Stability and free vibration analysis of tapered sandwich columns with functionally graded core and flexible connections

Abstract

In this study, the buckling and free vibration behavior of tapered functionally graded material (FGM) sandwich columns is explored. The connections are considered to be semi-rigid. The core material is functionally graded along the beam depth according to the simple power law form. Euler–Bernoulli beam theory and the Ritz method will be employed to derive the governing equations. Legendre polynomials are chosen as auxiliary functions. After reducing the order of Euler’s buckling equation, an Emden–Fowler differential equation will be obtained. To reach a closed-form solution, the flexural rigidity of the column will be approximated with an exponential function by enforcing least-squares method. Non-dimensional natural frequencies and critical buckling loads will be presented for various cross-sectional types. The effects of FGM power, taper ratio, and spring rigidities on the critical buckling loads, and natural frequencies will be also investigated. Numerical results for various boundary conditions and configurations reveal the high accuracy of authors’ scheme.

Appendices

Appendix A

The coefficients ${\eta _1}$–${\eta _4}$ of Eq. (12) are simplified and given in a closed form below:

$${\eta _1}= - \frac{2}{3}\frac{{{h_0}^{3}b{\Omega ^3}\left( {\left( {{p^3}+6{p^2}+11p} \right){E_{\text{c}}}+6{E_{\text{m}}}} \right)}}{{{L^3}\left( {v - 1} \right)\left( {v+1} \right)\left( {p+2} \right)\left( {p+3} \right)\left( {p+1} \right)}},$$

$${\eta _2}= - \frac{{2{h_0}^{2}{\Omega ^2}\left( {{E_{\text{c}}}\left( {b{h_0}+{h_{\text{f}}}} \right){p^3}+6{E_{\text{c}}}\left( {b{h_0}+{h_{\text{f}}}} \right){p^2}+11{E_{\text{c}}}\left( {b{h_0}+{h_{\text{f}}}} \right)p+6b{E_{\text{m}}}{h_0}+6{E_{\text{c}}}{h_{\text{f}}}} \right)}}{{\left( {v - 1} \right)\left( {v+1} \right){L^2}\left( {p+2} \right)\left( {p+3} \right)\left( {p+1} \right)}},$$

$$\begin{aligned} {\eta _3} & = - \frac{2}{3}\frac{{\left( {\frac{{3{E_{\text{c}}}{p^3}{h_0}^{3}\Omega }}{L}+\frac{{18{E_{\text{c}}}{p^2}{h_0}^{3}\Omega }}{L}+\frac{{33{E_{\text{c}}}p{h_0}^{3}\Omega }}{L}+\frac{{18{E_{\text{c}}}{h_0}^{3}\Omega }}{L}} \right)b}}{{\left( {{v^2}p+2{v^2} - p - 2} \right)\left( {p+3} \right)\left( {p+1} \right)}}+\frac{2}{3}\frac{{\left( {\frac{{18{E_{\text{c}}}{h_0}^{3}\Omega }}{L} - \frac{{18{E_{\text{m}}}{h_0}^{3}\Omega }}{L}} \right)b}}{{\left( {{v^2}p+2{v^2} - p - 2} \right)\left( {p+3} \right)\left( {p+1} \right)}} \\ & \quad +\frac{2}{3}\frac{{{E_{\text{c}}}\left( {\frac{{3{{\left( {{h_0}+{h_{\text{f}}}} \right)}^2}{h_0}\Omega }}{L} - \frac{{3{h_0}^{3}\Omega }}{L}} \right)}}{{ - {v^2}+1}}, \\ \end{aligned}$$

$$\begin{aligned} {\eta _4} & = - \frac{2}{3}\frac{{\left( {{E_{\text{c}}}{h_0}^{3}{p^3}+6{E_{\text{c}}}{h_0}^{3}{p^2}+11{E_{\text{c}}}{h_0}^{3}p+6{E_{\text{c}}}{h_0}^{3}} \right)b}}{{\left( {{v^2}p+2{v^2} - p - 2} \right)\left( {p+3} \right)\left( {p+1} \right)}}+\frac{2}{3}\frac{{\left( {6{E_{\text{c}}}{h_0}^{3} - 6{E_{\text{m}}}{h_0}^{3}} \right)b}}{{\left( {{v^2}p+2{v^2} - p - 2} \right)\left( {p+3} \right)\left( {p+1} \right)}} \\ & \quad +\frac{2}{3}\frac{{{E_{\text{c}}}\left( {{{\left( {{h_0}+{h_{\text{f}}}} \right)}^3} - {h_0}^{3}} \right)}}{{ - {v^2}+1}}. \\ \end{aligned}$$

