9/30/2013

3

rd_m

ech@

yaho

o.co

.in

Typical shapes of elements used in 3D analysis

y,v

x,u

z,w 4

12

3

Three DOF at each node u, v, w total four nodes => 3X4 = 12 DOF

Similar to triangle in 2Drd

_mec

h@ya

hoo.

co.in

Displacement functions of tetrahedralelement

zayaxaazyxwzayaxaazyxvzayaxaazyxu

1211109

8765

4321

),,(),,(),,(

+++=+++=+++=

{ } [ ] { } 11212313 = XU

Substitute displacements & nodal coordinates

=>

=>

{ } [ ] { } 1121212112 = Ad { } [ ] { }dA 1={ } [ ][ ] { } [ ] { } 1121231 =>= dNdAXU

rd_m

ech@

yaho

o.co

.in

Strain

=wvu

xz

yz

xy

z

y

x

zx

yz

xy

z

y

x

0

0

0

00

00

00

{ } [ ]{ }U=

9/30/2013

4

rd_m

ech@

yaho

o.co

.in

Expressing strain in terms of shape function derivatives

{ } [ ]{ }U= { } [ ]{ }dNU ={ } [ ][ ]{ } [ ] { } 112126 == dBdN

{ } [ ]{ } { } { } { } { }=

=n

iii

TTT PdVBUdSTUdVC12

1

Substitute strain in total PE expression

rd_m

ech@

yaho

o.co

.in

The stiffness matrix is given by,

Order of stiffness matrix [k] => 12X12 12 DOF 3 at each node No. of terms in a complete polynomial in 3D

(n+1)(n+2)(n+3)/6n order of polynomial.

[ ] [ ] [ ] [ ] dVBCBk TV

12666612

1212

=

rd_m

ech@

yaho

o.co

.in

Natural coordinate system Similar to that used in triangle element

N1 = V1/VN2 = V2/VN3 = V3/VN4 = V4/V

y,v

x,u

z,w 4

12

3P(x, y, z)

444

333

222

111

1111

6

zyxzyxzyxzyx

V =V1 = Volume P234