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Thus, the complete material law is
=
N
V
M
EA
0
0
0
x
γ
κ
0
k
s
GA
0
(1.110)
0
0
EI
s
=
E
or
1
EA
0
0
1
k
s
GA
=
s
(1.111)
0
0
1
EI
0
0
E
−
1
=
s
1.8.3 Equations of Equilibrium
Figure 1.16 shows a beam element isolating the internal forces. The sign convention for the
theory of elasticity stresses applies as well for the forces and moments on a beam cross-
section. Thus, the forces and moments shown in Fig. 1.16 are positive. Applied loads are
positive if their corresponding vectors lie in positive coordinate directions.
For the purpose of applying the condition
s
of equilibrium, the applied distributed load
p
z
(force/length) is replaced by its resultant
p
z
dx
. The summation of forces in the vertical
direction gives
+
∂
V
∂
0or
∂
V
∂
−
V
+
p
z
dx
+
V
dx
=
x
+
p
z
=
0
(1.112a)
x
Remember that a variable with an overbar, e.g.,
p
z
,
is an applied quantity. In a similar
fashion, it is found that
∂
N
x
+
p
x
=
0
(1.112b)
∂
is the equilibrium relation for the axial
direction. To establish the moment-shear equa-
tion, sum moments about the left face of the element
(
x
)
dx
V
dx
+
∂
M
∂
+
∂
V
dx
2
−
M
+
M
dx
−
−
p
z
dx
=
0
x
∂
x
FIGURE 1.16
Beam element with internal forces and applied loading in positive directions, all applied at the centroid of the
cross-section.
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