Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes

Tóm tắt Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes: ...)] {CL} . (21) Substituting (21) into (16) we obtain {zc(x)} = ( [G0(x)] + n ∑ j=1 [ G¯(x− ej) ] . [ χj ]) . {CL} = [GL (x,ω)] . {CL} , (22) Crack identification in multiple cracked beams made of functionally graded material. . . 111 where [GL (x,ω)] = [G0(x)] + n ∑ j=1 [ G¯(...e Nunber of measured points:200/span 10 % 20% 30% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 -4 SWT detail coefficients: Mode 2 L(m) A m p lit u d e Nunber of measured points:200/span 10 % 20% 30% (f) 116 Tran Van Lien, Ngo Trong Duc 11 ...nd third mode shapes if the SNR exceeds 75dB (Figs. 8d, 9d). a) b) c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x 10 -4 SWT of eigenmodes: 1 Depth Cr= 30% SNR= 75db L(m) D et ai l d iff er en ce dB4 0 0.2 0.4 0.6 0.8 1 -1 0 ...

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odes: 2 Depth Cr= 30% SNR= 80db
L(m)
D
e
ta
il 
d
if
fe
re
n
c
e
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
-4 SWT of eigenmodes: 2 Depth Cr= 30% SNR= 90db
L(m)
D
e
ta
il 
d
if
fe
re
n
c
e
dB4
(f)
 18 
g) 
h) 
i) 
Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 
crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db
L(m)
D
et
ai
l d
iff
er
en
ce
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db
L(m)
D
et
ai
l d
iff
er
en
ce
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db
L(m)
D
et
ai
l d
iff
er
en
ce
dB4
(g)
 18 
g) 
h) 
i) 
Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 
crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db
L(m)
D
e
ta
il 
d
iff
e
re
n
ce
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db
L(m)
D
e
ta
il 
d
if
fe
re
n
c
e
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db
L(m)
D
et
ai
l d
iff
er
en
ce
dB4
(h)
 18 
g) 
h) 
i) 
Fig. 8: Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1 
crack at 0.2m from the left node with the depth of 30% and noise level 75, 80 và 90dB 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 75db
L(m)
D
e
ta
il 
d
if
fe
re
n
c
e
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 80db
L(m)
D
e
ta
il 
d
if
fe
re
n
c
e
dB4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10
-4 SWT of eigenmodes: 3 Depth Cr= 30% SNR= 90db
L(m)
D
et
ai
l d
iff
er
en
ce
dB4
(i)
Fig. 7. Wavelet detail coefficients SWT of the first three mode shapes of FGM beam that has 1
crack at 0.2 m from the left node with the depth of 30% and noise level 75, 80 and 90 dB
shape if the SNR exceeds 80 dB (Figs. 7(b) and 8(b)), using the second and third mode
shapes if the SNR exceeds 75 dB (Figs. 7(d) and 8(d)).
Crack identification in multiple cracked beams made of functionally graded material. . . 121
19 
a) 
b) 
c) 
(a)
19 
a) 
b) 
c) (b)
19 
a) 
b 
c) 
(c)
20 
d) 
e) 
f) 
(d)
20 
d) 
e) 
f) (e)
20 
d) 
e) 
f) 
(f)
21 
g) 
h) 
i) 
Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM 
beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% 
and noise level 75, 80 and 90dB 
(g)
21 
g) 
h) 
i) 
Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM 
beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% 
and noise level 75, 80 and 90dB 
(h)
122 Tran Van Lien, Ngo Trong Duc
21 
g) 
h) 
i) 
Fig. 9: Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM 
beam that has 4 equivalent crack at 0.2m 0.4m 0.6m 0.8m from the left node with the depth of 30% 
and noise level 75, 80 and 90dB 
(i)
Fig. 8. Wavelet detail coefficients SWT of the first three mode shapes of the simple support FGM
beam that has 4 equivalent cracks at 0.2 m, 0.4 m, 0.6 m, 0.8 m from the left node with the depth
of 30% and noise level 75, 80 and 90 dB
4. CONCLUSIONS
In this paper, crack identification in a multiple cracked FGM beam by using SWT
of mode shapes and taking into account influence of Gaussian noise is addressed. Mode
shapes are obtained from the multiple cracked FGM beam element model using DSM and
spring model of cracks. Hence, SWT is applied for crack identification of the multiple
cracked FGM beam.
Numerical analysis was carried out to validate the proposed method and to investi-
gate effect of cracks, material properties and Gaussian noise on crack identification of a
multiple cracked FGM beam. The illustrating numerical results verify that the multiple
cracked FGM beam element model combined with the SWT can be reliably employed
for localization of cracks in beam-like structures with data contaminated with noise of
SNR from 75 dB. Moreover, crack position can be more clearly detected using only one
mode shape of the cracked beam. The investigated results show that proposed method
is efficient and realizable.
ACKNOWLEDGEMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.02-2017.301.
REFERENCES
[1] Z. H. Jin and R. C. Batra. Some basic fracture mechanics concepts in functionally graded
