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Equivalent continuum modeling of beam-like truss structures with flexible joints

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Abstract

The paper investigated the equivalent continuum modeling of beam-like repetitive truss structures considering the flexibility of joints, which models the contact between the truss member and joint by spring-damper with six directional stiffnesses and dampings. Firstly, a two-node hybrid joint-beam element was derived for modeling the truss member with flexible end joints, and a condensed model for the repeating element with flexible joints was obtained. Then, the energy equivalence method was adopted to equivalently model the truss structure with flexible joints and material damping as a spatial viscoelastic anisotropic beam model. Afterwards, the equations of motion for the equivalent beam model were derived and solved analytically in the frequency domain. In the numerical studies, the correctness of the presented method was verified by comparisons of the natural frequencies and frequency responses evaluated by the equivalent beam model with the results of the finite element method model.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11702146, 11732006 and 11827801) and the Equipment Pre-research Foundation (Grant 6140210010202).

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Authors and Affiliations

  1. College of Civil Engineering, Nanjing Forestry University, Nanjing, 210037, China

    Fushou Liu & Libin Wang

  2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Dongping Jin & Hao Wen

Authors
  1. Fushou Liu
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  2. Libin Wang
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  3. Dongping Jin
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  4. Hao Wen
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Corresponding author

Correspondence to Dongping Jin.

Appendix

Appendix

The correction matrix G in Eq. (9) is

$$ \varvec{G} = \left[ {\begin{array}{*{20}c} {g_{11} } & 0 & 0 & 0 & 0 & 0 & { - g_{11} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {g_{22} } & 0 & 0 & 0 & {g_{26} } & 0 & { - g_{22} } & 0 & 0 & 0 & {g_{212} } \\ 0 & 0 & {g_{33} } & 0 & {g_{35} } & 0 & 0 & 0 & { - g_{33} } & 0 & {g_{311} } & 0 \\ 0 & 0 & 0 & {g_{44} } & 0 & 0 & 0 & 0 & 0 & { - g_{44} } & 0 & 0 \\ 0 & 0 & {g_{53} } & 0 & {g_{55} } & 0 & 0 & 0 & { - g_{53} } & 0 & {g_{511} } & 0 \\ 0 & {g_{62} } & 0 & 0 & 0 & {g_{66} } & 0 & { - g_{62} } & 0 & 0 & 0 & {g_{612} } \\ { - g_{77} } & 0 & 0 & 0 & 0 & 0 & {g_{77} } & 0 & 0 & 0 & 0 & 0 \\ 0 & { - g_{88} } & 0 & 0 & 0 & {g_{86} } & 0 & {g_{88} } & 0 & 0 & 0 & {g_{812} } \\ 0 & 0 & { - g_{99} } & 0 & {g_{95} } & 0 & 0 & 0 & {g_{99} } & 0 & {g_{911} } & 0 \\ 0 & 0 & 0 & { - g_{1010} } & 0 & 0 & 0 & 0 & 0 & {g_{1010} } & 0 & 0 \\ 0 & 0 & { - g_{119} } & 0 & {g_{115} } & 0 & 0 & 0 & {g_{119} } & 0 & {g_{1111} } & 0 \\ 0 & { - g_{128} } & 0 & 0 & 0 & {g_{126} } & 0 & {g_{128} } & 0 & 0 & 0 & {g_{1212} } \\ \end{array} } \right] $$

