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Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams

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Abstract

In this paper, correspondence relations between the solutions of the static bending of functionally graded material (FGM) Reddy–Bickford beams (RBBs) and those of the corresponding homogenous Euler–Bernoulli beams are presented. The effective material properties of the FGM beams are assumed to vary continuously in the thickness direction. Governing equations for the titled problem are formulated via the principle of virtual displacements based on the third-order shear deformation beam theory, in which the higher-order shear force and bending moment are included. General solutions of the displacements and the stress resultants of the FGM RBBs are derived analytically in terms of the deflection of the reference homogenous Euler–Bernoulli beam with the same geometry, loadings and end conditions, which realize a classical and homogenized expression of the bending response of the shear deformable non-homogeneous FGM beams. Particular solutions for the FGM RBBs under specified end constraints and load conditions are given to validate the theory and methodology. The key merit of this work is to be capable of obtaining the high-accuracy solutions of thick FGM beams in terms of the classical beam theory solutions without dealing with the solution of the complicated coupling differential equations with boundary conditions of the problem.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China with Grant Numbers 11272278 and 11672260. The authors gratefully acknowledge the financial supports.

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Correspondence to Shi-Rong Li.

Appendix

Appendix

The dimensionless coefficients, \(\phi _i \), \(\phi _{xz0} \) and \(\phi _{xz2} \) for the Young’s moduli that vary as power-law function (64) are given as follows:

$$\begin{aligned} \phi _0= & {} 1+\frac{\eta }{p+1}, \quad \phi _1 =\frac{p\eta }{2(p+1)(p+2)} \end{aligned}$$
(A.1)
$$\begin{aligned} \phi _2= & {} 1+\frac{3\eta (p^{2}+p+2)}{(p+1)(p+2)(p+3)}, \phi _3 =\frac{\eta p(p^{2}+3p+8)}{8(p+1)(p+2)(p+3)(p+4)} \end{aligned}$$
(A.2)
$$\begin{aligned} \phi _4= & {} \frac{1}{80}+\frac{\eta (p^{4}+6p^{3}+23p^{2}+18p+24)}{16(p+1)(p+2)(p+3)(p+4)(p+5)} \end{aligned}$$
(A.3)
$$\begin{aligned} \phi _5= & {} \frac{\eta p(p^{4}+10p^{3}+55p^{2}+110p+184)}{32(p+1)(p+2)(p+3)(p+4)(p+5)(p+6)} \end{aligned}$$
(A.4)
$$\begin{aligned} \phi _6= & {} \frac{1}{448}+\frac{\eta (p^{6}+15p^{5}+115p^{4}+405p^{3}+964p^{2}+660p+720)}{64(p+1)(p+2)(p+3)(p+4)(p+5)(p+6)(p+7)} \end{aligned}$$
(A.5)
$$\begin{aligned} \phi _{zx0}= & {} \frac{1}{2(1+\nu )}\left( {\phi _0 -\frac{1}{3}\phi _2 } \right) , \quad \phi _{zx2} =\frac{1}{2(1+\nu )}\left( {\frac{1}{12}\phi _2 -4\phi _4 } \right) \end{aligned}$$
(A.6)

in which \(\eta =E_2 /E_1 -1\).

The dimensionless resultant forces and moments in terms of the dimensionless displacements are expressed as:

$$\begin{aligned} f_N= & {} \frac{12}{\delta ^{2}}\left( {\phi _0 \frac{\hbox {d}U}{\hbox {d}\xi }+\delta \phi _1 \frac{\hbox {d}\varphi }{\hbox {d}\xi }-\frac{4\delta }{3}\phi _3 \frac{\hbox {d}\gamma }{\hbox {d}\xi }} \right) \end{aligned}$$
(A.7)
$$\begin{aligned} m_{x1}= & {} \frac{12}{\delta }\left( {\phi _1 \frac{\hbox {d}U}{\hbox {d}\xi }+\frac{\delta \phi _2 }{12}\frac{\hbox {d}\varphi }{\hbox {d}\xi }-\frac{4\delta }{3}\phi _4 \frac{\hbox {d}\gamma }{\hbox {d}\xi }} \right) \end{aligned}$$
(A.8)
$$\begin{aligned} m_{x3}= & {} 12\delta \left( {\phi _3 \frac{\hbox {d}U}{\hbox {d}\xi }+\delta \phi _4 \frac{\hbox {d}\varphi }{\hbox {d}\xi }-\frac{4\delta }{3}\phi _6 \frac{\hbox {d}\gamma }{\hbox {d}\xi }} \right) \end{aligned}$$
(A.9)
$$\begin{aligned} f_{s0}= & {} \frac{1}{c_{s0} }\gamma , \quad f_{s2} =\frac{\delta ^{2}}{4c_{s2} }\gamma \end{aligned}$$
(A.10)

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Xia, YM., Li, SR. & Wan, ZQ. Bending Solutions of FGM Reddy–Bickford Beams in Terms of Those of the Homogenous Euler–Bernoulli Beams. Acta Mech. Solida Sin. 32, 499–516 (2019). https://doi.org/10.1007/s10338-019-00100-y

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  • DOI: https://doi.org/10.1007/s10338-019-00100-y

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