Problem 5: FEA Concepts (20 Points)

Nodal Interpolation

Consider a simple 3 node 2D triangular element with 2 DOFs for each node. The nodal displacements

are U₁ = V₁ = V₂ = 0; U₂ = 4 mm, U3 = 6 mm, and V3 = 2 mm.

E

The shape functions are:

|N₁ =

UI

N₂

N₂=

V3

P (5,5)

U3

U2

_a₁ +b₂x+c₁y

2A

a₂ + b₂x+c₂y

24

a₂+bx+c₂y

24

where A is the area of triangle. Calculate the displacement at point P (5, 5).

Node

1

2

3

a₁ = X₂Y₁X₂ Y₂

a₂ = X₂Y₁X₁Y₁

az = X₁ Y₂ - X₂Y₁

X Y

0

0

10

0

5 10

b₁ = y₁=Y₁

b₂ = y₁=Y₁

b₁ = y₁ - y₂

C₁ = X3 - X₂

C₂ = X₁ − X3

C₂ = x₂ −X₁