Dynamic analysis of fg stepped truncated conical shells surrounded by pasternak elastic foundations

Tóm tắt Dynamic analysis of fg stepped truncated conical shells surrounded by pasternak elastic foundations: ...nd Nxθ are the in-plane force resultants, Mx, Mθ and Mxθ are moment resultants, Qx, Qθ are transverse shear force resultants. The shear correction factor f is computed such that the strain energy due to transverse shear stresses in Eq. (10) are equals to the strain energy due to the true transver...631 0.1648 1.04 1× 105 0.1399 0.1418 1.39 0.1823 0.1836 0.70 0.1653 0.1669 0.96 2.5× 105 0.1426 0.1439 0.94 0.1856 0.1867 0.59 0.1684 0.1700 0.96 5× 105 0.1469 0.1491 1.53 0.1910 0.1930 1.02 0.1733 0.1752 1.11 Tabs. 2 and 3 presented the variations of the dimensionless fundamental natural frequ...GM2, FGM4 and FGM3 respectively. Six different values of kw (0, 2.5×10 4, 5×106, 2.5×107, 5×108, 109 N/m3) are taken for the study and results are illustrated in Fig.6. It is easy to remark that when kw2.5×107 N/m3 the effects of Winkler foundation stiffness on natural frequency are very s...

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stepped truncated conical shells
5.2.2. Influences of elastic foundations
It is necessary to examine the effects of different types of elastic foundations on the
free vibration of the FG stepped truncated conical shells. Consider now the above men-
tioned four-stepped conical shell surrounded by a Winkler foundation with various val-
ues of foundation stiffness kw. The parameters of the shell are as follows: h1 : h2 : h3 :
h4 = 1 : 2 : 1 : 2, h1 = 0.01 m, L1 : L2 : L3 : L4 = 1 : 1 : 1 : 1, R1 = 0.5 m; R2 = 1
m, α = 20◦, material properties at steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3,
respectively.
Six different values of kw (0, 2.5× 104, 5× 106, 2.5× 107, 5× 108, 109 N/m3) are taken
for the study and results are illustrated in Fig. 6. It is easy to remark that when kw ≤
2.5× 107 N/m3 the effects of Winkler foundation stiffness on natural frequency are very
small. When the stiffness of the Winkler foundation kw ≥ 5 × 108 N/m3, the natural
frequency of the shell increases as kw increases and then the effect of kw on the natural
frequency is obvious.
The effect of Pasternak foundations has been investigated in the next test case. The
same structure resting on a Pasternak foundation with kw = 5× 106 N/m3 and various
values of shear stiffness are chosen: kp = 102, 2.5 × 104, 5 × 106, 2.5 × 107, 108 (N/m).
Fig. 7 presents the variation of natural frequencies of the studied structure with respect
Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 147
to different values of the shear stiffness kp. It is observed from this figure that natural
frequencies increase rapidly as kp > 5× 106 N/m. When m increases, the influence of
Pasternak foundation on natural frequency becomes larger. With kp ≤ 2.5× 104 N/m,
Pasternak foundations have almost no effects on the natural frequencies of the shell.
 Dynamic analysis of FG stepped truncated conical shells with various properties and surrounded by 
Pasternak elastic foundations 13 
be seen that except the first three modes, the augmentation of the stepped thickness leads to the raise 
of natural frequencies of all other circumferential modes (m). In addition, the effect of the thickness of 
segments on the first mode is minimal. 
Fig.5. Effect of stepped thickness on the vibration of four-FG stepped truncated conical shells 
5.2.2 Influences of elastic foundations 
It is necessary to examine the effects of different types of elastic foundations on the free vibration 
of the FG stepped truncated conical shells. Consider now the above mentioned four-stepped conical 
shell surrounded by a Winkler foundation with various values of foundation stiffness kw with the 
parameters of the shell are as follows: h1:h2:h3:h4=1:2:1:2, 1=0.01m, L1:L2:L3:L4=1:1:1:1, R1=0.5m; 
R2=1m, α=20
o, m erial properties at steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3 
respectively. 
