Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures

cracked beam determined as solution of the equation 𝐹0(𝜆𝑘
0) = 0 (see Eq. 
(12)), 𝜙0(𝑥, 𝜆𝑘
0), 𝜙0
″(𝑥, 𝜆𝑘
0 ) are mode shape and curvature of intact beam determined as 
𝜙0(𝑥, 𝜆𝑘
0 ) = 𝐶𝑘
0[𝐿20
(𝑝1,𝑞1)(1, 𝜆𝑘
0)𝐿10(𝑥, 𝜆𝑘
0) − 𝐿10
(𝑝1,𝑞1)(1, 𝜆𝑘
0)𝐿20(𝑥, 𝜆𝑘
0)]; 
𝜙0
″(𝑥, 𝜆𝑘
0) = 𝐶𝑘
0[𝐿20
(𝑝1,𝑞1)(1, 𝜆𝑘
0 )𝐿10
″ (𝑥, 𝜆𝑘
0 ) − 𝐿10
(𝑝1,𝑞1)(1, 𝜆𝑘
0)𝐿20
″ (𝑥, 𝜆𝑘
0)] 
 (a) (b) (c) 
Fig. 1. Deviation of three mode shapes (a- first, b- second, c- third mode) due to 9 cracks at 0.1-0.9 of 
depths 10%; 30%; 50%; 60%. 
 (a) (b) (c) 
Fig. 2. Deviation of exact curvature of three modes (a- first, b- second, c- third mode) due to 9 cracks at 
0.1-0.9 with depth 10%;30%;50%;60%. 
The deviations (19) calculated for first three modes of a cantilever beam with 9 cracks (from 0.1 to 0.9) 
along the normalized beam length (horizontal axis) are shown in Fig. 1 and Fig. 2 for mode shapes and 
curvatures respectively. Every box in the Figures demonstrates four curves corresponding to various depth 
of the cracks from 10% to 60% beam thickness that show monotony increasing of the deviation magnitude 
with the crack depth. 
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1 -0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 0.2 0.4 0.6 0.8 1
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 0.2 0.4 0.6 0.8 1
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-20
-15
-10
-5
0
5
10
15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-60
-40
-20
0
20
40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(c) Third mode
Fig. 2. Deviation of exact curvature of thr e modes due to 9 cracks at 0.1–0.9
with depth 10%, 30%, 50%, 60%
Note that variation of mode shape due to cracks is visibly observed at the crack
positions (see Fig. 1), but magnitude of the variation is very small (within 10%). So that
cracks would be difficult to detect by mode shape measured with error of 10%. Deviation
of exact curvature caused by cracks is significantly magnified (see Fig. 2) in comparison
with mode shape variation. Nevertheless, the change in modal curvature is rather dis-
tributed than localized at the cracks positions so that cracks are also not easily localized
from measurement of curvature even if base-line data are available. This encourages us
to find another more efficient indicator for the crack detection, one of that is considered
in subsequent section.
Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures 129
4. SENSITIVITY OF LAPLACIAN APPROXIMATE CURVATURE DUE TO CRACK
Assume that mode shape and curvature of a beam have been measured at the mesh
(x0, x1, . . . , xn+1) with resolution h and x0 = 0, xn+1 = 1, i.e. there are given two sets of
data: φ
(
xj
)
, φ′′
(
xj
)
, j = 0, . . . , n+ 1. Let’s consider three subsequent points (xj−1, xj, xj+1)
of the mesh and suppose that each of the segments (xj−1, xj), (xj, xj+1)may contains only
one crack at position ej−1 ∈ (xj−1, xj), ej ∈ (xj, xj+1), respectively.
Taylor’s expansion of the function φ(x) at the points ej−1, ej yields
φ
(
xj+1 − 0
)
= φ
(
ej + 0
)
+ φ′
(
ej + 0
) (
xj+1 − ej
)
+ (1/2)φ′′
(
ej + 0
) (
xj+1 − ej
)2
+ . . . ,
φ(xj + 0) = φ(ej − 0) + φ′(ej − 0)(xj − ej) + (1/2)φ′′(ej − 0)(xj − ej)2 + . . . ,
φ(xj − 0) = φ(ej−1 + 0) + φ′(ej−1 + 0)(xj − ej−1) + (1/2)φ′′(ej−1 + 0)(xj − ej−1)2 + . . . ,
φ(xj−1 + 0) = φ(ej−1 − 0) + φ′(ej−1 − 0)(xj−1 − ej) + (1/2)φ′′(ej−1 − 0)(xj−1 − ej)2 + . . .
