Shape Functions in Natural Coordinates

To determine the shape functions, we evaluate T at the 4 nodes and use the following conditions :

,
	(24.8)

Substituting these conditions in the approximation (24.6), we get the following equations:

	(24.9)

Solving these equations, we obtain the following expressions for in terms of :

	(24.10)

Substituting the above expressions for in the approximation (24.6) and rearranging the terms, we get

(24.11)

where the shape functions are given by :

	(24.12)

Note that the shape factions satisfy the following property :

(24.13)

These are called as bi-linear Lagrangian shape functions and the corresponding rectangular element is called as the first order Lagrangian rectangular element . Note that, the shape functions can also be expressed as :

	(24.14)

where

	(24.15)

Here, and are the 1-D linear Lagrangian shape functions (eqs. 11.14-11.15) along the edges parallel to -axis. Similarly and are the 1-D linear Lagrangian shape functions along the edges parallel to -axis. Thus, the bi-linear Lagrangian shape functions (of the first order Lagrangian rectangular element) can be constructed as tensor products of the 1-D 1 st order (i.e., linear) Lagrangian shape functions along the edges parallel to and axes.