On the Performance of Steel Buildings with Skewed Beams against Progressive Collapse

: This research studies the performance of two-and four-story steel building models against progressive collapse in two cases: (1) skewed beam–column connections, and (2) straight cleated beam–beam connections. Nonlinear static and dynamic analyses were performed and then the time history analysis was performed under the simultaneous effect of the two horizontal components of three earthquake records and two column removal scenarios. The results showed that the 60-degree skewed connection had a weaker performance than the straight connection, and, according to the plastic hinge distribution, it was observed that the mentioned connection did not provide the life safety performance level.


Introduction
The progressive collapse of structures is generally caused by a local failure caused by accidental actions, such as fire, gas explosion, car collision, design and construction errors, or environmental corrosion, and leads to failure of a wide range of the building until its total failure.Khandelwal et al. [1] evaluated the progressive collapse of special and intermediate moment frames and observed that the configuration and strength of the system are more effective than the construction details for ductility in the resistance of the structure against the critical member removal scenario.Kim and Kim [2] investigated the seismic performance and progressive collapse of steel moment frames with three types of seismic connections and concluded that the progressive collapse potential of structures designed for moderate earthquake risk is significantly different according to the type of connection and structures designed for high earthquake levels, which are safer against progressive collapse caused by the sudden removal of a column.Sometimes, due to architectural issues, consideration of skewed beams in building plans are inevitable [3,4]; on the other hand, using skewed beams cause adjacent connections to be subjected to extra forces and moments, and this situation may have adverse effect on the performance of steel buildings against progressive collapse.In this regard, the present study investigates the resistance of two-and four-story building models against progressive collapse in which one group of models have skewed beam-column connections with the inclination angles of 15, 30, 45, and 60 degrees, and the second group of models have straight cleated beambeam connections corresponding to any considered angle.The performance of considered steel building models were studied using non-linear static (i.e., Push-Down) and nonlinear dynamic analyses according to the UFC regulations [5].It should be noted that the modeling methods used in this study are validated in two steps.The best and weakest modes are determined based on the results and finally, in order to evaluate the performance Eng.Proc.2023, 53, 11 2 of 6 of a building that loses one of its members during an earthquake, the two-story model was subjected to simultaneous analysis of the time history and column removal scenario.

Materials and Methods
To ensure the accuracy of the modeling procedure used in this study, the results of numerical modeling by ETABS were compared with the results of experimental and analytical studies performed by Sadek et al. [6] and Rezaei et al. [7]. Figure 1 shows the comparison of the results obtained from ETABS and existing studies [6,7].According to Figure 1a, it is seen that the experimental and numerical results are different in the range of inelastic behavior and hence, the amount of the dissipated energy for both diagrams was compared and it was concluded that the amount of this parameter is almost the same for both numerical and experimental diagrams, which approves the accuracy of the considered finite element modeling procedure for the pushdown analysis.It seems that one of the reasons for the difference of both diagrams in the range of inelastic behavior is that the assigned plastic hinges in ETABS are considered as concentrated hinges, while in reality, the yielded region of the members are expanded in a certain length.Also, in order to validate the considered modeling method for consideration of the nonlinear behavior of building models, the results obtained by ETABS were compared to the results obtained by Rezaei et al. [7] as depicted in Figure 1b, which are in a good agreement.
the modeling methods used in this study are validated in two steps.The best and weakest modes are determined based on the results and finally, in order to evaluate the performance of a building that loses one of its members during an earthquake, the two-story model was subjected to simultaneous analysis of the time history and column removal scenario.

