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or
∂
−
∂
M
∂
V
1
2
p
z
dx
x
−
−
=
V
dx
0
∂
x
∂
V
1
The terms
dx
and
2
p
z
dx
approach zero in the limit leaving
dM
dx
−
∂
x
V
=
0
(1.113)
The shear force
V
can be eliminated from Eqs. (1.112a) and (1.113) giving
d
2
M
dx
2
+
p
z
=
0
(1.114)
These equilibrium relations can be placed in the matrix form
d
x
N
M
p
x
p
z
0
+
=
0
(1.115)
d
x
0
D
s
s
+
p
=
0
if shear deformation effects are not considered. If shear deformation effects are included,
then
+
=
d
x
00
N
V
M
p
x
p
z
0
0
d
x
0
0
(1.116)
0
−
1
d
x
D
s
s
+
p
=
0
An important characteristic of
D
s
is its functional relationship to
D
u
. The operators
D
u
and
D
s
are formally adjoint in the sense that they satisfy the relationship
T
s
dV
u
T
u
D
T
s
dV
u
T
A
T
s
dS
u
T
D
s
s
dV
V
(
)
=
=
−
D
u
u
(1.117)
V
S
V
where the arbitrary vectors
u
and
s
are smooth in the region considered, and
A
T
is the
matrix of direction cosines in Eq. (1.57) for the tractions on an oblique surface. This ad-
joint relationship, which holds for both the case of general elasticity and for specialized
structures, is a member of a group of formulas referred to collectively as the divergence
theorem. An example is Appendix II, Eq. (II.7), a statement of the divergence theorem for
vectors. Equation (1.117) with
s
replaced by
σ
holds for the elasticity equations. In this case
D
T
of the equilibrium equation is denoted by
D
T
σ
and
D
of the kinematic equations by
D
u
.
to the left of the operator matrix
D
, i.e.,
u
of
u
D
T
, indicates that
the operator is applied to the preceding quantity, i.e., to
u
T
. Thus,
u
T
u
D
T
The subscript index
(
u
)
∗
. This
notation has the advantage of revealing clearly the structure of the resulting operator. For
example, the symmetry properties of the equations are apparent in the operator form.
T
=
(
D
u
u
)
The adjoint relationship of Eq. (1.117) can be demonstrated for particular cases using
integration by parts. Consider the two-dimensional elasticity equations for a rectangle of
∗
The transpose of a product of matrices is equal to the product of the transpose matrices in reversed order, i.e., if
H
=
AB
···
EF
then
H
T
F
T
E
T
B
T
A
T
=
···
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