
were extracted using a localized curve fit (i.e., smooth-
ing). Alvandi and Cremona [20] reviewed usual vibra-
tion-based damage identification techniques for struc-
tural damage evaluation. With the help of a simply sup-
ported beam with different damage levels, the reliability
of these techniques was investigated by using only few
mode shapes and/or modal frequencies of the structure
that can be easily obtained by dynamic tests and con-
cluded that broadly the detection judgement depends on
a threshold level of damage.
1.1. Detection of Damage Using Mode Shape
Information
From the review of literature, it is found that the vibra-
tion data such as frequency and mode shape are very
important parameters for detecting the damage in struc-
tures and a number of research works was carried out on
detection of damage using frequency or mode shape. But,
there is no confirmation on superiority of any method
over the others. Though, changes in mode shape are
much more sensitive to local damage compared to
changes in frequency, use of mode shape information is
restricted because 1) lower modes (usually measured
from vibration tests of large structure) may not signifi-
cantly reflect the local damage, 2) extracted mode shapes
are prone to environmental noise and 3) number of sen-
sors and the choice of sensor location may have a crucial
effect on accuracy of damage detection. So, a detailed
investigation has been carried out by the authors to as-
sess the influence of location and degree of damage on
mode shape. It is found that 1) displacement mode
shapes are sensitive to damage and the mode shape
changes with damage, 2) though higher modes are more
predominant in showing the shift in mode shape dis-
placements due to damage in the structure, lower modes
may not significantly reflect the damage, 3) shift in mode
shape largely depends on the location of damage and the
mode considered. Higher mode will magnify the shift in
mode shape, if the damage location does not fall near the
zero-displacement points, 4) any shift in mode shape of a
damaged structure with respect to the mode shape of
undamaged structure may lead to an interpretation of
damage in that location, and in most of the cases, it may
go wrong. Further, for higher modes, if the damage is
located at a location where zero displacement occurs in
that particular mode, shift in mode shape will be re-
flected in place other than the place where damage has
really taken place, 5) Shift in mode shape is predominant
in higher modes than in the lower modes. It may show a
number of locations with shift in mode shape with re-
spect to undamaged mode shape which may lead to mis-
interpretation of location of damage. So, it can be stated
that mode shape information alone can not provide cor-
rect information on detection of damage in the structure
unless it is treated otherwise, and 6) it is very difficult to
quantify damage accurately from mode shape information
alone. Further studies can be seen elsewhere [21,22].
Though significant damage might cause very small
changes in natural frequency (particularly for large
structures), natural frequencies are easy to be measured
and are less influenced by environmental noise. The
choice of using the natural frequency as a basic vibration

S. SASMAL AND K. RAMANJANEYULU169
characteristic for damage detection is the most attractive
one due to the fact that the natural frequencies of a
structure can be measured at one single location in the
structure, thus rendering a means for a rapid and global
technique. Further, it is observed that studies related to
the extension of transfer matrix method for detection of
damage are very few. Hence, in this study, a methodol-
ogy for detection of damage in structures using transfer
matrix technique has been proposed based on change in
natural frequency. The extent of research work carried
out towards quantification of damage is considerably less
compared to studies on localisation of damage. In view
of this, a methodology has been developed in this study
for detection and quantification of damage using transfer
matrix method based on modal frequencies obtained
from a damaged structure. Transfer matrix method [23]
is used in this study because of its versatility and ease
with which it can be applied to a structure of either uni-
form or non-uniform cross section and under a variety of
boundary conditions such as simple support, cantilever
support, and even for beam on elastic foundation. More-
over, for a methodology based on an iterative algorithm,
as proposed in this study, transfer matrix method is very
useful and easy to handle compared to FE formulation.
Theoretical developments of the methodology for detec-
tion and quantification of damage are presented first,
followed by detailed numerical studies to demonstrate
the efficacy of the proposed method.
2. Transfer Matrix Method for Obtaining
Modal Parameters
For computing plane flexural vibrations of a straight
beam using transfer matrix method, the beam section is
modelled by discrete uniform structural elements inter-
connected at the nodal points. Using the conventional
assumption of a mass-less beam, the inertia effects of the
beam element are dynamically represented by two
lumped masses at both ends of the element (as shown in
Figure 1).
Each individual beam is considered to be of individual
homogenous material property and geometry which can
be represented by area moment of inertia and Young’s
modulus of that particular element. Two displacements,
viz., vertical deflection () and rotation () and the cor-
Figure 1. Beam with concentrated masses.
X

