**6. FE evaluation process — Seismic response analysis**

The benefit of FE based evaluation is that a great bit of detail of the response of the system is made available through the analysis especially when the time history approach is taken. In essence all DOF selected in the modeling process are accessible over the full length of the simulated event. With further post-processing whether computational or graphical, additional response quantities and behavior can be accessed, plotted, and visualized.

In the MEMA UM campus study [17], it became particularly useful to examine hysteresis of the column section in the plastic hinge region. Figure 15 shows a typical plot of simulated moment-curvature response during the severe (*M 8.3*) event case. The results demonstrate that the yield limit state is achieved in both directions for a corner column, and the ultimate limit state is achieved in one direction. The latter result provided clear evidence of vulnerability and

**Figure 14.** Eigenvalue analysis results for the MEMA bridge study [16]; elevation views of 2D intermediate bent model showing deck transverse displacement (left) and rotation (right) modes; pile bending dominates response although

**Figure 13.** Eigenvalue analysis results for the MDOT bridge study; plan and isometric views of fixed base model trans‐ verse mode; lateral deck and column bending dominates response in this pair of adjacent bridge structures; some

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the possibility of complete failure or collapse.

combined action of bending and axial force in the piles is implied

bending and torsional coupling in the deck is evident

**Figure 13.** Eigenvalue analysis results for the MDOT bridge study; plan and isometric views of fixed base model trans‐ verse mode; lateral deck and column bending dominates response in this pair of adjacent bridge structures; some bending and torsional coupling in the deck is evident

a point on the bridge deck as a reference point, frequency response functions were derived that eliminated the influence of the excitation, and system response frequencies were extracted corresponding with excellent correlation to the 3D model SSI case without any model param‐ eter modification. Accelerometers were then moved to the abutments and frequency extraction performed [11] revealing evidence of the participation of the abutments in the transverse mode

**Figure 12.** Eigenvalue analysis results for the MEMA UM campus study bridge models [17]; top figures show plan and isometric views of fixed based model transverse mode causing bent and abutment column deformation; bottom fig‐ ures show comparable modes for SSI model; skew of roadway alignment introduces coupling of translation and rota‐ tion of the deck mass as well as bending and torsion of the deck; the resistance provided by the embankment is

In the MEMA bridge study, the preliminary analyses were again performed prior to time history analysis. Figure 14 shows that the fundamental mode of vibration for a typical intermediate bent in the interstate highway river crossing is one involving net translation of the deck and corresponding bending of the piles which were designed as axially loaded members. Consideration of the eccentricity of the deck mass with respect to the center of resistance of the soil-pile system provides for expectation of an overturning moment. Such a moment would generate an increase of axial force in one of the batter piles which would

The benefit of FE based evaluation is that a great bit of detail of the response of the system is made available through the analysis especially when the time history approach is taken. In essence all DOF selected in the modeling process are accessible over the full length of the simulated event. With further post-processing whether computational or graphical, additional

shape comparable to the one in Figure 13.

apparent from the contact developed during rotation

198 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

combine with the bending action.

**6. FE evaluation process — Seismic response analysis**

response quantities and behavior can be accessed, plotted, and visualized.

**Figure 14.** Eigenvalue analysis results for the MEMA bridge study [16]; elevation views of 2D intermediate bent model showing deck transverse displacement (left) and rotation (right) modes; pile bending dominates response although combined action of bending and axial force in the piles is implied

In the MEMA UM campus study [17], it became particularly useful to examine hysteresis of the column section in the plastic hinge region. Figure 15 shows a typical plot of simulated moment-curvature response during the severe (*M 8.3*) event case. The results demonstrate that the yield limit state is achieved in both directions for a corner column, and the ultimate limit state is achieved in one direction. The latter result provided clear evidence of vulnerability and the possibility of complete failure or collapse.

