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Conditions of Equilibrium
dV
dx
+
p
z
−
k
w
w
=
0
(1)
dM
dx
−
V
=
0
Material Law
M
=
EI
(κ
−
κ
T
)
(2)
V
=
GA
s
γ
Kinematics
d
dx
κ
=
(3a)
d
dx
γ
=
θ
+
If shear deformation is not taken into account
dx
d
2
w
dx
2
d
γ
=
θ
=−
κ
=−
0or
(3b)
Boundary Conditions
Force
V
=
VM
=
M
(4)
Displacement
w
=
wθ
=
θ
where the variables with superscript bars are specified quantities.
Equations (1), (2), and (3b) lead to the differential equation
dx
2
EI
d
2
dx
2
d
2
d
2
dx
2
(
w
+
k
w
w
−
p
z
+
EI
) κ
T
=
0
(5)
This is a rather simple fourth order differential equation, for which an exact transcenden-
tal solution is readily established if
EI
and
k
w
are constant (Hetenyi, 1946). This solution
takes the form
C
1
e
λ
x
cos
C
2
e
λ
x
sin
C
3
e
−
λ
x
cos
C
4
e
−
λ
x
sin
w(
x
)
=
λ
x
+
λ
x
+
λ
x
+
λ
x
(6)
√
k
w
/
where
4
EI
. However, in this example the principle of virtual work is utilized
to form an approximate solution for this beam element. The procedure is the same as that
developed in Section 4.4.2.
λ
=
4
Approximate Element Matrices Obtained with the Help of the Principle of Virtual Work
The stiffness matrix is obtained from the principle of virtual work and, in particular, from
the internal virtual work expressions, which for a beam element on an elastic foundation
appears as
δ
=
δκ
+
δγ
−
δw
p
z
−
δw(
−
k
w
w)
W
i
[
M
V
]
dx
(7)
x
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