materials. Journal of the Mechanics and Physics of Solids, 44, (8), (1996), pp. 1221–1235.
https://doi.org/10.1016/0022-5096(96)00041-5.
[2] F. Erdogan and B. H. Wu. The surface crack problem for a plate with function-
ally graded properties. Journal of Applied Mechanics, 64, (3), (1997), pp. 449–456.
https://doi.org/10.1115/1.2788914.
Crack identification in multiple cracked beams made of functionally graded material. . . 123
[3] L. L. Ke, J. Yang, S. Kitipornchai, and Y. Xiang. Flexural vibration and elastic buckling of a
cracked Timoshenko beam made of functionally graded materials. Mechanics of Advanced Ma-
terials and Structures, 16, (6), (2009), pp. 488–502. https://doi.org/10.1080/15376490902781175.
[4] J. Yang, Y. Chen, Y. Xiang, and X. L. Jia. Free and forced vibration of cracked inhomogeneous
beams under an axial force and a moving load. Journal of Sound and Vibration, 312, (1-2),
(2008), pp. 166–181. https://doi.org/10.1016/j.jsv.2007.10.034.
[5] K. Aydin. Free vibration of functionally graded beams with arbitrary number
of surface cracks. European Journal of Mechanics-A/Solids, 42, (2013), pp. 112–124.
https://doi.org/10.1016/j.euromechsol.2013.05.002.
[6] J. Yang and Y. Chen. Free vibration and buckling analyses of functionally
graded beams with edge cracks. Composite Structures, 83, (1), (2008), pp. 48–60.
https://doi.org/10.1016/j.compstruct.2007.03.006.
[7] D. Wei, Y. Liu, and Z. Xiang. An analytical method for free vibration analysis of functionally
graded beams with edge cracks. Journal of Sound and Vibration, 331, (7), (2012), pp. 1686–1700.
https://doi.org/10.1016/j.jsv.2011.11.020.
[8] K. Sherafatnia, G. Farrahi, and S. A. Faghidian. Analytic approach to free vibration and buck-
ling analysis of functionally graded beams with edge cracks using four engineering beam
theories. International Journal of Engineering-Transactions C: Aspects, 27, (6), (2013), pp. 979–
990. https://doi.org/10.5829/idosi.ije.2014.27.06c.17.
[9] N. T. Khiem and D. T. Hung. A closed-form solution for free vibration of multiple cracked
Timoshenko beam and application. Vietnam Journal of Mechanics, 39, (4), (2017), pp. 315–328.
https://doi.org/10.15625/0866-7136/9641.
[10] N. T. Khiem, L. K. Toan, and N. T. L. Khue. Change in mode shape nodes of multiple
cracked bar: I. The theoretical study. Vietnam Journal of Mechanics, 35, (3), (2013), pp. 175–
188. https://doi.org/10.15625/0866-7136/35/3/2486.
[11] N. T. Khiem, L. K. Toan, and N. T. L. Khue. Change in mode shape nodes of multiple cracked
bar: II. The numerical analysis. Vietnam Journal of Mechanics, 35, (4), (2013), pp. 299–311.
https://doi.org/10.15625/0866-7136/35/4/2487.
[12] S. Kitipornchai, L. L. Ke, J. Yang, and Y. Xiang. Nonlinear vibration of edge cracked function-
ally graded Timoshenko beams. Journal of Sound and Vibration, 324, (3-5), (2009), pp. 962–982.
https://doi.org/10.1016/j.jsv.2009.02.023.
[13] Z. Yu and F. Chu. Identification of crack in functionally graded material beams using the p-
version of finite element method. Journal of Sound and Vibration, 325, (1-2), (2009), pp. 69–84.
https://doi.org/10.1016/j.jsv.2009.03.010.
[14] Sá. D. Akbasá. Free vibration characteristics of edge cracked functionally graded beams by
using finite element method. International Journal of Engineering Trends and Technology, 4, (10),
(2013), pp. 4590–4597.
[15] A. Banerjee, B. Panigrahi, and G. Pohit. Crack modelling and detection in Timoshenko
FGM beam under transverse vibration using frequency contour and response surface
model with GA. Nondestructive Testing and Evaluation, 31, (2), (2016), pp. 142–164.
https://doi.org/10.1080/10589759.2015.1071812.
[16] K. V. Nguyen. Crack detection of a beam-like bridge using 3D mode shapes. Vietnam Journal
of Mechanics, 36, (1), (2014), pp. 13–25. https://doi.org/10.15625/0866-7136/36/1/2965.
[17] H. Su and J. R. Banerjee. Development of dynamic stiffness method for free vibration of
functionally graded Timoshenko beams. Computers & Structures, 147, (2015), pp. 107–116.
https://doi.org/10.1016/j.compstruc.2014.10.001.
124 Tran Van Lien, Ngo Trong Duc
[18] T. V. Lien, N. T. Duc, and N. T. Khiem. Free vibration analysis of multiple cracked function-
ally graded Timoshenko beams. Latin American Journal of Solids and Structures, 14, (9), (2017),
pp. 1752–1766. https://doi.org/10.1590/1679-78253693.
[19] T. V. Lien, N. T. Duc, and N. T. Khiem. Mode shape analysis of multiple cracked functionally
graded Timoshenko beams. Latin American Journal of Solids and Structures, 14, (7), (2017),
pp. 1327–1344. https://doi.org/10.1590/1679-78253496.
[20] N. T. Khiem and T. V. Lien. The dynamic stiffness matrix method in forced vibration anal-
ysis of multiple-cracked beam. Journal of Sound and Vibration, 254, (3), (2002), pp. 541–555.
