n
1 2 x 3 y 4 z u(x, y, z) Ni 1 2 xi 3 yi 4 zi
i 1
2
Analysis of Three – Dimensional Problems
n
1 2 x 3 y 4 z u(x, y, z) Ni 1 2xi 3 yi 4zi i 1
elements.
0
Analysis of Three – Dimensional Problems
Consider a three - dimensional element having n nodal points.
For the rigid - body movement in x - direction we require the
(5.27)
4
Analysis of Three – Dimensional Problems
5.3.2 Strain – Displacement Relations
For a three – dimensional analysis we require all the six
strain components, that is x , y , z , yz , zx and xy .
displacement relations are given by
x
u x
n i 1
Ni x
ui
y
v y
n i 1
Ni y
vi
z
w z
n i 1
Ni z
wi
yz
v z
w y
n i 1
Ni z
vi
n i 1
Ni y
wi
zx
w x
u z
n i 1
Ni x
wi
n i 1
Ni z
following:
u(x, y, z) 1 2x 3 y 4z
(5.23)
where all the alphas are constants. But for the 3D n - node
element we can say the x component of displacement at any point
n
Ni 1
i 1
n
Ni xi x
i 1
n
Ni yi y
i 1
n
Ni zi z
i 1
(5.26)
3
Analysis of Three – Dimensional Problems
n
Ni xi x
i 1
n
Ni yi y
i 1
n
Ni zi z
i 1
Above three conditions are the mapping formulations(映
象公式) for isoparametric elements and hence are satisfied
when we use isoparametric elements. We only need to
ensure that for the shape functions
n
Ni 1
i 1
This becomes
1
2
x
3
y
4z
n
N
i
1
n
Ni
xi
2
n
Ni
yi
3
n
Ni zi 4
i1
i1
i1
i1
Equating coefficients(使系数相等), we come up with necessary
conditions for rigid – body movement and constant strain, that is
smaller; ②The displacement field for an element must reflect rigid –
body motion when the nodal displacements are compatible
with rigid – body motion.
If these convergence requirements are satisfied in the parent elements, they will prevail(奏效) in the curved geometry of the
The element strains are obtained in terms of derivatives of element displacements with respect to the Cartesian coordinates.
5
For a n – nodeAntharelyesi-s odfimTehnrsieoena–l Deilmemeennst,iotnhael Pstrraoibn le–ms
ui
xy
u y
v x
n i 1
Ni y
ui
n i 1
Ni x
vi
(5.34)
u
v
n
ui
Ni ( ,,
) vi
w i1
wi
6
Analysis of Three – Dimensional Problems
Strain – Displacement Transformation
u(xi , yi , zi ) ui 1 2 xi 3 yi 4 zi
(5.25)
n
Now replacing the ui in u(x, y, z) Niui , we have i 1
n
u(x, y, z) Ni 1 2 xi 3 yi 4 zi i 1
Considering u 1 2 x 3 y 4 z ，thus
n
u(x, y, z) Niui i 1
where ui denotes the nodal value of u(x, y, z) .
(5.24)
1
Analysis of Three – Dimensional Problems

At any node i, we then require that
Analysis of Three – Dimensional Problems
A Comment on Convergence for 3D Curved Elements
As we know, the interpolation functions must be such that ① Constant strain is maintained as the elements are made