11.3 Analytical Methods for Predicting the Dynamic Responses …
385
M e
K e
v(t)
z(t)
l
l/2
Impluse
M e
Impluse
a
b
Fig. 11.14 Equivalent SDOF system a specimen loaded by blast impulse b equivalent SDOF
system (Remennikov and Uy 2014)
of the specimen, thus the deformation of the equivalent SDOF system can be used
to represent the mid-span deflection of the UHPCC-FST column specimen.
The SDOF system can be described by the following differential equation of
motion
K LM M ¨
z(t) + c˙ z(t) + R(z(t)) = 0
(11.11)
where M is mass of the column specimen, c is the damping of the system and it can
be ignored in elastic and plastic stages for the very short duration, R is the resistance
of the system and K LM is load-mass transformation factor of the equivalent SDOF
system defined in (Department of Defence 2008). Table 11.5 presents the values
of K LM for different structural modes (Department of Defence 2008). The initial
conditions (when t = 0) for Eq. (11.11) can be written as
z(0) = 0, v 0 = v(0) = ˙
z(0) =
dz
dt
=
I
M
(11.12)
Table 11.5 K LM , k M and R u values for different structural modes (Department of Defence 2008)
Structural modes
Deformation stage
K LM
k M
R u
F
l
Elastic
0.49
48EJ
l 3
4Mu
l
Plastic
0.33
0
4Mu
l
Resilient
0.49
48EJ
l 3
–
F
l
Elastic
0.37
192EJ
l 3
8Mu
l
Plastic
0.33
0
8Mu
l
Resilient
0.43
192EJ
l 3
–
F
l
Elastic
0.43
107EJ
l 3
16Mu
3l
Elastoplastic
0.49
48EJ
l 3
6Mu
l
Plastic
0.33
0
6Mu
l
Resilient
0.46
107EJ
l 3
–