**Figure 15.** Normalized moment-curvature hysteresis plots for a corner column of the MEMA UM campus study bridge (see Figs. 3, 8, 12) subject to an extreme event (see Fig. 6); nonlinear user-input curves developed separately using a fiber section model [12, 17]

In the MDOT study [14], with the availability of the USGS simulation tool for developing random realizations of input ground acceleration time histories (Fig. 6) at different intensities, limit state determination was enabled for the columns and piles over the full range of damaging events. Comparison of the peak and characteristic responses enabled a performance evaluation of the system based on critical material, section, or member limit states such as first cracking, first yield, plastic hinge formation, and plastic collapse mechanism. In the case of the bridge studied, it was learned that the piles at the abutments and the columns of the central bent provided substantial energy absorption in the extreme event case through ductile hysteretic response in these members. It was also learned that the pile system at the abutments and central bent adequately distributed lateral forces so that the soil remained linear throughout the event. While nonlinear slip at the superstructure to abutment pile cap bearing connection was attempted, this proved too difficult for the software to resolve and convergence was never reached. Ultimately, rigid connections were assumed and the slip mechanism was interpreted as another potential energy absorption source.

In the MEMA Coldwater River bridges study [16], linear dynamic response was performed for most of the analysis runs. An example of the response motion time histories at the level of a typical intermediate bent pile cap is shown in Figure 16. Peak internal force (axial force, shear, and bending moment) responses in the piles were obtained and compared with design values and pile test data.

In the critical case, a nonlinear static pushover analysis was performed to estimate the capacity of the pile system. In the FE model, both geometric and material nonlinear options for the software were used. In the latter option, a fiber representation of the cross-section was used that accounted for 1D nonlinear normal stress-normal strain behavior in the concrete and the steel reinforcement, enabling computation of the force-displacement behavior shown in Figure 17. A nonlinear time history was also run for this critical case.

In assessing the results of the time history analysis, a simplified analysis was performed using a single DOF equivalent model using a lumped stiffness based on a unit force lateral load analysis. When compared to the linear dynamic analysis of the full bent, this simplified analysis was able to demonstrate that the first mode of the system dominated the overall response and

**Figure 17.** Plots of pushover response to lateral load at bearing positions of intermediate bent pile cap for one of the Coldwater River bridges studied [16] (see Figs. 5 and 10); left plot shows force-displacement response; right plot

**0 5 10 15 20 25 30 35 40**

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**Time (s)**

**0 5 10 15 20 25 30 35 40**

**Time (s)**

**Figure 16.** Normalized time history plots of response at center of intermediate bent pile cap for one of the Coldwater River bridges studied [16] (see Figs. 5 and 10) subject to an extreme event (see Fig. 6 scaled to PGA=0.25g); top plot

could have been used as a predictive tool.

shows deformed shape and limit state condition for plastic hinge locations

**-0.6 -0.3 0 0.3 0.6**

**-1 -0.5 0 0.5 1**

shows acceleration; bottom plot shows displacement)

**a(t), g**

**d(t)/dmax**

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as another potential energy absorption source.

17. A nonlinear time history was also run for this critical case.


0


**κ / κmax**



0


**κ / κmax**

**M / M**

**Figure 15.** Normalized moment-curvature hysteresis plots for a corner column of the MEMA UM campus study bridge (see Figs. 3, 8, 12) subject to an extreme event (see Fig. 6); nonlinear user-input curves developed separately using a

In the MDOT study [14], with the availability of the USGS simulation tool for developing random realizations of input ground acceleration time histories (Fig. 6) at different intensities, limit state determination was enabled for the columns and piles over the full range of damaging events. Comparison of the peak and characteristic responses enabled a performance evaluation of the system based on critical material, section, or member limit states such as first cracking, first yield, plastic hinge formation, and plastic collapse mechanism. In the case of the bridge studied, it was learned that the piles at the abutments and the columns of the central bent provided substantial energy absorption in the extreme event case through ductile hysteretic response in these members. It was also learned that the pile system at the abutments and central bent adequately distributed lateral forces so that the soil remained linear throughout the event. While nonlinear slip at the superstructure to abutment pile cap bearing connection was attempted, this proved too difficult for the software to resolve and convergence was never reached. Ultimately, rigid connections were assumed and the slip mechanism was interpreted

In the MEMA Coldwater River bridges study [16], linear dynamic response was performed for most of the analysis runs. An example of the response motion time histories at the level of a typical intermediate bent pile cap is shown in Figure 16. Peak internal force (axial force, shear, and bending moment) responses in the piles were obtained and compared with design values

In the critical case, a nonlinear static pushover analysis was performed to estimate the capacity of the pile system. In the FE model, both geometric and material nonlinear options for the software were used. In the latter option, a fiber representation of the cross-section was used that accounted for 1D nonlinear normal stress-normal strain behavior in the concrete and the steel reinforcement, enabling computation of the force-displacement behavior shown in Figure

**max**

0.5

2 - dir

1

**M / M**

fiber section model [12, 17]

and pile test data.

