Biomedical Engineering Reference
In-Depth Information
Figure 18.3
Triangular and quadrilateral element distribution for the bar problem.
∼
=
B
∼
(18.69)
are constant per element. The bi-linear quadrilateral element, on the other hand,
is clearly enhanced. A typical shape function, of for example the first node in an
element, is given by (with respect to the local coordinate system)
1
4
(1
−
ξ
)(1
−
η
) .
N
1
=
(18.70)
Hence
1
4
(1
−
ξ
−
η
+
ξη
) ,
N
1
=
(18.71)
which means that an additional non-linear term is present in the shape functions.
Therefore a
linear
variation of the stress field within an element is represented.
Two remarks have to be made at this point:
•
The numerical analysis in this example was based on plane stress theory, while in the
chapter the equations were elaborated for a plane strain problem. How this elaboration
is done for a plane stress problem is discussed in Exercise
18
.1.
•
Strictly speaking, a concentrated force in one node of the mesh is not correct. Mesh
refinement in this case would lead to an infinite displacement of the node where the
force is acting. This can be avoided by applying a distributed load over a small part of
the beam.
Exercises
18.1 For Hooke's law the Cauchy stress tensor
σ
is related to the infinitesimal
strain tensor
ε
via
d
.
σ
=
K
tr(
ε
)
I
+
2
G
ε
(a) What is tr(
ε
) for the plane strain case?
(b) What is tr(
ε
) for the plane stress case?