M

V
Y
Figure 2. Sign convention for state array variables of beam
element.
Z
responding forces viz., shear force (V) and bending mo-
ment (M) are considered for describing the state array
variables at each section and the sign convention of the
state array variables is shown in Figure 2.
The equilibrium between sections i and i-1 of an ele-
ment will be maintained by
10
LR
ii
VV


 (1)
10
LRL
iiii
MM Vl


 (2)
where the superscript L and R stands for left and right
side of a section respectively.
Two more equations that are required for solving the
problem can be obtained from compatibility conditions
and the final equations can be expressed as
 
23
11
26
1
L
RR R ii
iiiii i
ii
ll
lM
EI EI
 
R
V


  (3)
 
2
11
21
L
RR ii
ii ii
ii
ll
R
M
V
EI EI



 
1
(4)
1
L
R
iiii
R
M
MlV



(5)
1
L
R
ii
VV

 (6)
and can be expressed in matrix form as,
L
i
M
V















=
23
2
1
126
01 2
00 1
00 01
R
i
i
ll
lEI EI
ll
M
EI EI
lV



















(7)
So, from Equation (7), the field matrix (Fi) connecting
with can be expressed as
L
i
ZR
i
Z1
1
L
R
iii
Z
Z F (8)
The point matrix (Pi) connecting with is
found by using continuity of deflection, slope and mo-
ment across the concentrated mass mi,
R
i
ZL
i
Z

S. SASMAL AND K. RAMANJANEYULU 170
Figure 3. Free-body diagram of mass mi.
R
L
ii

;
R
L
ii

 and
R
L
ii
M
M (9)
The vibrating mass, however, introduces the inertial
force which causes discontinuity in shear. The free-body
diagram shown in Figure 3 yields a relation from simple
equilibrium considerations as:
2RL
iii
VVm i

 (10)
(in formulation, a particular sign convention has been
followed)
Equations (9) and (10) can be expressed in matrix form
as:
12 14
32 34
nn
uu
nn
uu
R
i
M
V









=
2
1 000
0100
0010
001
L
ii
i
M
mV




















 
(11)
R
L
iii
Z
ZP (12)
By combining both field and point matrices, relation
between the state vectors of adjacent ends (i and i-1) can
be obtained as
1
R
R
iiii
Z
Z

PF (13)
2.1. Transfer Matrix for Frequency
Determinant
The transfer matrix method can be applied to solve more
complicated problems by considering a beam that is
made up of piecewise uniform mass-less elements, with
masses concentrated at discrete points. If a structural
element is made up of n segments (between the ends 0 to
n), relationship between the state vectors at the extreme
ends (0 and n) of the beam can be obtained as
nn-1144332211 0
FP FPFPFPFPF
nn
Z
Z

...........
0n
Z
ZU (14)
Equation (14) can be written in full, as
n
η
M
V








=
11 1213 14
21 2223 24
31 323334
41 4243 440
nnnn
nnnn
nnnn
nnnn
i
η
uuuu
uuuu
M
uuuu
V uuuu












where the coefficients to are functions of cir-
n
u11
n
u44
cular frequency

. Boundary conditions can be applied
to the equations formulated from Equation (15) to arrive
at the frequency determinant. For example, a beam (con-
sists of n segments) with simply supported ends can be-
solved as follows:
The boundary conditions at simply supported ends are
n

= 0, = 0,
n
M0

= 0, and = 0;
0
M
By substituting these boundary conditions in Equation
(15), the following relation can be obtained
12 01400
nn
uuV 

 (16a)
And,
32 034 00
nn
uuV 

 (16b)
where is element of ith row and jth column of the
transfer matrix which can be obtained by using Equation
(15) and superscript k denotes the number of segments.
The normal modes can be found for the system using the
following procedure.
k
ij
u
For a nontrivial solution of Equations (16a) and (16b),
the determinant of the coefficients must be zero, that is
12 14
32 34
nn
nn
uu
uu
=0 (17)
The same procedure can be followed for other bound-
ary conditions also. Since, the elements are func-
tions of the circular frequency
ij
u

, this determinant serves
to compute the natural circular frequencies. In view of
the fact that a beam which possesses n segments will
have n-1 discrete masses, the expansion of the frequency
determinant leads to an equation of n-1 degree in .
2

