$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

When the node of a deformable shell belongs to a rigid body, we need to substitute the nodal degrees of freedom with the rigid body degrees of freedom. The positions of the shell front face and back face nodes are $$\begin{equation} \begin{aligned}\mathbf{x}_{b} & =\mathbf{r}+\boldsymbol{\Lambda}\cdot\left(\mathbf{X}_{b}-\mathbf{R}\right)\equiv\mathbf{r}+\mathbf{a}_{b}\\ \mathbf{y}_{b} & =\mathbf{r}+\boldsymbol{\Lambda}\cdot\left(\mathbf{Y}_{b}-\mathbf{R}\right)\equiv\mathbf{r}+\mathbf{b}_{b} \end{aligned} \,,\label{eq:fbs-rigid-shell} \end{equation}$$ where $\mathbf{r}$ is the current position of the rigid body center of mass and $\mathbf{R}$ is its initial position; $\boldsymbol{\Lambda}$ is the rotation tensor for the rigid body. We assume that $\mathbf{x}_{b}$ and $\mathbf{y}_{b}$ are connected to the same rigid body. From these relations it follows that virtual displacements are $$\begin{equation} \begin{aligned}\delta\mathbf{u}_{a} & =\delta\mathbf{r}-\hat{\mathbf{a}}_{a}\cdot\delta\boldsymbol{\theta}\\ \delta\mathbf{w}_{b} & =\delta\mathbf{r}-\hat{\mathbf{b}}_{a}\cdot\delta\boldsymbol{\theta} \end{aligned} \,,\label{eq:fbs-rsi-virtual} \end{equation}$$ and incremental displacements are $$\begin{equation} \begin{aligned}\Delta\mathbf{u}_{b} & =\Delta\mathbf{r}-\hat{\mathbf{a}}_{b}\cdot\Delta\boldsymbol{\theta}\\ \Delta\mathbf{w}_{b} & =\Delta\mathbf{r}-\hat{\mathbf{b}}_{b}\cdot\Delta\boldsymbol{\theta} \end{aligned} \,,\label{eq:fbs-rsi-incremental} \end{equation}$$ where $\hat{\mathbf{a}}$ is the skew-symmetric tensor whose dual vector is $\mathbf{a}$ , such that $\hat{\mathbf{a}}\cdot\mathbf{v}=\mathbf{a}\times\mathbf{v}$ for any vector $\mathbf{v}$ . When nodes are flexible (when they do not belong to any rigid body), the virtual work has the general form $$\begin{equation} \delta W=\sum_{a=1}^{m}\delta\mathbf{u}_{a}\cdot\mathbf{f}_{a}^{u}+\delta\mathbf{w}_{a}\cdot\mathbf{f}_{a}^{w}+\delta p_{a}\,f_{a}^{p}=\sum_{a=1}^{m}\left[\begin{array}{ccc} \delta\mathbf{v}_{a} & \delta\mathbf{w}_{a} & \delta p_{a}\end{array}\right]\left[\begin{array}{c} \mathbf{f}_{a}^{u}\\ \mathbf{f}_{a}^{w}\\ f_{a}^{p} \end{array}\right]\,,\label{eq:fbs-rsi-work} \end{equation}$$ where $p$ denotes any additional degree-of-freedom at that node. If node $a$ is rigid we get $$\begin{equation} \left[\begin{array}{ccc} \delta\mathbf{u}_{a} & \delta\mathbf{w}_{a} & \delta p_{a}\end{array}\right]=\left[\begin{array}{ccc} \delta\mathbf{r} & \delta\boldsymbol{\theta} & \delta p_{a}\end{array}\right]\left[\begin{array}{ccc} \mathbf{I} & \mathbf{I} & \mathbf{0}\\ \hat{\mathbf{a}}_{a} & \hat{\mathbf{b}}_{a} & \mathbf{0}\\ \mathbf{0} & \mathbf{0} & 1 \end{array}\right]\,.\label{eq:fbs-rsi-a-rigid} \end{equation}$$ If node $b$ is rigid we get $$\begin{equation} \left[\begin{array}{c} \Delta\mathbf{u}_{b}\\ \Delta\mathbf{w}_{b}\\ \Delta p_{b} \end{array}\right]=\left[\begin{array}{ccc} \mathbf{I} & -\hat{\mathbf{a}}_{b} & \mathbf{0}\\ \mathbf{I} & -\hat{\mathbf{b}}_{b} & \mathbf{0}\\ \mathbf{0} & \mathbf{0} & 1 \end{array}\right]\left[\begin{array}{c} \Delta\mathbf{r}\\ \Delta\boldsymbol{\theta}\\ \Delta p_{b} \end{array}\right]\,.\label{eq:fbs-rsi-b-rigid} \end{equation}$$ When node $a$ belongs to a rigid body, the expression for $\delta W$ must be substituted with $$\begin{equation} \begin{aligned}\delta W & =\sum_{a=1}^{m}\left[\begin{array}{ccc} \delta\mathbf{u}_{a} & \delta\mathbf{w}_{a} & \delta p_{a}\end{array}\right]\left[\begin{array}{c} \mathbf{f}_{a}^{u}\\ \mathbf{f}_{a}^{w}\\ f_{a}^{p} \end{array}\right]\\ & =\sum_{a=1}^{m}\left[\begin{array}{ccc} \delta\mathbf{r} & \delta\boldsymbol{\omega} & \delta p_{a}\end{array}\right]\left[\begin{array}{c} \mathbf{f}_{a}^{u}+\mathbf{f}_{a}^{w}\\ \hat{\mathbf{a}}_{a}^{n+\alpha}\cdot\mathbf{f}_{a}^{u}+\hat{\mathbf{b}}_{a}^{n+\alpha}\cdot\mathbf{f}_{a}^{w}\\ f_{a}^{p} \end{array}\right] \end{aligned} \,.\label{eq:fbs-rsi-work-2} \end{equation}$$