with

$$ g_{11} = - \frac{{{\mkern 1mu} p_{uxi} }}{{1 + p_{uxi} + {\mkern 1mu} p_{uxj} }},g_{22} {\mkern 1mu} {\mkern 1mu} = - \frac{{12p_{uyi} }}{{\varDelta_{2} }}(1 + p_{\theta zi} + p_{\theta zj} ), $$
$$ g_{26} {\mkern 1mu} {\mkern 1mu} = \frac{{L_{m} }}{{\varDelta_{2} }}\left[ {(1 + 4p_{\theta zi} + 4p_{\theta zj} + 12p_{\theta zi} {\mkern 1mu} p_{\theta zj} {\mkern 1mu} )\tilde{e}_{1} {\mkern 1mu} } \right. + 12p_{uyj} ({\mkern 1mu} 1 + p_{\theta zi} + p_{\theta zj} )\tilde{e}_{1} \left. { - 6p_{uyi} (1 + 2p_{\theta zj} )} \right], $$
$$ g_{212} {\mkern 1mu} {\mkern 1mu} = - {\mkern 1mu} \frac{{6p_{{uyi{\mkern 1mu} }} L_{m} }}{{\varDelta_{2} }}\left[ {2{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \tilde{e}_{2} (1 + p_{\theta zi} + p_{\theta zj} ) + 2p_{\theta zi} + 1} \right], $$
$$ g_{33} = - \frac{{12p_{uzi} }}{{\varDelta_{3} }}(1 + {\mkern 1mu} p_{\theta yi} + {\mkern 1mu} p_{\theta yj} ), $$
$$\begin{aligned} g_{35} {\mkern 1mu} {\mkern 1mu} &= - \frac{{L_{m} }}{{\varDelta_{3} }}\left[ {(1 + 4p_{\theta yi} + 4p_{\theta yj} + 12p_{\theta yi} p_{\theta yj} )\tilde{e}_{1} {\mkern 1mu} }\right.\\ & \quad \left.{+ 12p_{uzj} {\mkern 1mu} ({\mkern 1mu} {\mkern 1mu} 1 + p_{\theta yi} + {\mkern 1mu} p_{\theta yj} )\tilde{e}_{1} {\mkern 1mu} - 6p_{uzi} (1 + 2p_{\theta yj} )} \right], \end{aligned}$$
$$ g_{311} {\mkern 1mu} {\mkern 1mu} = {\mkern 1mu} \frac{{6p_{{uzi{\mkern 1mu} }} L_{m} }}{{\varDelta_{2} }}\left[ {2{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \tilde{e}_{2} (1 + p_{\theta yi} + p_{\theta yj} ) + 2p_{\theta yi} + 1} \right], $$
$$ g_{44} = - \frac{{{\mkern 1mu} p_{\theta xi} }}{{1 + {\mkern 1mu} p_{\theta xi} + p_{\theta xj} }}, $$
$$ \begin{aligned} g_{53} {\mkern 1mu} {\mkern 1mu} &= \frac{{6{\mkern 1mu} }}{{\varDelta_{3} L_{m} }}(p_{\theta yi} + 2p_{\theta yi} p_{\theta yj} ), \\ g_{55} {\mkern 1mu} {\mkern 1mu} & = - \frac{{2p_{\theta yi} }}{{\varDelta_{3} }}\left[ {{\mkern 1mu} {\mkern 1mu} 3{\mkern 1mu} \tilde{e}_{1} ({\mkern 1mu} {\mkern 1mu} 1 + 2p_{\theta yj} ){\mkern 1mu} + {\mkern 1mu} 6({\mkern 1mu} p_{uzi} + p_{uzj} + {\mkern 1mu} {\mkern 1mu} p_{\theta yj} ){\mkern 1mu} + 2} \right],\end{aligned} $$