Materials and Methods
To ensure the accuracy of the modeling procedure used in this study, the results of numerical modeling by ETABS were compared with the results of experimental and analytical studies performed by Sadek et al. [6] and Rezaei et al. [7]. Figure 1 shows the comparison of the results obtained from ETABS and existing studies [6,7].According to Figure 1a, it is seen that the experimental and numerical results are different in the range of inelastic behavior and hence, the amount of the dissipated energy for both diagrams was compared and it was concluded that the amount of this parameter is almost the same for both numerical and experimental diagrams, which approves the accuracy of the considered finite element modeling procedure for the pushdown analysis.It seems that one of the reasons for the difference of both diagrams in the range of inelastic behavior is that the assigned plastic hinges in ETABS are considered as concentrated hinges, while in reality, the yielded region of the members are expanded in a certain length.Also, in order to validate the considered modeling method for consideration of the nonlinear behavior of building models, the results obtained by ETABS were compared to the results obtained by Rezaei et al. [7] as depicted in Figure 1b, which are in a good agreement.In order to investigate the progressive collapse behavior of the steel structures with a skewed beam-column connection, two-and four-story building models were considered in two cases: (1) skewed beam-column connections at angles of 15, 30, 45, and 60 degrees, and (2) straight cleated beam-beam connections corresponding to any considered angle.To this aim, nonlinear static (push-down) and nonlinear dynamic analyses were performed based on the UFC standard.According to the UFC regulations, for the push-down analysis, the gravity load is multiplied in the spans around the removed column by a coefficient called the dynamic increase factor, and then the top node of the removed column structure is pushed down in the form of a displacement control procedure.In this method, which is suitable for obtaining the capacity of the structure, the structure is pushed up to its final capacity and finally, the force-displacement curve is obtained.In order to evaluate the performance of the structure against progressive collapse, according to Equation (1), a factor called the load proportion factor (LPF) is defined.Obtaining values greater than or equal to one for this coefficient indicates that the structure has the ability to withstand the gravity load due to the removal of the column and progressive In order to investigate the progressive collapse behavior of the steel structures with a skewed beam-column connection, two-and four-story building models were considered in two cases: (1) skewed beam-column connections at angles of 15, 30, 45, and 60 degrees, and (2) straight cleated beam-beam connections corresponding to any considered angle.To this aim, nonlinear static (push-down) and nonlinear dynamic analyses were performed based on the UFC standard.According to the UFC regulations, for the push-down analysis, the gravity load is multiplied in the spans around the removed column by a coefficient called the dynamic increase factor, and then the top node of the removed column structure is pushed down in the form of a displacement control procedure.In this method, which is suitable for obtaining the capacity of the structure, the structure is pushed up to its final capacity and finally, the force-displacement curve is obtained.In order to evaluate the performance of the structure against progressive collapse, according to Equation (1), a factor called the load proportion factor (LPF) is defined.Obtaining values greater than or equal to one for this coefficient indicates that the structure has the ability to withstand the gravity load due to the removal of the column and progressive collapse does not occur, and if this coefficient is smaller than one, there is a possibility of progressive collapse.
Load proportion factor = (Balance load corresponding to the failure displacement)/(Total gravity load) (1) In the nonlinear dynamic analysis method, the structure is analyzed under the effect of the gravity load combination introduced in the UFC regulation, and then the column forces are determined according to the desired column removal scenario; the calculated values of internal forces and moments are applied in a concentrated form at the top node of the removed column, and during the history analysis, the equivalent forces of the desired column are dynamically removed.The considered building plans with skewed beamcolumn connections and corresponding straight cleated beam-beam connections are shown in Figure 2a,b.Based on the specifications of the UFC guideline, the considered scenarios for removing columns were also selected as presented in Table 1.
collapse does not occur, and if this coefficient is smaller than one, there is a possibility of progressive collapse.
Load proportion factor = (Balance load corresponding to the failure displacement)/(Total gravity load) In the nonlinear dynamic analysis method, the structure is analyzed under the effect of the gravity load combination introduced in the UFC regulation, and then the column forces are determined according to the desired column removal scenario; the calculated values of internal forces and moments are applied in a concentrated form at the top node of the removed column, and during the history analysis, the equivalent forces of the desired column are dynamically removed.The considered building plans with skewed beam-column connections and corresponding straight cleated beam-beam connections are shown in Figure 2a,b.Based on the specifications of the UFC guideline, the considered scenarios for removing columns were also selected as presented in Table 1.Finally, the time history analysis was performed considering the simultaneous effect of the horizontal components of three earthquake records and two column removal scenarios to evaluate the behavior of the two-story building model.The characteristics of the selected records are shown in Table 2.In this research, different parameters such as the load proportion factor (LPF), the vertical displacement time history of the joint above the removed column, distribution of plastic hinges, column axial forces, and rotation of plastic hinges were compared with each other to investigate the performance of the considered building models against progressive collapse.Finally, the time history analysis was performed considering the simultaneous effect of the horizontal components of three earthquake records and two column removal scenarios to evaluate the behavior of the two-story building model.The characteristics of the selected records are shown in Table 2.In this research, different parameters such as the load proportion factor (LPF), the vertical displacement time history of the joint above the removed column, distribution of plastic hinges, column axial forces, and rotation of plastic hinges were compared with each other to investigate the performance of the considered building models against progressive collapse.

