Civil Engineering Reference
In-Depth Information
P
L
∆L
Figure 4.26.
Elastic stiffness.
The fixed-end forces on a member can also be found by similar methods.
When discussing the flexibility and stiffness of a member, an elastic spring
that is axially loaded can be considered. This is equivalent to the axial
properties of a linear element used in structures. Figure 4.26 shows an
elastic spring in an un-deformed and deformed position.
The displacement of the spring is directly proportional to the applied
axial load. This is the basic flexibility equation and is written as follows:
∆=
fP
In this equation,
f
is the flexibility of the spring and
P
is the applied axial
load in the direction of the length of the spring. This can also be written in
terms of stiffness using a displacement of 1 unit.
P
=∆
This equation is the general equation for stiffness.
P
represents all the
known forces, ∆
represents the unknown rotations and deflections, and
K
is the stiffness matrix for a member or structure. The entire system of a
structure can be modeled into a set of simultaneous equations written in
the following form and will be expanded in the next section and chapter:
[[]
=
[]
K
∆
P
The derivation of elastic member stiffness in the X-Z system will be
derived in the two following examples using the conjugate beam method
and area moment method.
Example 4.15
q
iy
stiffness
Derive the
q
iy
stiffness using the conjugate beam method for a linear
member.
A free-body diagram is shown in Figure 4.27 with an imposed rota-
tion of 1 unit on the
i
i-end of the member. The moments are assumed in the
