EMA 405
3-D Elements
Introduction
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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
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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
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Boundary Conditions continued
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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
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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