Civil Engineering Reference
In-Depth Information
Example 5.15
Geometric stiffness
Derive the
q
iz
stiffness using Castigliano's theorems for a linear member
including the geometric effects.
In this case, consider a beam with a known rotation while the deflec-
tion is held to zero. Figure 5.18 shows the deformed beam with applied
end forces. Also shown is a left-hand free-body of the beam cut at any
distance
x
from the
i-
end.
M
iz
P
ix
θ
iz
P
iy
L
M
iz
M
P
P
ix
y
θ
iz
P
iy
V
x
Figure 5.18.
Example 5.15
Geometric stiffness.
The internal bending moment in the beam is found from equilibrium.
MPxM Py
x
=−−
iy
iz
ix
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.
yax x xd
y
yL
y
yL
=+++
()
=
()
=
()
=
()
=
3
2
00
0
'
'
0
q
iz
0
3
2
q
x
L
2
x
L
y
=
−
+
x
iz
2
x
L
3
2
x
L
2
MPxM P
=−+
ix
q
− +−
x
x
iy
iz
2
