6.3 Comparison of Results from 2-D and 3-D Analyses
A 3-D model of the Ohio Union building was developed, and progressive collapse analysis was performed using this model. Detailed procedures for modeling and SAP200 analysis were presented in Chapter 4. Figure 6.13 shows the deformed shapes obtained from 3-D linear static analysis after the removal of each column. As shown, the building was significantly deformed as more columns were removed. Larger deformations were observed in the beams and columns in the higher stories and next to the removed columns, which trends were consistent with DCR values calculated from 3-D models (Figure 6.14).
In particular, transverse beams connected to the removed columns significantly deformed, indicating that their contribution to overall resistance of the frame was significant. The deformations in the 3-D model were smaller mainly due to contribution of slab and transverse beams, which were discussed below.
6.3.1 Calculated DCR Values
Figure 6.15 shows a comparison of DCR values for moments determined from 2-D and 3-2-D models after four columns were removed. As mentioned in Section 6.2.1, columns were more impacted than beams in 2-D linear static analysis. Five columns
exceeded the DCR criteria of 2.0 (GSA, 2003), but none for the beams after four columns were removed (Figure 6.6). The DCR values of all beams were less than 1.0, and the maximum DCR value observed in beams was 0.94. However, DCR values calculated from 3-D linear static analysis showed an opposite trend compared to 2-D results. Beams were more influenced by a column loss. As shown in Table 6.2, the maximum DCR value of beams was 1.49 (Beam 65 in the top story and next to the removed Column 22, in Figure 4.2) while that of columns was 0.96 (Column 23 right above removed Column 22, in Figure 4.2). The reason that beams had higher DCR values than columns in the 3-D model was probably due to the larger deformation and participation of beams in the transverse directions. As can be seen in Figure 6.13, beams, especially in the top story, were significantly deformed in the transverse direction after each column removal. It can be concluded that 2-D model may lead to limited and underestimated demands for beams.
More interestingly, it was observed that DCR values calculated from 3-D model were smaller than those from 2-D model for columns and most beams. As shown in Figure 6.15, all members had DCR values of less than 1.5, and satisfied GSA acceptance criteria of 2.0 for columns and 3.0 for beams. This could be mainly due to contribution of transverse beams. The transverse beams can more distribute loads to the connected columns and beams in the transverse direction, leading to a decrease of force demands in structural members.
6.3.2 Comparison of Calculated and Measured Strains
Table 6.3 shows changes in strain (Δε) obtained from the field test, compared
changed as each column was torched and removed (see Section 5.4). Most of the strain values dropped more when the last column was torched than when it was removed.
Therefore, Δε (Field Test) reported in Table 6.3 are the changes in strain values recorded from the strain gauges in the field after last column torching. Δε (Computational Model) are the changes in strain values after last column removal. Δε is calculated by considering the combined effect of axial load and a bending moment, both of which were determined from the SAP2000 analysis:
SE ΔΜ AE
Δε ΔΡ [6.1]
where, ΔP is the change in axial force determined from SAP 2000, ΔM is the change in bending moment determined from SAP 2000, A is the cross sectional area of the column or beam, E is the elastic modulus for a given structural member, and S is the section modulus. Equation 6.1 is based on the assumption that the section stays elastic under the applied loads. This assumption is reasonable because almost all measured and calculated
strain values were under or near the yield strain
The strain gauge 15, attached on Beam 67, was selected from the experimental study to compare the results from 2-D and 3-D models because it was the only strain gauge left in the perimeter frame in the 2-D model after the four columns were removed (location of strain gauges are shown in Figure 5.7). Strain gauges 1, 4, 8, 9 and 11 were attached on the interior columns. As shown in Table 6.3, for strain gauge 15, Δε calculated from the 3-D model was closer to the experimental result than that from the 2-D model. After the column was removed, the loads carried by that column were
transferred to neighboring columns and beams. 3-D model can account for redistribution of the building’s weight to both exterior and interior columns and beams while only exterior members were considered in the 2-D model. All of Δε values calculated from the 3-D models were very comparable to the measured strains. Figures 6.16 to 6.19 compare strain values (Δε) measured in the field and calculated from linear static and nonlinear dynamic analyses in 3-D models. For the interior columns (i.e., Strain gauge 4, 9, and 11) and the beam (i.e, Strain gauge 15), the measured strains were closer to the Δε values calculated from the 3-D nonlinear dynamic analysis. The strain increments (Δε) calculated from the linear static analysis were much larger than the measured values.
6.3.3 Vertical Displacements
Figure 6.20 shows changes in maximum joint displacement calculated from the 3-D linear static analysis during the entire column removal process. After all four columns were removed, the maximum calculated displacements were 3.09, 3.13, 3.95, and 2.90 in.
at the joints above the first (Column 27), second (Column 22), third (Column 2), and fourth (Column 7) removed columns, respectively (Table 6.4). Each maximum displacement was shown by a horizontal line in the Figure 6.20 since the linear static analysis procedure did not consider dynamic behavior and vibration of structural members.
Figures 6.21 and 6.22 show changes of the maximum joint displacements calculated from 2-D and 3-D nonlinear dynamic analysis, respectively. Whenever a column was removed, the building was allowed to deform until it settled, showing
dynamic response effects. Both models show similar trends; Joint 3 above the third removed column had the largest maximum and permanent vertical displacement.
