EMA 405
3-D Elements
Introduction
3-D elements have 3 degrees of freedom per
node (ux, uy, uz)
The two fundamental shapes are hexahedral and
tetrahedral elements
Comments
Mesh generation is easier with
tetrahedral elements
Tetrahedral elements tend to produce
more degrees of freedom in a given
model
Try mapped meshing if you want to use
hex elements
4-node Tetrahedral element
u a1 a2 x a3 y a4 z
v a5 a6 x a7 y a8 z
w a9 a10 x a11 y a12 z
Constant Strain
10-node Tetrahedral element
u a1 a2 x a3 y a4 z
a5 xy a6 xz a7 yz a8 x
2
a9 y a10 z
2
Linear Strain
2
8-node Hexahedral element
u a1 a2 x a3 y a4 z
a5 xy a6 xz a7 yz a8 xyz
v a9 a10 x a11 y a12 z
a13 xy a14 xz a15 yz a16 xyz
w a17 a18 x a19 y a20 z
a21 xy a22 xz a23 yz a24 xyz
Linear Strain
20-node Hexahedral element
u a1 a2 x a3 y a4 z
a5 xy a6 xz a7 yz a8 x 2
a9 y 2 a10 z 2 a11 x 2 y a12 x 2 z
a13 y 2 x a14 y 2 z a15 z 2 x a16 z 2 y
a17 x yz a18 y xz a19 z xy a20 xyz
2
2
2
Quadratic Strain
Not compatible with 10-node tetrahedral elements
Boundary Conditions
We have to restrict 3 translational rigid
body modes and 3 rotational rigid body
modes
We can restrict a single node in all 3
directions to take care of the translational
modes
Rotations are trickier
Boundary Conditions continued
Consider a 2-D case
With one node restricted in all directions, rotation
about z-axis is possible
Restricting one node on x-axis in y-direction will
prevent rotation about z
Do similar things to restrict rotations about x and y
Restrict in x and y
y
Restrict in y
x
Example – hollow
cylinder with hole
E=100 Gpa, =0.3
L=80 mm
Pressure load=1 Mpa
(applied on end faces)
Inner radius=7.5 mm
Outer radius=10 mm
Radius of hole=2.5
mm
Theory says peak
stress is 3.65 MPa
0
