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A new absolute nodal coordinate formulation beam element with multilayer circular cross section

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Abstract

A systematic numerical integration method is applied to the absolute nodal coordinate formulation (ANCF) fully parameterized beam element with smooth varying and continuous cross section. Moreover, the formulation for the integration points and weight coefficients are given in the method which is used to model the multilayer beam with a circular cross section. To negate the effect of the bending stiffness for the element used to model the high-voltage electrical wire, the general continuum mechanical approach is adjusted. Additionally, the insulation cover for some particular types of the wire is described by the nearly incompressible Mooney–Rivlin material model. Finally, a static problem is presented to prove the accuracy and convergence properties of the element, and a dynamic problem of a flexible pendulum is simulated whereby the balance of the energy can be ensured. An experiment is carried out in which a wire is released as a pendulum and falls on a steel rod. The configurations of the wire are captured by a high-speed camera and compared with the simulation results. The feasibility of the wire model can therefore be demonstrated.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 11802072) and the Fundamental Research Funds for the Central Universities (Grant HIT. NSRIF 2018032). The experiment was completed with the help of China Electric Power Research Institute.

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Authors and Affiliations

  1. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, China

    Peng Lan, Qinglong Tian & Zuqing Yu

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  1. Peng Lan
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  2. Qinglong Tian
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  3. Zuqing Yu
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Corresponding author

Correspondence to Zuqing Yu.

Appendix

Appendix

The global position vector is

$${\mathbf{r}} = {\mathbf{Se}}.$$

The position vector gradients are defined as

$$\left\{ \begin{aligned} \frac{{\partial {\mathbf{r}}}}{\partial x} = {\mathbf{r}}_{x} = \frac{{\partial {\mathbf{S}}(x,y,z) \cdot {\mathbf{e}}}}{\partial x} = {\mathbf{S}}_{x} \cdot {\mathbf{e}}, \hfill \\ \frac{{\partial {\mathbf{r}}}}{\partial y} = {\mathbf{r}}_{y} = \frac{{\partial {\mathbf{S}}(x,y,z) \cdot {\mathbf{e}}}}{\partial y} = {\mathbf{S}}_{y} \cdot {\mathbf{e}}, \hfill \\ \frac{{\partial {\mathbf{r}}}}{\partial z} = {\mathbf{r}}_{z} = \frac{{\partial {\mathbf{S}}(x,y,z) \cdot {\mathbf{e}}}}{\partial z} = {\mathbf{S}}_{z} \cdot {\mathbf{e}}. \, \hfill \\ \end{aligned} \right.$$

Cr is the right Cauchy–Green strain tensors

$${\mathbf{C}}_{r} = {\mathbf{J}}^{\text{T}} {\mathbf{J}},$$

where J is the matrix of position vector gradients, which is defined as

$${\mathbf{J}} = [{\mathbf{r}}_{x} ,{\mathbf{r}}_{y} ,{\mathbf{r}}_{z} ].$$

Because the right Cauchy–Green strain tensors are symmetric, one can identify six independent strain components that can be used to define the following strain vector

$${\mathbf{C}}_{rv} = [{\mathbf{C}}_{r} (1,1),{\mathbf{C}}_{r} (2,2),{\mathbf{C}}_{r} (3,3),{\mathbf{C}}_{r} (1,2),{\mathbf{C}}_{r} (1,3),{\mathbf{C}}_{r} (2,3)]^{\text{T}} .$$

The three invariants I1, I2 and I3 are defined as

$$\left\{ {\begin{array}{l} {I_{1} = \text{tr}\left( {{\mathbf{C}}_{r} } \right) \, } ,\\ {I_{2} = \frac{1}{2}\left\{ { \, \left[ {\text{tr}\left( {{\mathbf{C}}_{r} } \right)} \right]^{2} - \text{tr}\left( {{\mathbf{C}}_{r}^{2} } \right)} \right\}} \\ {I_{3} = \det \left( {{\mathbf{C}}_{r} } \right)\,{=}\left| {{\mathbf{C}}_{r} }\right|. \, } \\ \end{array} } \right.,$$

The three invariants \(I_{1}\), \(I_{2}\) and \(I_{3}\) can be written more explicitly as

