Computer Aided Engineering Design

FINITE ELEMENT METHOD 331

Solving for displacements gives U =^15
16

× 10 –3× [0 0 0.18 –0.24 0.18 –0.24 0 0]T. The displaced

plate is shown in Figure 11.14 (dashed lines). The vector containing the applied and support loads can
be determined as KU = 10^3 × [0 0 2 –1 0 0 –2 1]T.

0 0.5 1 1.5 2 2.5

1

0.8

0.6

0.4

0.2

0

• 0.2


• 0.4

Figure 11.14 Resultant displacements (scaled) for Example 11.4

11.7 Four-Node Elements

Triangular elements are easy to implement with the drawback that the strain throughout the element
is a constant. To get better stress field approximations, we may have to use a fine mesh of triangular
elements. Otherwise, four-node quadrilateral elements, shown in Figure 11.15, may be employed.
The procedure to determine the finite element stiffness matrix is similar to that for a triangular
element. The first step is to determine the interpolation or shape functions. Like in case of a triangular
element, the displacement u would depend on the nodal displacements along the x direction. Thus, u
would depend on u 1 ,... , u 4 which could be modeled using the polynomial approximation

y

x

v 4

v 3

u 4

3 u 3

v 1

4

v

u

P

v 2

u 1

1

2

u 2

η

4 (–1, 1) (1, 1)

ξ

1 (–1, –1) 2 (1, –1)

Figure 11.15 A four-node quadrilateral element