Analysis of a cantilever subject to earthquake motion

The modal dynamic analysis results agree exactly with the benchmark solution, since the linear variation of the inputs over each increment results in exact integration.

The dynamic analysis is run for 10 sec (1000 increments). The displacement, velocity, and acceleration at the top of the column are plotted as functions of time using the Visualization module in Abaqus/CAE. The response quantities in these plots are all measured relative to the base of the structure. The Fortran program that calculates the benchmark solution writes its results to various files, so that Abaqus/CAE can be used to plot the benchmark solution on the same graphs as the Abaqus results.

Figure 3 shows the displacement of the top of the column, relative to its base, for the first 2 seconds of response. The approximate and benchmark solutions agree well on this plot. The relative velocity and acceleration of the column top for the first 2 sec are shown in Figure 4 and Figure 5. The difference between the benchmark and the approximate solutions is now more apparent, especially in the acceleration trace. Figure 6 through Figure 8 show the response from 8 to 10 seconds after the start of the event. The higher mode content of the approximate solution now shows a significant phase error in the relative displacement trace (Figure 6), and the acceleration solution is quite seriously in error.

The source of this phase error is the phase error inherent in the time integration operator, shown in Figure 2. It is a simple matter to estimate the error and its effect on each mode after 10 seconds of response. Such a calculation is summarized in Table 2. As shown in the table, with the 0.01 second time step chosen, the error in the first and second modes is about 4% and 46%, respectively; for all other modes the errors are well in excess of 100%, so the effect shown in Figure 6 is entirely predictable: with a 0.01 second time step, the errors are very large for all but the first mode response. It is interesting to observe that, to achieve less than a 5% phase error after 10 seconds in Mode 6, the phase error would have to be less than 8 × 10⁻⁵ per cycle, implying a time step that is not larger than about 10⁻⁵ seconds.

Figure 9 shows the displacement of the undamped system during the entire 10-second analysis. Even without damping, the first mode response so dominates the solution that the predicted tip displacement response after 10 seconds is not grossly in error. In reality, there will always be some damping; if the structure is undergoing large motion, it is likely that the damping will be enough to remove most of the response above the second mode in this period of time. The common design approach is to incorporate all dissipation of energy as equivalent linear viscous damping—typically assumed to be a certain fraction (2–6%) of critical damping in each mode when modal dynamics is used. This approach cannot be used in direct integration analysis since the modes are not extracted. Instead, damping can be used to introduce mass and stiffness proportional damping into models that are integrated directly. We have not used this option here. In calculations for extremely large input motions this linearized approach is usually replaced with a nonlinear analysis in which the damping mechanisms are modeled explicitly.

The spectra for response spectrum analysis are obtained by integrating 10 seconds of the acceleration record using the Fortran program given in the file cantilever_spectradata.f. By varying the frequency range and the damping values, several different response spectra can be obtained. Figure 10 through Figure 12 depict the displacement and velocity spectra for the frequency ranges 0.1–30 Hz and 0.01–5.0 Hz with no damping and with damping chosen as 2% and 4% of critical damping.

The response spectrum procedure estimates the response at each frequency either as the sum of the absolute values of the modal responses (the absolute summation, or ABS, method) or as the square root of the sum of the squares of the modal responses (SRSS method). The absolute summation method is always conservative, in the sense that it overpredicts the response.

Since the natural modes of the cantilever are well separated in this case, the ten-percent summation method will give the same results as the SRSS method. The complete quadratic combination method will also give these same results, and the Naval Research Laboratory method will give values close to those provided by ABS summation. A comparison of these methods in a more complex case is provided in Response spectra of a three-dimensional frame building.

We can compare the response estimates provided by the response spectrum analysis with the exact values by examining the predictions of response quantities at the top of the column. The exact peak displacement is 59.2 mm (2.33 in), and the peak velocity is 0.508 m/sec (20 in/sec). The comparison is based on the response spectrum values obtained with the assumption of no damping and is shown in Table 3. We see that, using the displacement spectrum, the ABS summation method overestimates the peak displacement by 14% and the peak velocity by 28%, whereas the SRSS method underestimates the peak displacement by 3% and the peak velocity by 22%. Using the velocity spectrum, the ABS method overestimates the peak displacement by 20% and the peak velocity by 27%, whereas the SRSS method overpredicts the peak displacement by 4% and underpredicts the peak velocity by 22%. Despite these rather large errors, the method is commonly used because of its simplicity and ready application to design cases. The response spectra results found in Table 3 can be obtained by executing the Fortran program given in the file cantilever_spectradata.f and then running the Abaqus input given in cantilever_responsespec.inp. To obtain results using the ABS summation method, two response spectrum steps must be added to sum the directional excitation components algebraically and to sum the absolute values of the responses in each natural mode.

Baseline correction adds a piecewise quadratic correction to the acceleration record to minimize the mean square velocity of the motion. This correction will change the displacement quite substantially (the corrected base displacement will tend to zero at the end of the motion), but the change in the acceleration record will not be very large. As a result, the relative displacement between the tip and the base of the cantilever will be affected very little, but the absolute displacement will change substantially if significant baseline correction is added.

Baseline correction can be applied in Abaqus as a piecewise quadratic correction through the time domain. In this example we apply two corrections: one done for the entire period of time (here 25 sec) and one done using three intervals: 0.0–8.3 sec, 8.3–16.7 sec, and 16.7–25 sec. Figure 13 shows the total (not relative) displacement of the tip of the cantilever with and without these corrections, and Figure 14 shows the base displacement with and without baseline correction. Figure 14 was produced by running all three analyses for 25 seconds (the duration of the acceleration record) and then plotting the total displacement of the base of the cantilever. The effect of the correction on displacement is clear from Figure 14; as more intervals are used for the correction, the base displacement at the end of the analysis tends more toward zero.