Period T, sec

Figure 2.23. Unique site-specific design spectra.

The response spectrum tells us that the forces experienced by buildings during an earthquake are not just a function of the quake, but are also their dynamic response characteristics to the quake. The response primarily depends on the period of the building being studied. A great deal of single-mode information can be read directly from the response spectrum. Referring to Fig. 2.25, the horizontal axis of the response spectrum expresses the period of the building affected by the quake. The vertical axis shows the velocity attained by this building during the quake. The diagonal axis running up toward the left-hand corner reads the maximum accelerations to which the building is subjected. The axis at right angles to this will read the displacement of the building in relation to the support. Superimposed on these tripartite scales are the response curves for an assumed 5% damping of critical. Now let us see how various buildings react during an earthquake described by these curves.

If the building to be studied had a natural period of 1 second, we would start at the bottom of the chart and reference vertically until we intersect the response curve. From this intersection, point A, we travel to the extreme right and read a velocity of 16-in. per second. Following a displacement line diagonally down to the right, we find a displacement

of 2.5-in. Following an acceleration line down to the left, we see that it will experience an acceleration of 0.25g. If we then move to the 2-second period, point B, in the same sequence, we find that we will have the same maximum velocity of 16-in. per second, a displacement of 4-in., and a maximum acceleration of 0.10g. If we then move to 4 seconds, point C, we see a velocity of 16-in. per second, a displacement of 10-in., and an acceleration of 0.06g. If we run all out to 10 seconds, point D, we find a velocity of 7 in. per second, a displacement of 10-in. the same as for point C, and an acceleration of 0.01g. Notice that the values vary widely, as stated earlier, depending on the period of the building exposed to this particular quake.

The mode superposition, or spectrum method, outlined in the previous section is a useful technique for the elastic analysis of structures. It is not directly transferable to inelastic analysis because the principle of superposition is no longer applicable. Also, the analysis is subject to uncertainties inherent in the modal superimposition method. The actual process of combining the different modal contributions is, after all, a probabilistic technique and

in certain cases, may not be entirely representative of the actual behavior of the structure. Time-history analysis overcomes these two uncertainties, but it requires a large computational effort. It is not normally employed as an analysis tool in practical design of buildings. The method consists of a step-by-step direct integration in which the time domain is discretized into a number of small increments St; for each time interval, the equations of motion are solved with the displacements and velocities of the previous step serving as initial functions. The method is applicable to both elastic and inelastic analyses. In elastic analysis, the stiffness characteristics of the structure are assumed to be constant for the duration of the earthquake. In inelastic analysis, however, the stiffness is assumed to be constant through the incremental time St only. Modifications to structural stiffness caused by cracking, formation of plastic hinges, etc., are incorporated between the incremental solutions. A brief outline of the analysis procedure applicable to both elastic and inelastic analysis is given in the following discussion.

Analysis Procedure. The method consists of applying a specific earthquake motion directly to the base of a computer model of a structure. Instantaneous stresses throughout the structure are calculated at small intervals of time for the duration of the

earthquake or a significant portion of it. The maximum stresses that occur within the entire analysis period are found by scanning the computer results.

The procedure includes the following steps:

1. An earthquake record representing the design earthquake is selected.

2. The record is digitized as a series of small time intervals of about 1/50 to 1/100 of a second.

3. A mathematical model of the building is set up, usually consisting of a lumped mass at each floor.

4. The digitized record is applied to the model as accelerations at the base of the structure.

5. The computer integrates the equations of motions and gives a complete record of the acceleration, velocity, and displacement of each lumped mass at each interval.

The accelerations and relative displacements of the lumped masses are translated into member stresses. The maximum values are found by scanning the output record.

Period t, sec

Figure 2.25. Velocity, displacement, and acceleration readout from response spectra.

Period t, sec

Figure 2.25. Velocity, displacement, and acceleration readout from response spectra.

This procedure automatically includes various modes of vibration by combining their effect as they occur, thus eliminating the uncertainties associated with modal combination methods.

The time-history technique represents one of the most sophisticated methods of analysis used in building design. However, it has the following sources of uncertainty:

2. If the analysis used unchanging values for stiffness and damping (i.e., linear analysis), it will not reflect the cumulative effects of stiffness variation and progressive damage.

3. There are uncertainties related to the erratic nature of earthquakes. By pure coincidence, the maximum response of the calculated time-history could fall at either a peak or a valley of the digitized spectrum.

4. Small inaccuracies in estimating properties of the structure will have considerable effect on the maximum response.

5. Errors latent in the magnitude of the time step chosen are difficult to assess unless the solution is repeated with several time steps.

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