2.2. Numerical Procedure for Solution of
Frequency Equation
In the preceding section, the matrix multiplications have
been made by treating as a free parameter. After
applying the boundary conditions the resulting frequency
equations are solved for . For complicated systems,
the algebraic solution would become complicated and
furthermore, it would be very cumbersome to extract the
roots. In such cases, it is advantageous to replace alge-
braic solution with numerical computation. For system
with 'n' segments with simply supported ends, the fre-
quency determinant (as described in Equation 17) would
become
2

2




(15) 12 14
22 24
nn
nn
uu
uu
 =0 (18)
If the matrix multiplication is carried out algebraically,

S. SASMAL AND K. RAMANJANEYULU171
then the coefficients , , and and con-
sequently the frequency condition would be complicated
functions of . The procedure adopted in practice,
however, is to choose certain values for and com-
pute the corresponding values of the frequency determi-
nant
n
u12
n
u14
n
u22
n
u24
2

2

)(

. The value of the determinant is then
plotted against


, the zero values of occur at the
natural circular frequencies of the system. This proce-
dure has been adopted in the study for tracking of fre-
quencies.

3. Determination of Frequency of a
Structure Using Transfer Matrix Method
In this study, a computer program called FREQ has been
developed based on the formulation presented in the
preceding sections and the flow-chart of the program for
obtaining frequencies of a structure, is shown in Figure 4.
The formulations and the computer program have been
validated by comparing the results of this study with
those obtained using Finite Element Analysis (FEA).
Table 1 gives the comparison of frequencies obtained by
using transfer matrix method and FEA. From this table, it
can be seen that the results of this study are in good
agreement with those obtained using satndard FEA
package. For the validation study, a beam with 90
elements have been considered with Young's modulus
(E)= 25106 kN/m2, moment of inertia (I)= 0.001333 m4
and cross sectional area (A) = 0.1 m2.
As discussed in the preceding section, the determinant
for the whole beam after incorporating the boundary
conditions is computed for an assumed (initial) natural
frequency. Then, an iterative procedure has been carried
out by incrementing natural frequency to get the deter-
minant of the transfer matrix. The frequency for which
the determinant value is nearly zero, has been assigned
as the natural frequency of the beam. The variation of the
determinant of the transfer matrix for different modes of
the beam is shown in Figure 5. For clarity, the determi-
nant value () has been scaled down suitably after
reaching a particular frequency. For example, for first,
second and third natural frequencies, the determinant ()
of the transfer matrix is scaled down to 1/10th, 1/100th
Table 1. Comparison of frequency obtained using transfer
matrix method and FEA.
Modes Frequency () in Hz
First mode 5.648 (5.670)
Second mode 22.564 (22.557)
Third mode 50.478 (50.306)
Fourth mode 88.108 (88.352)
Note: Results obtained from FEA are presented in
brackets
Figure 4. Flow-chart of computer program (FREQ).
and 1/500th respectively. The frequencies corresponding
to zero values of the determinant () represent the natu-
ral frequencies () of the beam for different modes (as
shown in Figure 5).
The central philosophy of detection of damage of
beam like structure using transfer matrix formulation
presented here, is to determine the reduction in flexural
rigidity of one or more elements of the beam which
would signify the existence of damage in the structure. In
this context, question may arise that how far the frequen-
cies of a structure are influenced by the damage in a par-
ticular element(s), in other words, what is the change in
the determinant of transfer matrix with the change in
flexural rigidity in one or more elements of the beam. In
view of this, a study has been carried out to evaluate the
frequency determinant by changing the magnitude and
locations of the damaged element(s) to evaluate the in-
fluence of location and magnitude of damage on fre-
quency of a structure. It is noticed that the frequencies
corresponding to higher modes are influenced predomi-
nantly by change in flexural rigidity of one or more ele-
ments of the beam. For clarity, the changes in determi-
nant values for the first two frequencies are shown in
Figure 6. It is observed from the figure that by reducing
End
Start
Input - geometry and details for dynamic analysis
Idealization of structure into 2D beam
Discretization of the beam into no. of elements
Assume initial value of

for first fundamental mode
Formation of Field (F) and Point (P) Transfer Matrices
Calculation of Global Transfer Matrix for the beam
Determinant (

) of frequency matrix for the beam
incorporating the boundary conditions
If   0
yes
no

=

+0.1
Calculation of fundamental frequency of that mode
no If No. of modes 
required