Six different values of kw (0, 2.5×10
4, 5×106, 2.5×107, 5×108, 109 N/m3) are taken for the study 
and results are illustrated in Fig.6. It is easy to remark that when kw2.5×107 N/m3 the effects of 
Winkler foundation stiffness on natural frequency are very small. When the stiffness of the Winkler 
foundation kw≥5×108 N/m3, the natural frequency of the shell increases as kw increases and then the 
effect of kw on the natural frequency is obvious. 
Fig.6. Influence of Winkler foundations on natural 
frequencies of four-FGMI(a=1/b=0.5/c=4/p=2) stepped 
truncated conical shell with F-C boundary 
conditions 
Fig.7. Influence of Pasternak foundations on natural 
frequencies of four-FGMI(a=1/b=0.5/c=4/p=2) 
 stepped truncated conical shell with F-C boundary 
conditions 
Fig. 6. Influence of Winkler founda-
tions on natural frequencies of four-
FGMI(a=1/b=0.5/c=4/p=2) stepped truncated
conical shell with F-C boundary conditions
 Dynamic analysis of FG stepped truncated conical shells with various properties and surrounded by 
Pasternak elastic foundations 13 
be seen that except the first three modes, the augmentation of the stepped thickness leads to the raise 
of natural frequencies of all other circumferential modes (m). In addition, the effect of the thickness of 
segments on the first mode is minimal. 
Fig.5. Effect of stepped thickness on the vibration of four-FG stepped truncated conical shells 
5.2.2 Influences of elastic f undations 
It is necessary to examine the effects of different types of elastic f undations on the free vibration 
of the FG stepped truncated conic l shells. Consider now the above mentioned four-stepped conical 
shell s rrounded by Winkler foundation with various values of foundation stiffness kw with the 
parameters of the shell are as follows: h1:h2:h3:h4=1:2:1:2, h1=0.01m, L1:L2:L3:L4=1:1:1:1, R1=0.5m; 
R2=1m, α=20
o, material properties at steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3 
respectively. 
Six different values of kw (0, 2.5×10
4, 5×106, 2.5×107, 5×108, 109 N/m3) are taken for the study 
and results are illustrated in Fig.6. It is easy to remark that when kw2.5×107 N/m3 the effects of 
Winkler foundation stiffness on natural frequency are very small. When the stiffness of the Winkler 
foundation kw≥5×108 N/m3, the natural frequency of the shell increases as kw increases and then the 
effect of kw on the natural frequency is obvious. 
Fig.6. Influence of Winkler foundations on natural 
frequencies of four-FGMI(a=1/b=0.5/c=4/p=2) stepped 
truncated conical shell with F-C boundary 
conditions 
Fig.7. Influence of Pasternak foundations on natural 
frequencies of four-FGMI(a=1/b=0.5/c=4/p=2) 
 stepped truncated conical shell with F-C boundary 
conditions 
Fig. 7. Influence of Pasternak founda-
tions on natural frequencies of four-
FGMI(a=1/b=0.5/c=4/p=2) stepped truncated
conical shell with F-C boundary conditions
14 
Effect of Pasternak foundations has been investigated in the next test case. The same structure 
rests on a Pasternak foundation with kw = 5×106 N/m3 a d vari us values of shear stiffness are chosen: 
kp = 10
2, 2.5×104, 5×106, 2.5×107, 108 (N/m). Fig.7 present the varia ion of natural frequencies of the 
studied structure with respect to shear stiffness kp. It is observed from this figure that natural 
frequencies increase rapidly as kp > 5×10
6 N/m. When m increases, the influence of Pasternak 
foundation on natural frequency becomes larger. With kp  2.5×10
4 N/m, Pasternak foundations have 
almost no effects on the natural frequencies of the shell. 
