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FIGURE 4.1
Sign conventions for a beam element.
and
=
.
s
a
···
p
a
···
−
I
0
···
···
(4.3b)
.
s
b
0
I
p
b
where
I
is the unit diagonal matrix. For the axial force
N
a
Sign Convention 2
.
Deflections and slopes remain the same according to both sign conventions, and, hence,
no special displacement transformation is required.
|
=−
N
a
|
Sign Convention 1
4.2
Fundamental Relations for a Beam Element
There are several possibilities for developing relations between the forces and displace-
ments on both ends of a beam segment. We first consider the pure bending of a beam
element with no loads applied between the ends. Begin with the element of Fig. 4.1a in
which the net forces (and moments) are shown on ends
a
and
b
using Sign Convention 1.
4.2.1 The Equations of Equilibrium
From Fig. 4.1a, the conditions of equilibrium appear as
M
x
=
b
=
0:
M
b
=
V
a
+
M
a
F
z
=
0:
V
b
=
V
a
=
=
.
If the axial effects are taken into account,
F
x
0:
N
b
N
a
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