SAP000 EXAMPLE 6-00 LINK LINEAR LINK WITH RAMP LOADING PROBLEM DESCRIPTION In this example a ramp load is applied to an undamped single degree of freedom structure. The ramp loading has a finite rise time (ramp time), t r, but remains constant thereafter. The displacements of the structure at two selected times are compared with hand calculated results based on theory presented in Chopra 995. The output times used for comparison are selected such that one is less than t r and the other is greater than t r. The following properties are assumed for the model: k = spring stiffness = 4 k/in m = mass = k-sec /in From those two properties the expected period of the structure, T, is calculated as: m T = π = π = π seconds k 4 The analysis is performed using two different ramp loading rise times, t r, in two different load cases. In case A, t r is equal to the period of the structure, t r = T = π. In case B, t r is equal to one-half the period of the structure, t r = T/ = π/. The SAP000 model consists of a two-joint linear link element that is fixed at one end and free at the other end. As shown in the figure on the following page, the link is modeled in the XZ plane and is oriented such that its length (and thus its local axis) is parallel to the global Z axis. The link element length is arbitrarily chosen as ten inches. The only active degree of freedom in the model is U z. Similarly, the only degree of freedom in the linear link element with nonzero properties is the U (axial) degree of freedom. The mass, m, and the ramp loading, p(t), are applied to the joint at the free end of the link, which is joint in the figure. EXAMPLE 6-00 -
SAP000 GEOMETRY, PROPERTIES AND LOADING p(t) p(t) Mass m applied at joint m u(t) Z Y Linear link element with stiffness, k, defined for the U degree of freedom k X Loading p(t) = p o (t / t r ), p o, t = t r t = t r p p o p(t) Properties k = 4 k / in m = k-sec / in p o = 8 kips π sec, (Case A) t r = π / sec, (Case B) t r t SUMMARY OF LOAD CASES IN THIS EXAMPLE The following table summarizes the load cases that are used in this example. Load Case P MODAL MHISTA Description Linear static load case with an 8 kip load applied in the positive global Z direction at joint. An eigenvector-type modal load case. An undamped modal time history with a ramp loading rise time of t r = T = π seconds. EXAMPLE 6-00 -
SAP000 Load Case DHISTA DHISTA MHISTB DHISTB DHISTB MHISTC DHISTC Description An undamped direct integration time history with a ramp loading rise time of t r = T = π seconds using a 0.5 second time step (approximately /0 of system natural period). An undamped direct integration time history with a ramp loading rise time of t r = T = π seconds using a 0.005 second time step. An undamped modal time history with a ramp loading rise time of t r = T = π/ seconds. An undamped direct integration time history with a ramp loading rise time of t r = T = π/ seconds using a 0.5 second time step. An undamped direct integration time history with a ramp loading rise time of t r = T = π/ seconds using a 0.005 second time step. A modal time history with a ramp loading rise time of t r = T = π/ seconds and very large damping. This load case is described in the Discussion section of this example. A direct integration time history with a ramp loading rise time of t r = T = π/ seconds using a 0.5 second time step and very large damping. This load case is described in the Discussion section of this example. The ramp loading rise times are carefully chosen in this example for easy comparison with normalized results presented in Figure 4.5. of section 4.5 on pages 7 through 9 in Chopra 995. As discussed in Chopra 995, when the ramp rise time is an even multiple of the structure period, the velocity at the end of the ramp u ( t r ) is zero and the system does not vibrate during the constantforce phase. Conversely, if the ramp rise time is not an even multiple of the structure period, the system does vibrate during the constant-force phase. EXAMPLE 6-00 - 3
SAP000 Thus for load cases MHISTA, DHISTA and DHISTA the displacement after the ramp rise time where t t r is expected to be constant. The displacement after the ramp rise time for models MHISTB and DHISTB is expected to vary. Models MHISTC and DHISTC are described later. TECHNICAL FEATURES OF SAP000 TESTED Linear links Modal load case for eigenvectors Modal time history load case Direct integration time history load case Ramp loading RESULTS COMPARISON Independent results are hand calculated using theory presented in Section 4.5 on pages 6 through 9 of Chopra 995. In particular, equations 4.5. and 4.5.4b are used. Ramp rise time t r = π seconds Output Parameter Load Case SAP000 Independent MHISTA 0.3478 Percent Difference 0% U z (jt. ) in at t = sec DHISTA dt = 0.5 sec DHISTA dt = 0.005 sec 0.3409 -.5% 0.3478 0.3478 0% MHISTA.0000 0% U z (jt. ) in at t = 4 sec DHISTA dt = 0.5 sec DHISTA dt = 0.005 sec.9979-0.%.0000.0000 0% EXAMPLE 6-00 - 4
