A Variational Formulation to Find Finite Element Bending, Buckling and Vibration Equations of Nonlocal Timoshenko Beams

Abstract

A variational approach is developed to obtain bending, buckling and vibration finite element equations of nonlocal Timoshenko beams in this study. The reason for using the finite element method in this research is to investigate the behavior of nano-beams with complex geometry, material property and different boundary conditions. Weak forms of governing equations are derived, and the nonlocal differential elasticity theory is used to find the finite element formulation of nonlocal Timoshenko beams. In deriving the weak formulations, it is seen that it is impossible to construct the quadratic functional form due to non-symmetric bilinear property. Using the developed concepts and formulations, the bending and buckling of nonlocal Timoshenko beams with four classical boundary conditions are analyzed and the obtained results are compared with those reported in the literature. In order to show the capabilities of the proposed formulation in comparison with exact methods, the simply supported stepped nonlocal Timoshenko beam is selected and bending and buckling analyses are performed as well.

Appendix: Nonlocal Timoshenko Beam element matrices

The element matrices for nonlocal Timoshenko beam are presented as

$$\begin{aligned} \left\{ U \right\} & = \left\{ {\begin{array}{*{20}c} {w_{1} } & {w_{2} } & {w_{3} } & {w_{4} } & {\phi_{1} } & {\phi_{2} } & {\phi_{3} } & {\phi_{4} } \\ \end{array} } \right\}^{T} \\ \left[ K \right] & = \frac{{K_{s} GA}}{1680l}\left[ {\begin{array}{*{20}c} {42\left[ {C_{1} } \right]} & {21l\left[ {C_{2} } \right]} \\ {21l\left[ {C_{2} } \right]^{T} } & {l^{2} \left[ {C_{3} } \right]} \\ \end{array} } \right] + \frac{EI}{40l}\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & {\left[ {C_{1} } \right]} \\ \end{array} } \right] \\ \left[ M \right] & = \frac{\rho l}{1680}\left[ {\begin{array}{*{20}c} {A\left[ {C_{3} } \right]} & 0 \\ 0 & {I\left[ {C_{3} } \right]} \\ \end{array} } \right] + \frac{{\eta^{2} \rho }}{80l}\left[ {\begin{array}{*{20}c} 0 & 0 \\ {Al\left[ {C_{2} } \right]} & {2I\left[ {C_{1} } \right]} \\ \end{array} } \right] \\ \left[ B \right] & = \frac{{\bar{P}}}{40l}\left[ {\begin{array}{*{20}c} {\left[ {C_{1} } \right]} & 0 \\ 0 & 0 \\ \end{array} } \right] + \frac{9}{8}\frac{{\eta^{2} \bar{P}}}{{l^{2} }}\left[ {\begin{array}{*{20}c} 0 & 0 \\ {\left[ {C_{4} } \right]} & 0 \\ \end{array} } \right] \\ \left\{ Q \right\} & = \frac{{q_{0} l}}{8}\left\{ {\begin{array}{*{20}c} 1 \\ 3 \\ 3 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right\} + q_{0} \eta^{2} \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ { - 1} \\ 0 \\ 0 \\ 1 \\ \end{array} } \right\};\quad \left\{ F \right\} = \left\{ {\begin{array}{*{20}c} { - \left. V \right|_{{\bar{x} = 0}} } \\ 0 \\ 0 \\ {\left. V \right|_{{\bar{x} = l}} } \\ { - \left. M \right|_{{\bar{x} = 0}} } \\ 0 \\ 0 \\ {\left. M \right|_{{\bar{x} = l}} } \\ \end{array} } \right\} \\ \end{aligned}$$

where $\left[ {C_{1} } \right]$, $\left[ {C_{2} } \right]$, $\left[ {C_{3} } \right]$ and $\left[ {C_{4} } \right]$ are defined as

$$\begin{aligned} \left[ {C_{1} } \right] & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {148} & { - 189} \\ { - 189} & {432} \\ \end{array} } & {\begin{array}{*{20}c} {54} & { - 13} \\ { - 297} & {54} \\ \end{array} } \\ {\begin{array}{*{20}c} {54} & { - 297} \\ { - 13} & {54} \\ \end{array} } & {\begin{array}{*{20}c} {432} & { - 189} \\ { - 189} & {148} \\ \end{array} } \\ \end{array} } \right] \\ \left[ {C_{2} } \right] & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - 40} & { - 57} \\ {57} & 0 \\ \end{array} } & {\begin{array}{*{20}c} {24} & { - 7} \\ { - 81} & {24} \\ \end{array} } \\ {\begin{array}{*{20}c} { - 24} & {81} \\ 7 & { - 24} \\ \end{array} } & {\begin{array}{*{20}c} 0 & { - 57} \\ {57} & {40} \\ \end{array} } \\ \end{array} } \right] \\ \left[ {C_{3} } \right] & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {128} & {99} \\ {99} & {648} \\ \end{array} } & {\begin{array}{*{20}c} { - 36} & {19} \\ { - 81} & { - 36} \\ \end{array} } \\ {\begin{array}{*{20}c} { - 36} & { - 81} \\ {19} & { - 36} \\ \end{array} } & {\begin{array}{*{20}c} {648} & {99} \\ {99} & {128} \\ \end{array} } \\ \end{array} } \right] \\ \left[ {C_{4} } \right] & = \left[ {\begin{array}{*{20}c} {13} & { - 31} & {23} & { - 5} \\ { - 9} & {27} & { - 27} & 9 \\ { - 9} & {27} & { - 27} & 9 \\ 5 & { - 23} & {31} & { - 13} \\ \end{array} } \right] \\ \end{aligned}$$