Civil Engineering Reference
In-Depth Information
If the lateral deformation,
y
, is assumed to be a general cubic function,
the four known boundary conditions can be used to find the particular
solution.
3
2
yax x xd
y
yL
y
yL
=+++
()
=
()
=
()
=
()
=
0
∆
iy
0
00
0
'
'
3
2
1
23
x
L
x
L
y
∆
+
=
−
iy
3
2
Substituting this equation into the internal moment equation yields the
following:
23
x
L
3
x
L
2
MPxM P
=−+
∆
−
+
x
iy
iz
ixiy
3
2
It is assumed that axial shortening is caused only by the axial force,
P
ix
.
This is the same assumption used for the ordinary elastic stiffness deriva-
tion. The geometric stiffness derivation considers the lateral and rotational
deformations, ∆
iy
and
q
iz
. From Castigliano's theorem, the general deflec-
tion and rotation of the free end are as follows:
d
d
M
P
dx
EI
∫
x
∆
iy
=
M
x
iy
z
d
d
M
M
dx
EI
∫
q
=
M
x
iz
x
iz
z
The partial derivatives of the internal moment equation with respect to
applied shear force and moment are the following:
d
d
d
d
M
P
x
=
x
iy
M
M
x
=−1
iz