$$ g_{511} {\mkern 1mu} {\mkern 1mu} = - \frac{{2p_{\theta yi} }}{{\varDelta_{3} }}\left[ {3{\mkern 1mu} \tilde{e}_{2} + 6{\mkern 1mu} {\mkern 1mu} \tilde{e}_{2} {\mkern 1mu} p_{\theta yj} + 1 - 6p_{uzi} - {\mkern 1mu} 6{\mkern 1mu} p_{{uzj{\mkern 1mu} }} } \right], $$
$$\begin{aligned} g_{62} {\mkern 1mu} {\mkern 1mu}& = - \frac{{6{\mkern 1mu} }}{{\varDelta_{2} L_{m} }}(p_{\theta zi} + 2p_{\theta zi} p_{\theta zj} ),\,\\ g_{66} {\mkern 1mu} {\mkern 1mu} &= - \frac{{2p_{\theta zi} }}{{\varDelta_{2} }}\left[ {{\mkern 1mu} 3{\mkern 1mu} \tilde{e}_{1} ({\mkern 1mu} 1 + 2p_{\theta zj} ){\mkern 1mu} + {\mkern 1mu} {\mkern 1mu} 6({\mkern 1mu} {\mkern 1mu} {\mkern 1mu} p_{uyi} {\mkern 1mu} + {\mkern 1mu} {\mkern 1mu} p_{uyj} + {\mkern 1mu} {\mkern 1mu} p_{\theta zj} ) + 2} \right], \end{aligned}$$
$$ g_{612} {\mkern 1mu} {\mkern 1mu} = - \frac{{2p_{\theta zi} }}{{\varDelta_{2} }}\left[ {3{\mkern 1mu} \tilde{e}_{2} ({\mkern 1mu} 1 + 2{\mkern 1mu} p_{\theta zj} ) - {\mkern 1mu} 6{\mkern 1mu} (p_{uyi} + p_{{uyj{\mkern 1mu} }} ) + 1} \right], $$
$$ g_{77} = - \frac{{{\mkern 1mu} p_{uxj} }}{{1 + p_{uxi} + {\mkern 1mu} p_{uxj} }}, $$
$$ \begin{aligned} g_{86} {\mkern 1mu} {\mkern 1mu} & = {\mkern 1mu} \frac{{6p_{{uyj{\mkern 1mu} }} L_{m} }}{{\varDelta_{2} }}\left[ {2{\mkern 1mu} {\mkern 1mu} \tilde{e}_{1} (1 + p_{\theta zi} + p_{\theta zj} ) + 2p_{\theta zj} + 1} \right],\\ g_{88} {\mkern 1mu} {\mkern 1mu} & = - \frac{{12p_{uyj} }}{{\varDelta_{2} }}(1 + p_{\theta zi} + p_{\theta zj} ),\end{aligned} $$
$$\begin{aligned} g_{812} {\mkern 1mu} {\mkern 1mu} &= - \frac{{L_{m} }}{{\varDelta_{2} }}\left[ {(1 + 4p_{\theta zi} + 4p_{\theta zj} }\right.\\ & \quad \left.{+ 12p_{\theta zi} {\mkern 1mu} p_{\theta zj} {\mkern 1mu} )\tilde{e}_{2} + 12p_{uyi} (1 + {\mkern 1mu} p_{\theta zi} + p_{\theta zj} ){\mkern 1mu} \tilde{e}_{2} } \right. \\ & \quad \left. { -\, 6p_{uyj} (1 + 2p_{\theta zi} )} \right], \end{aligned}$$
$$\begin{aligned} g_{95} {\mkern 1mu} {\mkern 1mu} &= - {\mkern 1mu} \frac{{6p_{{uzj{\mkern 1mu} }} L_{m} }}{{\varDelta_{2} }}\left[ {2{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \tilde{e}_{1} (1 + p_{\theta yi} + p_{\theta yj} ) + 2p_{\theta yj} + 1} \right],\,\\ g_{99} &= - \frac{{12p_{uzj} }}{{\varDelta_{3} }}(1 + {\mkern 1mu} p_{\theta yi} + {\mkern 1mu} p_{\theta yj} ), \end{aligned}$$
