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the following equation)
M
=
0:
−
M
+
(
M
+
dM
)
−
Vdx
+
H
(
d
w
+
d
w
0
)
=
0
(11.13a)
dM
−
Vdx
+
H
(
d
w
+
d
w
)
=
0
(11.13b)
0
dM
dx
H
d
w
+
d
w
0
dx
−
Vdx
+
dx
=
0
(11.13c)
dx
V
H
d
dx
+
w
dM
dx
−
d
0
dx
−
=
0
(11.13d)
where, from Eq. (11.12)
V
H
d
dx
+
d
w
0
Q
=
−
(11.14)
dx
The equilibrium of the forces in the horizontal and vertical directions
F
hor
0:
dH
=
dx
−
p
x
=
0
(11.15)
F
vert
0:
dV
=
dx
+
p
z
=
0
(11.16)
The term
H
d
dx
in Eq. (11.13d) introduces a nonlinearity with respect to the longitudinal
force, which for compressive loads may lead to stability failure. The other difference as
compared with the usual equations f
or
a beam (for instance those g
iv
en in Chapter 4) is the
inclusion of the initial imperfections
0
(given rotations) that
describe the initial deviations from perfect geometry. Because these initial imperfections are
difficult to avoid in practice, they need to be considered in describing the realistic behavior
of most structures. The kinematical equations and the material law are assumed to be linear.
Additional nonlinearities may be introduced in the formulation, but are omitted here.
w
0
(given displacements) and
θ
Kinematics
From Chapter 1 the kinematical equations for a beam are
d
θ
dx
κ
=
du
dx
ε
=
(11.17)
dx
+
θ
For an Euler-Bernoulli beam, the shear deformation effects are ignored so that
d
γ
=
γ
=
0, and
the kinematical relations are reduced to
d
2
d
dx
w
dx
2
θ
=−
and
κ
=−
(11.18)
Material Law
In the theory of second order it is presumed that the unknown longitudinal strains contained
in
N
are known quantities, obtained from the linear analysis as the approximations
N
0
=
ε
≈
N
EA
(11.19)
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