Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions

Abstract

This paper presents an analytical approach to analyze the free vibration of stiffened cylindrical shells made of porous functionally graded (FGP) materials. The governing equations are derived by using Hamilton’s principle based on the first-order shear deformation theory (FSDT) in conjunction with Lekhnitsky smeared technique. Two types of porosity distributions including symmetric and non-symmetric are considered. Applying the Galerkin method and the axial displacement field functions, the natural frequencies of the cylindrical shell under different boundary conditions are determined. The accuracy of the present formulation is made by comparing the obtained results with those available in published reports. The influences of material parameters such as porosity coefficient, porosity distribution types, and shell geometrical parameters on the natural frequencies are investigated and discussed.

Appendix A

$$ \begin{array}{*{20}l} {\chi _{{11}} = \bar{A}_{{11}} \frac{{\partial ^{2} }}{{\partial x^{2} }} + \bar{A}_{{66}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }};\;\;\chi _{{12}} = \frac{{\bar{A}_{{12}} + \bar{A}_{{66}} }}{R}\frac{{\partial ^{2} }}{{\partial x\partial \theta }};\;\;\chi _{{13}} = \bar{A}_{{12}} \frac{\partial }{{R\partial x}};\;\;\chi _{{14}} = \bar{B}_{{11}} \frac{{\partial ^{2} }}{{\partial x^{2} }} + \bar{B}_{{66}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }};} \hfill \\ {\chi _{{15}} = \frac{{\bar{B}_{{12}} + \bar{B}_{{66}} }}{R}\frac{{\partial ^{2} }}{{\partial x\partial \theta }};} \hfill \\ {\chi _{{21}} = \chi _{{12}} ;\;\;\chi _{{22}} = \bar{A}_{{66}} \frac{{\partial ^{2} }}{{\partial x^{2} }} + \bar{A}_{{22}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }} - k_{s} \frac{{\bar{A}_{{44}} }}{{R^{2} }};\;\;\chi _{{23}} = \left( {\bar{A}_{{22}} + k_{s} \bar{A}_{{44}} } \right)\frac{\partial }{{R^{2} \partial \theta }};} \hfill \\ {\chi _{{24}} = \left( {\bar{B}_{{12}} + \bar{B}_{{66}} } \right)\frac{{\partial ^{2} }}{{R\partial x\partial \theta }};\;\;\chi _{{25}} = \bar{B}_{{66}} \frac{{\partial ^{2} }}{{\partial x^{2} }} + \bar{B}_{{22}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }} + k_{s} \frac{{\bar{A}_{{44}} }}{R};} \hfill \\ {\chi _{{31}} = - \chi _{{13}} ;\;\;\chi _{{32}} = - \chi _{{23}} ;} \hfill \\ {\chi _{{33}} = k_{s} \bar{A}_{{55}} \frac{{\partial ^{2} }}{{\partial x^{2} }} + \frac{{k_{s} \bar{A}_{{44}} }}{{R^{2} }}\frac{{\partial ^{2} }}{{\partial \theta ^{2} }} - \frac{{\bar{A}_{{22}} }}{{R^{2} }};\;\;\chi _{{34}} = \left( {k_{s} \bar{A}_{{55}} - \frac{{\bar{B}_{{12}} }}{R}} \right)\frac{\partial }{{\partial x}};\;\;\chi _{{35}} = \left( {k_{s} \bar{A}_{{44}} - \frac{{\bar{B}_{{22}} }}{R}} \right)\frac{\partial }{{R\partial \theta }};} \hfill \\ {\chi _{{41}} = \chi _{{14}} ;\;\;\chi _{{42}} = \chi _{{24}} ;\;\;\chi _{{43}} = - \chi _{{34}} ;\;\;\chi _{{44}} = \bar{D}_{{11}} \frac{{\partial ^{2} }}{{\partial x^{2} }} - k_{s} \bar{A}_{{55}} + \bar{D}_{{66}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }};\;\;\chi _{{45}} = \frac{{\bar{D}_{{12}} + \bar{D}_{{66}} }}{R}\frac{{\partial ^{2} }}{{\partial x\partial \theta }};} \hfill \\ {\chi _{{51}} = \chi _{{15}} ;\;\;\chi _{{52}} = \chi _{{25}} ;\;\;\chi _{{53}} = - \chi _{{35}} ;\;\;\chi _{{54}} = - \chi _{{45}} ;\;\;\chi _{{55}} = \bar{D}_{{66}} \frac{{\partial ^{2} }}{{\partial x^{2} }} - k_{s} \bar{A}_{{44}} + \bar{D}_{{22}} \frac{{\partial ^{2} }}{{R^{2} \partial \theta ^{2} }}} \hfill \\ \end{array} $$