$$\begin{aligned} g_{911} {\mkern 1mu} {\mkern 1mu} &= \frac{{L_{m} }}{{\varDelta_{3} }}\left[ {(1 + 4p_{\theta yi} + 4p_{\theta yj} + 12p_{\theta yi} p_{\theta yj} )\tilde{e}_{2} {\mkern 1mu} }\right. \\ & \quad \left.{+ 12p_{uzi} {\mkern 1mu} ({\mkern 1mu} {\mkern 1mu} 1 + p_{\theta yi} + {\mkern 1mu} p_{\theta yj} )\tilde{e}_{2} {\mkern 1mu} - 6p_{uzj} (1 + 2p_{\theta yi} )} \right], \end{aligned}$$
$$ g_{1010} = - \frac{{{\mkern 1mu} p_{\theta xj} }}{{1 + {\mkern 1mu} p_{\theta xi} + p_{\theta xj} }}, $$
$$ \begin{aligned} g_{115} {\mkern 1mu} {\mkern 1mu} & = - \frac{{2p_{\theta yj} }}{{\varDelta_{3} }}\left[ {3{\mkern 1mu} \tilde{e}_{1} + 6{\mkern 1mu} {\mkern 1mu} \tilde{e}_{1} {\mkern 1mu} p_{\theta yi} + 1 - 6p_{uzi} - {\mkern 1mu} 6{\mkern 1mu} p_{{uzj{\mkern 1mu} }} } \right],\\ g_{119} {\mkern 1mu} {\mkern 1mu} & = - \frac{{6{\mkern 1mu} }}{{\varDelta_{3} L_{m} }}(p_{\theta yj} + 2p_{\theta yi} p_{\theta yj} ),\end{aligned} $$
$$ g_{1111} {\mkern 1mu} {\mkern 1mu} = - \frac{{2p_{\theta yj} }}{{\varDelta_{3} }}\left[ {{\mkern 1mu} {\mkern 1mu} 3{\mkern 1mu} \tilde{e}_{2} ({\mkern 1mu} {\mkern 1mu} 1 + 2p_{\theta yi} ){\mkern 1mu} + {\mkern 1mu} 6({\mkern 1mu} p_{uzi} + p_{uzj} + {\mkern 1mu} {\mkern 1mu} p_{\theta yi} ){\mkern 1mu} + 2} \right], $$
$$ \begin{aligned} g_{126} {\mkern 1mu} {\mkern 1mu} & = - \frac{{2p_{\theta zj} }}{{\varDelta_{2} }}\left[ {3{\mkern 1mu} \tilde{e}_{1} ({\mkern 1mu} 1 + 2{\mkern 1mu} p_{\theta zi} ) - {\mkern 1mu} 6{\mkern 1mu} (p_{uyi} + p_{{uyj{\mkern 1mu} }} ) + 1} \right],\\ g_{128} {\mkern 1mu} {\mkern 1mu} &= \frac{{6{\mkern 1mu} }}{{\varDelta_{2} L_{m} }}(p_{\theta zj} + 2p_{\theta zi} p_{\theta zj} ),\end{aligned} $$
$$ g_{1212} {\mkern 1mu} {\mkern 1mu} = - \frac{{2p_{\theta zj} }}{{\varDelta_{2} }}\left[ {{\mkern 1mu} 3{\mkern 1mu} \tilde{e}_{2} ({\mkern 1mu} 1 + 2p_{\theta zi} ){\mkern 1mu} + {\mkern 1mu} {\mkern 1mu} 6({\mkern 1mu} {\mkern 1mu} {\mkern 1mu} p_{uyi} {\mkern 1mu} + {\mkern 1mu} {\mkern 1mu} p_{uyj} + {\mkern 1mu} {\mkern 1mu} p_{\theta zi} ) + 2} \right], $$

