Civil Engineering Reference
In-Depth Information
Consider that after the beam yielded as shown in Figure 2.1(b), the applied moments are
removed. Figure 2.1(e) shows the beam with plastic rotations remaining at the two ends.
The beam returns to a straightened position after the applied moments have been removed,
and therefore the beam is in equilibrium within itself. However, with the plastic rotations at
the plastic hinges, it requires that Joint '
i
' and Joint '
j
' be rotated with
x
i
and
x
j
(i.e. the
i
th
and
j
th degrees of freedom are rotations) while the connecting members of these joints require
that these joint rotations be zero. Therefore, Figure 2.1(e) violates the compatibility condition
when equilibrium is maintained.
On the other hand, Figure 2.1(f ) shows the beam with plastic rotations at the two ends while
imposing zero joint rotations (i.e.
x
i
=
x
j
= 0). This satisfies the compatibility condition because
the connecting members at Joint '
i
' and Joint '
j
' also require that these joint rotations be zero.
However, because the beam is now bent, it induces moments
m
1
and
m
2
at the two ends while
the connecting members remain straight and induce no moment at their ends. Therefore,
Figure 2.1(f ) violates the equilibrium when the compatibility condition is satisfied.
It is therefore the objective of the force analogy method and this chapter to discuss the
technique of analyzing the nonlinear structure while maintaining equilibrium and satisfying
the compatibility condition.
2.2 Force Analogy Method for Static Single-
Degree-of-Freedom Systems
2.2.1 Inelastic Displacement
The FAM begins with the concept of inelastic displacement. Consider a single degree of
freedom (SDOF) system with a force versus displacement curve shown in Figure 2.2. In this
figure,
K
e
denotes the initial stiffness,
K
t
denotes the varying post-yield stiffness, and
F
y
and
x
y
represents the yield force and yield displacement, respectively. Assume that a static force of
F
s
is applied at the only degree of freedom, the system reaches a displacement at Point C as shown
in Figure 2.2. Let
x
represents and total displacement in the system, and note in the figure that
x
>
x
y
(i.e. the system has yielded).
The concept of the FAM is to extend the initial stiffness line OA until it reaches the force
F
s
at
Point B. By defining the displacement with respect to Point B as the elastic displacement
x
0
, the
difference between the total displacement
x
and the elastic displacement
x
0
is the inelastic
F
B
C
F
s
K
t
A
F
y
K
e
x”
x
O
x
y
x'
x
Figure 2.2
Inelastic displacement in the force-displacement relationship.