https://doi.org/10.1006/jsvi.2001.4109.
[21] T. V. Lien, N. T. Duc, and N. T. Khiem. Mode shape analysis of multiple cracked functionally
graded beam-like structures by using dynamic stiffness method. Vietnam Journal of Mechanics,
39, (3), (2017), pp. 215–228. https://doi.org/10.15625/0866-7136/8631.
[22] F. Nazari and M. H. Abolbashari. Double cracks identification in functionally graded beams
using artificial neural network. Journal of Solid Mechanics, 5, (1), (2013), pp. 14–21.
[23] N. T. Khiem and N. N. Huyen. A method for crack identification in functionally graded
Timoshenko beam. Nondestructive Testing and Evaluation, 32, (3), (2017), pp. 319–341.
https://doi.org/10.1080/10589759.2016.1226304.
[24] H. Sohn, C. R. Farrar, F. M. Hemez, and J. Czarnecki. A review of structural health monitoring
literature: 1996–2001. Los Alamos National Laboratory, USA, (2003).
[25] C. Surace and R. Ruotolo. Crack detection of a beam using the wavelet transform. In
Proceedings-Spie The International Society For Optical Engineering, (1994), pp. 1141–1147.
[26] K. M. Liew and Q. Wang. Application of wavelet theory for crack identifica-
tion in structures. Journal of Engineering Mechanics, 124, (2), (1998), pp. 152–157.
https://doi.org/10.1061/(asce)0733-9399(1998)124:2(152).
[27] Q. Wang and X. Deng. Damage detection with spatial wavelets. International Journal of Solids
and Structures, 36, (23), (1999), pp. 3443–3468. https://doi.org/10.1016/s0020-7683(98)00152-
8.
[28] E. Douka, S. Loutridis, and A. Trochidis. Crack identification in beams using wavelet
analysis. International Journal of Solids and Structures, 40, (13-14), (2003), pp. 3557–3569.
https://doi.org/10.1016/s0020-7683(03)00147-1.
[29] C. C. Chang and L. W. Chen. Detection of the location and size of cracks in the multiple
cracked beam by spatial wavelet based approach. Mechanical Systems and Signal Processing,
19, (1), (2005), pp. 139–155. https://doi.org/10.1016/j.ymssp.2003.11.001.
[30] W. Zhang, Z. Wang, and H. Ma. Crack identification in stepped cantilever beam combining
wavelet analysis with transform matrix. Acta Mechanica Solida Sinica, 22, (4), (2009), pp. 360–
368. https://doi.org/10.1016/s0894-9166(09)60285-8.
[31] S. Zhong and S. O. Oyadiji. Crack detection in simply supported beams without baseline
modal parameters by stationary wavelet transform. Mechanical Systems and Signal Processing,
21, (4), (2007), pp. 1853–1884. https://doi.org/10.1016/j.ymssp.2006.07.007.
[32] H. Gửkdag˘ and O. Kopmaz. A new damage detection approach for beam-type structures
based on the combination of continuous and discrete wavelet transforms. Journal of Sound
and Vibration, 324, (3-5), (2009), pp. 1158–1180. https://doi.org/10.1016/j.jsv.2009.02.030.
[33] T. V. Lien, N. T. Khiem, and T. A. Hao. Crack identification in frame structures using the
stationary wavelet transform of mode shapes. Jokull, 64, (6), (2014), pp. 251–262.
[34] J. C. Goswami and A. K. Chan. Fundamentals of wavelets: theory, algorithms, and applications.
John Wiley & Sons, (2011).
Crack identification in multiple cracked beams made of functionally graded material. . . 125
[35] M. Misiti, Y. Misiti, G. Oppenheim, and J. M. Poggi. Matlab wavelet toolbox TM 4 user’s guide.
The MathWorks, Inc. Natick, Massachusetts, (2009).
[36] A. V. Ovanesova and L. E. Suarez. Applications of wavelet transforms to dam-
age detection in frame structures. Engineering Structures, 26, (1), (2004), pp. 39–49.
https://doi.org/10.1016/j.engstruct.2003.08.009.
[37] X. H. Wang, R. S. H. Istepanian, and Y. H. Song. Microarray image enhancement by denois-
ing using stationary wavelet transform. IEEE Transactions on Nanobioscience, 2, (4), (2003),
pp. 184–189. https://doi.org/10.1109/tnb.2003.816225.
APPENDIX 1
Frequency equation of (3) is
η3 + aη2 + bη + c = 0.
In case of A12 = 0, coefficients of above equation are
a = ω2
[
I11
A33
+
I11 A22 + I22A11
A11A22
]
, b = ω4
[
I11 I22 − I212
A11A22
+
I11
A33
I11 A22 + I22A11
A11A22
]
−ω2 I11
A22
,
c = ω4
[
ω2 I11
A33
I11 I22 − I212
A11A22
− I
2
11
A11A22
]
.
Roots of cub algebraic equation are η1(ω), η2(ω), η3(ω)
η1 = −a/3+ u− b1/u, η2,3 = −a/3− (u− b1/u)/2± i
√
3(u + b1/u)/2,
where
u = (a1+
√
b31 + c
2
1− a3/27)1/3, a1 = ab/6− c/2, b1 = b/3− a2/9, c1 = a3/27− a1.
Constants in formula (6) are
λ1,4 = ±k1, λ2,5 = ±k2, λ3,6 = ±k3, k j = √ηj, j = 1, 2, 3,
αj =
ω2 I12
ω2 I11 + λ2j A11
, β j =
λj A33
(ω2 I11 + λ2j A33)
, j = 1, 2, . . . , 6.
APPENDIX 2
Differential matrix operators
- Simply supported (S): u(x, t) = M(x, t) = w(x, t) = 0
 24 
2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−=  
where 
1
3
1
2
11
3/132
1
3
11 27/;9/3/;2/6/;)27/( aacabbcabacbau −=−=−=−++= 
Constants in formula (6) are 
3,2,1,;;; 36,325,214,1 ===== jkkkk jj  
6,...,2,1;
)(
;
33
2
11
2
33
11
2
11
2
12
2
=
+
=
+
= j
AI
A
AI
I
j
j
j
j
j