in which

$$ \begin{aligned} \tilde{e}_{1} = & \frac{{e_{1} }}{{L_{m} }},\,\tilde{e}_{2} = \frac{{e_{2} }}{{L_{m} }},\,p_{{uxi{\mkern 1mu} }} = \frac{EA}{{k_{{uxi{\mkern 1mu} }} L_{m} }},\\ p_{{uxj{\mkern 1mu} }} & = \frac{EA}{{k_{{uxj{\mkern 1mu} }} L_{m} }},\,p_{{\theta xi{\mkern 1mu} }} = \frac{GJ}{{k_{{\theta xi{\mkern 1mu} }} }},\,p_{{\theta xj{\mkern 1mu} }} = \frac{GJ}{{k_{{\theta xj{\mkern 1mu} }} }},\end{aligned} $$
$$ \begin{aligned} p_{{uyi{\mkern 1mu} }} & = \frac{{EI_{z} }}{{k_{{uyi{\mkern 1mu} }} L_{m}^{3} }},\,p_{{uyj{\mkern 1mu} }} = \frac{{EI_{z} }}{{k_{{uyj{\mkern 1mu} }} L_{m}^{3} }},\\ p_{{uzi{\mkern 1mu} }} & = \frac{{EI_{y} }}{{k_{{uzi{\mkern 1mu} }} L_{m}^{3} }},\,p_{{uzj{\mkern 1mu} }} = \frac{{EI_{y} }}{{k_{{uzj{\mkern 1mu} }} L_{m}^{3} }},\end{aligned} $$
$$ \begin{aligned} p_{\theta yi} & = \frac{{EI_{y} }}{{k_{\theta yi} L_{m} }},\,p_{\theta yj} {\mkern 1mu} = \frac{{EI_{y} }}{{k_{\theta yj} L_{m} }},\\ p_{\theta zi} & = \frac{{EI_{z} }}{{k_{\theta zi} L_{m} }},\,p_{\theta zj} {\mkern 1mu} = \frac{{EI_{z} }}{{k_{\theta zj} L_{m} }},\end{aligned} $$
$$\begin{aligned} \varDelta_{2} &= {\mkern 1mu} \left( {1 + 4{\mkern 1mu} p_{\theta zi} + 4p_{\theta zj} + 12p_{{\theta zi{\mkern 1mu} }} p_{\theta zj} {\mkern 1mu} } \right) \\ & \quad + 12\left( {p_{uyi} {\mkern 1mu} + p_{uyj} {\mkern 1mu} {\mkern 1mu} } \right)\left( {1 + {\mkern 1mu} {\mkern 1mu} p_{\theta zi} + p_{\theta zj} } \right), \end{aligned}$$
$$\begin{aligned} \varDelta_{3} &= {\mkern 1mu} \left( {{\mkern 1mu} 1 + {\mkern 1mu} 4p_{\theta yi} + {\mkern 1mu} 4p_{\theta yj} + 12p_{{\theta yi{\mkern 1mu} }} p_{\theta yj} {\mkern 1mu} } \right) \\ & \quad + 12\left( {p_{uzi} {\mkern 1mu} + p_{uzj} {\mkern 1mu} {\mkern 1mu} } \right)\left( {1 + {\mkern 1mu} {\mkern 1mu} p_{\theta yi} + {\mkern 1mu} p_{\theta yj} {\mkern 1mu} } \right). \end{aligned}$$

The stiffness matrix of the hybrid joint-beam element is

$$ \varvec{K}_{s} = \frac{E}{{L_{m} }}\left[ {\begin{array}{*{20}c} {A\phi_{1} } & 0 & 0 & 0 & 0 & 0 & { - A\phi_{1} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {\frac{{12I_{z} \phi_{2} }}{{L_{m}^{2} }}} & 0 & 0 & 0 & {\frac{{6I_{z} \phi_{3} }}{{L_{m} }}} & 0 & { - \frac{{12I_{z} \phi_{2} }}{{L_{m}^{2} }}} & 0 & 0 & 0 & {\frac{{6I_{z} \phi_{4} }}{{L_{m} }}} \\ {} & {} & {\frac{{12I_{y} \phi_{5} }}{{L_{m}^{2} }}} & 0 & { - \frac{{6I_{y} \phi_{6} }}{{L_{m} }}} & 0 & 0 & 0 & { - \frac{{12I_{y} \phi_{5} }}{{L_{m}^{2} }}} & 0 & { - \frac{{6I_{y} \phi_{7} }}{{L_{m} }}} & 0 \\ {} & {} & {} & {\frac{{GJ\phi_{8} }}{E}} & 0 & 0 & 0 & 0 & 0 & { - \frac{{GJ\phi_{8} }}{E}} & 0 & 0 \\ {} & {} & {} & {} & {4I_{y} \phi_{9} } & 0 & 0 & 0 & {\frac{{6I_{y} \phi_{6} }}{{L_{m} }}} & 0 & {2I_{y} \phi_{10} } & 0 \\ {} & {} & {} & {} & {} & {4I_{z} \phi_{11} } & 0 & {\frac{{6I_{z} \phi_{3} }}{{L_{m} }}} & 0 & 0 & 0 & {2I_{z} \phi_{12} } \\ {} & {} & {} & {} & {} & {} & {A\phi_{1} } & 0 & 0 & 0 & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {} & {\frac{{12I_{z} \phi_{2} }}{{L_{m}^{2} }}} & 0 & 0 & 0 & { - \frac{{6I_{z} \phi_{4} }}{{L_{m} }}} \\ {} & {} & {symm} & {} & {} & {} & {} & {} & {\frac{{12I_{y} \phi_{5} }}{{L_{m}^{2} }}} & 0 & {\frac{{6I_{y} \phi_{7} }}{{L_{m} }}} & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {\frac{{GJ\phi_{8} }}{E}} & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {4I_{y} \phi_{13} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {} & {4I_{z} \phi_{14} } \\ \end{array} } \right] $$

