Transfer
Function
Frequency
Response
Dr. Layer 1.0 Exercise 8- Natural Period of Homogeneous Soil Deposits
Background
In lesson No.4 (Transfer Functions) expressions for the transfer function and amplification factor for a uniform undamped soil layer on rigid rock were presented. In particular, the amplification function, |F|, was expressed by:
 |
(1)
|
where w is the frequency of the excitation, H is the thickness of the soil layer, and vs is the layer shear wave velocity. A simple inspection of Equation (1) shows that as wH/vs approaches p/2+np, the denominator approaches zero, which implies that infinite amplification, or resonance, will occur. The frequencies at which resonance occur are usually referred to as the resonance frequencies. The highest resonance frequency is usually referred as the natural frequency of the soil profile, wo. The inverse to the natural frequency is the natural period of the soil profile, To.
 |
(2)
|
 |
(3)
|
It is important to note that the natural frequency depends
on geometry (thickness) and material properties (s-wave velocity) of the
soil layer. It is also important to distinguish between natural frequency
of the soil profile, wo,
and frequency of the excitation w, which
is implemented on the slider bar in Dr. Layer. The frequency of the excitation
represents the number of completed loading cycles per second applied at
the bottom of the layer. In contrast, the natural frequency, wo,
is a property of the soil layer.
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Objective
The objective of this exercise is to illustrate the concept of natural frequency in soil deposits. Different soil properties, harmonic loading conditions, and layered thicknesses are used to study resonance in soil layers.Things to Do
- Open the
Dr. Layer program. By default we get twelve layers. The top six layers
are hardwired into the system with a very fast velocity. Select the
bottom six layers and assign them a very slow wave velocity.
- Starting
with the set up above, and using the slider bars, increase the loading
amplitude and frequency to a 50% relative to the maximum values. For
this exercise use the default harmonic loading condition.
- Using
the plot box tool,
,
create a displacement time history plot at the top of the soil profile.
By default a displacement time history box is already attached to the
bottom of the soil profile. Use this plot to check the loading condition
that is applied to the soil layer.
-
Move the cursor to the bottom left portion of the screen and push the time increment button,
to set up the wave motion. Observe the wave motion for a couple of
minutes. Check the displacement plot box at the top of the layer and
check if the displacement increases or decreases with time. If the
displacement continuously increases with time you have reached resonance.
If the displacement slows down with time and then increases again
you have not reached resonance. Notice that the displacement time
history plot at the bottom of the soil layer shows constant amplitude
and frequency.
Figure 1
- Notice
that the rigid block moves little. Most of the deformation recorded
in the upper plot is due to the deformation of the bottom soft layer.
If we consider the upper layer as rigid, it is possible to approximately
estimate the natural frequency of the soil layer. By measuring the height
of the bottom soft layer and the frequency of the excitation, and using
the wave velocity for a soft material and Equation 2 it is possible
to calculate the natural frequencies for the system. Check if the natural
frequency of the system,
,
coincides with the frequency of the excitation, w.
To evaluate the height of the layer use the axis bars. To evaluate the
shear wave velocity refer to exercise 1.
- Repeat the analysis with an upper layer formed by four.
and eight, top rigid sublayers as shown in Figure 2. Use the same loading
condition in all cases. For each case evaluate if resonance is reached.
Figure 2
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Observation
- Equations 2 shows that for the same height, H, and shear wave velocity,
, resonance may be reached for each positive value of n. That
is, an infinite number of loading frequencies, w,
may induce resonance.
- The natural frequency depends on the material properties of the layer
(i.e. )
and layer geometry (i.e. height H).
On Your Own
- Repeat the analysis with different loading amplitude and frequencies, and different sublayers wave speeds.
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Last
Updated:
12/27/00
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