Some systems may react more or less quickly than others to changing loads. The implication
of this is illustrated by the following examples.
Consider a system where the restraint loads respond fully to (and therefore completely
counteract) an imposed load in 25 milliseconds. For the imposed load profile and configuration shown in Figures 4-9a and 4-9b, respectively, the restraint loads would follow a force versus time profile similar to that shown in Figure 4-9c. This is due to the fact that before the restraints can fully react to the applied load, the load has been removed. Therefore, the induced reaction loads (and by extension, the member forces and moments, and stresses) are much lower than would occur under a static load of the same magnitude (each restraint under a static load would see a reaction equal to -P/2, for a total of-P).
Now consider a system identical in all respects except that its restraint loads respond fully
to any imposed load in only 1 millisecond. For the imposed load profile and configuration
shown in Figures 4-9a and 4-9b, the restraint loads would follow a force versus time profile
similar to that shown in Figure 4-10 — virtually identical to that of the applied load.
Therefore, when a system responds rapidly to applied loads, the induced reaction loads,
member forces, etc. are approximately the same as those which would occur under the same
static load.
The pertinent question is then, what is a fast response time, and what is a slow response time. In truth, there is no absolute answer—what is really important is the relative response time of the system as compared to the rate of change of the applied load. For example, what if the load applied to the system in Figure 4-9 had a duration of 25 seconds, instead of 10 milliseconds? The restraint loads would have sufficient time to fully respond to the applied load, and the reactions would be the same as for a static load. In fact, a static load is simply a dynamic load with a duration long enough that all systems have the opportunity to respond fully to it.
From the above, it is evident that system response to dynamic loads can produce at least two possible results, based upon the system response time. Slowly responding systems result in response loads lower than the applied loads, while rapidly responding systems result in response loads approximately the same as the applied loads. What happens when the system response is somewhere in between?
Consider the systems described in Figure 4-9 and 4-10 (i.e., with response times of 25
milliseconds and 1 millisecond, respectively), this time loaded with a harmonic load, cycling
between P and -P, with frequency of 1 cycle per 25 milliseconds. The responses of the two
systems are shown in Figures 4-lla and 4-llb. The first system again lags behind and fails
to fully develop response loads. The second system responds almost instantly and just about
fully responds to the applied load, as before.
Now consider a system which has a response time somewhere in the middle — about 12.5
milliseconds. Upon initial loading, the system initially attempts to respond to the load P,
with restraint loads each equal to -P/2. Since the system response lags, it does not fully
develop these restraint loads, but, after 12.5 milliseconds, will have total system response
(restraint loads) of somewhere around -0.7P. Considering the cyclic load, the applied load
on the system will be, at 12.5 milliseconds, -P, for a net load on the system of-1.7P (see Figure 4-12).
The system now attempts to resolve the net load of-1.7P with two restraint loads of+0.85P.
Assuming that at time T=25 milliseconds, these loads have actually reached only +0.6P (due
to the response lag), or a total of+1.2P, the external load will now be +P, so the net system
load will be +2.2P, as shown in Figure 4-13.
This net load will then be resisted by total restraint loads (system response) of-2.2P, which
will have reached approximately -1.5P by T=37.5 milliseconds, at which time the load will
have reversed again, creating a net load on the system of-2.5P. Continuing in this way, the
net load on the system will be approximately 2.8P at 50 milliseconds, -3.0P at 62.5
milliseconds, 3. IP at 75 milliseconds, and so forth. The total developed load (total restraint
loads) is shown as a function of time in Figure 4-14.
This may continue until the developed load spirals out of control, and the structure fails.
From this example, it is clear that there is a third possibility for a system response under
dynamic loading — the induced load may far exceed the applied load. This is the type of
response with which the engineer must normally be concerned.
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