Results and Discussion
As the value of the load proportion factor (LPF) is increased compared to 1, the performance of the building against progressive collapse improves and the building can tolerate the gravity load.According to the diagrams shown in Figure 3a-d, in the models with both skewed beam-column connections (A) or straight cleated beam-beam connections (B), the column removal scenario of 4 has a better performance compared to other scenarios.In general, the corner column removal scenarios have the weakest performance against progressive collapse; however, the LPF ratio is greater than 1 in all column removal scenarios for the considered two-story building models.As shown in Figure 3e,f, after the removal of the column, despite the increase in the value of the axial force of all the columns, major changes and a redistribution of the axial forces occur in the columns around the removed column, which reveals the importance of paying attention to these columns.

Results and Discussion
As the value of the load proportion factor (LPF) is increased compared to 1, the performance of the building against progressive collapse improves and the building can tolerate the gravity load.According to the diagrams shown in Figure 3a-d, in the models with both skewed beam-column connections (A) or straight cleated beam-beam connections (B), the column removal scenario of 4 has a better performance compared to other scenarios.In general, the corner column removal scenarios have the weakest performance against progressive collapse; however, the LPF ratio is greater than 1 in all column removal scenarios for the considered two-story building models.As shown in Figure 3e,f, after the removal of the column, despite the increase in the value of the axial force of all the columns, major changes and a redistribution of the axial forces occur in the columns around the removed column, which reveals the importance of paying attention to these columns.According to Figure 4, it can be seen that in four-story buildings, the scenarios of removing the internal columns have the best performance from the point of view of the LPF, while the scenarios of removing the corner and side columns have a poor performance.So, the scenario number of 4 shows the best performance against progressive collapse.The diagrams depicted in Figures 5 and 6 show the changes in the vertical displacement value at the joint above the removed column for the two-story and four-story buildings under the nonlinear dynamic analysis.According to Figure 4, it can be seen that in four-story buildings, the scenarios of removing the internal columns have the best performance from the point of view of the LPF, while the scenarios of removing the corner and side columns have a poor performance.So, the scenario number of 4 shows the best performance against progressive collapse.

