1.4.13 Analysis of a cantilever subject to earthquake motion

Product: ABAQUS/Standard  

This example demonstrates the use of ABAQUS in a seismic analysis where the forcing function is given by the time history of acceleration at an anchor point of the structure. Three types of analyses are illustrated: modal dynamics in the time domain, direct time integration, and response spectrum analysis.

In problems such as this one, the *MODAL DYNAMIC option is the analysis method of choice because it is computationally inexpensive and it is very accurate (provided that enough modes are extracted), since the integration of the modal amplitudes (the “generalized coordinates”) is exact. Direct time integration (the *DYNAMIC option) is also used in this problem to illustrate the accuracy of the time integration operator. Response spectrum analyses, based on spectra calculated from the same earthquake record, are also performed and compared with the exact solution.

Examples are also included to illustrate the use of the *BASELINE CORRECTION option. The *BASELINE CORRECTION option is used to modify the acceleration record by adding a correction to the acceleration record to minimize the mean square velocity over the time of the event. The correction to the acceleration record is piecewise quadratic in time. In this example the analyses are first performed without baseline correction. Two different baseline corrections are then applied, and the results with and without baseline correction are compared.

Problem description

Time domain analysis

The seismic analysis is performed using the El Centro N-S acceleration history, which is discretized every 0.01 second. An exact benchmark solution is readily obtained by integrating the eigenvalues and eigenvectors of the structure exactly in time over the first 10 seconds of the acceleration input (see, for example, Hurty and Rubinstein, 1964). (This solution is calculated using the FORTRAN program contained in the file cantilever_exact.f.) The number of modes included in this solution has been found by trial, which has shown that using the six lowest modes (up to 61.9 Hz) gives displacements that are accurate to 0.01%. The higher modes have a negligible effect since the earthquake acceleration input is discretized every 0.01 second.

The analysis using the *MODAL DYNAMIC option is identical to the benchmark solution, except for the spatial discretization, since ABAQUS integrates the response of the generalized coordinates exactly for inputs that vary linearly during each time increment.

The direct integration analysis is run using the Hilber-Hughes operator with the operator parameter set to 0.0, which gives the standard trapezoidal rule. This operator is unconditionally stable and has no numerical damping, but it exhibits a phase error. Figure 1.4.13–2, taken from Hilber et al. (1977), shows how this error grows with the ratio of the time step to the oscillator period. Automatic time stepping would normally be chosen, with ABAQUS adjusting the time step to achieve the accuracy specified by the choice of the HAFTOL parameter on the *DYNAMIC procedure option. In this case we choose instead to use a fixed time step of 0.01 seconds so that the integration errors are readily illustrated.

For both of these time history analyses the base motion is read from the given acceleration history by using the *AMPLITUDE option. For direct integration this base motion is prescribed by using the *BOUNDARY option, whereas for the *MODAL DYNAMIC procedure it must be given using the *BASE MOTION option.

Response spectrum analysis

Response spectrum analysis provides an inexpensive technique for estimating the peak (linear) response of a structure to a dynamic excitation. The spectrum is first constructed for the given acceleration history by integrating the equation of motion of a damped single degree of freedom system. This provides the maximum displacement, velocity, and acceleration response of such a system. Plots of these responses as functions of the natural frequency of the single degree of freedom system are known as displacement, velocity, and acceleration spectra. The maximum response of the structure is then estimated from these spectra by the *RESPONSE SPECTRUM procedure.

Results and discussion

Input files

References

Tables

Table 1.4.13–1 Natural frequencies in Hertz.

ModeExactFinite Element
B23 elementsB21 elements
1020501020
1.729.729.729.729.726.728
24.5674.5674.5674.5674.5194.554
312.78712.79112.78712.78712.62312.740
425.05825.08225.05925.05824.77424.961
541.42341.52941.43041.42341.22241.288
661.87862.22061.90161.87962.32861.767
786.42587.31786.48886.42688.45386.472
8115.060117.040115.210115.070119.210115.510
9147.790151.470148.100147.800151.380141.010
10148.610169.170169.170169.170168.990169.120

Table 1.4.13–2 Estimated phase errors after 10 seconds of response, using a time step of 0.01 second (based on Figure 1.4.13–2).

ModePeriod, T, (seconds) Phase error per periodPhase error after 10 seconds
11.37.007.005%3.6%
20.219.046.01%46%
30.078.128.05%600%
40.040.251.17%4000%
50.024.414.4%16000%
60.016.619.6%37000%

Table 1.4.13–3 Estimates of maximum displacement and velocity at the top of the column provided by response spectrum analysis.

 DisplacementVelocity
Exact value59.2 mm (2.33 in)0.508 m/sec (20 in/sec)
Displacement spectrum:  
ABS summation67.3 mm (2.65 in)0.641 m/sec (25.22 in/sec)
SRSS summation57.1 mm (2.25 in)0.392 m/sec (15.45 in/sec)
Velocity spectrum:  
ABS summation70.9 mm (2.79 in)0.642 m/sec (25.28 in/sec)
SRSS summation61.0 mm (2.40 in)0.395 m/sec (15.57 in/sec)


Figures

Figure 1.4.13–1 Vertical cantilever beam.

Figure 1.4.13–2 Relative period error (phase error) versus for Hilber-Hughes, Wilson, Newmark, and Houbolt methods (from Hilber et al., 1977).

Figure 1.4.13–3 Relative displacement at the top of the column for the first 2 seconds of response.

Figure 1.4.13–4 Relative velocity at the top of the column for the first 2 seconds of response.

Figure 1.4.13–5 Relative acceleration at the top of the column for the first 2 seconds of response.

Figure 1.4.13–6 Relative displacement at the top of the column for the period 8–10 seconds.

Figure 1.4.13–7 Relative velocity at the top of the column for the time period 8–10 seconds.

Figure 1.4.13–8 Relative acceleration at the top of the column for the time period 8–10 seconds.

Figure 1.4.13–9 Relative displacement at the top of the column for the time period 1–10 seconds.

Figure 1.4.13–10 Displacement spectra for the frequency range 0.1–30 Hz.

Figure 1.4.13–11 Velocity spectra for the frequency range 0.1–30 Hz.

Figure 1.4.13–12 Displacement and velocity spectra for the frequency range 0.01–5.0 Hz.

Figure 1.4.13–13 Absolute displacement of the cantilever's tip with and without baseline correction.

Figure 1.4.13–14 Base displacement with and without baseline correction.