Fig.8. Influences of both Winkler stiffness and Pasternak stiffness to natural frequencies of four- 
FGMI(a=1/b=0.5/c=1/p=4) stepped truncated conical shell with S-S boundary conditions 
Next, effects of both Winkler stiffness and Pasternak stiffness on natural frequencies of four-FG 
stepped truncated conical shells will be studied. The parameters of the shell are as follows: 
h1:h2:h3:h4=1:2:3:4, h1=0.01m, L1:L2:L3:L4=1:1:1:1, R1=0.5m; R2=1m, α=20o, material properties at 
steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3 respectively. The values of Winkler stiffness and 
Pasternak stiffness are kw = 10
-2, 102, 104, 106, 107, 5.107, 108, 2.5.108, 5.108, 109, 2.5×109, 5×109, 1010, 
5×1010, 1011, 5×1011, 1012 N/m3 and kp = 0, 10
6, 2.5×106, 5×106, 107 N/m, respectively. From Fig.8, it 
can be seen that the effect of Winkler stiffness and Pasternak stiffness on natural frequencies is 
important only on a certain range (kw from 10
7 to 1011 N/m3, kp from 5×10
6 to 108 N/m). When kw 
reaches to the limit value kw = 10
12 N/m3, Pasternak stiffness values have less effect on natural 
frequencies. 
5.2.3 Influence of the power-law p and various values of the parameter b 
In Fig. 9 the first four frequencies of four-step functionally graded conical shell (F-C) versus the 
power-law index p for two power-law distributions and for various values of the parameter b(b is 
contained in the interval [0,1]) are presented. The parameters of the shell are as follows: 
h1:h2:h3:h4=1:2:3:4, h1=0.01m, L1:L2:L3:L4=1:1:1:1, R1=0.5m; R2=1m, α=30
o, FGMI (a=1/0b1/c=3/p),
material properties at steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3 respectively. As can be 
seen from Fig. 9, natural frequencies of FGM shells often present an intermediate value between the 
natural frequencies of the limit cases of homogeneous shells of zirconia p = 0 and of aluminum p=, 
as expected. However, natural frequencies sometimes exceed limit cases, this fact can depend on 
various parameters, such as the geometry of the shell, the boundary conditions, the power-law 
distribution profile, etc. 
Fig. 8. nfluences of both Winkler stiffness and Pasternak stiffness to natural frequencies of four-
FGMI(a=1/b=0.5/c=1/p=4) stepped truncated conical shell with S-S boundary conditions
Next, effects of both Winkler stiffness and Pasternak stiffness on natural frequencies
of four-FG stepped truncated conical shells will be studied and illustrated in Fig. 8. The
parameters of the shell are as follows: h1 : h2 : h3 : h4 = 1 : 2 : 3 : 4, h1 = 0.01 m,
L1 : L2 : L3 : L4 = 1 : 1 : 1 : 1, R1 = 0.5 m; R2 = 1 m, α = 20◦, material properties at
steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3, respectively. Th values of Winkler
148 Le Quang Vinh, Nguyen Manh Cuong
stiffness and Pasternak stiffness are kw = 10−2, 102, 104, 106, 107, 5× 107, 108, 2.5× 108, 5×
108, 109, 2.5× 109, 5× 109, 1010, 5× 1010, 1011, 5× 1011, 1012 N/m3 and kp = 0, 106, 2.5×
106, 5× 106, 107 N/m, respectively. From Fig. 8, it can be seen that the effect of Winkler
stiffness and Pasternak stiffness on natural frequencies is important only on a certain
range (kw from 107 to 1011 N/m3, kp from 5 × 106 to 108 N/m). When kw reaches to
the limit value kw = 1012 N/m3, Pasternak stiffness values have less effect on natural
frequencies.