(20)
Using the expressions (20) with neglected terms of order higher 2 gives
φ
(
xj+1
)− 2φ (xj)+ φ (xj−1) = φ′′ (xj) h2 + φ′′ (ej) αj + φ′′ (ej−1) αj−1,
with
αj = γj
(
xj+1 − ej
)
+ h
(
x¯j − ej
)
, αj−1 = γj−1
(
ej−1 − xj−1
)
+ h
(
ej−1 − x¯j−1
)
,
x¯j = (xj+1 + xj)/2, x¯j−1 = (xj + xj−1)/2.
(21)
Recalling the notations introduced for approximate curvature one gets finally
φ̂′′
(
xj
)− φ′′ (xj) = β jφ′′ (xj) , (22)
where
β j =
φ̂′′
(
xj
)
φ′′
(
xj
) − 1 = φ′′ (ej) αj + φ′′ (ej−1) αj−1
φ′′
(
xj
)
h2
' φ
′′ (ej) γj + φ′′ (ej−1) γj−1
2φ′′
(
xj
)
h
+O
(
h2
)
.
(23)
In case of no crack surrounding the mesh point xj, one has got φˆ′′
(
xj
)− φ′′ (xj) =
O
(
h2
)
, that implies negligible difference between approximate and exact curvatures at
an intact section, i.e.,
φˆ′′0
(
xj
)− φ′′0 (xj) = O (h2) . (24)
On the other hand, if both the crack locations coincide with xj, i.e., ej−1 = ej = xj,
γj−1 = γj, Eq. (22) gives
φˆ′′
(
xj
)− φ′′ (xj) = γjφ′′ (xj) /h. (25)
The latter equation shows that miscalculation of the Laplacian curvature at a crack
position depends on the crack magnitude, value of curvature at the crack and resolu-
tion step. Namely, the miscalculation gets to be increasing with reduction of the step h
and grow with the crack magnitude γj. Also, crack appeared at the node of curvature
(where curvature vanishes) makes no effect on the mode shape, curvature including the
approximate one.
In general, Eqs. (22), (24) allow one to obtain
∆φ̂′′
(
xj
)
= φ̂′′
(
xj
)− φ̂′′0 (xj) = ∆φ′′ (xj)+ β jφ′′ (xj) , (26)
130 Nguyen Tien Khiem
where
∆φ′′
(
xj
)
= φ′′
(
xj
)− φ′′0 (xj) . (27)
It can be seen from Eq. (26) that the miscalculation of the approximate curvature
increases its sensitivity to crack in comparison with exact curvature. For illustration of
the fact, deviation of the Laplacian curvature due to multiple cracks is calculated by using
expression (16) for three lowest modes of cantilever beam and results are demonstrated
in Fig. 3.
node of curvature (where curvature vanishes) makes no effect on the mode shape, curvature including the 
approximate one. 
In general, Equations (22), (24) allow one to obtain 
𝛥�̂�″(𝑥𝑗) = �̂�
″(𝑥𝑗) − �̂�0
″(𝑥𝑗) = 𝛥𝜙
″(𝑥𝑗) + 𝛽𝑗𝜙
″(𝑥𝑗), 
(26) 
where 
𝛥𝜙″(𝑥𝑗) = 𝜙
″(𝑥𝑗) − 𝜙0
″(𝑥𝑗). 
It can be seen from Eq. (26) that the miscalculation of the approximate curvature increases its sensitivity to 
crack in comparison with exact curvature. For illustration of the fact, deviation of the Laplacian curvature 
due to multiple cracks is calculated by using expression (16) for three lowest modes of cantilever beam and 
results are demonstrated in Fig. 3. 
Fig. 3. Deviation of approximate curvature of first three modes due to 9 cracks at 0.1-0.9 with equal 
depth 10%; 30%; 50%; 60% 
Graphs shown in Fig. 3 demonstrate strong sensitivity of approximate curvature to either magnitude or 
position of cracks that confirms theoretically once more the usefulness of the approximate curvature in 
crack localization for beam that was only numerically acknowledged in a number of previous studies. 
V. CONCLUDING REMARKS 
The main results of this study can be summarized as follow: 
1. An expression for exact mode shapes and mode shape curvatures have been obtained for multiple 
cracked beams that provides an efficient tool for analysis and identification of the beam structures. 
2. Using the obtained expression, it was shown that mode shape curvature is really more sensitive to 
cracks than the mode shape itself, however, the exact curvature sensitivity to crack is much less than 
that of approximate curvature calculated by the finite difference approximation. 
