144
CHAPTER 8 NONLINEAR DYNAMIC TIME - HISTORY ANALYSIS - RESULTS AND COMPARISON
8.1
OVERVIEW The inelastic time-history analysis is the most accurate method to
predict the force and deformation demands at various components of the structure. The use of the inelastic time history analysis is limited because the dynamic response is very sensitive to the modeling and ground motion characteristics. It requires proper modeling of the cyclic load-deformation characteristics, and careful consideration of the deterioration properties of all the important components. The computation time, the time required for input preparation, and interpreting the voluminous output, makes the use of the inelastic time history analysis difficult for seismic performance evaluation. In the present study, SAP2000 was used in performing the nonlinear dynamic time-history analysis on the three-dimensional model of the bridge, with the El Centro Earthquake ground motion. The classical Newmark integration method was used ( =0.5, =0.25), with a time step of t=0.01s with a total of 4000 steps (input time: 40s). In line with most previous studies, it was deemed necessary to compare the results of the modal inelastic pushover analysis, with those of the nonlinear time-history analysis. Nonlinear time-history analysis was performed in both longitudinal and transverse directions of the bridge structure using the El Centro Earthquake, and the results are compared with the modal pushover analyses results.
145
8.2
NONLINEAR TIME-HISTORY AND MODAL PUSHOVER ANALYSES – RESULTS AND COMPARISON
8.2.1
Displacement of the Deck at Each Bent Location in the Transverse Direction The displacement of the deck calculated at each bent location, when
the modal pushover analysis and the time-history analysis were carried out in the transverse direction of the bridge structure, is shown in Figure 8.1. From the pushover analysis results, it was found that for the fundamental mode, the center of the mass of the superstructure directly above bent B4 experienced a maximum deck displacement of 87mm, whereas in the higher mode (eighth mode), the center of the mass of the superstructure underwent a maximum displacement of 84mm. As both the fundamental mode and higher mode experienced more or less the same deck displacement the total responses of the deck at each bent location, by using the modal combination rule (SRSS), was found to be of a considerably larger value. The results of the modal pushover analysis, which accounts for the two transverse modes (fundamental mode and eighth mode), were not closer to those of the time-history analysis, due to the estimation of the total response by using the modal combination rule (SRSS) (Figure 8.1).
146
140
Mode#1
Mode#8
Time-history
SRSS
120 100 80 60 40 20 0 B1
B2
B3
B4 B5 Bent Number
B6
B7
Figure 8.1 Displacement of the deck in the transverse direction From Figure 8.1, it was observed that the SRSS overestimates the transverse displacement of the deck of the bridge (120mm), compared to the more accurate approach of the nonlinear time-history (78.3mm). On the other hand, from both the results of the independent pushover analysis of mode#1 and mode#8, it was found that the displacement of the deck at all the bent locations in the higher mode (mode#8) were much closer to the nonlinear time-history analysis results, indicating the significance of the higher mode. 8.2.2
Bent top Displacement in the Transverse Direction The bent top displacements determined by the standard pushover
analysis (SPA) for the fundamental mode, modal pushover analyses (mode#1 and mode#8) and the SRSS results, were compared with those from the nonlinear time history analysis, and are shown in Figure 8.2.
From the
147
pushover analysis results it was found that for the fundamental mode, the middle bent B4 experienced a maximum displacement of 79.2mm, whereas in the higher mode (eighth mode), the middle bent underwent a maximum displacement of 75.9mm. The comparison of the results of mode#1, mode#8, the SRSS and the time-history analysis are shown in Figure 8.2.
Mode#1
Mode#8
Time-history
SRSS
120 100 80 60 40 20 0 B1
B2
B3
B4
B5
B6
B7
Bent Number
Figure 8.2 Bent top displacement in the transverse direction From the results of the independent pushover analysis of mode#1 and mode#8, it was found that the bent top displacements observed in both the fundamental mode and the higher mode (mode#8) overestimate the displacement observed with the nonlinear time-history analysis. The bent top displacement calculated using the SRSS overestimates the results compared to mode#1, mode#8 and the time-history results.
148
8.2.3
Top Drift (%) for Bridge Bents and Decks in the Transverse Direction The top drift (%) for each bent has been calculated by dividing the
maximum bent top displacement by the height of the bridge bent, and multiplied by hundred. Each bent top displacement was found from the nonlinear static pushover analyses and time history analysis. The comparison of each bent top drift in the transverse direction obtained, is shown in Figure 8.3.
