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As indicated in the following chapter, the inclusion of applied loadings in the transfer
matrix form of the response can be accomplished in several other ways. A simple approach
which is particularly useful if the transfer matrix elements are known analytically will be
discussed briefly here.
It is convenient to introduce a general notation for a transfer matrix:
0
b
w
V
M
U
U
U
U
w
V
M
w
ww
wθ
w
V
w
M
b
a
0
b
V
b
M
b
U
U
U
U
θ
θw
θθ
θ
V
θ
M
=
+
U
V
w
U
V
θ
U
VV
U
VM
(4.117)
U
M
w
U
M
θ
U
MV
U
MM
z
i
U
i
z
b
=
()
z
a
+
where
,V,M,
represents a transfer matrix element. For
example, by comparison of Eqs. (4.86) and (4.117),
U
=
x
b
−
x
a
and
U
ij
,i
and
j
=
w
,
θ
for a simple beam segment is
−
.
wθ
b
,V
b
,
and
M
b
are defined in Eq. (4.87) for a simple uniform
beam segment or by Eq. (4.90b) for a more general element.
The form of a transfer matrix is such that the effect of various types of loading can be
identified readily. For example, it is apparent from Eq. (4.117) that a shear force
V
at
x
0
0
The loading functions
w
b
,
θ
=
a
contributes the magnitude
VU
An applied concentrated
load would have the effect of a s
he
ar force. Thus, the effect on the defl
ec
tion at
x
()
to the deflection
w
at
x
=
b
.
w
V
=
b
of
a downward concentrated force
P
at
x
=
a
would be expressed as
−
PU
()
for Sign
w
V
Convention 1. This same reasoning may be applied to the other responses,
θ
,V,
and
M,
giving the loading function vector for a concentrated load at
x
=
a
as
0
b
w
−
P
U
w
V
()
0
b
V
b
M
b
θ
−
P
U
()
θ
V
=
(4.118)
−
P
U
VV
()
−
PU
MV
()
This procedure is readily extended [Pilkey and Chang, 1978] to distributed loads with
the loading functions given by the Duhamel
8
or convolution integral
j
b
=−
p
z
(
x
)
U
jV
(
−
x
)
dx
=−
p
z
(
−
x
)
U
jV
(
x
)
dx
(4.119)
0
0
with
j
Two forms of the convolution integral are provided in Eq. (4.119) since
one or the other can be more convenient for a particula
r
problem. Since many types of
loading can be expressed in terms of the loading intensity
p
z
(
=
w
,
θ
,V,M
.
,
including both distributed
and concentrated loadings, e.g., applied forces, moments, and thermal loading, Eqs. (4.90b)
or (4.119) are very versatile for computing the loading terms in the transfer matrix.
x
)
EXAMPLE 4.9 Loading Functions for a Linearly Varying Load
Find the loading function component
0
b
w
for the linearly varying distributed applied load
of Fig. 4.14.
8
Jean Marie Constant Duhamel (1797-1872) was a French professor of higher algebra at the Sorbonne University.
Hermite was his successor in this position. Using Poisson's theory of elasticity, Duhamel investigated mathemat-
ically the influence of temperature on stress.
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