$$\left\{ \begin{aligned} I_{1} =\, & {\text{tr}}\left( {{\mathbf{C}}_{r} } \right) \\ \, =\, & {\mathbf{C}}_{rv} (1) + {\mathbf{C}}_{rv} (2) + {\mathbf{C}}_{rv} (3), \\ I_{2} =\, & \frac{1}{2}\left\{ { \, \left[ {{\text{tr}}\left( {{\mathbf{C}}_{r} } \right)} \right]^{2} - {\text{tr}}\left( {{\mathbf{C}}_{r}^{2} } \right)} \right\} \\ {=}\, & {\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (2) + {\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (3) + {\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (3) - {\mathbf{C}}_{rv}^{2} (4) - {\mathbf{C}}_{rv}^{2} (5) - {\mathbf{C}}_{rv}^{2} (6), \\ I_{3} =\, & \det \left( {{\mathbf{C}}_{r} } \right){ = }\left| {{\mathbf{C}}_{r} } \right| \\ {=}\, & {\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (3) + 2{\mathbf{C}}_{rv} (4){\mathbf{C}}_{rv} (5){\mathbf{C}}_{rv} (6) - {\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv}^{2} (6) - {\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv}^{2} (5) - {\mathbf{C}}_{rv} (3){\mathbf{C}}_{rv}^{2} (4), \\ \end{aligned} \right.$$
$$\begin{aligned} {\mathbf{C}}_{rv} (1) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{x} {\mathbf{e}},{\mathbf{C}}_{rv} (2) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{\text{y}}^{\text{T}} {\mathbf{S}}_{y} {\mathbf{e}},{\mathbf{C}}_{rv} (3) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}}, \hfill \\ {\mathbf{C}}_{rv} (4) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{y} {\mathbf{e}},{\mathbf{C}}_{rv} (5) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}},{\mathbf{C}}_{rv} (6) = {\mathbf{e}}^{\text{T}} {\mathbf{S}}_{\text{y}}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}}. \hfill \\ \end{aligned}$$

\({{\partial I_{1} } \mathord{\left/ {\vphantom {{\partial I_{1} } {\partial {\mathbf{e}}}}} \right. \kern-0pt} {\partial {\mathbf{e}}}}\), \({{\partial I_{2} } \mathord{\left/ {\vphantom {{\partial I_{2} } {\partial {\mathbf{e}}}}} \right. \kern-0pt} {\partial {\mathbf{e}}}}\) and \({{\partial I_{3} } \mathord{\left/ {\vphantom {{\partial I_{3} } {\partial {\mathbf{e}}}}} \right. \kern-0pt} {\partial {\mathbf{e}}}}\) can be written as

$$\frac{{\partial I_{1} }}{{\partial {\mathbf{e}}}} = \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}} + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}} + \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}},$$
$$\begin{aligned} \frac{{\partial I_{2} }}{{\partial {\mathbf{e}}}} = & \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (2) + {\mathbf{C}}_{rv} (1)\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}} +\, \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (3) \\ & + {\mathbf{C}}_{rv} (1)\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}} + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (3) + {\mathbf{C}}_{rv} (2)\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}} \\ & -\, 2{\mathbf{C}}_{rv} (4)\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}} -\, 2{\mathbf{C}}_{rv} (5)\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}} - 2{\mathbf{C}}_{rv} (6)\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}, \\ \end{aligned}$$
$$\begin{aligned} \frac{{\partial I_{3} }}{{\partial {\mathbf{e}}}} = & \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (3) + {\mathbf{C}}_{rv} (1)\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (3) + {\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (2)\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}} \, \\ \, & \quad +\, 2\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (5){\mathbf{C}}_{rv} (6) + 2{\mathbf{C}}_{rv} (4)\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv} (6) + 2{\mathbf{C}}_{rv} (4){\mathbf{C}}_{rv} (5)\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}} \\ \, & \quad -\, \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv}^{2} (6) - 2{\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (6)\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}} \\ \, & \quad -\, \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv}^{2} (5) - 2{\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (5)\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}} \\ \, & \quad -\, \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}{\mathbf{C}}_{rv}^{2} (4) - 2{\mathbf{C}}_{rv} (3){\mathbf{C}}_{rv} (4)\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}. \\ \end{aligned}$$

\({{\partial^{2} I_{1} } \mathord{\left/ {\vphantom {{\partial^{2} I_{1} } {\partial {\mathbf{e}}^{2} }}} \right. \kern-0pt} {\partial {\mathbf{e}}^{2} }}\), \({{\partial^{2} I_{2} } \mathord{\left/ {\vphantom {{\partial^{2} I_{2} } {\partial {\mathbf{e}}^{2} }}} \right. \kern-0pt} {\partial {\mathbf{e}}^{2} }}\) and \({{\partial^{2} I_{3} } \mathord{\left/ {\vphantom {{\partial^{2} I_{3} } {\partial {\mathbf{e}}^{2} }}} \right. \kern-0pt} {\partial {\mathbf{e}}^{2} }}\) can be written as