Appendix B

Herein, the Bessel functions appeared in Eq. (23) are presented:

$${J_n}(x)=\sum\limits_{{m=0}}^{\infty } {\frac{{{{( - 1)}^m}}}{{m!\Gamma (m+n+1)}}{{\left( {\frac{x}{2}} \right)}^{2m+n}}} ,$$

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$${Y_n}(x)=\frac{{{J_n}(x)\cos \;(n\pi ) - {J_{ - n}}(x)}}{{\sin \;(n\pi )}}.$$

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In the Bessel relations, $\Gamma$ denotes the gamma function. It is obvious that these harmonic functions lead to sinusoidal and cosinusoidal functions under certain circumstances.

Appendix C

The determinant of coefficients matrix given in Eq. (25) in a closed form is:

$${\varphi _1}=\frac{{2\mu }}{{L\beta }},$$

$${\varphi _2}=\frac{{{L^2}\psi }}{\alpha },$$

$${\varphi _3}={L^2},$$

$${\varphi _4}=\beta {\varphi _3},$$

$${\varphi _5}={\beta ^2}(\beta {\xi _0}{\varphi _3}+L{\varphi _2} - {\mu ^2} - {\xi _0}{\varphi _4})\varphi _{1}^{2}+4{\varphi _2}{\xi _0},$$

$${\varphi _6}=2\beta {\varphi _1}{\xi _0}\left( { - \frac{1}{4}{\beta ^2}\varphi _{1}^{2}{\varphi _3}+L{\varphi _2}} \right),$$

$${\varphi _7}=8{\xi _0}{\xi _1}{\varphi _2}{e^{ - \frac{1}{2}L\beta }},$$

$${\varphi _8}=\beta {\varphi _1}\left( {{e^{ - L\beta }}} \right),$$

$$\begin{aligned} det[A]&=\frac{{\beta {\varphi _1}}}{{16}}\left[\vphantom{\left.\left.\left.- 4{\varphi _2}{\xi _0})({J_0}({\varphi _1}){Y_1}({\varphi _1}) - {Y_0}({\varphi _1}){J_0}({\varphi _1})\right)\right)\right]}{J_0}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\left({\varphi _7}{Y_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\right.\right. \hfill \\ &\quad\left.\left.- {\varphi _8}({\varphi _5}{Y_0}({\varphi _1})+{\varphi _6}{Y_1}({\varphi _1})) - 4{\xi _1}{\varphi _2}(\beta {\varphi _1}{Y_0}({\varphi _1})\right.\right. \hfill \\ &\quad \left.\left.\vphantom{\left({\varphi _7}{Y_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\right.}+2{\xi _0}{Y_1}({\varphi _1}))\right)+{Y_0}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\left( - {\varphi _7}{J_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\right.\right. \hfill \\ &\quad \left.\left.+{\varphi _8}({\varphi _5}{J_0}({\varphi _1})+{\varphi _6}{J_1}({\varphi _1}))+4{\xi _1}{\varphi _2}(\beta {\varphi _1}{J_0}({\varphi _1})\right.\right. \hfill \\ &\quad \left.\left.{\vphantom{\left( - {\varphi _7}{J_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\right.}}+2{\xi _0}{J_1}({\varphi _1}))\right) - 2{\xi _1}\left( - {e^{ - \tfrac{{\beta L}}{2}}}{J_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)({\varphi _5}{Y_0}({\varphi _1})\right.\right. \hfill \\ &\quad \left.\left.+{\varphi _6}{Y_1}({\varphi _1}))+{e^{ - \tfrac{{\beta L}}{2}}}{Y_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)({\varphi _5}{J_0}({\varphi _1})+{\varphi _6}{J_1}({\varphi _1}))\right. \right.\hfill \\ &\quad \left.\left.+\left( - \frac{{{\beta ^4}\varphi _{1}^{4}{\varphi _3}}}{4}+{\beta ^2}\varphi _{1}^{2}( - \beta {\xi _0}{\varphi _3}+{\mu ^2}+{\xi _0}{\varphi _4})\right. \right.\right.\hfill \\ &\quad \left.\vphantom{\left[{J_0}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)({\varphi _7}{Y_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)\right.}\left.\vphantom{\left( - {e^{ - \tfrac{{\beta L}}{2}}}{J_1}\left( {{\varphi _1}{e^{ - \tfrac{{\beta L}}{2}}}} \right)({\varphi _5}{Y_0}({\varphi _1})\right.}\left.\vphantom{\left( - \frac{{{\beta ^4}\varphi _{1}^{4}{\varphi _3}}}{4}+{\beta ^2}\varphi _{1}^{2}( - \beta {\xi _0}{\varphi _3}+{\mu ^2}+{\xi _0}{\varphi _4})\right.}- 4{\varphi _2}{\xi _0})({J_0}({\varphi _1}){Y_1}({\varphi _1}) - {Y_0}({\varphi _1}){J_0}({\varphi _1})\right)\right)\right]. \hfill \\ \end{aligned}$$