where the coefficients \( \phi_{1} {\mkern 1mu} {\mkern 1mu} \) ~ \( \phi_{14} {\mkern 1mu} {\mkern 1mu} \) are given as

$$\begin{aligned} \phi_{1} {\mkern 1mu} {\mkern 1mu} & = \frac{1}{{(1 + {\mkern 1mu} p_{uxi} + p_{uxj} )}},\\ \phi_{2} {\mkern 1mu} & = \frac{1}{{\varDelta_{2} }}(1 + p_{\theta zi} + p_{\theta zj} ){\mkern 1mu},\end{aligned} $$
$$ \begin{aligned} \phi_{3} & = \frac{{{\mkern 1mu} 1}}{{\varDelta_{2} }}\left[ {1 + 2{\mkern 1mu} {\mkern 1mu} p_{\theta zj} {\mkern 1mu} + 2{\mkern 1mu} \tilde{e}_{1} (1 + p_{\theta zi} + {\mkern 1mu} p_{\theta zj} {\mkern 1mu} )} \right],\\ \phi_{4} {\mkern 1mu} {\mkern 1mu} & = \frac{{{\mkern 1mu} 1}}{{\varDelta_{2} }}\left[ {1 + 2{\mkern 1mu} {\mkern 1mu} p_{\theta zi} + 2{\mkern 1mu} \tilde{e}_{2} ({\mkern 1mu} 1 + p_{\theta zi} + {\mkern 1mu} p_{\theta zj} )} \right],\end{aligned} $$
$$ \begin{aligned} \phi_{5} & = \frac{{{\mkern 1mu} 1}}{{\varDelta_{3} }}(1 + p_{\theta yi} + p_{\theta yj} ),\\ \phi_{6} & = \frac{1}{{\varDelta_{3} }}\left[ {1 + 2{\mkern 1mu} {\mkern 1mu} p_{\theta yj} + 2{\mkern 1mu} \tilde{e}_{1} ({\mkern 1mu} 1 + p_{\theta yi} + {\mkern 1mu} p_{\theta yj} )} \right],\end{aligned} $$
$$ \begin{aligned} \phi_{7} & = \frac{1}{{\varDelta_{3} }}\left[ {1 + 2{\mkern 1mu} {\mkern 1mu} p_{\theta yi} + 2{\mkern 1mu} \tilde{e}_{2} ({\mkern 1mu} 1 + p_{\theta yi} + {\mkern 1mu} p_{\theta yj} {\mkern 1mu} )} \right],\\ \phi_{8} {\mkern 1mu} {\mkern 1mu} & = \frac{1}{{(1 + {\mkern 1mu} p_{\theta xi} + p_{\theta xj} )}},\end{aligned} $$
$$\begin{aligned} \phi_{9} {\mkern 1mu} {\mkern 1mu} &= \frac{1}{{\varDelta_{3} }}\left[\vphantom{\tilde{e}_{1}^{2}\!}{1 + 3{\mkern 1mu} {\mkern 1mu} (\tilde{e}_{{1{\mkern 1mu} }} + 2p_{\theta yj} \tilde{e}_{{1{\mkern 1mu} }} + {\mkern 1mu} p_{\theta yj} {\mkern 1mu} ){\mkern 1mu} }\right. \\ & \quad \left.{+ 3{\mkern 1mu} {\mkern 1mu} \tilde{e}_{1}^{2} (1 + {\mkern 1mu} p_{\theta yi} + p_{\theta yj} ) + 3(p_{uzi} + p_{uzj} )} \right], \end{aligned}$$
$$\begin{aligned} \phi_{10} {\mkern 1mu} &= \frac{1}{{\varDelta_{3} }}\left[ {1 + {\mkern 1mu} {\mkern 1mu} 3(\tilde{e}_{{1{\mkern 1mu} }} + \tilde{e}_{2} {\mkern 1mu} {\mkern 1mu} + 2p_{\theta yi} \tilde{e}_{1} {\mkern 1mu} {\mkern 1mu} + 2{\mkern 1mu} p_{\theta yj} \tilde{e}_{2} ) }\right. \\ & \quad \left.{+ 6{\mkern 1mu} \tilde{e}_{1} {\mkern 1mu} \tilde{e}_{2} (1{\mkern 1mu} + p_{\theta yi} + {\mkern 1mu} p_{\theta yj} ) - 6(p_{uzi} + p_{uzj} )} \right], \end{aligned}$$
