Bending analysis of functionally graded sandwich beams with general boundary conditions using a modified Fourier series method

Abstract

A modified Fourier method and six-parameter constrained model are employed to investigate the static bending characteristics of functionally graded sandwich beams under classical and non-classical boundary conditions based on the first-order shear deformation theory. Three types of sandwich beams including isotropic hardcore, functionally graded core, and isotropic softcore are considered. The effective material properties of functionally graded materials are assumed to vary according to power law distribution of volume fraction of constituents by Voigt model. The governing equations and boundary conditions are derived from the principle of minimum potential energy and are solved using the modified Fourier series method which includes the standard Fourier cosine series together with two auxiliary polynomials terms. The high convergence rate, availability and accuracy of the formulation are verified by comparisons with results of other methods. Moreover, numerous new bending results for functionally graded sandwich beams with general boundary conditions are presented. The significant effects of various boundary conditions, different types of sandwich beams, power-law index, span-to-height ratio and skin–core-skin thickness ratio on the displacements, axial stresses, and shear stresses of the sandwich beams with symmetrical and unsymmetrical forms are also investigated.

Appendix A: Expansion coefficients of Fourier cosine series

The coefficients of Fourier cosine series generated from the supplementary functions and their derivatives, sine and transverse load terms as follows

$$f_{1} \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{1}^{0} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{1}^{0} \; = \left\{ {\;\begin{array}{*{20}l} {\frac{L}{12}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;m = 0} \hfill \\ {\frac{{2\,L\,\,\left[ {6 - 6\;( - 1\,)\,^{m} - m^{\,2} {\kern 1pt} \pi^{\,2} } \right]}}{{m^{\,4} \;\pi^{\,4} }}\quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.1)

$$f_{2} \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{2}^{0} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{2}^{0} \; = \left\{ {\;\begin{array}{*{20}l} { - \frac{L}{12}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\quad \quad m = 0} \hfill \\ {\frac{{2\,L\,\,\left[ {6 - 6\;( - 1\,)\,^{m} + m^{\,2} {\kern 1pt} \pi^{\,2} \;( - 1\,)\,^{m} } \right]}}{{m^{\,4} \;\pi^{\,4} }}\quad \;m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.2)

$$f\,_{1}^{\prime } \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{1}^{1} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{1}^{1} \; = \left\{ {\;\begin{array}{*{20}l} {0\quad \quad \quad \quad \quad \quad \;m = 0} \hfill \\ {\frac{{4\,\,\left[ {2 + \;( - 1\,)\,^{m} } \right]}}{{m^{\,2} \;\pi^{\,2} }}\quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.3)

$$f\,_{2}^{\prime } \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{2}^{1} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{2}^{1} \; = \left\{ {\;\begin{array}{*{20}l} {0\quad \quad \quad \quad \quad \quad \;\;\;m = 0} \hfill \\ {\frac{{4\,\,\left[ {1 + \;2\;( - 1\,)\,^{m} } \right]}}{{m^{\,2} \;\pi^{\,2} }}\quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.4)

$$f\,_{1}^{{\prime {\kern 1pt} \prime }} \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{1}^{2} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{1}^{2} \; = \left\{ {\;\begin{array}{*{20}l} { - \frac{1}{L}\quad \quad \quad \quad \quad \quad m = 0} \hfill \\ {\frac{{ - 12\,\,\left[ {1 - \;( - 1\,)\,^{m} } \right]}}{{L\;m^{\,2} \;\pi^{\,2} }}\quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.5)

$$f\,_{2}^{{\prime {\kern 1pt} \prime }} \,\,(x)\,\, = \,\sum\limits_{m\,\; = \;0}^{M} {\alpha \,_{2}^{2} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\; \Rightarrow \,\;\alpha \,_{2}^{2} \; = \left\{ {\;\begin{array}{*{20}l} {\frac{1}{L}\quad \quad \quad \quad \quad \quad \;\;\;m = 0} \hfill \\ {\frac{{ - 12\,\,\left[ {1 - \;( - 1\,)\,^{m} } \right]}}{{L\;m^{\,2} \;\pi^{\,2} }}\quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.6)

$${\text{s}} \,i\,n\,\,(\lambda {\kern 1pt}_{m} \,x)\,\, = \,\sum\limits_{k\,\; = \;0}^{M} {\beta \,_{k}^{m} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{k} \,x\,)\;\; \Rightarrow \,\;\beta \,_{k}^{m} \; = \left\{ {\;\,\begin{array}{*{20}l} \begin{gathered} 0\quad \quad \quad \quad \quad \quad \;\;\;\quad \quad \;m = 0 \hfill \\ 0\quad \quad \quad \quad \quad \quad \;\;\;\quad \quad \;m = k \hfill \\ \end{gathered} \hfill \\ \begin{gathered} \frac{{\,\,1 - \;( - 1{\kern 1pt} \,)\,^{m} }}{m\;\pi }\quad \quad \quad \quad \;\quad m \ne 0\;,\;\,k = 0 \hfill \\ \frac{{\,2\;m\;\,\left[ {\;( - 1\,)\,^{m\;\, + \;k} - 1} \right]}}{{\left( {k^{\,2} - m^{\,2} } \right)\;\pi }}\quad \quad m \ne 0\;,\;\,k \ne 0 \hfill \\ \end{gathered} \hfill \\ \end{array} } \right.\quad$$

(A.7)

$$q\,\,(x)\,\, = \,\,q\,_{0} \; = \,\,\sum\limits_{m\,\; = \;0}^{M} {\;Q\,_{m} } \;{\text{c}} {\kern 1pt} \,o\,s\;(\lambda {\kern 1pt}_{m} \,x\,)\;\, \Rightarrow \,\;Q\,_{m} \; = \left\{ {\;\begin{array}{*{20}l} {q\,_{0} \quad \;\;m = 0} \hfill \\ {0\quad \quad m \ne 0} \hfill \\ \end{array} } \right.\quad$$

(A.8)