Dynamics of Structures

There is a fundamental difference between the usual gravity (self-weight) loads that act on a structure, which are quasi-static (i.e., they change very slowly, or not at all with time), and those produced by hurricanes, blasts, and earthquakes, which are extremely dynamic in nature. In the case of hurricanes and other wind loads, it is possible to model their effects as equivalent static pressures in the laboratory as the frequency of the winds is very long compared to the fundamental natural frequency of the typical structure. Important exceptions to this include flexible structures, such as long-span cable-stayed and suspension bridges, tall masts, and wind turbine structures, where the natural frequency of the structure can match that of the wind gusts or straight winds. In the case of earthquakes, the loads are primarily inertial as the ground moves, and the structure tends to stay still. In this case, the loading depends on the actual mass, stiffness, and damping of the structure, and the quantities of interest are the accelerations, velocities, and displacements around the structure. This second set of quantities is very difficult to reproduce accurately in the laboratory if shake tables are not available.

Using basic physics, like Newton's Second Law, one can simplify the problem of the equilibrium of a structure (such as a bridge or a frame with rigid beam), which is subject to ground motions (u_g), to that of a single degree-of-freedom mass (m) with stiffness (k) and damping (c) characteristics. The latter two can be represented by a spring in which the force is proportional to the displacement (u) as well as a dashpot in which the forces are proportional to the velocity (v) (Figure 1). These components can be combined in parallel and/or series to model different structural configurations.

Stiffness is defined as the force required to deform the structure by a unit amount. Suppose that one loads a cantilever beam with a known force (P) and measure its elastic deformation at the tip (). The stiffness is defined as k = P/. For the simple elastic cantilever system shown, k= L³/3EI, where L is the length of the cantilever, I is its moment of inertia, and E is Young's modulus for the material used. Next, imagine what happens if one removes the force suddenly, thereby allowing the cantilever to vibrate. One intuitively will expect the amplitude of vibrations to begin to decrease with every cycle. This phenomenon is called damping and refers to a series of complex internal mechanisms, such as friction, that tend to reduce the oscillations. The quantification of damping is described later in this lab, but it is important to note that at this point, not much is known about these mechanisms from either a theoretical or practical standpoint. A useful concept is to visualize the critical damping coefficient (c_cr), which corresponds to the case where the cantilever will come to rest after just one complete oscillation.

Figure 1: Single degree of freedom system model.

Writing an equation of horizontal equilibrium of forces for the system pictured in Figure 1 leads to:

  (Eq. 1)

If we look at a simpler case for a moment, where we can ignore damping because its effects are negligible, and there is no external forcing function, Equation 1 becomes the linear homogeneous second-order differential equation:

  (Eq. 2)

whose solution is of the form:

  (Eq. 3)

Differentiating twice will give us:

  (Eq. 4)

Substituting Equation 4 into Equation 2, yields:

  (Eq. 5)

The general solution is:

  (Eq. 6)

Where  is the undamped natural frequency of the system.

If this system is given an initial displacement () and/or an initial velocity (), Equation 6 becomes:

  (Eq. 7)

If we add the effect of damping (c) and define , the damped natural frequency of the system becomes  and the equivalent to Equation 7 is:

 (Eq. 8)

For the case of an initial displacement u₀, Figure 2 shows the behavior for several values of .

Figure 2: Effect of damping on free vibrations: definition of critical damping (upper); calculation of damping from logarithmic decrement (lower).

If in Figure 2, one defines , where u_n and u_n+1 are the displacement in successive cycles, then:

  (Eq. 9)

Going back to Equation 1, if the ground motion is taken as the sinusoidal function , the analogue of Equation 8 is:

  (Eq. 10)

Where  is the phase lag, and R_a is the amplification response factor, whose plots are illustrated in Figure 3. Figure 3 shows that for low values of damping ( <0.2), as the frequency of the forcing function approaches the natural frequency of the system, the response of the system becomes unstable, a phenomenon that is commonly referred to as resonance.

Figure 3: Displacement, velocity and acceleration response.

In this lab, we will experimentally investigate the concepts and derivations behind Equations 1- 10 in the context of the dynamics of structures using a shake table.

First, determine the frequency (ω) at which the maximum displacement occurred for each model. The original simple formula discussed above, , needs to be modified because the mass of the beam itself (m_b = W_beam/g), which is distributed over its height, is not negligible compared with the mass at the top (m = W_block/g). The equivalent mass for the case of a cantilever beam is (m+0.23m_b), where m is the mass at the top and m_b is the distributed mass of the beam. The stiffness k is given by the reciprocal of the deformation () caused at the top of the cantilever by a unit force:

  (Eq. 11)

where L is the length of the beam, E is the modulus of elasticity, and I is the moment of inertia. I is given by , where b is the width and h is the thickness of the beam. Thus, the natural circular frequency of a cantilever beam, including its self-weight is:

  (Eq.12)   

Based on this equation, the predicted natural frequencies are calculated in the Table 1.

Beam Number	Length (in)	Width (in.)	Thick. (in.)	I (in.4)	E (ksi)	Weight (lbs)	Beam Weight (lbs.)	Effective Mass (lbs-sec.2/in)	Natural frequency (cycles per second)
1	12.0	1.002	0.124	1.59E-04	10200	0.147	0.149	4.70E-04	2.45
2	16.0	1.003	0.124	1.59E-04	10200	0.146	0.199	4.97E-04	1.55
3	20.0	1.002	0.125	1.63E-04	10200	0.146	0.251	5.28E-04	1.09
4	24.0	1.003	0.125	1.63E-04	10200	0.148	0.301	5.63E-04	0.80
5	28.0	1.001	0.125	1.63E-04	10200	0.144	0.350	5.82E-04	0.62
6	32.0	1.000	0.124	1.59E-04	10200	0.146	0.397	6.15E-04	0.49
7	36.0	1.002	0.126	1.67E-04	10200	0.147	0.455	6.52E-04	0.41
8	40.00	1.000	0.125	1.63E-04	10200	0.148	0.500	6.81E-04	0.34

Table 1: Natural frequencies of the cantilever beams tested.