Table 6.4 shows the comparison of maximum vertical displacements calculated from linear static analysis and nonlinear dynamic analysis. Linear static analysis resulted in higher maximum vertical displacements than nonlinear dynamic analysis in both 2-D and 3-D models. For example, the maximum vertical displacement calculated from 2-D linear static analysis was 12.1 in. at Joint 3 while that from the 2-D nonlinear dynamic analysis was 8.06 in. It seems that the impact factor of 2 (i.e., dead loads multiplied by 2) in linear static analysis led to very conservative results. This observation is consistent with the strain results. Ruth et al. (2006) found that an impact factor of 1.5 better represented the dynamic effect, especially for steel moment frames. Marjanishvili (2004) also reported that a more complicated analysis method such as nonlinear dynamic analysis may result in less severe structural response, due to more accurate estimates of load distribution and less stringent evaluation criteria.
Another observation from Table 6.4 was that 3-D models showed lower maximum displacements than 2-D models whether linear static analysis or nonlinear dynamic analysis. Similar to the DCR result, the transverse beams connected to the interior columns and the beams increased the overall resistance of structure, leading to smaller deformation in the 3-D model.
6.3.4 Plastic Hinge Rotations
The acceptance criteria for nonlinear dynamic analysis are determined by plastic hinge rotations or displacement ductility (GSA, 2003). Plastic hinge rotation was chosen
as performance evaluation criteria for the nonlinear dynamic analysis. The plastic hinge locations in the 3-D model are shown in Figure 6.23. The default hinge properties provided in the SAP2000 program were used (see Figure 4.10), which corresponds to the hinge definitions in FEMA 356 (FEMA-356, 2000).
As addressed in Equation 4.4, plastic hinge rotations can be calculated as the ratio of horizontal line and tangent to the maximum joint displacement. Table 6.5 shows plastic hinge rotations at the location where columns were removed after four columns removal. Hinge rotations calculated from the 3-D model were smaller than those from the 2-D model, because of lower maximum displacement values in 3-D nonlinear dynamic analysis. In spite of that, the maximum plastic hinge rotation was only 1.80o at the hinge above third removed column (Column 2) in linear static analysis. For both 2-D and 3-D nonlinear dynamic procedure, the values of plastic hinge rotation were much smaller than 12o of GSA (2003) criteria. Thus, the building was considered not susceptible to progressive collapse according to GSA guidelines (GSA, 2003). As shown in Figure 6.6, however, several columns were subjected to demands larger than the GSA (2003) DCR limit of 2.0. Considering that no significant deformations were observed during field testing, GSA criteria for plastic deformations or hinge rotations may be more realistic than the GSA criteria for force demands or DCR values.
Table 6.1 DCR values calculated from 2-D models for selected frame members.
Table 6.2 DCR values calculated from 3-D model for selected frame members.
Table 6.3 Comparison of change in Strain (Δε) obtained from the field test after last column torching with that calculated from 2-D and 3-D analyses after all columns removal.
Table 6.4 Comparison of vertical displacement (in.) after all columns removal.
Table 6.5 Plastic hinge rotations (θ, degree) at the location where each column was removed after all columns removal.
Removed Columns 2-D Nonlinear Dynamic Analysis
Figure 6.1 Bending moment diagram after removal of first column (Column 27).
Figure 6.2 Bending moment diagram after removal of second column (Column 22).
unit: kip-ft unit: kip-ft
Figure 6.3 Bending moment diagram after removal of third column (Column 2).
Figure 6.4 Bending moment diagram after removal of fourth column (Column 7).
unit: kip-ft
unit: kip-ft
Figure 6.5 Moment diagram and corresponding DCR values after loss of four columns in the Ohio Union building.
Figure 6.6 Change in DCR values of each frame member for all cases.
0.32
Figure 6.7 Column removal procedure for dynamic analysis.
Figure 6.9 Displacement of the joint above the first removed column (Joint 1) after the first column removal.
Figure 6.10 Displacement of the joints above the first and second removed columns (Joint 1 and 2, respectively) after the second column removal.
Maximum Vertical Displacement
Permanent Vertical Displacement
Figure 6.11 Displacement of the joints above the first, second, and third removed columns (Joint 1, 2, and 3, respectively) after the third column removal.
Figure 6.13 Deformed shape of the 3-D Ohio Union building model after each column removal as part of linear static analysis.
Figure 6.14 Deformed shape of 3-D model with corresponding DCR values after the loss of four columns.
Figure 6.15 Comparison of DCR values determined from 2-D and 3-D linear static analysis after four columns removal.
Figure 6.16 Comparison of measured and calculated strain values for Strain Gauge 4.
Figure 6.17 Comparison of measured and calculated strain values for Strain Gauge 9.
Figure 6.18 Comparison of measured and calculated strain values for Strain Gauge 11.
Figure 6.20 Changes in maximum joint displacement calculated from the 3-D linear static analysis during the entire column removal process.
Figure 6.21 Time history of joint displacements calculated from 2-D nonlinear dynamic analysis.
Figure 6.22 Time history of joint displacements calculated from 3-D nonlinear dynamic analysis.
CHAPTER 7