5.2.3. Influence of the power-law p and various values of the parameter b
In Fig. 9 the variation of first four frequencies of four-step functionally graded con-
ical shell (F-C) versus the power-law index p for two power-law distributions and for
various values of the parameter b (b is contained in the interval [0, 1]) are presented. The
parameters of the shell are as follows: h1 : h2 : h3 : h4 = 1 : 2 : 3 : 4, h1 = 0.01 m,
L1 : L2 : L3 : L4 = 1 : 1 : 1 : 1, R1 = 0.5 m; R2 = 1 m, α = 30◦, FGMI(a=1/0≤b≤1/c=3/p),
material properties at steps 1, 2, 3, 4 are FGM1, FGM2, FGM4 and FGM3, respectively. As
can be seen from Fig. 9, natural frequencies of FGM shells often present an intermediateDynamic analysis of FG stepped truncated conical shells with various properties and surrounded by 
Pasternak elastic foundations 15 
Fig.9. First four frequencies of four-FG stepped truncated conical shell (F-C) versus the power-law exponent p 
for various values of the parameter b 
6. CONCLUSION
This research has succeded in constructing a Continuous Element model for Functional Graded 
stepped truncated conical shells made of various materials and surrounded by Winkler and Pasternak 
elastic foundations. The effect of the Pasternak elastic foundation and of Function Graded Material has 
been well integrated into the presented element. Very good agreements are noticed between the results 
obtained by our approach and those of other methods. Various numerical results have confirmed that 
Continuous Element model is accurate and economies the storage capacity of computers by using a 
minimum meshing. The effects of various parameters on vibration behavior of the stepped shell are 
also investigated. From the above results, it can be concluded that: 
1. The ratio thickness-to-radius has larger effect on natural frequencies when m increases (m > 1).
2. The stiffness parameters of the elastic foundation have a significant effect on the vibration of the FG
stepped truncated conical shells. As the stiffness parameters of the elastic foundation are greater, the 
frequencies are higher. 
3. For the FG stepped truncated conical shells surrounded by elastic foundation, the effect of Winkler
stiffness and Pasternak stiffness on natural frequency is noticeable in a certain range. When the 
Winkler stiffness reaches a limited value (as kw = 10
12 N/m3), the influence of shearing layer elastic 
stiffness parameter in natural frequency is hardly recognized. 
Fig. 9. First four frequencies of four-FG stepped truncated conical shell (F-C) versus the power-
law exponent p for various values of the parameter b
Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 149
value between the natural frequencies of the limit cases of homogeneous shells of zirco-
nia p = 0 and of aluminum p = ∞, as expected. However, natural frequencies sometimes
exceed limit cases, this fact can depend on various parameters, such as the geometry of
the shell, the boundary conditions, the power-law distribution profile, etc.
6. CONCLUSIONS
This research has succeed in constructing a Continuous Element model for Func-
tional Graded stepped truncated conical shells made of various materials and surrounded
by Winkler and Pasternak elastic foundations. The effect of the Pasternak elastic foun-
dation and of Function Graded Material have been well integrated into the presented
element. Good agreements are noticed between the results obtained by our approach
and those of other methods. Numerical results have confirmed that Continuous Element
model is accurate and economies the storage capacity of computers by using a minimum
meshing. The effects of various parameters on vibration behavior of the stepped shell are
also investigated. From the above results, it can be concluded that:
1. The ratio thickness-to-radius has a larger effect on natural frequencies when m
increases (m > 1).
2. The stiffness parameters of the elastic foundation have a significant effect on the vi-
bration of the FG stepped truncated conical shells. As the stiffness parameters are greater,
the frequencies are higher.
3. For the FG stepped truncated conical shells surrounded by elastic foundation, the
effect of Winkler stiffness and Pasternak stiffness on natural frequency is noticeable in a
certain range. When the Winkler stiffness reaches a limited value (as kw = 1012 N/m3),
the influence of shearing layer elastic stiffness parameter in natural frequency is hardly
recognized.