3. The paradox can be explained by the fact that sensitivity of the approximate curvature to crack is 
magnified by its miscalculation, that is also strongly depended upon crack magnitude and resolution 
step. 
4. Finally, the approximate Laplacian curvature would be a useful indicator for multiple-crack detection, 
if the base-line mode shape has been measured with sufficient accuracy. 
5. The effect of noise in measurement of mode shape on the sensitivity of the approximate curvature to 
crack is not yet considered in the present paper, it would be a topic for further study of the author. 
-5
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
-150
-100
-50
0
50
100
150
0 0.2 0.4 0.6 0.8 1
-400
-300
-200
-100
0
100
200
300
400
0 0.2 0.4 0.6 0.8 1
(a)
node of curvature (where curvature vanishes) makes no effect on the mode shape, curvature including the 
approximate one. 
In general, Equations (22), (24) allow one to obtain 
𝛥�̂�″(𝑥𝑗) = �̂�
″(𝑥𝑗) − �̂�0
″(𝑥𝑗) = 𝛥𝜙
″(𝑥𝑗) + 𝛽𝑗𝜙
″(𝑥𝑗), 
(26) 
where 
𝛥𝜙″(𝑥𝑗) 𝜙
″(𝑥𝑗) 𝜙0
″(𝑥𝑗). 
It can be seen from Eq. (26) that the miscalculation of the approxi ate curvature increases its sensitivity to 
crack in comparison with exact curvature. For illustration f t e fact, e iati f t e a lacia c r ature 
due to multiple cracks is calculated by using expression (1 ) f r t r l st s f til r and 
results are demonstrated in Fig. 3. 
Fig. 3. Deviation of a proximate curvature of first three es e t crac s at . - . it e al 
depth 10 ; 30 ; 50 ; 60 
Graphs shown in Fig. 3 demonstrate strong sensitivity of approxi ate curvature to either agnitude or 
position of cracks that confirms theoretically once ore the usefulness of the approxi ate curvature in 
crack localization for beam that was only numerically acknowledged in a number of previous studies. 
V. CONCLUDING RE ARKS 
The main results of this study can be summarized as follow: 
1. An expression for exact mode shapes and mode shape curvatures have been obtained for multiple 
cracked beams that provides an efficient tool for analysis and identification of the beam structures. 
2. Using the obtained expression, it was shown that mode shape curvature is really more sensitive to 
cracks than the mode shape itself, however, the exact curvature sensitivity to crack is much less than 
that of approximate curvature calculated by the finite difference approximation. 
3. The paradox can be explained by the fact that sensitivity of the approximate curvature to crack is 
magnified by its miscalculation, that is also strongly depended upon crack magnitude and resolution 
step. 
4. Finally, the approximate Laplacian curvature would be a useful indicator for multiple-crack detection, 
if the base-line mode shape has been measured with sufficient accuracy. 
5. The effect of noise in measurement of mode shape on the sensitivity of the approximate curvature to 
crack is not yet considered in the present paper, it would be a topic for further study of the author. 
-5
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
-150
-1 0
-50
0
50
1 0
150
0 0.2 0.4 0.6 0.8 1
-400
-
-
- . 0.8 1
(b)
 t r r t r is s) a es effect on the ode shape, curvature including the 
i t . 
 l, ti ( ), ( ) ll e t tai 
𝛥𝜙″ 𝑥𝑗 �̂�
″(𝑥𝑗) �̂�0
″(𝑥𝑗) 𝛥𝜙
″(𝑥𝑗) 𝛽𝑗𝜙
″(𝑥𝑗), 
(26) 
𝛥𝜙″(𝑥𝑗) = 𝜙
″(𝑥𝑗) − 𝜙0
″(𝑥𝑗). 
 . ( ) t t t is l lati f t e approximate curvature increases it iti it t 
 i it t r t r . r ill strati of the fact, deviation of the a l i t 
 lti l i l l t si re si ( 6) for three lowest odes of ca til 
 tr t i i . . 
i . . i ti f r i t r t re f first t ree modes due to 9 cracks at . - . it e al 
t %; %; 0%; 60% 
 i i . str t str se siti it of approximate curvature to either agnitude or 
 t t fir t r ti ll ce m re the usefulness of the approxi ate curvature in 
 li ti r t t s l ericall ac no ledged in a nu ber of previous studies. 
. I S 
 i lt f t i t s arize as follo : 
. i f r t s s e s ape curvatures have been obtained for ultiple 
 t t r i ffi i t t l f r a al sis and identification of the bea structures. 