Appendix D

The entries of the coefficient matrix of Eq. (25) together with the correlation coefficient are expressed by the following:

$${A_{11}}={\mu ^2},$$

$${A_{12}}=\frac{{{L^2}\psi }}{\alpha },$$

$${A_{13}}=\frac{{{L^2}{J_0}\left( {\frac{{2\mu }}{{L\beta }}} \right)\psi }}{\alpha },$$

$${A_{14}}=\frac{{{L^2}{Y_0}\left( {\frac{{2\mu }}{{L\beta }}} \right)\psi }}{\alpha },$$

$${A_{21}}=L,$$

$${A_{22}}=1,$$

$${A_{23}}={J_0}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right),$$

$${A_{24}}={Y_0}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right),$$

$${A_{31}}={\xi _0},$$

$${A_{32}}=0,$$

$${A_{33}}=\frac{{L{J_1}\left( {\frac{{2\mu }}{{L\beta }}} \right)\mu {\xi _0}+{J_0}\left( {\frac{{2\mu }}{{L\beta }}} \right){\mu ^2}}}{{{L^2}}},$$

$${A_{34}}=\frac{{L{Y_1}\left( {\frac{{2\mu }}{{L\beta }}} \right)\mu {\xi _0}+{Y_0}\left( {\frac{{2\mu }}{{L\beta }}} \right){\mu ^2}}}{{{L^2}}},$$

$${A_{41}}={\xi _1},$$

$${A_{42}}=0,$$

$${A_{43}}=\frac{{{e^{ - \frac{1}{2}L\beta }}\mu {\xi _1}L{J_1}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right) - {e^{ - L\beta }}{\mu ^2}{J_0}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right)}}{{{L^2}}},$$

$${A_{44}}=\frac{{{e^{ - \frac{1}{2}L\beta }}\mu {\xi _1}L{Y_1}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right) - {e^{ - L\beta }}{\mu ^2}{Y_0}\left( {\frac{{2\mu {e^{ - \frac{1}{2}L\beta }}}}{{L\beta }}} \right)}}{{{L^2}}}.$$

Appendix E

The shifted Legendre polynomials are given by the following:

$${L_j}(2\bar {x} - 1)={( - 1)^j}\sum\limits_{{k=0}}^{j} {\left( \begin{gathered} j \hfill \\ k \hfill \\ \end{gathered} \right)\left( \begin{gathered} j+k \hfill \\ k \hfill \\ \end{gathered} \right)} {( - \bar {x})^k}.$$

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