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0 and
V
0
,M
0
are given by (3), all of the elements of
z
a
ar
e
known. Then
the deflection, slope, shear, and moment at
x
Since
w
=
θ
=
0
0
z
i
,
where
U
i
U
i
=
L
are given by
z
b
=
z
a
+
is
z
i
taken from Eq. (4.86). The response
z
at
x
less than
L
is evaluated by assigning
x
b
in
U
i
z
a
+
to be any location along the beam at which the state variables are sought. Of course,
is
chosen to correspond to this location also, i.e.,
=
x
b
−
x
a
=
x
b
.
4.5.4 Inclusion of Axial Extension and Torsion
The transfer matrix can be expanded to include the effects of axial extension. If the axial
displacement is
u,
the governing differential equations for extension of a bar of cross-
sectional area A are (Chapter 1)
du
dx
=
N
EA
,
dN
dx
=−
p
x
(4.120)
In transfer matrix form, the solution to these equations is
x
x
EA
−
p
x
(τ )
EA
u
=
u
0
+
N
0
d
τ
0
x
(4.121)
N
=
N
0
−
p
x
(τ )
d
τ
0
or, in terms of
x
a
,x
b
,
and
=
x
b
−
x
a
,
u
b
N
b
u
N
1
u
N
EA
01
/
b
=
a
+
(4.122)
U
i
z
i
z
b
=
z
a
+
If the effects of extension and bending are combined, then, for the sign convention of Fig.
4.1a (Sign Convention 1),
u
w
/
EA
u
w
1
1
−
3
2
−
/
6
EI
−
/
2
EI
1
2
/
2
EI
/
E I
(4.123)
=
1
N
V
M
N
V
M
1
1
b
a
U
i
z
b
=
z
a
Table 4.5 provides loading vectors
z
i
for several common applied forces.
The governing first order equations for the torsion of a shaft are (Chapter 1)
d
dx
=
M
t
GJ
,
dM
t
dx
=−
m
x
(4.124)
where
is the angle of twist,
M
t
i
s t
he twisting moment,
G
is the shear modulus of elasticity,
J
is the torsional constant, and
m
x
(force-length/length) is the magnitude of the applied
distributed torque. Because of the similarity of Eq. (4.124) with the governing equations
[Eq. (4.120)] for the extension of a bar, t
he
transfer matrices of Eqs. (4.122)
an
d (4.123) apply
to the torsion of a bar if
u, N, E A,
and
p
x
are replaced by
φ
φ
,M
t
,GJ,
and
m
x
,
respectively.
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