The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 433

FIGURE P6.26

FIGURE P6.27

6.27 Derive the stiffness matrix of the triangular element shown in Fig. P6.27 by mapping
it to a four-node square element.

Hint: The coordinate transformation is

1
4 [

x = ( 1 − ξ)( 1 − η) x 1 + ( 1 + ξ)( 1 − η) x 2 + 2 ( 1 + η) x 3 ]

1
4 [

y = ( 1 − ξ)( 1 − η) y 1 + ( 1 + ξ)( 1 − η) y 2 + 2 ( 1 + η) y 3 ]

For this coordinate transformation line 3-4 of the square element corresponds to
point 3 of the triangle. Use N 1

1
4

1
4

1
2

= ( 1 − ξ)( 1 − η) ,N 2 = ( 1 + ξ)( 1 − η) ,N 3 = ( 1 + η).

2

6.28 Construct the shape function N i ] for the truss element shown

in Fig. P6.28a. Use this shape function and two elements to calculate the displacement
at x

(ξ ) = sin[ π( a i + b i ξ + c i ξ )

2 of the bar shown in Fi g . P6.28b. Compare the results to the exact solution
which is obtained from du

= L /

/ =

p x /(

).

dx EA