SAP000 Ramp rise time t r = π/ seconds Output Parameter Load Case SAP000 Independent MHISTB 0.69436 Percent Difference 0% U z (jt. ) in at t = sec DHISTB dt = 0.5 sec DHISTB dt = 0.005 sec 0.6849 -.5% 0.69436 0.69436 0% MHISTB 0.74030 0% U z (jt. ) in at t = 4 sec DHISTB dt = 0.5 sec DHISTB dt = 0.005 sec 0.7764 -.7% 0.7403 0.74030 0% DISCUSSION For this single degree of freedom problem it is expected that the modal time history results will compare exactly with the theoretical results, and that the direct integration results converge to the theoretical results as the time step used in the analysis is reduced. A typical rule of thumb for direct integration time histories is that the time step used should not be larger than one-tenth of the smallest structural period that is of interest. The 0.5 second time step used in load cases DHISTA, DHISTB and DHISTC was selected using this rule of thumb. The following figure shows a plot of joint displacement versus time for the load cases where the ramp loading rise time, t r, is equal to the structure period, π. As expected, and as discussed in Chopra 995, the displacement is constant when the time exceeds t r. Load cases MHISTA and DHISTA show an exact comparison with the theoretical results while load case DHISTA shows an acceptable comparison with the theoretical results. Note that for DHISTA, the displacement when the time exceeds t r is almost, but not quite, constant. EXAMPLE 6-00 - 5
SAP000 Decreasing the time step used in the direct integration analysis improves the accuracy of the results as demonstrated by load case DHISTA..5 Ramp Rise Time = π, Undamped Joint Displacement (in).5 0.5 Ramp Case MHISTA Case DHISTA Case DHISTA 0 0 3 4 5 6 7 8 9 0 Ramp time = π sec Time (sec) The following figure shows a plot of joint displacement versus time for the load cases with no damping and the ramp loading rise time, t r, equal to one-half the structure period. As expected, and as discussed in Chopra 995, the displacement varies when the time exceeds t r. Again load cases MHISTB and DHISTB show an exact comparison with the theoretical results while load case DHISTB shows an acceptable comparison with the theoretical results. EXAMPLE 6-00 - 6
SAP000 3.5 Ramp Rise Time = π/, Undamped 3 Joint Displacement (in).5.5 0.5 Ramp Case MHISTB Case DHISTB Case DHISTB 0 0 3 4 5 6 7 8 9 0 Ramp time = π/ sec Time (sec) Often times ramp loads are used in models that are subjected to time history analysis. The ramp loads are used to apply gravity loads to the structure before applying other time history loads, such as earthquake, wind, blast, and similar loads. In those cases it is desirable to have the load remain constant after the ramp loading rise time, t r, is exceeded. We recommend that this be achieved by using high damping values, such as 99.9% of critical damping, in the portion of the analysis where the ramp load is used to apply the gravity load. Load cases MHISTC and DHISTC demonstrate this. Load case MHISTC is identical to load case MHISTB, except that the modal damping has been changed from 0% of critical damping to 99.9% of critical damping. Similarly, load case DHISTC is identical to load case DHISTB, except that 99.9% of critical damping has been added. EXAMPLE 6-00 - 7
SAP000 The proportional damping coefficients input for the DHISTC load case to simulate 99.9% of critical damping are calculated as follows. The damping is assumed to be 0.999 for two different frequencies. Those frequencies are the natural frequency of the single degree of freedom system, /π, and an arbitrarily selected frequency of. The mass and stiffness proportional coefficients, α and β respectively, are then calculated as shown, where d and d are damping values and f and f are frequencies: α = 4π ( f f d f f f f d ) = 4π π ** 0.999 π * * 0.999 π α = 3.03 (mass proportional coefficient) f d β = π ( f f d f ) * 0.999 * 0.999 = π π π β = 0.4 (stiffness proportional coefficient) The following figure shows a plot of joint displacement versus time for the load cases with 99.9% of critical damping and the ramp loading rise time, t r, equal to one-half the structure period. The ramp loading behaves as desired in that it remains constant once the maximum value is reached. Note that because of the damping, the maximum value is reached at a time significantly larger than the ramp time. Thus when using this method to apply gravity load, check that the analysis was run for a period of time, beyond the ramp time, that is long enough for the gravity load to have stabilized. No independent theoretical results are provided for these load cases. EXAMPLE 6-00 - 8
SAP000.5 Ramp Rise Time = π/, 99.9 % Critical Damping Joint Displacement (in).5 0.5 Ramp Case MHISTC Case DHISTC 0 0 3 4 5 6 7 8 9 0 Ramp time = π/ sec Time (sec) COMPUTER FILE: Example 6-00 CONCLUSION The SAP000 results show an exact comparison with the independent results for the modal time history analyses and an acceptable comparison with the independent results for the direct integration time history analyses. The results for the direct integration time history analyses converge to the independent and modal time history result as the time step used in the direct integration time history analysis is decreased. EXAMPLE 6-00 - 9
SAP000 HAND CALCULATION EXAMPLE 6-00 - 0
SAP000 EXAMPLE 6-00 -