1.80
Mode#1
Mode#8
Time-history
1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 B1
B2
B3
B4 B5 Bent Number
B6
B7
Figure 8.3 Bent top drift (%) Similarly, the deck drift (%) at each bent location was calculated, and is shown in Figure 8.4. It was observed that the maximum deck top drift was 1.1 times greater than the bent top drift obtained from the pushover analysis, and 1.4 times greater than that obtained from the time history analysis.
149
2.00 Mode#1
Mode#8
Time-history
1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 B1
B2
B3
B4 B5 Bent Number
B6
B7
Figure 8.4 Deck drift (%) 8.2.4
Drift Capacity and Demand in the Transverse Direction Drift capacity is defined as the global drift of the bent, which is
obtained from the pushover analysis. The drift demand is defined as the average maximum bent top drift, when subjected to an earthquake load. The global drift capacity and demand of the bent in the transverse direction is shown in Table 8.1. The global drift of the bent was greater than the drift demand. Table 8.1 Drift capacity and demand in the transverse direction Sl. No.
Drift capacity
Drift demand
1.
1.79
1.61
150
8.2.5
Control Node Displacement in the Transverse and Longitudinal Directions In the study bridge, the center of the mass of the superstructure was
assumed as the control node. The control node displacements obtained from the longitudinal and transverse pushover analysis were compared with the time-history analysis results. The control node displacements of the bridge structure exhibited by the time-history analysis in the transverse and longitudinal directions are shown in Figures 8.5 and 8.6. The comparison of the modal pushover analysis results with the time–history analysis results in the transverse and longitudinal directions, are given in Tables 8.2 and 8.3 repectively.
Figure 8.5 Control node displacement in the transverse direction From Table 8.2, it is found that modal pushover analyses and the SRSS overestimate the control node displacement, when compared with the time-history analysis.
151
Table 8.2 Control node displacement in the transverse direction Sl. No.
1.
Pushover analysis mode#1 mode#8 (mm) (mm) 87 84
SRSS (mm)
Time-history analysis (mm)
121
78
Figure 8.6 Control node displacement in the longitudinal direction Table 8.3 Control node displacement in the longitudinal direction
Sl. No.
1.
Pushover analysis
Time-history
(mode#2)
analysis
(mm)
(mm)
22
26
From the longitudinal pushover analysis results, it was observed that there was an equal deck displacement of 22mm in the longitudinal direction. The expansion bearing, which is the compression only element has 25.4mm thickness. The expansion bearing which is modeled as gap element
152
will contribute resistance, when the relative displacement between the adjacent spans is more than the initial gap of 25.4mm. When the gap closes, pounding occurs and the gap element offers infinite stiffness. The longitudinal displacements of each span of the bridge structure obtained from pushover and time-history analyses are given in Table 8.4. From the pushover analysis results it was found, that the span displacements have not exceeded 25.4mm, indicating that pounding damage would not occur in the bridge. Table 8.4 Longitudinal displacement of each span of the bridge
Span Number
End of the deck
Pushover analysis (mode#2) (mm)
I II III IV V VI VII
VIII
Time-history analysis (mm)
left
7.80
8.5
right
9.80
10.84
left
14.50
16.33
right
16.00
18.2
left
19.20
22.2
right
20.10
23.4
left
21.70
25.6
right
22.00
26
left
22.00
26
right
21.70
25.6
left
20.10
23.4
right
19.20
22.2
left
16.00
18.2
right
14.50
16.33
left
9.80
10.8
right
7.80
8.5
153
From the time history analysis results it was found that spans IV and V undergo a maximum displacement of 26mm, which was greater than the thickness of the expansion joint indicating that pounding damage would occur in the bridge structure. Thus, the pushover analysis had failed to predict the pounding damage that could occur in the bridge. 8.2.6
Base Shear in the Transverse Direction Base shear is the shearing force (Vb) developed at the base of a
structure by the tendency of its upper mass to remain at rest, while the base is translated by ground motion during an earthquake. The base shear obtained when the bridge is subjected to the El Centro Earthquake in the transverse direction is shown in Figure 8.7. The comparison of the base shear values obtained from the fundamental mode (mode#1), the higher mode (mode#8), SRSS and the nonlinear time-history analysis are given in Table 8.5. The nonlinear time-history analysis estimated a maximum lateral force value of 29070kN that had occurred due to seismic ground motion at the base of the structure during the time period of 0.24s. The maximum base shear value obtained from the time-history analysis was 2.79 times of the base shear value obtained from the fundamental mode pushover analysis, and 4.91 times that of the base shear value obtained from the higher mode. Thus, the pushover analysis results and SRSS underestimated the base shear value in the transverse direction compared to time-history analysis.
