Geometric Description

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Geometric Description

The geometry of a typical plane shell element is also defined using the Cartesian coordinates of the top and bottom surface corresponding to each node, thus

$\displaystyle \left\{\begin{array}{c} x\\ y \end{array}\right\}$	=	$\displaystyle \sum_{i=1}^{n} N_i(\xi) \left[ \frac{1+\eta}{2} \left\{\begin{arr... ...c{1-\eta}{2} \left\{\begin{array}{c} x_i^b\\ y_i^b \end{array}\right\} \right]$	(51)
	=	$\displaystyle \sum_{i=1}^{n} N_i(\xi) \left\{\begin{array}{c} x_i\\ y_i \end{array}\right\}_\eta$	(52)

or consizly as

$\begin{displaymath}{\bf x} = \sum_{i=1}^{n} N_i(\xi) {\bf x}_{i \eta} \end{displaymath}$

(53)

where the index i refer to the i'th node. In order to construct the Jacobian matrix, the natural coordinate derivatives are obtained as

$\displaystyle \frac{\partial {\bf x}}{\partial \xi}$	=	$\displaystyle \sum_{i=1}^{n}\frac{\partial N_i(\xi)}{\partial \xi} {\bf x}_{i \eta}$	(54)
$\displaystyle \frac{\partial {\bf x}}{\partial \eta}$	=	$\displaystyle \frac{1}{2}\sum_{i=1}^{n} N_i(\xi) \left[ {\bf x}_i^t - {\bf x}_i^b \right]$	(55)

The Jacobian matrix required to transfer derivatives between the global and natural coordinate systems is then given by

$\begin{displaymath}{\bf J}= [\frac{\partial {\bf x}}{\partial \xi} \;\; \frac{\partial {\bf x}}{\partial \eta}]^T \end{displaymath}$

(56)

A. Zeiny
2000-09-06