$$ \begin{aligned} \phi_{11} {\mkern 1mu} {\mkern 1mu} &= \frac{1}{{\varDelta_{2} }}\left[\vphantom{\tilde{e}_{1}^{2}\!}{1 + 3(\tilde{e}_{{1{\mkern 1mu} }} + 2p_{\theta zj} \tilde{e}_{1} {\mkern 1mu} + p_{\theta zj} ) }\right. \\ & \quad \left.{+ 3{\mkern 1mu} {\mkern 1mu} \tilde{e}_{1}^{2} (1{\mkern 1mu} + {\mkern 1mu} p_{\theta zi} + p_{\theta zj} ){\mkern 1mu} {\mkern 1mu} + 3(p_{uyi} + p_{uyj} )} \right], \end{aligned} $$
$$\begin{aligned} \phi_{12} {\mkern 1mu} {\mkern 1mu} &= \frac{1}{{\varDelta_{2} }}\left[ {1 + 3(\tilde{e}_{{1{\mkern 1mu} }} + \tilde{e}_{2} {\mkern 1mu} {\mkern 1mu} + 2p_{\theta zi} \tilde{e}_{1} {\mkern 1mu} + 2p_{\theta zj} \tilde{e}_{2} ) }\right. \\ & \quad \left.{+ 6{\mkern 1mu} \tilde{e}_{1} {\mkern 1mu} \tilde{e}_{2} (1{\mkern 1mu} {\mkern 1mu} + p_{\theta zi} {\mkern 1mu} + {\mkern 1mu} p_{\theta zj} ) - 6(p_{uyi} {\mkern 1mu} + p_{uyj} )} \right], \end{aligned} $$
$$\begin{aligned} \phi_{13} {\mkern 1mu} {\mkern 1mu} &= \frac{1}{{\varDelta_{3} }}\left[\vphantom{\tilde{e}_{1}^{2}\!}{1 + 3(\tilde{e}_{{2{\mkern 1mu} }} + 2p_{\theta yi} \tilde{e}_{2} {\mkern 1mu} + {\mkern 1mu} p_{\theta yi} ){\mkern 1mu} {\mkern 1mu} }\right. \\ & \quad \left.{ + 3{\mkern 1mu} {\mkern 1mu} \tilde{e}_{2}^{2} (1 + {\mkern 1mu} p_{\theta yi} + p_{\theta yj} ) + 3(p_{uzi} + p_{uzj} )} \right], \end{aligned} $$
$$ \begin{aligned} \phi_{14}& = \frac{1}{{\varDelta_{2} }}\left[ {1 + 3(\tilde{e}_{2} + 2p_{\theta zi} \tilde{e}_{2} + {\mkern 1mu} p_{\theta zi} ) }\right. \\ & \quad \left.{ + 3{\mkern 1mu} {\mkern 1mu} \tilde{e}_{2}^{2} (1{\mkern 1mu} {\mkern 1mu} + {\mkern 1mu} p_{\theta zi} {\mkern 1mu} {\mkern 1mu} + p_{\theta zj} ) + 3(p_{uyi} + p_{uyj} )} \right]. \end{aligned} $$

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Liu, F., Wang, L., Jin, D. et al. Equivalent continuum modeling of beam-like truss structures with flexible joints. Acta Mech. Sin. 35, 1067–1078 (2019). https://doi.org/10.1007/s10409-019-00872-z

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