Information Technology Reference
In-Depth Information
2.4
Engineering Beam Theory
The classical and generalized variational principles apply for beam theory if the notation
of Section 2.3 is changed from that of three-dimensional elasticity to the notation of beam
theory. Often, for example, terms such as
σ
, the stress vector for elasticity, are replaced by
s
, the vector containing the shear force and bending moment. It is instructive, however, to
know how the variational principles for beams can be derived by following the procedures
used in deriving the three-dimensional elasticity versions of the principles.
The beam theory relations of Chapter 1, Section 1.8, will be utilized. The conversion of
the local governing differential equations for beams to their equivalent global (integral)
forms will be discussed first.
2.4.1
Equations of Equilibrium and Force Boundary Conditions (A)
The conditions of equilibrium for a beam [Chapter 1, Eq. (1.116)]
D
s
s
+
p
=
0
(2.110a)
and the force boundary conditions [Chapter 1, Eq. (1.122)]
s
=
s
on
S
p
(2.110b)
can be placed in the equivalent global form
L
0
δ
u
T
D
s
s
p
dx
=
δ
)
L
0
+
u
T
(
s
−
s
(2.111) or (A)
on
S
p
where, from Chapter 1, Section 1.8, if axial deformation terms are ignored,
V
M
w
θ
∂
x
p
z
0
0
s
=
u
=
D
s
=
p
=
−
1
∂
x
and
L
is the length of the beam. In component form, this appears as
L
L
0
V
+
M
−
[
(
p
z
)δw
+
(
V
)δθ
]
dx
=
(
V
−
V
)δw
+
(
M
−
M
)δθ
(2.112) or (A)
0
on
S
p
2.4.2 Strain-Displacement Relations and Displacement Boundary Conditions (B)
The local (differential) form of the strain-displacement relations [Chapter 1, Eq. (1.102)]
=
D
u
u
(2.113a)
and the displacement boundary conditions [Chapter 1, Eq. (1.123)]
u
=
u
on
S
u
(2.113b)
are equivalent to the global (integral) form
L
0
δ
=
δ
)
L
0
s
T
s
T
(
D
u
u
−
)
dx
(
u
−
u
(2.114) or (B)
on
S
u
Search WWH ::
Custom Search