Biomedical Engineering Reference
In-Depth Information
Fig. 2.6
Essential boundary condition nonaligned with the global axis
loads (external and body forces) and to the unknown reactions due the imposed
displacement constrains. With the Eq. (
2.104
) it is assumed that the displacement
components considered are axial aligned with the prescribed displacements. If this
is not the case it is required the identification of all prescribed displacement
orientations and transform locally the discrete equilibrium equations to correspond
to the global axis. Thus,
u ¼ T u
ð
2
:
105
Þ
where u is the vector of nodal point degrees of freedom in the required directions.
The transformation matrix T is defined by Eq. (
2.106
) and Fig.
2.6
, which is a
typical representation of the constrained displacements in 2D and 3D analysis.
2
3
u
x
v
x
w
x
u
x
v
x
4
5
T
2D
¼
and
T
3D
¼
u
y
v
y
w
y
ð
2
:
106
Þ
u
y
v
y
u
z
v
z
w
z
Using Eqs. (
2.105
) and (
2.106
) it is possible to write,
K u
þ
M u ¼ f
ð
2
:
107
Þ
where,
M ¼ T
T
MT
ð
2
:
108
Þ
K
¼
T
T
KT
ð
2
:
109
Þ
f ¼ T
T
f
ð
2
:
110
Þ
Notice that the matrix multiplications in Eqs. (
2.108
), (
2.109
) and (
2.110
)
involve changes only in those columns and rows of M, K and f that are actually
