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Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation

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Abstract

In this paper, an analytical method is used to study the nonlinear primary resonance of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal stiffeners. The SSFG cylindrical shell is surrounded by linear and nonlinear elastic foundation and the effect of structural damping on the system response is also considered. The material properties of the shell and stiffeners are assumed to be continuously graded in the thickness direction. Three-parameter nonlinear elastic foundation model is consists of two-parameter linear elastic foundation (Winkler and Pasternak) and one hardening/softening cubic nonlinearity parameter. Based on the von Kármán nonlinear equations and the classical plate theory of shells, the strain–displacement relations are derived. The smeared stiffener technique is used to the model of the internal stiffeners. Using the Galerkin method, the partial differential equations of motion are discretized. The nonlinear primary resonance is analyzed by means of the multiple scales method. The effects of various geometrical characteristics, material parameters and elastic foundation coefficients are investigated on the nonlinear primary resonance.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

    Habib Ahmadi

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  1. Habib Ahmadi
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Appendix

Appendix

$$\begin{aligned} A & =J_{{11}}^{*}{m^4}{\pi ^4}+\left( {J_{{33}}^{*} - J_{{12}}^{*} - J_{{21}}^{*}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{22}}^{*}{n^4}{\lambda ^4} \\ B & =J_{{21}}^{{**}}{m^4}{\pi ^4}+\left( {J_{{11}}^{*}+J_{{22}}^{*} - 2J_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{12}}^{{**}}{n^4}{\lambda ^4} - \frac{{{L^2}}}{R}{m^2}{n^2} \\ {B^*} & =A_{{21}}^{*}{m^4}{\pi ^4}+\left( {A_{{11}}^{*}+A_{{22}}^{*} - 2J_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{12}}^{{**}}{n^4}{\lambda ^4} - \frac{{{L^2}}}{R}{m^2}{n^2} \\ D & =A_{{11}}^{{**}}{m^4}{\pi ^4}+\left( {A_{{12}}^{{**}}+A_{{21}}^{{**}}+4A_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+A_{{22}}^{{**}}{n^4}{\lambda ^4} \\ G & =\left( {\frac{{{n^4}{\lambda ^4}}}{{16J_{{11}}^{*}}}+\frac{{{m^4}{\pi ^4}}}{{16J_{{22}}^{*}}}} \right) \\ \lambda & =\frac{L}{R} \\ \end{aligned}$$

where

$$\begin{aligned} \Delta & ={J_{11}}{J_{22}} - {J_{12}}{J_{21}},~~J_{{22}}^{*}=\frac{{{J_{22}}}}{\Delta },~~J_{{12}}^{*}=\frac{{{J_{12}}}}{\Delta },\,\,\,J_{{11}}^{*}=\frac{{{J_{11}}}}{\Delta },~~~J_{{21}}^{*}=\frac{{{J_{21}}}}{\Delta },~~~J_{{33}}^{*}=\frac{1}{{{J_{33}}}},~~~J_{{36}}^{*}=\frac{{{J_{36}}}}{{{J_{33}}}} \\ J_{{11}}^{{**}} & =J_{{22}}^{*}{J_{14}} - J_{{12}}^{*}{J_{24}},~~~J_{{12}}^{{**}}=J_{{22}}^{*}{J_{15}} - J_{{12}}^{*}{J_{25}},\,\,\,J_{{21}}^{{**}}=J_{{11}}^{*}{J_{24}} - J_{{21}}^{*}{J_{14}},~~~J_{{22}}^{{**}}=J_{{11}}^{*}{J_{25}} - J_{{21}}^{*}{J_{15}} \\ A_{{11}}^{*} & =J_{{22}}^{*}{J_{14}} - J_{{21}}^{*}{J_{15}},~~~A_{{21}}^{*}=J_{{11}}^{*}{J_{15}} - J_{{12}}^{*}{J_{14}},\,\,\,A_{{12}}^{*}=J_{{22}}^{*}{J_{24}} - J_{{21}}^{*}{J_{25}},~~~A_{{22}}^{*}=J_{{11}}^{*}{J_{25}} - J_{{12}}^{*}{J_{24}} \\ A_{{11}}^{{**}} & =J_{{11}}^{{**}}{J_{14}} - J_{{21}}^{{**}}{J_{15}} - {J_{41}},\,\,\,A_{{12}}^{{**}}=J_{{12}}^{{**}}{J_{14}} - J_{{22}}^{{**}}{J_{15}} - {J_{42}},\,\,\,A_{{21}}^{{**}}=J_{{11}}^{{**}}{J_{24}} - J_{{21}}^{{**}}{J_{25}} - {J_{51}} \\ A_{{22}}^{{**}} & =J_{{12}}^{{**}}{J_{24}} - J_{{22}}^{{**}}{J_{25}} - {J_{52}},\,\,\,A_{{36}}^{{**}}=J_{{36}}^{{**}}{J_{36}} - {J_{63}} \\ \end{aligned}$$