154
Figure 8.7 Base Shear in the transverse direction Table 8.5 Base Shear in the transverse direction Pushover Analysis (mode#1)
(mode#8)
SRSS (kN)
10417.61
5911.78
11978.14
Sl.No. 1.
8.2.7
Time-history analysis (kN) 29070.00
Base Shear in the Longitudinal Direction The base shear obtained when the bridge is subjected to the El
Centro earthquake in the longitudinal direction is shown in Figure 8.8. Table 8.6 compares the base shear values obtained from the mode (mode#2) which participated in the vibration of the structure in the longitudinal direction, with the results of the time-history analysis. Similar to the results obtained in the transverse direction; in the longitudinal direction also, the pushover analysis underestimated the base shear value.
155
Figure 8.8 Base shear in the longitudinal direction Table 8.6 Base shear in the longitudinal direction
Sl.No. 1.
8.2.8
Pushover analysis Time-history analysis (kN)
(kN)
38619.86
41670.00
Overturning Moment The comparison of the overturning moment obtained from the
modal pushover analysis and the time-history analysis when performed in the transverse direction with the MPA procedure (SRSS) is shown in Figure 8.9. The use of MPA procedure (SRSS) rule results in considerable overestimation of the column moments. The modal pushover analyses results match very well with the time-history analysis results in the intermediate bents, while they overestimate the result at the exterior bents.
156
5000 Mode#1
Mode#8
Time-history
SRSS
4500 4000 3500 3000 2500 2000 1500 1000 500 0 B1
B2
B3
B4
B5
B6
B7
Bent Number
Figure 8.9 Overturning moment at each bent 8.2.9
Base Shear at Each Bent The comparison of the base shear values obtained from the modal
pushover and time history analyses, SRSS in the transverse direction is shown in Figure 8.10. It was observed that, the difference between the base shear values calculated from the time-history analysis and those from the nonlinear static analysis, is more near the abutments of the bridge. The base shear values obtained from the modal pushover analyses were a little greater than the time-history analysis results in the intermediate bents, while they overestimated the base shear at the exterior bents. The SRSS rule results in a considerable overestimation of the column’s shear.
157
1800
Mode#1
Mode#8
Time-history
SRSS
1600 1400 1200 1000 800 600 400 200 0 B1
B2
B3
B4
B5
B6
B7
Bent Number
Figure 8.10 Base shear at each bent 8.2.10
Summary The nonlinear time-history analysis was performed in both
longitudinal and transverse directions of the bridge structure using El Centro Earthquake record and results are compared with modal pushover analyses results. Modal pushover analysis and the SRSS have overestimated the bent top displacements compared to time-history analysis. The modal combination rule (SRSS) overestimates the displacement of each span at all the bent locations in the transverse direction. The transverse displacement of each span at all the bent locations in the higher mode (mode#8) was much closer to the nonlinear time-history analysis results, indicating the significance of the higher mode.
158
Modal pushover analyses overestimate the control node displacement in the transverse direction, when compared with the time-history analysis. Modal pushover analysis underestimates the control node displacement in the longitudinal direction, when compared with the time-history analysis. The maximum base shear value obtained from the time-history analysis was compared with the modal pushover analysis and SRSS rule. It was found that modal pushover analysis results as well as the SRSS value underestimated the base shear, when compared with the time-history analysis. In the intermediate bents, at each bent location the overturning moment and shear forces calculated were found to match well in both modal pushover analyses and time-history analysis. At the exterior bents a larger difference in results was observed. From the pushover analysis results performed in the longitudinal direction, it was found that the displacement of the deck panels have not exceeded the expansion joint thickness, indicating that pounding damage would not occur in the bridge. From the time-history analysis results, it was found that spans IV and V undergo a maximum displacement, greater than the thickness of the expansion joint indicating that pounding damage would occur in the bridge structure. Thus, the pushover analysis had failed to predict the pounding damage that could occur in the bridge.
