# 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 kw2.5×107 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 kw2.5×107 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 kw2.5×107 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=1/0b1/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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