 
A2. Differential matrix operators 
- Simply supported (S): 0),(),(),( === txwtxMtxu 
 










=
100
0
001
2212 xxS AAB 
- Pined (P): 0),(),(),( === txwtxMtxN 
 











−
=
100
0
0
2212
1211
xx
xx
P AA
AA
B 
- Clamped (C): 0),(),(),( === txwtxtxu  
 










=
100
010
001
CB 
- Free (F): 0),(),(),( === txQtxMtxN 
  










−

−
=
x
xx
xx
AA
AA
AA
3333
2212
1211
0
0
0
FB 
• Simply supported beam ends (SS):      SBBB L ==0 
• Pined ends (PP):      PBBB L ==0 
• Simple beam (SP):        PBBBB LS == ;0 
• Cantilevered beam (CF):        FC BBBB L == ;0 
• 
Clamped beam (CC):      CBBB L ==0 
[BS] =
 1 0 0A12∂x A22∂x 0
0 0 1
.
- Pined (P): N(x, t) = M(x, t) = w(x, t) = 0
126 Tran Van Lien, Ngo Trong Duc
 24 
2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−=  
where 
1
3
1
2
11
3/132
1
3
11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= 
Constants in formula (6) are 
3,2,1,;;; 36,325,214,1 ===== jkkkk jj  
6,...,2,1;
)(
;
33
2
11
2
33
11
2
11
2
12
2
=
+
=
+
= j
AI
A
AI
I
j
j
j
j
j





