Civil Engineering Reference
In-Depth Information
EXAMPLE 2.1
Consider the two element system depicted in Figure 2.2 given that
Node 1 is attached to a fixed support, yielding the displacement constraint
U
1
=
0.
k
1
=
50 lb./in.,
k
2
=
75 lb./in.,
F
2
=
F
3
=
75 lb.
for these conditions determine nodal displacements
U
2
and
U
3
.
■
Solution
Substituting the specified values into Equation 2.15 yields
=
50
−
50
0
0
U
2
U
3
F
1
75
75
−
50
125
−
75
0
−
75
75
and we note that, owing to the constraint of zero displacement at node 1, nodal force
F
1
becomes an unknown reaction force. Formally, the first algebraic equation represented in
this matrix equation becomes
F
1
and this is known as a
constraint equation,
as it represents the equilibrium condition
of a node at which the displacement is constrained. The second and third equations
become
−
50
U
2
=
125
U
2
U
3
75
75
−
75
=
−
75
75
which can be solved to obtain
U
2
=
4 in. Note that the matrix equations
governing the unknown displacements are obtained by simply striking out the first row
and column of the 3
3 in. and
U
3
=
3 matrix system, since the constrained displacement is zero.
Hence, the constraint does not affect the values of the
active
displacements (we use the
term
active
to refer to displacements that are unknown and must be computed). Substitu-
tion of the calculated values of
U
2
and
U
3
into the constraint equation yields the value
F
1
=
−
×
150 lb., which value is clearly in equilibrium with the applied nodal forces of
75 lb. each. We also illustrate element equilibrium by writing the equations for each
element as
f
(1)
1
f
(1)
2
50
0
3
−
lb
−
50
150
150
=
=
.
for element 1
−
50
50
f
(2)
2
f
(2)
3
75
3
4
−
lb
−
75
75
75
=
=
.
for element 2
−
75
75
Example 2.1 illustrates the general procedure for solution of finite element mod-
els: Formulate the system equilibrium equations, apply the specified constraint
conditions, solve the reduced set of equations for the “active” displacements, and
substitute the computed displacements into the constraint equations to obtain the
unknown reactions. While not directly applicable for the spring element, for
