Exact stiffness matrix for nonlocal bars embedded in elastic foundation media: the virtual-force approach

Abstract

This paper proposes a solution to the exact bar-foundation element that includes the bar nonlocal effect. The exact element stiffness matrix and fixed-end force vector are derived based on the exact element flexibility equation using the so-called natural approach. The virtual-force principle is employed to work out the governing differential compatibility equation as well as the associated end-boundary compatibility conditions. Exact force interpolation functions are used to derive the exact element flexibility equation and can be obtained as the analytical solution of the governing differential compatibility equation of the problem. A numerical example of a nanowire-elastic substrate system is used to verify the accuracy and efficiency of the natural nonlocal bar-foundation model and to demonstrate the superiority over its counterpart, a displacement-based model. The effects of material nonlocality on the system responses are also discussed in the example.

Appendices

Appendix 1: Axial-force interpolation functions, foundation-force interpolation functions, and nodal displacements due to $p_x (x)$

The axial-force interpolation functions may be written as

$$\begin{aligned} N_{\textit{BB}1} (x)=\frac{1}{2}\Big ( {\mathrm{e}^{\lambda x}-\mathrm{e}^{\lambda ( {2L-x} )}} \Big )\Big ( {\coth \lambda L-1} \Big ), \quad N_{BB2} (x)=\frac{\sinh \lambda x}{\sinh \lambda L}. \end{aligned}$$

The foundation-force interpolation functions may be written as

$$\begin{aligned} N_{sB1} (x)=\frac{\lambda }{2}\Big ( {\mathrm{e}^{\lambda x}+\mathrm{e}^{\lambda ( {2L-x} )}} \Big )\Big ( {\coth \lambda L-1} \Big ), \quad N_{sB2} (x)=\frac{\lambda \cosh \lambda x}{\sinh \lambda L}. \end{aligned}$$

The second derivative of the axial-force interpolation functions may be written as

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\mathrm{d}^{2}N_{BB1} (x)}{\mathrm{d}x^{2}}=\frac{\lambda ^{2}}{2}\Big ( {\mathrm{e}^{\lambda x}-\mathrm{e}^{\lambda ( {2L-x} )}} \Big )\Big ( {\coth \lambda L-1} \Big ) \quad \frac{\mathrm{d}^{2}N_{BB2} (x)}{\mathrm{d}x^{2}}=\frac{\lambda ^{2}\sinh \lambda x}{\sinh \lambda L}. \\ \end{array} \end{aligned}$$

The nodal displacements due to the uniformly distributed load $p_x (x)=p_{x0}$ may be written as

$$\begin{aligned} U_{1p_x } =\frac{p_{x0} }{k_s }\quad \hbox {and}\quad U_{2p_x } =\frac{p_{x0} }{k_s }. \end{aligned}$$

Appendix 2: Nonlocal bar embeded in elastic foundation flexibility matrix

The bar contribution to the element flexibility matrix may be written as

$$\begin{aligned} \mathbf{F}_{\textit{BB}}&= \Bigg ( {{\begin{array}{l@{\quad }l} {F_{11}^{BB} }&{} {F_{12}^{BB} } \\ \mathrm{symm.}&{} {F_{22}^{BB} } \\ \end{array} }} \Bigg ) \end{aligned}$$

where

$$\begin{aligned} F_{11}^{BB}&= F_{22}^{BB} =\frac{1}{2EA}\Bigg ( {\frac{\coth \lambda L}{\lambda }-\frac{L}{\sinh ^{2}\lambda L}} \Bigg ),\\ F_{12}^{BB}&= F_{21}^{BB} =\frac{1}{2EA\sinh \lambda L}\Bigg ( {\frac{1}{\lambda }-\frac{L}{\tanh \lambda L}} \Bigg ). \end{aligned}$$

The foundation contribution to the element flexibility matrix may be written as

$$\begin{aligned} \mathbf{F}_{ss}&= \Bigg ( {{\begin{array}{l@{\quad }l} {F_{11}^{ss} }&{} {F_{12}^{ss} } \\ \mathrm{symm.}&{} {F_{22}^{ss} } \\ \end{array} }} \Bigg ) \end{aligned}$$

where

$$\begin{aligned} F_{11}^{ss}&= F_{22}^{ss} =\frac{\lambda ^{2}}{2k_s }\Bigg ( {\frac{\coth \lambda L}{\lambda }+\frac{L}{\sinh ^{2}\lambda L}}\Bigg ),\\ F_{12}^{ss}&= F_{21}^{ss} =\frac{\lambda }{2k_s \sinh \lambda L}\Bigg ( {1+\frac{\lambda L}{\tanh \lambda L}} \Bigg ). \end{aligned}$$

The nonlocal contribution to the element flexibility matrix may be written as

$$\begin{aligned} \mathbf{F}_{\textit{BB}}^\mathrm{Nonlocal}&= \Bigg ( {{\begin{array}{l@{\quad }l} {F_{11}^\mathrm{Nonlocal} }&{} {F_{12}^\mathrm{Nonlocal} } \\ \mathrm{symm.}&{} {F_{22}^\mathrm{Nonlocal} } \\ \end{array} }} \Bigg ) \end{aligned}$$

where

$$\begin{aligned} F_{11}^\mathrm{Nonlocal}&= F_{22}^\mathrm{Nonlocal} =\frac{\lambda ( {\mathrm{e}_0 a} )^{2}}{2EA}\Bigg ( {\coth \lambda L-\frac{\lambda L}{\sinh ^{2}\lambda L}} \Bigg ),\\ F_{12}^\mathrm{Nonlocal}&= F_{21}^\mathrm{Nonlocal} =\frac{\lambda ( {\mathrm{e}_0 a} )^{2}}{2EA\sinh \lambda L}\Bigg ( {1-\frac{\lambda L}{\tanh \lambda L}} \Bigg ). \end{aligned}$$