Total Lagrangian Description for Two-Dimensional Shells

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Total Lagrangian Description for Two-Dimensional Shells

In case of plane and axisymmetric shells, the formulation is deduced from the three-dimensional shell formulation. The strain vector is given by

$\displaystyle {\mbox{\boldmath$\epsilon$ }}$	=	$\displaystyle \left\{\begin{array}{c} \epsilon_{x'}\\ \gamma_{x'y'}\\ \epsilo... ...}\left[\frac{\partial v'}{\partial x'}\right] ^2\\ 0\\ 0 \end{array} \right\}$	(91)
	=	$\displaystyle \sum_{i=1}^n \left( {\bf B}_{o} + {\bf B}_{l} \right)_i \left\{\begin{array}{c} \overline{{\bf u }}_i \\ \theta_i \end{array}\right\}$	(92)

where $\epsilon_{z'}=0$ in case of plane shell. The nonlinear strain component is rewritten as

$\begin{displaymath}\mbox{{\boldmath$\epsilon$ }}_l = \frac{1}{2} \frac{\partial ... ...{array} \right\} =\frac{1}{2} \epsilon_{c} \left\{ D \right\} \end{displaymath}$

(93)

The coupling strain is related to the nodal degrees of freedom by

$\begin{displaymath}\epsilon_{c} = \sum_{i=1}^n \left[\begin{array}{cc} {\bf G... ...egin{array}{c} {\bf u}_i \\ \theta_i \end{array} \right\} \end{displaymath}$

(94)

The matrix ${\bf B}_l$ is then given by

$\displaystyle \left[B_l\right]_i$	=	$\displaystyle \left\{D\right\}_i \left[\begin{array}{cc} {\bf G}^u& G^\theta \end{array} \right]_i$
	=	$\displaystyle {\bf D}_i \; {\bf G}_i$	(95)
$\displaystyle {\bf G}^u$	=	$\displaystyle \left[ \begin{array}{cc} 0 & \frac{\partial N_i}{\partial x'} \end{array}\right] \left [ q \right]$	(96)
$\displaystyle G^\theta$	=	$\displaystyle \frac{t_i}{2} \left[ \begin{array}{cc} 0 & \frac{ \partial \left(... ...i \right) }{\partial x'} \end{array}\right] \left [ q \right]\left\{R_i\right\}$	(97)

The stiffness matrix is then given by

$\begin{displaymath}\left[ K \right] =\int_{\Omega} \left [ \left[ B_o+B_l \right... ...igma_{x'} \left[ G \right]^T \left[ G \right] \right] d\Omega \end{displaymath}$

(98)

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A. Zeiny
2000-09-06