The developed continuous element model with its powerful assembling procedure
can be expanded to study more complex shell structures such as: joined cylindrical-
conical shells, combined cylindrical-conical shell and annular plates, ring-stiffened shells
and those structures surrounded by elastic foundations and fluid.
ACKNOWLEDGEMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under Grant number: 107.02-2018.07.
REFERENCES
[1] P. L. Pasternak. On a new method of analysis of an elastic foundation by means of two foun-
dation constants. Gos. Izd. Lit. po Stroit I Arkh. Moscow, USSR, (1954). (in Russian).
[2] A. D. Kerr. Elastic and viscoelastic foundation models. Journal of Applied Mechanics, 31, (3),
(1964), pp. 491–498. https://doi.org/10.1115/1.3629667.
[3] A. H. Sofiyev and N. Kuruoglu. Vibration analysis of FGM truncated and complete conical
shells resting on elastic foundations under various boundary conditions. Journal of Engineering
Mathematics, 77, (1), (2012), pp. 131–145. https://doi.org/10.1007/s10665-012-9535-3.
150 Le Quang Vinh, Nguyen Manh Cuong
[4] A. H. Sofiyev and E. Schnack. The vibration analysis of FGM truncated conical shells resting
on two-parameter elastic foundations. Mechanics of Advanced Materials and Structures, 19, (4),
(2012), pp. 241–249. https://doi.org/10.1080/15376494.2011.642934.
[5] H. L. K. Dung, Dao Van and N. T. Nga. On the stability of functionally graded truncated con-
ical shells reinforced by functionally graded stiffeners and surrounded by an elastic medium.
Composite Structures, 108, (2014), pp. 77–90. https://doi.org/10.1016/j.compstruct.2013.09.002.
[6] K. Xie, M. Chen, and Z. Li. An analytic method for free and forced vibration analysis of
stepped conical shells with arbitrary boundary conditions. Thin-Walled Structures, 111, (2017),
pp. 126–137. https://doi.org/10.1016/j.tws.2016.11.017.
[7] Y. Qu, Y. Chen, Y. Chen, X. Long, H. Hua, and G. Meng. A domain decomposition method
for vibration analysis of conical shells with uniform and stepped thickness. Journal of Vibration
and Acoustics, 135, (1), (2013). https://doi.org/10.1115/1.4006753.
[8] L. Q. Vinh, N. M. Cuong, and L. T. B. Nam. Dynamic analysis of stepped composite conical
shells via continuous element method. In 2nd National Conference on Mechanical Engineering
and Automation, Hanoi, Vietnam, (2016), pp. 338–344.
[9] L. T. B. Nam, N. M. Cuong, T. I. Tran, and L. Q. Vinh. Dynamic analysis of stepped
composite cylindrical shells surrounded by Pasternak elastic foundations based on the
continuous element method. Vietnam Journal of Mechanics, 40, (2), (2018), pp. 105–119.
https://doi.org/10.15625/0866-7136/9832.
Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations 151
APPENDIX
Matrix [A (ω)]10×10:
Axm =

A11 A12 A13 A14 A15 A16 A17 A18 A19 A110
A21 A22 A23 A24 A25 A26 A27 A28 A29 A210
A31 A32 A33 A34 A35 A36 A37 A38 A39 A310
A41 A42 A43 A44 A45 A46 A47 A48 A49 A410
A51 A52 A53 A54 A55 A56 A57 A58 A59 A510
A61 A62 A63 A64 A65 A66 A67 A68 A69 A610
A71 A72 A73 A74 A75 A76 A77 A78 A79 A710
A81 A82 A83 A84 A85 A86 A87 A88 A89 A810
A91 A92 A93 A94 A95 A96 A97 A98 A99 A910
A101 A102 A103 A104 A105 A106 A107 A108 A109 A1010

,
A11 = c4 sin α , A12 = mc4, A13 = c4 cos α, A14 = c5 sin α,
A15 = mc5 , A16 =
D11
c1
, A17 = 0 , A18 = 0 A19 = −B11c1 , A110 = 0 ,
A21 =
m
R(x)
, A22 =
sin α
R(x)
, A23 = 0 , A24 = 0 , A25 = 0 , A26 = 0 ,
A27 = −D66c10 , A28 = 0 , A29 = 0 , A210 = −
B66
c10
,
A31 = 0, A32 = 0, A33 = 0, A34 = −1, A35 = 0, A36 = 0,
A37 = 0, A38 =
1
f F55
, A39 = 0, A310 = 0,
A41 = c2 sin α , A42 = mc2 , A43 = c2 cos α , A44 = c3 sin α , A45 = mc3 ,
A46 = −B11c1 , A47 = 0 , A48 = 0 , A49 =
A11
c1
, A410 = 0,
A51 = 0 , A52 = 0 , A53 = 0 , A54 =
m
R(x)
, A55 =
sin α
R(x)
, A56 = 0 ,
A57 =
B66
c10
, A58 = 0 , A59 = 0 , A510 = −A66c10 ,
A61 = c6 sin α− I0ω2 , A62 = mc6 sin α , A63 = c6 sin α cos α ,
A64 = c7 sin2 α− I1ω2 , A65 = mc7 sin α , A66 = −
(
c4 +
1
R(x)
)
sin α ,
A67 = − mR(x) , A68 = 0 , A69 = −c2 sin α , A610 = 0 ,
A71 = mc6 sin α , A72 = m2c6 +
f F44 cos α
R(x)2
− I0ω2 , A73 = m cos α
(
c6 +
f F44
R(x)2
)
,
A74 = mc7 sin α , A75 = m2c7 − f F44 cos αR(x)2 − I1ω
2 , A76 = −mc4 ,
152 Le Quang Vinh, Nguyen Manh Cuong
A77 = −2 sin αR(x) , A78 = 0 , A79 = −mc2 , A710 = 0 ,
A81 = c13
(
c1 sin α+
A11
R(x)2
cos α+ kpc2 sin α
)
,
A82 = mc13
(
f F44
R2
cos α+ c11 +
A11
R(x)2
cos α+ kpc2
)
,
A83 = c13
(
m2 f F44
R(x)2
+ c11 cos α+ kpc2 cos α− I0ω2 + kw +
m2kp
R(x)2
)
,
A84 = c13
(
c12 sin α+
B22
R(x)2
cos α+ kp
sin α
R(x)
+ kpc3 sin α
)
,
A85 = mc13
(
− f F44
R(x)
+ c12 +
B22
R(x)2
cos α+ kpc3
)
,
A86 = c13
(
A12
R(x)
D11
c1
cos α− B12
R(x)
B11
c1
cos α− kp B11c1
)
, A87 = 0 ,
A88 = − sin αR(x) , A89 = c13
(
− A12
R(x)
B11
c1
cos α+
B12
R(x)
A11
c1
cos α+ kp
A11
c1
)
, A810 = 0,
A91 = 2c8 sin2 α− I1ω2 , A92 = 2mc8 sin α , A93 = 2c8 sin α cos α ,
A94 = 2c9 sin2 α− I2ω2 , A95 = 2mc9 sin α , A96 = −2c5 sin α , A97 = 0 ,
A98 = 1 , A99 = −
(
c3 +
1
R(x)
)
2 sin α , A910 = − mR(x) ,
A101 = mc8 sin α , A102 = m2c8 − f F44 cos αR(x) − I1ω
2 , A103 = m
(
c8 cos α− f F44R(x)
)
,
A104 = mc9 sin α , A105 = m2c9 + f F44 − I2ω2 , A106 = −mc5 , A107 = 0 ,
A108 = 0 , A109 = −mc3 , A1010 = −2 sin αR(x) .

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