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Similarly, use the unit load method to find
w
b
and
θ
b
in terms of
V
a
and
M
a
.
This leads to
V
a
M
a
−
w
b
θ
b
w
b
θ
b
3
2
12
EI
/
−
6
EI
/
=
=
k
ab
(6)
2
6
EI
/
2
EI
/
and
V
b
M
b
12
EI
w
b
θ
b
w
b
θ
b
3
2
/
6
EI
/
=
=
k
bb
(7)
2
6
EI
/
4
EI
/
Equations (4), (5), (6), and (7) form the stiffness matrix of Eq. (5) of Example 3.11 or
k
aa
v
k
ab
p
=
(8)
k
ba
k
bb
EXAMPLE 3.15 Truss Analysis
Find the horizontal displacement of point
a
of the truss in Fig. 3.12a. All members have
equal lengths, cross-sectional areas, and moduli of elasticity.
The unit load method proceeds with the application of a virtual (unit) force at
a
in the
direction of the desired displacement. Equation (3.15), appropriately modified to account
for axial extension, leads to an expression for the displacement
V
a
,
giving
L
EA
N
V
a
=
δ
N
(1)
All bars
where
A, E ,
and
L
are the respective area, modulus of elasticity, and length of the
j
th
member;
N
is the axial force in the
j
th member due to the applied loading (Fig. 3.12b);
N
is
the axial force in the
j
th member due to the virtual (unit) force at
a
applied to the structure
without the actual loadings (Fig. 3.12c); and the summation is taken over all members. The
N
and
δ
N
can be determined from equilibrium alone since the truss is statically determinate.
The necessary calculations are readily performed in tabular for
m
(Fig. 3.13). Refer to Fig.
3.12b for the member numbering scheme. Thus,
V
a
=−
δ
9
√
3
EA
2
PL
/(
).
The negative sign
indicates that
V
a
is in the direction opposite to the applied unit force.
Equivalence of the Unit Load Method and Castigliano's Theorem, Part II
In Example 3.7, it was shown that the application of Castigliano's theorem, part II to a truss
can be equivalent to the use of the unit load method. This equivalence, of course, applies
to other structures as well, as is illustrated in the following table.
Unit Load Method
Castigliano's Theorem, Part II
=
δ
=
(∂
Extension
u
N
(
N
/
EA
)
dx
u
N
/∂
P
)(
N
/
EA
)
dx
w
=
δ
w
=
(∂
Bending
M
(
M
/
EI
)
dx
M
/∂
P
)(
M
/
EI
)
dx
φ
=
δ
φ
=
(∂
Torsion
M
t
(
M
t
/
GJ
)
dx
M
t
/∂
P
)(
M
t
/
GJ
)
dx
In this table,
N
is the axial force in a member due to a virtual (unit) force applied at the
point and in the direction of the desired displacement. The other variables are defined
δ
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