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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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