The measured and the theoretical values of the normal frequency for our model systems are compared in the Table 2. The actual natural frequencies were computed by carefully displacing the cantilever beam by 1 inch and then looking at the displacement vs. time response. The comparison below are made in terms of periods (T_{d ,} in sec.) as these were determined from T_d = u₀-u₁, as shown in Figure 2(b). This requires care and patience to obtain reliable results. The demonstrations shown were only meant to give an overall illustration of the system behavior.

Beam Number	Natural frequency (cycles per second)	Predicted Period (sec.)	Actual Period (sec.)	Error (%)
1	2.45	2.56	2.65	-3.33%
2	1.55	4.06	4.23	-4.22%
3	1.09	5.78	6.79	-17.52%
4	0.80	7.84	8.04	-2.54%
5	0.62	10.06	10.63	-5.70%
6	0.49	12.79	13.04	-1.97%
7	0.41	15.32	16.78	-9.50%
8	0.34	18.59	20.56	-10.59%

Table 2. Comparison of results.

The differences stem primarily from the fact that the beams are not rigidly attached to the wooden base, and the added flexibility at the base increases the period of the structure. Another source of error is that the damping was not accounted for in the calculations, because damping is very difficult to measure and amplitude dependent.

Next, from each of the displacement vs. time histories, extract the maximum value for each frequency and plot the magnitude of the displacement vs. normalized frequency like that in Figure 3. An example is shown in Figure 4, where we have normalized frequency versus the first natural frequency (Beam Number 1) and plotted the maximum displacement of that beam when the shake table was subjected to a varying sinusoidal deformation with amplitude of 1 in.

Figure 4: Deformation of Beam #1 vs. normalized table frequency. 

Initially, when the ratio of ω/ω_n is small, there is not much response as the energy input from the table motion does not excite the model. As ω/ω_n approaches 1, there is a very significant increase in the response, with the deformations becoming quite large. The maximum response is reached when ω/ω_n is very close to 1. As the normalized frequency increases beyond ω/ω_n = 1, the dynamic response begins to die down; when ω/ω_n becomes large we are in a situation where the load is being applied very slowly with respect to the natural frequency of the structure, and the deformation should become equal to that from a statically applied load.

The intent of these experiments is primarily to show the changes in behavior qualitatively, as shown in the demonstrations for the two frame structures. Obtaining results similar to those in Figures 3 and 4 requires great care and patience as sources of friction and similar will affect the amount of damping and thus shift the curves similar to those in Figure 3(c) to the left or right as the actual damped frequency, , changes.

In this experiment, the natural frequency and damping of a simple cantilever system were measured by using shake tables. Although the frequency content of an earthquake is random and covers a large bandwidth of frequencies, frequency spectra can be developed by translating the acceleration time history into the frequency domain through the use of Fourier transforms. If the predominant frequencies of the ground motion match that of the structure, it is likely that the structure will undergo large displacement and consequently be exposed to great damage or even collapse. Seismic design looks at the acceleration levels expected form an earthquake at a given location based on historical records, distance to the earthquake source, the type and size of the earthquake source, and the attenuation of the surface and body waves to determine a reasonable level of acceleration to be used for design.

What the general public often does not realize is that current seismic design provisions are only intended to minimize the probability of collapse and loss of life in the case that a maximum credible earthquake occurs to an acceptable level (around 5% to 10% in most cases). While structural designs to obtain lower probabilities of failure are possible, they begin to become uneconomical. Minimizing losses and improving resilience after such an event are not explicitly considered today, although such considerations are becoming more common, as many times the contents of a building and its functionality may be much more important than its safety. Consider for example the case of a nuclear power plant (like Fukushima in the 2011 Great Kanto Earthquake), a residential ten-story building in Los Angeles, or a computer chip manufacturing facility in Silicon Valley and their exposure and vulnerability to seismic events.

In the case of the nuclear power plant, it may be desirable to design the structure to minimize any damage given that the consequence of even a minimal failure can have very dire consequences. In this case, we should try to locate this facility as far away as possible from earthquake sources to minimize exposure, because minimizing vulnerability to the desired level is very difficult and expensive. The reality is that it is prohibitively expensive to do this given the public's desire to avoid not only a Fukushima-type incident, but also even a more limited one, like the nuclear disaster on Three Mile Island.

For the multi-story building in Los Angeles, it is more difficult to minimize exposure because a large network of seismic faults with somewhat unknown return periods is nearby, including the San Andreas Fault. In this case, the emphasis should be on robust design and detailing to minimize the structure's vulnerability; the owners of the residences should be conscious that they are taking a significant risk should an earthquake occur. They should not expect the building to collapse, but the building may be a complete loss if the earthquake is of a large enough magnitude.

For the computer chip plant, the problems may be completely different because the structure itself may be quite flexible and outside the frequency range of the earthquake. Thus, the structure may not suffer any damage; however, its contents (chip manufacturing equipment) may be severely damaged, and chip production could be disrupted. Depending on the specific set of chips being manufactured at the facility, the economic damage both to the owner of the facility and to the industry as a whole can be tremendous.

These three examples illustrate why one needs to develop resilient design strategies for our infrastructure. To reach this goal we need to understand both the input (ground motion) and output (structural response). This issue can only be addressed through a combined analytical and experimental approach. The former is reflected in the equations listed above, while the latter can only be achieved through the experimental work done through quasi-static, pseudo-dynamic, and shake table approaches.