Civil Engineering Reference
In-Depth Information
M
L
E, I
L
p
PHL
Figure 3.5
A SDOF system with one PHL.
substituting the result into Eq. (3.61) gives the equation for calculating the absolute acceleration
response:
y
k
= −
M
−1
Cx
k
−
M
−1
Kx
k
−
x
0
k
ð3
:
62Þ
This completes the calculation of all response quantities in a nonlinear dynamic analysis.
Example 3.2 Force Analogy Method with Input Displacement
Unlike static analysis in Chapter 2 where the input is the applied force, dynamic analysis
requires an input displacement to the FAM as shown in Eq. (3.59). Consider a SDOF system
as shown in Figure 3.5 with a plastic hinge at the base of the column that exhibits elastic-plastic
behavior with a yield moment of
m
p
. The stiffness matrices for this column and the moment
versus plastic rotation relationship can be calculated as:
,
2
K
=
3
EI
L
3
K
0
=
3
EI
L
3
K
00
=
3
EI
L
3
,
L
−
L
p
L
−
L
p
ð3
:
63aÞ
θ
00
ðÞ=0
m
ðÞ=
m
p
m
ðÞ≤
m
p
m
ðÞ>
m
p
,
if
then
ð3
:
63bÞ
Now let the lateral displacement of the mass be monotonically applied (i.e. applying a dis-
placement to the system rather than a force, or sometimes referred to as a displacement control
type of loading) and be equal to
x
k
=
m
p
L
2
/4
EI
at time step
k
. Also for simplicity, let
L
p
=0.
It follows from Eqs. (3.43) and (3.63a) that
m
p
L
2
4
EI
m
k
+
3
EI
L
θ
0
k
=
3
EI
ð3
:
64Þ
L
2
First, assume that
θ
0
k
= 0, i.e. the column remains linear. It follows from Eq. (3.64) that
m
k
=3
m
p
=
4
ð3
:
65Þ
Since the calculated
m
k
is less than
m
p
, the assumption that the column remains linear is correct,
and the moment is therefore given in Eq. (3.65).