 
A2. Differential matrix operators 
- Simply supported (S): 0),(),(),( === txwtxMtxu 
 










=
100
0
001
2212 xxS AAB 
- Pined (P): 0),(),(),( === txwtxMtxN 
 











−
=
100
0
0
2212
1211
xx
xx
P AA
AA
B 
- Clamped (C): 0),(),(),( === txwtxtxu  
 










=
100
010
001
CB 
- Free (F): 0),(),(),( === txQtxMtxN 
  










−

−
=
x
xx
xx
AA
AA
AA
3333
2212
1211
0
0
0
FB 
• Simply supported beam ends (SS):      SBBB L ==0 
• Pined ends (PP):      PBBB L ==0 
• Simple beam (SP):        PBBBB LS == ;0 
• Cantilevered beam (CF):        FC BBBB L == ;0 
• 
Clamped beam (CC):      CBBB L ==0 
[BP] =
 A11∂x −A12∂x 0A12∂x A22∂x 0
0 0 1
.
- Clamped (C): u(x, t) = θ(x, t) = w(x, t) = 0
 24 
2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−=  
where 
1
3
1
2
11
3/132
1
3
11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= 
Constants in formula (6) are 
3,2,1,;;; 36,325,214,1 ===== jkkkk jj  
6,...,2,1;
)( 33
2
11
2
33
11
2
11
2
12
2
=
+
=
+
= j
AI
A
AI
I
j
j
j
j
j





 
A2. Differential matrix operators 
- Simply supported (S): 0),(),(),( === txwtxMtxu 
 










=
100
0
001
2212 xxS AAB 
- Pined (P): 0),(),(),( === txwtxMtxN 
 











−
=
100
0
0
2212
1211
xx
xx
P AA
AA
B 
- Clamped (C): 0),(),(),( === txwtxtxu  
 










=
100
010
001
CB 
- Free (F): 0),(),(),( === txQtxtxN 
  










−

−
=
x
xx
xx
AA
AA
AA
3333
2212
1211
0
0
0
FB 
• Simply supported beam ends (SS):      SBBB L ==0 
• Pined ends (PP):      PBBB L ==0 
• Simple beam (SP):        PBBBB LS == ;0 
• Cantilevered beam (CF):        FC BBBB L == ;0 
• 
Clamped beam (CC):      CBBB L ==0 
[BC] =
 1 0 00 1 0
0 0 1
.
- Free (F): N(x, t) = M(x, t) = Q(x, t) = 0
 24 
2/)/(32/)/(3/;/3/ 113,211 ubuiubuaubua +−−−=−+−=  
where 
1
3
1
2
11
3/132
1
3
11 27/;9/3/;2/6/;)27/( aacabbcabaacbau −=−=−=−++= 
Constants in formula (6) are 
3,2,1,;;; 36,325,214,1 ===== jkkkk jj  
6,...,2,1;
)(
;
33
2
11
2
33
11
2
1
2
12
2
=
+
=
+
= j
AI
A
AI
I
j
j
j
j
j





 
A2. Differential matrix operators 
- Simply supported (S): 0),(),(),( === txwtxMtxu 
  





=
100
0
001
2212 xxS AAB 
- Pined (P): 0),(),(),( === txwtxMtxN 
  






−
=
100
0
0
2212
1211
xx
xx
P AA
AA
B 
- Clamped (C): 0),(),(),( === txwtxtxu  
 










=
100
010
001
CB 
- Free (F): 0),(),(),( === txQtxMtxN 
  










−

−
=
x
xx
xx
AA
AA
AA
3333
2212
1211
0
0
0
FB 
• Simply supported bea ends (SS):      SBBB L ==0 
• Pined ends (P ):     PBB =0 
• Simple beam (SP):        PBBBB LS == ;0 
• Cantilevered beam (CF):        FC BBBB L == ;0 
• 
Clamped beam (CC):      CBBB L ==0 
[BF] =
 A11∂x −A12∂x 0A12∂x 2∂ 0
0 −A33 A33∂x
.
- Simply supported beam ends (SS): [B0] = [BL] = [BS].
- Pined ends (PP): [B0] = [BL] = [BP].
- Simple a (SP): [B0] = [ S] , [BL] = [BP].
- Cantilevered beam (CF): [B0] = [BC] , [BL] = [BF].
- Clamped beam (CC): [B0] = [BL] = [BC].

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