Results and Discussion
As the value of the load proportion factor (LPF) is increased compared to 1, the performance of the building against progressive collapse improves and the building can tolerate the gravity load.According to the diagrams shown in Figure 3a-d, in the models with both skewed beam-column connections (A) or straight cleated beam-beam connections (B), the column removal scenario of 4 has a better performance compared to other scenarios.In general, the corner column removal scenarios have the weakest performance against progressive collapse; however, the LPF ratio is greater than 1 in all column removal scenarios for the considered two-story building models.As shown in Figure 3e,f, after the removal of the column, despite the increase in the value of the axial force of all the columns, major changes and a redistribution of the axial forces occur in the columns around the removed column, which reveals the importance of paying attention to these columns.According to Figure 4, it can be seen that in four-story buildings, the scenarios of removing the internal columns have the best performance from the point of view of the LPF, while the scenarios of removing the corner and side columns have a poor performance.So, the scenario number of 4 shows the best performance against progressive collapse.The diagrams depicted in Figures 5 and 6 show the changes in the vertical displacement value at the joint above the removed column for the two-story and four-story buildings under the nonlinear dynamic analysis.The diagrams depicted in Figures 5 and 6 show the changes in the vertical displacement value at the joint above the removed column for the two-story and four-story buildings under the nonlinear dynamic analysis.Figure 7 shows the vertical displacement time-history of the joint above the removed column for the two-story building models with skewed beam-column connections with angles of 30 and 60 degrees under the simultaneous effect of two horizontal components of the earthquake record number 181.According to Figure 7, it is seen that due to the column removal during the earthquake, there was a sudden increase in the vertical displacement of the joint, which can have adverse effects on the seismic performance of the structure.These results highlight the necessity of considering the resistance of each building against progressive collapse.As shown in Figure 8, it can be seen that in the model with a 30-degree connection in column removal scenario number 4, the value of the interstory drift ratio in X direction is lower than the limit imposed by the Standard 2800 [8], which is equal to 0.025.While in the X direction of the two-story building with a skewed connection of 60 degrees with the corner column removal scenario number 3, the maximum value of the interstory drift ratio has far exceeded 0.025.Moreover, according to the distribution of plastic hinges (Figure 8), it is seen that in the model with a 30-degree skewed connection, the life safety performance level was provided, but the performance level of the model with the 60-degree skewed connection exceeded the collapse prevention performance level.Therefore, it is concluded that the buildings with skewed connections are more sensitive to progressive collapse, especially when the corner columns are removed.Considering the importance of the investigation of buildings' performance against progressive collapse, it is suggested that this issue should be studied in future research for the novel beam-column connections [9,10].Figure 7 shows the vertical displacement time-history of the joint above the removed column for the two-story building models with skewed beam-column connections with angles of 30 and 60 degrees under the simultaneous effect of two horizontal components of the earthquake record number 181.According to Figure 7, it is seen that due to the column removal during the earthquake, there was a sudden increase in the vertical displacement of the joint, which can have adverse effects on the seismic performance of the structure.These results highlight the necessity of considering the resistance of each building against progressive collapse.Figure 7 shows the vertical displacement time-history of the joint above the removed column for the two-story building models with skewed beam-column connections with angles of 30 and 60 degrees under the simultaneous effect of two horizontal components of the earthquake record number 181.According to Figure 7, it is seen that due to the column removal during the earthquake, there was a sudden increase in the vertical displacement of the joint, which can have adverse effects on the seismic performance of the structure.These results highlight the necessity of considering the resistance of each building against progressive collapse.As shown in Figure 8, it can be seen that in the model with a 30-degree connection in column removal scenario number 4, the value of the interstory drift ratio in X direction is lower than the limit imposed by the Standard 2800 [8], which is equal to 0.025.While in the X direction of the two-story building with a skewed connection of 60 degrees with the corner column removal scenario number 3, the maximum value of the interstory drift ratio has far exceeded 0.025.Moreover, according to the distribution of plastic hinges (Figure 8), it is seen that in the model with a 30-degree skewed connection, the life safety performance level was provided, but the performance level of the model with the 60-degree skewed connection exceeded the collapse prevention performance level.Therefore, it is concluded that the buildings with skewed connections are more sensitive to progressive collapse, especially when the corner columns are removed.Considering the importance of the investigation of buildings' performance against progressive collapse, it is suggested that this issue should be studied in future research for the novel beam-column connections [9,10].As shown in Figure 8, it can be seen that in the model with a 30-degree connection in column removal scenario number 4, the value of the interstory drift ratio in X direction is lower than the limit imposed by the Standard 2800 [8], which is equal to 0.025.While in the X direction of the two-story building with a skewed connection of 60 degrees with the corner column removal scenario number 3, the maximum value of the interstory drift ratio has far exceeded 0.025.Moreover, according to the distribution of plastic hinges (Figure 8), it is seen that in the model with a 30-degree skewed connection, the life safety performance level was provided, but the performance level of the model with the 60-degree skewed connection exceeded the collapse prevention performance level.Therefore, it is concluded that the buildings with skewed connections are more sensitive to progressive collapse, especially when the corner columns are removed.Considering the importance of the investigation of buildings' performance against progressive collapse, it is suggested that this issue should be studied in future research for the novel beam-column connections [9,10].

Conclusions
Based on the obtained results, it was concluded that with the increase in the number of floors of the building, due to the increase in the degree of indeterminacy and the redistribution of forces and moments, a better performance against progressive collapse is ob-

Conclusions
Based on the obtained results, it was concluded that with the increase in the number of floors of the building, due to the increase in the degree of indeterminacy and the redistribution of forces and moments, a better performance against progressive collapse is observed.Moreover, the corner column removal scenarios have a weaker performance compared to the internal column against progressive collapse and are more likely to fail in providing the life safety performance level.Therefore, even though the buildings have a good performance against progressive collapse if they meet the design codes' specifications, it is recommended that after designing the building, in order to ensure its resistance against progressive collapse, at least one scenario of removing the corner column should be investigated.Moreover, in the case of a skewed connection or a straight cleated beam-beam connection, due to the possibility of weaker performance, a column removal scenario in columns around that area based on the UFC regulation should be checked.Based on a considered weighting system in this research, it is recommended that if there are skewed beam-column connections with an angle of 60 degrees or more, straight cleated beam-beam connections should be replaced.