=

+1
yes

S. SASMAL AND K. RAMANJANEYULU 172
-1500
-1000
-500
0
500
1000
0102030405060708090100110 120 130140 150160 170180 190200 210220 230240 250260 270280 290300310 32
0
Frequency (Hz)
Determinant value
First frequency
Second frequency
Third frequency
Fourth frequency









 



Figure 5. Variation of determinant of transfer matrix for
different modes.
-200
-150
-100
-50
0
50
100
150
200
1 2 3 4 5 67 8 910111213
Frequencies
Determinant value
EI=100 EI=200EI=300 EI=400
EI=500 EI=600EI=700 EI=800
EI=900 EI=1000 EI=2000 EI=3000
EI=4000 EI=5000 EI=6000 EI=7000
EI=8000 EI=9000 EI=10000

1
(a) For first fundamental frequency.
-1000
0
1000
2000
3000
4000
5000
6000
7000
13 14 1516 17 1819 20 2122 23 2425 26 27 2829 30 3132 33 34 3536 37 3839 40 41 42 43 44 45 46 4748 49 50
Fre
q
uencies
Determinant value
EI=100 EI=200 EI=300 EI=400
EI=500 EI=600 EI=700 EI=800
EI=900EI=1000 EI=2000 EI=3000
EI=4000 EI=5000 EI=6000 EI=7000
EI=8000 EI=9000 EI=10000

2
(b) For second fundamental frequency.
Figure 6. Variation of determinant with degree of damage
(EI in kNm2).
flexural rigidity of a particular element of the beam con-
sidered in this study, frequency of the second mode var-
ies over a wider range than that of the first mode. This
signifies that the shift in frequency of second mode due
to damage is more predominant than that in the first
mode frequency. It is also noted from the study that this
phenomenon is valid for next higher modes.
4. Results and Discussions
Though the transfer matrix technique can easily be ap-
plied to any type of structure with appropriate boundary
conditions, a beam like structure with simply supported
ends is considered in this study to demonstrate the effi-
cacy of the methodology and its accuracy. The material
N
ode number
Figure 7. A typical beam like structure with elements and
node numbers.
and sectional properties of the beam considered in this
study are same as that mentioned for validation study. It
is true that a finer division of a structure would lead to a
more precise result, but for demonstrating the methodol-
ogy proposed in this study, a beam like structure with 10
elements (as shown in Figure 7) has been considered for
better representation, faster computation and clarity. An-
other reason behind considering less number of elements
in this study is that for single-spread damage case,
coarser mesh can occupy maximum amount of damage
in minimum number of elements which would reduce the
computation time without sacrificing the efficiency.
4.1. Solution Procedure for Detection of Damage
Using Change in Frequencies.
The methodology proposed in this study, uses natural
frequency information obtained from the transfer matrix
formulations, for detection, quantification and localiza-
tion of damage. A beam with known location and mag-
nitude of damage has been analysed for extracting the
natural frequencies. The existence of orthogonal damage
in a beam structure can be simulated numerically via a
change in flexural rigidity (EI) in a particular beam ele-
ment. Such changes or reduction in flexural rigidity
would result in change or decrease in the natural fre-
quencies of the system. Through the measurement of the
system natural frequencies of the structure, the location
and magnitude of the damage can be determined. As-
suming that flexural rigidity of all the segments of the
system are known, the dynamics of the system can be
obtained by the numerical model described in the pre-
ceding section.
When damage has occurred in a certain beam segment,
it can be detected through the changes in the system
natural frequencies. For the system containing damage,
the iterative procedure starts with an assumption that the
damage is located at the first beam element. The corre-
sponding flexural rigidity of the element is adjusted until
the first natural frequency of the system is matched with
the measured one. The process is then continued with the
second segment of the structure and the first natural fre-
quency of the system is again matched by adjusting the
flexural rigidity of the second element. The process is
repeated for all the segments of the structure. The same
1 23456 7 8 9 1011
L
1
2
3
4
5
6
7
8
9
1
0
D
a
m
age
l
ocat
i
o
n
Element nu
m
b
e
r

