2.2 Constant Material and Geometry Parameters
17
u 6 = u 4 +
2F 0 X
E A
.
(2.21)
This result can be introduced into Eq. (2.19) to receive the required expression to
substitute u 5 into Eq. (2.12) by given values:
u 5 = u 4 +
F 0 X
E A
.
(2.22)
Thus, the final system of equations under consideration of the force boundary condition is given as:
E A
X
⎡
⎣
2 −1 0
−1 2 −1
0 −1 1
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ =
⎡
⎣
0
0
F 0
⎤
⎦ .
(2.23)
It should be mentioned here that the same result can be obtained in this case based
on the finite element method if the equation for node 5 is eliminated. Solution of this
system of equations gives
⎡
⎢
⎢
⎣
u 2
u 3
u 4
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
X F 0
E A
2X F 0
E A
3X F 0
E A
⎤
⎥
⎥
⎦ ,
(2.24)
which gives the same result as the analytical solution under consideration of X =
L
4
,
see [6]. Alternatively, we may write the system of equations under consideration of
the displacement at node 5 as:
E A
X
⎡
⎢
⎢
⎣
2 −1 0 0
−1 2 −1 0
0 −1 2 −1
0 0 −1 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
u 2
u 3
u 4
u 5
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
0
0
0
F 0
⎤
⎥
⎥
⎦ ,
(2.25)
or solved for the nodal unknowns:
⎡
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
X F 0
E A
2X F 0
E A
3X F 0
E A
4X F 0
E A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
(2.26)
Substituting X =
L
4
gives the exact analytical solution as presented in [6].