Civil Engineering Reference
In-Depth Information
moment in the beam is a function of only the end shears and moments, as
given by the following equation:
MPxM
x
=−
iy
iz
The second beam in Figure 5.16 shows the bending deformations. In this
case, the internal moment is a function of not only the end shears and
moments, but also a function of the axial force multiplied by the beam's
lateral deflection,
y
.
MPxM Py
x
=−+
iy
iz
ix
This additional moment, the product of axial force,
P
ix
, and lateral deflec-
tion,
y
, is usually called the “P-delta effect.” To derive a stiffness matrix
that includes the P-delta effect, equilibrium of the deformed beam must
be considered.
Example 5.14
Geometric stiffness
Derive the ∆
iy
stiffness using Castigliano's theorems for a linear member
including the geometric effects.
Using the principle of superposition, consider a beam with an applied
deflection while the rotation is held to zero. Figure 5.17 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.
The internal bending moment in the beam is found from equilibrium.
(
)
MPxM P
=−+
∆
−
y
x
iy
iz
ix
iy
M
iz
P
ix
V
P
iy
∆
iy
L
M
iz
M
P
ix
P
∆
iy
-y
P
iy
V
x
Figure 5.17.
Example 5.14 Geometric stiffness.
