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Note that these displacement relations were derived following the procedure used in
Section 1.6.1 to find the displacement formulation governing differential equations for an
elastic solid. That is, the strain displacement relations were substituted into the constitu-
tive equations. The resulting stress (force)-displacement functions were then placed in the
equilibrium equations.
A more general form of the displacement formulation equations is obtained if shear
deformation effects are retained. Again, substitute the strain-displacement relations of Eq.
(1.102),
γ
=
w
+
θ
κ
=
θ
=
γ
d
x
and
d
x
,
in the constitutive relations of Eq. (1.110),
V
k
s
GA
and
M
. If the resulting force-displacement equations are placed in the conditions of
equilibrium of Eq. (1.116),
=
EI
κ
k
s
GA
d
dV
dx
=
d
dx
dx
+
θ
−
p
z
or
−
=
p
z
(1.128a)
EI
d
k
s
GA
d
dM
dx
=
d
dx
θ
dx
dx
+
θ
V
or
=
(1.128b)
or in matrix form
d
x
k
s
GAd
x
w
θ
p
z
0
d
x
k
s
GA
+
=
0
(1.129)
k
s
GAd
x
−
d
x
EId
x
+
k
s
GA
In matrix notation these beam equations appear as
D
s
ED
u
u
+
p
=
0
(1.130)
which corresponds to Eq. (1.63) for the general elastic solid.
1.8.6 Mixed Form of the Governing Differential Equations
The other most frequently used form of the governing equations is a mixed form involving
both forces and displacements [Wunderlich, 1977]. As in the case of the general elastic solid,
these relations for a beam are found in a straight forward fashion by eliminating
between
the strain-displacements of Eq. (1.102) and the constitutive relationships of Eq. (1.111). That
is, upon equating the strains
and ignoring the axial relations,
d
dx
=−
θ
+
V
k
s
GA
(1.131a)
d
θ
dx
=
M
EI
(1.131b)
or in matrix form
d
x
w
θ
1
V
M
/
k
s
GA
0
1
=
(1.132)
0
d
x
0
1
/
EI
These are then placed together with the equilibrium conditions of Eq. (1.116) in the form
dx
=−
θ
+
d
V
k
s
GA
(1.133a)
d
θ
dx
=
M
EI
(1.133b)
dV
dx
=−
p
z
(1.133c)
dM
dx
=
V
(1.133d)
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