Biomedical Engineering Reference
In-Depth Information
Fig. 5.37
1D finite element discretisation of a solid structure treated as a spring-mass system
−
==
du
dx
uu
(5.64)
2
1
ε
x
L
and the stress-strain relation is taken from Hooke's law, giving
uu
EE
L
−
2
1
(5.65)
σε
==
x
x
and finally we relate the stress to the applied load,
AE
(5.66)
P
=
σ
A
=
(
uu
−
)
x
2
1
L
Applying Eq. (5.66) to relate the nodal forces,
f
1
and
f
2
to the nodal displacements
u
1
and
u
2
, we get
AE
L
uu f
AE
L
uu
(5.67)
f
=−
(
−
)
=
(
−
)
1
2
1
2
2
1
and in matrix form we get
11
11
−
u
f
u
f
AE
[
]
1
1
1
1
= → =
k
(5.68)
e
−
u
f
u
f
L
2
2
2
2
stiffness matrix
Worked Example
Consider the displacement field variable for a spring-mass sys-
tem is due to an applied force (Fig.
5.37
).
For a single element the net displacement and its corresponding force is given by
duuF kuu F
= −
=−
(
−
)
=−
kuu
(
−
)
2
1
1
2
1
2
2
1
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