. i t t i r i , it s s t at ode shape curvature is rea ly ore sensitive to 
 t t its lf, er, t e e act curvature sensitivity to crack is uch less than 
t i t r t r l l t t e fi ite difference approxi ation. 
. l i t f ct t at se sitivity of the approxi ate curvature to crack is 
i i it i l l ti , t t is ls str l depended upon crack agnitude and resolution 
. 
. ll , t r i t l i r t re l e a useful indicator for ultiple-crack detection, 
i t li s s re it s fficient accuracy. 
. t i i r t f e s a e the sensitivity of the approxi ate curvature to 
 i t t i r i t r s t a er, it ld be a topic for further study of the author. 
-
. . 0.6 0.8 1
-150
-100
-50
0
50
100
150
0 0.2 0.4 0.6 0.8 1
-400
-3 0
-2 0
-1 0
0
1 0
2 0
3 0
4 0
0 0.2 0.4 0.6 .
(c)
Fig. 3. Deviation of approximate curvature of first three modes due to 9 cracks at 0.1–0.9
wi h equal depth 10%, 30%, 50%, 60%
Gr phs shown in Fig. 3 demo strate strong sensitivity of approximate c rvature to
either magnitude or position of cracks that confirms theoretically once more the useful-
ness of the approximate curvature in crack localization for beam that was only numeri-
cally acknowledged in a number of previous studies.
5. CONCLUDING REMARKS
The main results of th s study can be sum arized as follow:
- An expr ssi n for xact mode shap s and mode shape curvatu es have been ob-
tained for multiple cracked beams that provides an efficient tool for analysis and identi-
fication of the beam structures.
- Using the obtained expression, it was shown that mode shape curvature is really
more sensitive to cracks than the mode shape itself, however, the exact curvature sen-
sitivity to crack is much less than that of approximate curvature calculated by the finite
difference ap r tion.
- The parad x can be explained by the fact that sensitivit of the approximate cur-
vature to crack is magnified by its iscalculation, that is also strongly depended upon
crack magnitude and resolu ion step.
- Finally, the approximate Laplacian curvature would be a useful indicator for
multiple-crack detection, if the base-line mode shape has been measured with sufficient
accuracy.
Mode shape curvature of multiple cracked beam and its use for crack identification in beam-like structures 131
- The effect of noise in measurement of mode shape on the sensitivity of the approxi-
mate curvature to crack is not yet considered in the present paper, it would be a topic for
further study of the author.
ACKNOWLEDGEMENT
The author is thankful to Vietnam Academy of Science and Technology for its sup-
port under grant of ID: NVCC03.02/20-20.
REFERENCES
[1] S. W. Doebling, C. R. Farrar, M. B. Prime, and D. W. Shevitz. Damage identification and health
monitoring of structural and mechanical systems from changes in their vibration characteris-
tics: a literature review. Technical report, No. LA-13070-MS, Los Alamos National Lab., NM
(United States), (1996).
[2] S. W. Doebling, C. R. Farrar, and M. B. Prime. A summary review of vibration-based
damage identification methods. Shock and Vibration Digest, 30, (2), (1998), pp. 91–105.
https://doi.org/10.1177/058310249803000201.
[3] E. P. Carden and P. Fanning. Vibration based condition monitoring: a review. Structural
Health Monitoring, 3, (4), (2004), pp. 355–377. https://doi.org/10.1177/1475921704047500.
[4] W. Fan and P. Qiao. Vibration-based damage identification methods: a review
and comparative study. Structural Health Monitoring, 10, (1), (2011), pp. 83–111.
https://doi.org/10.1177/1475921710365419.
[5] O. S. Salawu. Detection of structural damage through changes in frequency: a review. Engi-
neering Structures, 19, (9), (1997), pp. 718–723. https://doi.org/10.1016/s0141-0296(96)00149-
6.
[6] P. F. Rizos, N. Aspragathos, and A. D. Dimarogonas. Identification of crack location and
magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 138,
(3), (1990), pp. 381–388. https://doi.org/10.1016/0022-460x(90)90593-o.
[7] J.-T. Kim, Y.-S. Ryu, H.-M. Cho, and N. Stubbs. Damage identification in beam-type struc-
tures: frequency-based method vs mode-shape-based method. Engineering Structures, 25, (1),
(2003), pp. 57–67. https://doi.org/10.1016/s0141-0296(02)00118-9.
[8] K. R. P. Babu, B. R. Kumar, K. L. Narayana, and K. M. Rao. Multiple crack detection in beams
from the differences in curvature mode shapes. ARPN Journal of Engineering and Applied Sci-
ences, 10, (4), (2015).
