Civil Engineering Reference
In-Depth Information
T= t
T=0
0.5
0.5
Figure 8.29
5. The rod shown in Figure 8.30 has an initially triangular temperature distribution
varying linearly from zero degrees at each end to 100 degrees at the centre. The rod
also has a variable diffusivity property as indicated. The rod is allowed to cool while
maintaining the ends at zero degrees. Using a single finite element, and a “lumped
mass” discretisation, estimate the temperature after 0.2 s at
x
=
2and
x
=
4 along
the rod.
Use one time step of
t
=
0
.
2 with a time scaling parameter
θ
=
0
.
5.
(Ans: 2-element solution, 47.2, 88.6; 1-element solution, 46.4, 92.8)
T
100
temperature distribution
at time t = 0
rod
x
0
2
4
6
8
c
x
variation of c
x
along rod
4
2
rod
x
0
2
4
6
8
Figure 8.30
6. Starting with the governing 2D diffusion equation, go through the Galerkin weighted
residual approach to derive terms
k
34
and
m
34
of the conductivity and “mass” matrices
of the element shown in Figure 8.31.
(Ans:
k
34
=
c
x
b/(
6
a)
−
c
y
a/(
3
b) m
34
=
ab
/
18)