where

$$\begin{aligned} {J_{11}} & =\frac{{{E_1}}}{{1 - {\nu ^2}}}+{Z_1}{E_{1s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,~{J_{12}}=\frac{{{E_1}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{1s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{14}} & =\frac{{{E_2}}}{{1 - {\nu ^2}}}+{Z_1}{E_{2s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,{J_{15}}=\frac{{{E_2}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{2s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{21}} & =\frac{{{E_1}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{1s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{22}}=\frac{{{E_1}}}{{1 - {\nu ^2}}}+{Z_2}{E_{1s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{24}} & =\frac{{{E_2}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{2s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{25}}=\frac{{{E_2}}}{{1 - {\nu ^2}}}+{Z_2}{E_{2s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{33}} & =\frac{{{E_1}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{1s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right),\,\,\,{J_{36}}=\frac{{{E_2}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{2s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right) \\ {J_{41}} & =\frac{{{E_3}}}{{1 - {\nu ^2}}}+{Z_1}{E_{3s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,{J_{42}}=\frac{{{E_3}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{3s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{51}} & =\frac{{{E_3}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{3s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{55}}=\frac{{{E_3}}}{{1 - {\nu ^2}}}+{Z_2}{E_{3s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{63}} & =\frac{{{E_3}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{3s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right) \\ \end{aligned}$$

where

$$\begin{aligned} {E_1} & =\mathop \int \limits_{{ - h/2}}^{{h/2}} {E_{{\text{sh}}}}\left( z \right){\text{d}}z=\left( {{E_m}+\frac{{{E_c} - {E_m}}}{{k+1}}} \right)h,\,\,\,{E_2}=\mathop \int \limits_{{ - h/2}}^{{h/2}} z{E_{{\text{sh}}}}\left( z \right){\text{d}}z=\frac{{\left( {{E_c} - {E_m}} \right)k{h^2}}}{{2\left( {k+1} \right)\left( {k+2} \right)}} \\ {E_3} & =\mathop \int \limits_{{ - h/2}}^{{h/2}} {z^2}{E_{{\text{sh}}}}\left( z \right){\text{d}}z=\left[ {\frac{{{E_m}}}{{12}}} \right.~\left. {+\left( {{E_c} - {E_m}} \right)\left( {\frac{1}{{k+3}}+\frac{1}{{k+2}}+\frac{1}{{4k+4}}} \right)} \right]{h^3} \\ {E_{1s}} & =\mathop \int \limits_{{h/2}}^{{h/2+{h_s}}} {E_s}\left( z \right){\text{d}}z=\left( {{E_c}+\frac{{{E_m} - {E_c}}}{{{k_s}+1}}} \right){h_s} \\ {E_{2s}} & =\mathop \int \limits_{{h/2}}^{{h/2+{h_s}}} z{E_s}\left( z \right)dz \\ & =\frac{{{E_c}}}{2}~h{h_s}\left( {\frac{{{h_s}}}{h}+1} \right)+\left( {{E_m} - {E_c}} \right)~h{h_s}\left( {\frac{1}{{{k_s}+2}}\frac{{{h_s}}}{h}+\frac{1}{{2{k_s}+2}}} \right) \\ {E_{3s}} & =\mathop \int \limits_{{\frac{h}{2}}}^{{\frac{h}{2}+{h_s}}} {z^2}{E_s}\left( z \right){\text{d}}z \\ & =\frac{{{E_c}}}{3}h_{s}^{3}\left( {\frac{3}{4}\frac{{{h^2}}}{{h_{s}^{2}}}+\frac{3}{2}\frac{h}{{{h_s}}}+1} \right)+\left( {{E_m} - {E_c}} \right)h_{s}^{3}\left[ {\frac{1}{{{k_s}+3}}+\frac{1}{{{k_s}+2}}\frac{h}{{{h_s}}}+\frac{1}{{4\left( {{k_s}+1} \right)}}\frac{{{h^2}}}{{h_{s}^{2}}}} \right] \\ \end{aligned}$$

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Ahmadi, H. Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation. Engineering with Computers 35, 1491–1505 (2019). https://doi.org/10.1007/s00366-018-0679-2

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