[9] E. Sazonov and P. Klinkhachorn. Optimal spatial sampling interval for damage detection
by curvature or strain energy mode shapes. Journal of Sound and Vibration, 285, (4-5), (2005),
pp. 783–801. https://doi.org/10.1016/j.jsv.2004.08.021.
[10] M. Cao, M. Radzien´ski, W. Xu, and W. Ostachowicz. Identification of multiple damage in
beams based on robust curvature mode shapes. Mechanical Systems and Signal Processing, 46,
(2), (2014), pp. 468–480. https://doi.org/10.1016/j.ymssp.2014.01.004.
[11] D. Dessi and G. Camerlengo. Damage identification techniques via modal curvature analy-
sis: overview and comparison. Mechanical Systems and Signal Processing, 52, (2015), pp. 181–
205. https://doi.org/10.1016/j.ymssp.2014.05.031.
[12] J. Ciambella and F. Vestroni. The use of modal curvatures for damage localiza-
tion in beam-type structures. Journal of Sound and Vibration, 340, (2015), pp. 126–137.
https://doi.org/10.1016/j.jsv.2014.11.037.
132 Nguyen Tien Khiem
[13] G. Raju and L. Ramesh. Crack detection in structural beams by using curvature mode shapes.
IJIRST–International Journal for Innovative Research in Science & Technology, 3, (2), (2016),
pp. 282–289.
[14] A. C. Altunıs¸ık, F. Y. Okur, S. Karaca, and V. Kahya. Vibration-based damage de-
tection in beam structures with multiple cracks: modal curvature vs. modal flex-
ibility methods. Nondestructive Testing and Evaluation, 34, (1), (2019), pp. 33–53.
https://doi.org/10.1080/10589759.2018.1518445.
[15] A. K. Pandey, M. Biswas, and M. M. Samman. Damage detection from changes in
curvature mode shapes. Journal of Sound and Vibration, 145, (2), (1991), pp. 321–332.
https://doi.org/10.1016/0022-460x(91)90595-b.
[16] M. M. A. Wahab and G. De Roeck. Damage detection in bridges using modal curvatures:
application to a real damage scenario. Journal of Sound and Vibration, 226, (2), (1999), pp. 217–
235. https://doi.org/10.1006/jsvi.1999.2295.
[17] C. P. Ratcliffe. Damage detection using a modified Laplacian operator on
mode shape data. Journal of Sound and Vibration, 204, (3), (1997), pp. 505–517.
https://doi.org/10.1006/jsvi.1997.0961.
[18] C. P. Ratcliffe. A frequency and curvature based experimental method for locating
damage in structures. Journal of Vibration and Acoustics, 122, (3), (2000), pp. 324–329.
https://doi.org/10.1115/1.1303121.
[19] M. Cao and P. Qiao. Novel Laplacian scheme and multiresolution modal curvatures for struc-
tural damage identification. Mechanical Systems and Signal Processing, 23, (4), (2009), pp. 1223–
1242. https://doi.org/10.1016/j.ymssp.2008.10.001.
[20] M. Chandrashekhar and R. Ganguli. Structural damage detection using modal cur-
vature and fuzzy logic. Structural Health Monitoring, 8, (4), (2009), pp. 267–282.
https://doi.org/10.1177/1475921708102088.
[21] M. M. R. Taha, A. Noureldin, J. L. Lucero, and T. J. Baca. Wavelet transform for structural
health monitoring: a compendium of uses and features. Structural Health Monitoring, 5, (3),
(2006), pp. 267–295. https://doi.org/10.1177/1475921706067741.
[22] T. V. Lien and N. T. Duc. Crack identification in multiple cracked beams made of functionally
graded material by using stationary wavelet transform of mode shapes. Vietnam Journal of
Mechanics, 41, (2), (2019), pp. 105–126. https://doi.org/10.15625/0866-7136/12835.
[23] N. G. Jaiswal and D. W. Pande. Sensitizing the mode shapes of beam towards damage detec-
tion using curvature and wavelet transform. Int. J. Sci. Technol. Res., 4, (4), (2015), pp. 266–272.
[24] N. T. Khiem and H. T. Tran. A procedure for multiple crack identification in beam-like struc-
tures from natural vibration mode. Journal of Vibration and Control, 20, (9), (2014), pp. 1417–
1427.
[25] T. G. Chondros, A. D. Dimarogonas, and J. Yao. A continuous cracked beam
vibration theory. Journal of Sound and Vibration, 215, (1), (1998), pp. 17–34.
https://doi.org/10.1006/jsvi.1998.1640.