**max**

0.5

1 - dir

1

200 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**Figure 16.** Normalized time history plots of response at center of intermediate bent pile cap for one of the Coldwater River bridges studied [16] (see Figs. 5 and 10) subject to an extreme event (see Fig. 6 scaled to PGA=0.25g); top plot shows acceleration; bottom plot shows displacement)

**Figure 17.** Plots of pushover response to lateral load at bearing positions of intermediate bent pile cap for one of the Coldwater River bridges studied [16] (see Figs. 5 and 10); left plot shows force-displacement response; right plot shows deformed shape and limit state condition for plastic hinge locations

In assessing the results of the time history analysis, a simplified analysis was performed using a single DOF equivalent model using a lumped stiffness based on a unit force lateral load analysis. When compared to the linear dynamic analysis of the full bent, this simplified analysis was able to demonstrate that the first mode of the system dominated the overall response and could have been used as a predictive tool.

A decision was made not to depend on nonlinear dynamic response analysis for all the bridges. This was in part due to the lack of confidence in the soil properties at the site from which reasonable assumptions could be made for simulation of nonlinear soil response and in part due to the scope of the work which was limited as has been mentioned previously. A limited attempt was made to verify at least the elastic properties of the soil adjacent to the sites using seismic refraction tests performed near the embankments which were accessible on dry land.

has been considered in these studies and FE based time history response analysis has formed the basis of the performance evaluation. In one case, the objective was to validate results of a regional study that did not consider many of the key details of the structural system using FE based analysis. In this case, the hazard and ground motion intensity were selected to be

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Examples of 2D and 3D structural models are presented that incorporate a wide variety of finite element and material types and consider the effects of soil-foundation-structure inter‐ action which, in the view of the author, is an essential part of reliably establishing the per‐ formance of bridge structural systems. The incorporation of such interaction presents challenges to the analyst which are not well represented in standard textbooks on highway bridge design and even some of the FE literature. To complement such reference works, a portion of the chapter is devoted to discussion of preliminary analyses that are quite naturally performed while the more complex models for final evaluation are constructed. The discussion highlights the importance of first capturing behavioral aspects of the system revealed by static response analysis under gravity or idealized lateral loads and subsequent examination of vibration mode shapes and natural frequencies obtained by eigenvalue analysis. These preliminary analyses provide not only quality assurance but also insight that may guide expectations for the results of the more complex models. The information obtained from subsequent static, nonlinear, and dynamic response analysis is then maximized so that the most useful or telling information is extracted from the analysis under seismic excitation.

The FE based approach to vulnerability assessment ensures that quantitative data formulated on basic mechanics principles is generated for consideration during the assessment. Extracting the data and using it to establish measures of performance remains somewhat of an art. In the study cases, a range of measures has been adopted including peak dynamic response acceler‐ ation and displacement as well as maximum internal forces in critical members and damage distribution in major subsystems. It is hoped that an appreciation of the complexity of highway bridge systems has been provided through the description of the many details of the FE models

Application of the results of FE analysis to a specific vulnerability assessment requires consideration of the objectives and end-user needs. A range of complexity in successive models used in the evaluation may be appropriate depending on the sensitivity of the evaluation on the outcomes of the analysis. Furthermore, the availability of powerful analysis tools should not overshadow lack of confidence in data provided to the analysis. In particular, soil property

In regions of moderate seismic hazard it may prove difficult to establish a sense of urgency for action on the basis of the results of a vulnerability analysis whether or not is based on FE modeling and considered highly accurate. In such a context it may be useful to incorporate the seismic vulnerability assessment in a broader one considering multiple hazards exhibiting

and earthquake intensity and motion characteristics are often not known precisely.

and the results obtained from analysis of response to seismic excitation.

comparable levels of risk.

consistent with that used in the regional study.
