Geometric Description

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Geometric Description

The geometry of a typical three dimensional shell element is defined using the Cartesian coordinates of the top and bottom surfaces corresponding to each node, thus

$\displaystyle \left\{\begin{array}{c} x\\ y\\ z \end{array}\right\}$	=	$\displaystyle \sum_{i=1}^{n} N_i(\xi,\eta) \left[ \frac{1+\zeta}{2} \left\{\beg... ...{2} \left\{\begin{array}{c} x_i^b\\ y_i^b\\ z_i^b \end{array}\right\} \right]$	(11)
	=	$\displaystyle \sum_{i=1}^{n} N_i(\xi,\eta) \left\{\begin{array}{c} x_i\\ y_i\\ z_i \end{array}\right\}_\zeta$	(12)

or consizly as

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

(13)

where the index i refer to the i'th node and n is the number of nodes per element. In order to construct the Jacobean matrix, the natural coordinate derivatives are obtained as

$\displaystyle \frac{\partial {\bf x}}{\partial \xi}$	=	$\displaystyle \sum_{i=1}^{n}\frac{\partial N_i(\xi,\eta)}{\partial \xi} {\bf x}_{i \zeta}$	(14)
$\displaystyle \frac{\partial {\bf x}}{\partial \eta}$	=	$\displaystyle \sum_{i=1}^{n}\frac{\partial N_i(\xi,\eta)}{\partial \eta} {\bf x}_{i \zeta}$	(15)
$\displaystyle \frac{\partial {\bf x}}{\partial \zeta}$	=	$\displaystyle \frac{1}{2}\sum_{i=1}^{n} N_i(\xi,\eta) \left[ {\bf x}_i^t - {\bf x}_i^b \right]$	(16)

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

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

(17)

Next: Kinematic Description Up: Three Dimensional Element Mechanics Previous: Coordinate Systems

A. Zeiny
2000-09-06