$$\frac{{\partial^{2} I_{1} }}{{\partial {\mathbf{e}}^{2} }} = \frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }} + \frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }} + \frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }},$$
$$\begin{aligned} \frac{{\partial^{2} I_{2} }}{{\partial {\mathbf{e}}^{2} }} = & \frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (2) + \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} + {\mathbf{C}}_{rv} (1)\frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }} + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad +\, \frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} + {\mathbf{C}}_{rv} (1)\frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }} + \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad +\, \frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} + {\mathbf{C}}_{rv} (2)\frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }} + \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad -\, 2\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} - 2\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} - 2\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad -\, 2{\mathbf{C}}_{rv} (4)\frac{{\partial^{2} {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}^{2} }} - 2{\mathbf{C}}_{rv} (5)\frac{{\partial^{2} {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}^{2} }} - 2{\mathbf{C}}_{rv} (6)\frac{{\partial^{2} {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}^{2} }}, \\ \end{aligned}$$
$$\begin{aligned} \frac{{\partial^{2} I_{3} }}{{\partial {\mathbf{e}}^{2} }} = & \frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (2) \\ \, & \quad +\, \frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (3) + \frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (1) \\ \, & \quad +\, \frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (2) + \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (2) + \frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (1) \\ \, & \quad +\, 2\frac{{\partial^{2} {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (5){\mathbf{C}}_{rv} (6) + 2\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (6) + 2\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (5) \\ \, & \quad +\, 2\frac{{\partial^{2} {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (4){\mathbf{C}}_{rv} (6) + 2\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (6) + 2\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (4) \\ \, & \quad +\, 2\frac{{\partial^{2} {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv} (4){\mathbf{C}}_{rv} (5) + 2\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (5) + 2\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (4) \\ \, & \quad -\, \frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv}^{2} (6) - 2\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (6) - 2{\mathbf{C}}_{rv} (6)\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad -\, 2\varvec{C}_{rv} (1)\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} -\, 2{\mathbf{C}}_{rv} (1){\mathbf{C}}_{rv} (6)\frac{{\partial^{2} {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}^{2} }} - \frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv}^{2} (5) \\ \, & \quad -\, 2\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (5) - 2{\mathbf{C}}_{rv} (5)\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} - 2{\mathbf{C}}_{rv} (2)\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} \\ \, & \quad -\, 2{\mathbf{C}}_{rv} (2){\mathbf{C}}_{rv} (5)\frac{{\partial^{2} {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}^{2} }} - \frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }}{\mathbf{C}}_{rv}^{2} (4) - 2\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} {\mathbf{C}}_{rv} (4) \\ \, & \quad -\, 2{\mathbf{C}}_{rv} (4)\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} -\, 2{\mathbf{C}}_{rv} (3)\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}\left( {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}}} \right)^{\text{T}} -\, 2{\mathbf{C}}_{rv} (3){\mathbf{C}}_{rv} (4)\frac{{\partial^{2} {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}^{2} }}. \\ \end{aligned}$$
$$\begin{array}{*{20}c} {\left\{ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}}} = 2{\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{x} {\mathbf{e}},} \\ {\frac{{\partial {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}}} = 2{\mathbf{S}}_{\text{y}}^{\text{T}} {\mathbf{S}}_{y} {\mathbf{e}},} \\ {\frac{{\partial {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}}} = 2{\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}},} \\ {\frac{{\partial {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}}} = {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{y} {\mathbf{e}} + {\mathbf{S}}_{y}^{\text{T}} S_{x} {\mathbf{e}},} \\ {\frac{{\partial {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}}} = {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}} + {\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{x} {\mathbf{e}},} \\ {\frac{{\partial {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}}} = {\mathbf{S}}_{y}^{\text{T}} {\mathbf{S}}_{z} {\mathbf{e}} + {\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{y} {\mathbf{e}}.} \\ \end{array} } \right.} & {\left\{ {\begin{array}{*{20}c} {\frac{{\partial^{2} {\mathbf{C}}_{rv} (1)}}{{\partial {\mathbf{e}}^{2} }} = 2{\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{x} ,} \\ {\frac{{\partial^{2} {\mathbf{C}}_{rv} (2)}}{{\partial {\mathbf{e}}^{2} }} = 2{\mathbf{S}}_{\text{y}}^{\text{T}} {\mathbf{S}}_{y} ,} \\ {\frac{{\partial^{2} {\mathbf{C}}_{rv} (3)}}{{\partial {\mathbf{e}}^{2} }} = 2{\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{z} ,} \\ {\frac{{\partial^{2} {\mathbf{C}}_{rv} (4)}}{{\partial {\mathbf{e}}^{2} }} = {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{y} + {\mathbf{S}}_{y}^{\text{T}} {\mathbf{S}}_{x} ,} \\ {\frac{{\partial^{2} {\mathbf{C}}_{rv} (5)}}{{\partial {\mathbf{e}}^{2} }} = {\mathbf{S}}_{x}^{\text{T}} {\mathbf{S}}_{z} + {\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{x} ,} \\ {\frac{{\partial^{2} {\mathbf{C}}_{rv} (6)}}{{\partial {\mathbf{e}}^{2} }} = {\mathbf{S}}_{y}^{\text{T}} {\mathbf{S}}_{z} + {\mathbf{S}}_{z}^{\text{T}} {\mathbf{S}}_{y} .} \\ \end{array} } \right.} \\ \end{array}$$

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Lan, P., Tian, Q. & Yu, Z. A new absolute nodal coordinate formulation beam element with multilayer circular cross section. Acta Mech. Sin. 36, 82–96 (2020). https://doi.org/10.1007/s10409-019-00897-4

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