Page 52 - Dynamic Loading and Design of Structures

Page 36

Thus, from eqn (2.3) the system experiences an instantaneous displacement y(t) equal to

(2.14)

Finally, the complete displacement history is evaluated by integrating from time t=0 to the
present time t as

(2.15)

If the static displacement due to the load magnitude F1 is

(2.16)

then

(2.17)

If we finally add the effect of initial conditions at t=0, then we have a generel, closed form
expression for the dynamic displacement of the SDOF system in the form of Duhamel’s
integral as

(2.18)

(b) Suddenly applied load of duration t d
Here we have a combination of constant load f(t)=1 until time t=td and free vibrations past
t>t with initial conditions y(t =t )\hbox and y(t=t ). The resulting DLF factors are:
d
d
d
(2.19)

(2.20)

where y F /k. Figure 2.6 plots the above results for two cases, where we observe an intense
st= 1
response when the duration of the load on the oscillator is large (td=1.2T). If the load is on the
oscillator for a short time (td=0.1T), the dynamic response is less than the static one.

(c) Constant load with rise time t r