Civil Engineering Reference
In-Depth Information
The element-to-global displacement relations are
u
(1)
1
u
(1)
2
=
u
(2)
1
u
(2)
2
=
u
(3)
1
u
(3)
2
=
U
4
Proceeding as in the previous example, we then write the individual element equations as
=
U
1
=
U
2
=
U
3
f
(1
1
f
(1
2
0
0
3
k
−
3
k
00
U
1
U
2
U
3
U
4
−
3
k
3
k
00
=
(1)
0
0
0
0
0
0
0
0
0
f
(2
1
f
(2
2
0
00
00
U
1
U
2
U
3
U
4
0
k
−
2
k
0
=
(2)
0
−
2
k
2
k
0
00
00
0
0
f
(3
1
f
(3
2
00 0 0
00 0 0
00
k
−
k
00
−
kk
U
1
U
2
U
3
U
4
=
(3)
Adding Equations 1-3, we obtain
3
−
30 0
−
35
−
20
0
−
23
−
1
00
−
11
U
1
U
2
U
3
U
4
F
1
W
W
W
k
=
(4)
where we utilize the fact that the sum of the element forces at each node must equal the
applied force at that node and, at node 1, the force is an unknown reaction.
Applying the displacement constraint
U
1
=
0
(
this is also preprocessing
), we obtain
−
3
kU
2
=
F
1
(5)
as the constraint equation and the matrix equation
=
−
20
5
U
2
U
3
U
4
W
W
W
k
−
−
(6)
23
1
0
−
11
for the active displacements. Again note that Equation 6 is obtained by eliminating the
constraint equation from 4 corresponding to the prescribed zero displacement.
Simultaneous solution (
the solution step
) of the algebraic equations represented by
Equation 6 yields the displacements as
W
k
2
W
k
3
W
k
U
2
=
U
3
=
U
4
=
and Equation 5 gives the reaction force as
F
1
=−
3
W
(This is
postprocessing.
)
