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k
i
1
become
k
11
=
V
a
,
21
=
M
a
,
31
=
V
b
,
41
=
M
b
The beam of Fig. 4.6 can be used to find
V
a
,M
a
,V
b
,
and
M
b
, based on the specified conditions
w
Integrate
d
2
dx
2
=
1
,
θ
=
0
,
w
=
0
,
and
θ
=
0
.
w/
=−
M
/
EI
with
M
=−
M
a
−
V
a
x
a
a
b
b
(Sign Convention 2), giving
M
a
x
V
a
x
2
2
d
dx
=
1
EI
+
+
C
1
M
a
x
2
V
a
x
3
6
1
EI
w
=
2
+
+
C
1
x
+
C
2
Apply the boundary conditions
w
a
=
1
,
θ
a
=
0
,
w
b
=
0
,
and
θ
b
=
0tofix the arbitrary
constants
C
1
and
C
2
and then solve for
M
a
and
V
a
.From
θ
a
=
0 and
w
b
=
0
,C
1
=
0 and
C
2
=
2
3
−
M
a
/
2
EI
−
V
a
/
6
EI
. Apply
θ
=
0tothefirst equation and
w
=
1 to the second giving
b
a
3
2
V
a
=
12
EI
/
=
k
11
,
M
a
=−
6
EI
/
=
k
21
The conditions of equilibrium applied to the beam of Fig. 4.6 yield
M
b
and
V
b
.Wefind
3
,
2
k
31
=
V
b
=−
V
a
=−
12
EI
/
k
41
=
M
b
=−
V
a
−
M
a
=−
6
EI
/
It is evident from this derivation of one column of stiffness coefficients that
k
i
1
,i
1
,
2
,
3
,
4
are a set of equilibrated forces on the element. In a sense,
V
a
and
M
a
are the force and the
moment required to generate the unit displacement
=
w
=
θ
=
0, whereas
V
b
and
M
b
are the reactive forces for this configuration. Each column has a similar interpreta-
tion, with the coefficients satisfying equilibrium. In the case of the first column, note that
1 with slope
a
a
3
F
z
=
0
(
V
a
+
V
b
=
k
11
+
k
31
=
(
12
−
12
)
EA
/
=
0
)
and the moments about any point must
2
be zero (e.g.,
).
The beam configurations for computing the second, third, and fourth columns of the
stiffness matrix are shown in Figs. 4.7a, b, and c, respectively.
M
|
x
=
b
=
0
,M
a
+
V
a
+
M
b
=
k
21
+
k
11
+
k
41
=
(
−
6
+
12
−
6
)
EI
/
=
0
4.4.2
Stiffness Matrices Based on Polynomial Trial Functions
For each of the above methods, and those of Chapter 3, for finding the stiffness matrix for a
beam segment, the exact solution of the engineering theory of beams has been arranged in a
stiffness matrix format. For other structural elements, it is not always possible to find an ex-
act solution to place in stiffness matrix form. In such cases, a method involving an assumed
or trial series solution leading to an approximate solution can be employed. Typically, this
approach is used to evaluate stiffness matrices within the finite element method.
Even though the exact stiffness matrix is readily derived for a beam, it is useful to employ
the beam element to illustrate the general procedure for using trial-functions. As will be
seen, for an Euler-Bernoulli beam element, a judiciously chosen series will result in the exact
rather than an approximate stiffness matrix. We continue to employ Sign Convention 2.
Derivation of Interpolation Functions
For a beam element extending from
x
=
a
to
x
=
b
, assume the deflection can be approxi-
mated by a polynomial
C
3
x
2
C
4
x
3
3
x
2
4
x
3
w
=
+
+
+
+···=
w
+
w
+
w
+
w
+···
C
1
C
2
x
2
x
(4.40)
1
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