Civil Engineering Reference
In-Depth Information
stiffness matrix) to upper-triangular form; that is, all terms below the main
diagonal become zero.
Step 1.
Multiply the first equation (row) by 12, multiply the second equation (row) by
16, add the two and replace the second equation with the resulting equation
to obtain
=
−
12
−
30
−
16
0
U
2
U
3
U
4
−
0 6
48
360
50
0
−
33
Step 2.
Multiply the third equation by 32, add it to the second equation, and replace
the third equation with the result. This gives the triangularized form desired:
=
−
12
−
30
−
16
0
U
2
U
3
U
4
−
0 6
48
00 8
360
1240
In this form, the equations can now be solved from the “bottom to the top,” and it will be
found that, at each step, there is only one unknown. In this case, the sequence is
1240
48
U
4
=
=
25
.
83 mm
1
96
[
U
3
=
−
360
+
48(25
.
83)]
=
9
.
17 mm
1
16
[
U
2
=
−
30
+
12(9
.
17)]
=
5
.
0mm
The reaction force at node 1 is obtained from the constraint equation
F
1
=−
4
U
2
=−
4(5
.
0)
=−
20 N
and we observe system equilibrium since the external forces sum to zero as required.
2.5 MINIMUM POTENTIAL ENERGY
The first theorem of Castigliano is but a forerunner to the general principle of
minimum potential energy
. There are many ways to state this principle, and it has
been proven rigorously [2]. Here, we state the principle without proof but expect
the reader to compare the results with the first theorem of Castigliano. The prin-
ciple of minimum potential energy is stated as follows:
Of all displacement states of a body or structure, subjected to external loading,
that satisfy the geometric boundary conditions (imposed displacements), the dis-
placement state that also satisfies the equilibrium equations is such that the
total
potential energy is a minimum for stable equilibrium.