Figure 1 .
Figure 1.(a) The model of the experimental specimen tested by Sadek et al. [6] and the comparison of the vertical force-displacement (solid lines) and energy (dashed lines) diagrams of the span midpoint under push-down analysis.(b) The five-story building model with a moment-resisting frame and the comparison of the capacity curves obtained via pushover analysis.

Figure 1 .
Figure 1.(a) The model of the experimental specimen tested by Sadek et al. [6] and the comparison of the vertical force-displacement (solid lines) and energy (dashed lines) diagrams of the span mid-point under push-down analysis.(b) The five-story building model with a moment-resisting frame and the comparison of the capacity curves obtained via pushover analysis.

Figure 2 .
Figure 2. Plans of models to study the effect of the skewed connection in progressive collapse: (a) skewed beam-column connection; (b) straight cleated beam-beam connection (All dimensions are in meters.).

Figure 2 .
Figure 2. Plans of models to study the effect of the skewed connection in progressive collapse: (a) skewed beam-column connection; (b) straight cleated beam-beam connection (All dimensions are in meters).

Figure 3 .
Figure 3. LPF_displacement diagrams of two-story buildings with skewed beam-column connection (A) and corresponding straight cleated beam-beam connection (B) at angles of (a) 15, (b) 30, (c) 45, and (d) 60 degrees and values of axial forces for the scenario number of 1 in the two-story model with a 30° skewed beam-column connection (e) before removing the column and (f) after removing the column (The unit of forces is in kg).

Figure 3 .
Figure 3. LPF_displacement diagrams of two-story buildings with skewed beam-column connection (A) and corresponding straight cleated beam-beam connection (B) at angles of (a) 15, (b) 30, (c) 45, and (d) 60 degrees and values of axial forces for the scenario number of 1 in the two-story model with a 30 • skewed beam-column connection (e) before removing the column and (f) after removing the column (The unit of forces is in kg).

Figure 3 .
Figure 3. LPF_displacement diagrams of two-story buildings with skewed beam-column connection (A) and corresponding straight cleated beam-beam connection (B) at angles of (a) 15, (b) 30, (c) 45, and (d) 60 degrees and values of axial forces for the scenario number of 1 in the two-story model with a 30° skewed beam-column connection (e) before removing the column and (f) after removing the column (The unit of forces is in kg).

Figure 7 .
Figure 7. Vertical displacement-time diagrams of the two-story building models with a (a) 30 degrees skewed connection and the scenario number 4; (b) 60 degrees skewed connection and the scenario number 3.

Figure 7 .
Figure 7. Vertical displacement-time diagrams of the two-story building models with a (a) 30 degrees skewed connection and the scenario number 4; (b) 60 degrees skewed connection and the scenario number 3.

Figure 7 .
Figure 7. Vertical displacement-time diagrams of the two-story building models with a (a) 30 degrees skewed connection and the scenario number 4; (b) 60 degrees skewed connection and the scenario number 3.

Figure 8 .
Figure 8. Maximum interstory drift ratio diagram and distribution of plastic hinges of the two-story models with a (a) 30-degree skewed connection and the column removal scenario number 4, and a (b) 60-degree skewed connection and the column removal scenario number 3.

Figure 8 .
Figure 8. Maximum interstory drift ratio diagram and distribution of plastic hinges of the two-story models with a (a) 30-degree skewed connection and the column removal scenario number 4, and a (b) 60-degree skewed connection and the column removal scenario number 3.

Table 1 .
The considered column removal scenarios.

Table 2 .
Specification of pulse near-field records based on FEMA P695.

Table 1 .
The considered column removal scenarios.

Table 2 .
Specification of pulse near-field records based on FEMA P695.