S. SASMAL AND K. RAMANJANEYULU173
technique is followed for other modes which can be
measured through vibration testing. The location and
magnitude of the damage of the structure can be identi-
fied by the intersection of various rigidity-versus- dam-
aged beam element location curves. The intersection of
the curves obtained for different modes represent damage
locations and magnitudes (flexural rigidity) which
caused the changes in the system natural frequencies.
Flow-chart of the computer program developed in this
study based on the formulation described above for de-
tection and quantification of structural damage is shown
in Figure 8.
4.2. Case Study
For a numerical simulation, a beam is considered where
the geometric and material properties are same as that
mentioned for validation study. It is significant to men-
tion that, in this study, 1) single damage does not repre-
sent only one damage (one crack) in the entire structure
which is not practical in real structure too. As the formu-
lation states, an element in the structure can be chosen to
take a considerable length of the structure. The proposed
methodology would show the location and magnitude of
damage in an element considering all the damages oc-
curred in that particular element which can be used for
further discretisation, if required, to arrive at more par-
ticular locations. 2) It is also noticed that the most of the
reported methodologies for damage detection perform
well when degree of damage is very severe. But, in real
practice, when large damages are already included in the
structure, a sophisticated methodology for damage detec-
tion is not required, rather it can be located either by
visual observation or simple inspection techniques. So, in
this study, low levels of damages are considered to illus-
trate the methodology and to check its acceptability. 3)
For all the case studies presented here, frequencies cor-
responding to only first four modes are considered be-
cause more number of modes may not be available from
the field experiments. It is always a challenging problem
to detect and quantify damage from less number of
modes. Further, consideration of more number of modes
is computationally expensive too.
Three levels of damage in two different locations have
been studied separately, i.e, a beam with 10%, 20% and
30% damage in an element near support (3rd element as
shown in Figure 7) and near centre (5th element as shown
in Figure 7) respectively. These studies have been con-
sidered to examine the performance of the proposed
methodology because it is known that the change in fre-
quency with damage (reduction in flexural rigidity) of a
structure greatly depends on the degree and location of
damage.
Using the proposed methodology and computer pro-
gram developed based on the flow-chart shown in Figure
8, iterative study has been carried out for satisfying the
frequencies corresponding to different modes of a dam-
aged beam. Final flexural rigidities of each element
along the length of the beam are obtained from the com-
puter program and plotted for the cases mentioned above.
It is observed that the true location and magnitude of the
damage are identified by the intersection of the various
rigidity versus element location curves. Cases with
damage of 10% (remaining flexural rigidity of 29993
kNm2) in 3rd and 5th element are shown in Figure 9 and
Figure 10 respectively. It is observed from Figures 9
and 10 that intersections of curves for different modes
correctly indicate the damage locations (in 3rd and 5th
element) with a remaining flexural rigidity of 30000
kNm2.
Start
Input- geometric and details for dynamic analysis
Input- measured frequency and mode shapes of beam
Figure 8. Flow chart for detection and localisation of struc-
tural damage.
Flexural rigidity of ith element (EIi)=100 kNm2
with the other se
g
ments as undama
g
ed sections
Calculate frequency of jth mode (ji) from ‘FREQ’ for
the beam with assu
m
ed ri
g
idit
y
(
EI
i
)
for ith element
For no. of modes
available
(
j
)
= 1 to
m
For no. of element
(
i
)
= 1 to n
Increase rigidity
(
i
=
i
+
1)
If

ji

measured
fre
q
uenc
y
no
yes
yes
If element i  n
no
If modes j  m
no
yes
(
j
=
j
+
1)
Plot the rigidity versus element diagram for all the
modes available from ex
p
erimen
t
Intersection of results for different modes represent the prob-
able location of the structural damage and corresponding
ri
g
idit
y
value
(
EI
)
denotes the ma
g
nitude of dama
g
e
End

S. SASMAL AND K. RAMANJANEYULU 174
0
5000
10000
15000
20000
25000
30000
35000
123456789
Element nummer
Flexural rigidity (kNm
2)
10
First mode
Second mode
Third mode
Fourth mode
Figure 9. Flexural rigidity versus element diagram for 10%
damage in 3rd element.
0
5000
10000
15000
20000
25000
30000
35000
12345678910
Element nummer
Flexural rigidity (kNm
2
)
First mode
Second mode
Third mode
Fourth mode
Figure 10. Flexural rigidity versus element diagram for
10% damage in 5th element.
0
5000
10000
15000
20000
25000
30000
35000
12345678910
Element nummer
Flexural rigidity (kNm
2
)
First mode
Second mode
Third mode
Fourth mode
Figure 11. Flexural rigidity versus element diagram for
20% damage in 3rd element.
Similarly, Cases with damage of 20% (remaining flex-
ural rigidity of 26660 kNm2) and 30% (remaining flex-
ural rigidity of 23328 kNm2) in 3rd and 5th element are
shown in Figures 11-12 and Figure 13-14 respectively
which indicate damage in the correct elements with a
magnitude of 26500 kNm2 (as shown in Figure 11 and
Figure 12) and 23500 kNm2 (as shown in Figure 13 and
Figure 14), respectively.
It is important to note that the evaluated magnitudes of
damage are quite close to the actual values.
In these studies discussed above, known degrees and
locations of damages have been considered for validating
0
5000
10000
15000
20000
25000
30000
35000
12345678910
Element nummer
Flexural rigidity (kNm
2
)
First mode
Second mode
Third mode
Fourth mode
Figure 12. Flexural rigidity versus element diagram for
20% damage in 5th element.
0
5000
10000
15000
20000
25000
30000
35000
1234567891
Element nummer
Flexural rigidity (kNm
2)
0
First mode
Second mode
Third mode
Fourth mode
Figure 13. Flexural rigidity versus element diagram for
30% damage in 3rd element.
0
5000
10000
15000
20000
25000
30000
35000
1234567891
Element nummer
Flexural rigidity (kNm
2
)
0
First mode
Second mode
Third mode
Fourth mode
Figure 14. Flexural rigidity versus element diagram for
30% damage in 5th element.
the methodology for detection and localisation of dam-
age. It is found that the procedure is able to identify the
location and magnitude of damage. Hence, this proce-
dure can be adopted for detection and quantification of
damage of structures using measured frequencies of first
few modes. In this study, the problems are selected in
such a way that both strengths and limitations of the
proposed methodology can be examined. From the re-
sults shown in Figures 9-14, a few observations can be
made as: 1) frequency based methodology proposed in
this study can be used for localisation as well as quanti-
fication of damage, 2) since, the proposed methodology
is based on only frequency information, structures with
symmetrical boundary condition would always show two

S. SASMAL AND K. RAMANJANEYULU175
possible locations of damage, and, 3) it is desirable to
obtain the lowest measured frequency of a damaged
structure with maximum possible accuracy to get an im-
proved and more accurate estimation.
During the study, it is further observed that the pro-
posed methodology is able to provide information about
the state of damage and its location in a damaged struc-
ture, but the accuracy and reliability of the results (both
localisation and quantification) also depends on correct-
ness of information on the undamaged state. So, the
proposed methodology would perform satisfactorily with
a condition of availability of information (flexural rigid-
ity) in its undamaged state. Hence, the study is further
being extended to formulate a procedure which can be
used for identification of damage when information
about the undamaged state of a structure is not available,
and it is being explored to check the efficacy and the
suitable solutions (if any) for the proposed methodology
with various levels of noise in modal data.
5. Concluding Remarks
The present paper addresses the methodology for
detection, localisation and quantification of damage
based on the formulations made using transfer matrix
technique. First, the formulations and the computer
program have been developed for obtaining the vibration
characteristics of beam-like structures. The computer
program has been validated by comparing the results of
this study with those obtained using Finite Element
Analysis (FEA) package. The results of this study are in
good agreement with those obtained using standard FEA
package. From the existing studies, it is noted that dis-
placement mode shapes are sensitive to damage and
higher modes show predominant shift in mode shape
displacements due to damage in the structure. But, shift
in mode shape largely depends on the location of damage
and the mode considered and it is difficult to quantify
damage from mode shape information. Hence, a meth-
odology for detection, localisation and quantification of
damage in structures has been proposed based on change
in natural frequency obtained from transfer matrix tech-
nique. The existence of orthogonal damage in a beam
structure can be simulated numerically through change in
flexural rigidity (EI) in a particular beam element. For
the system containing damage, an iterative procedure has
been adopted by adjusting the flexural rigidity of the
element such that computed frequency matches with the
measured values. The location and magnitude of the
damage of the structure can be identified by the intersec-
tion of the various rigidity-versus-element location
curves. Studies have been presented by considering
single spread-damage cases with different degrees and
locations of damage to validate the accuracy, reliability
and to identify the possible limitation of the proposed
methodology. It is found that the proposed methodology
can localise and quantify damage in a structure with con-
siderable accuracy.
6. Acknowledgements
This paper is being published with the kind permission of
the Director, Structural Engineering Research Centre,
Chennai, India.
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