Abstract

Progressive collapse is a catastrophic structural failure mechanism, triggering by an abnormal event such as explosion that makes local damage of a key member in the structure. After that, other failures which are not directly affected by initial abnormal event, occur in the structure; eventually result in the collapse of the whole structure or to an extent disproportionate to the original failure. In this paper, the intermediate steel moment frames structures with different levels of height designed for moderate and very high level of seismic zones of Iran are studied. For evaluating the potential of progressive collapse, the alternate path method in accordance with both GSA and UFC guidelines is used in two different methods; linear static analysis and nonlinear static analysis. The results show that, generally, for the steel structures designed for higher seismicity there is higher capacity for progressive collapse and in the low height steel structures, there is not enough redundancy to redistribute loads of the failed elements so, the potential of progressive collapse increases with decreasing the height of the structure.

Keywords: progressive collapse, alternate path method, steel moment frames, seismic zones.

1 Introduction After the Northridge and Kobe earthquakes and the failure of moment resisting steel frames in these events, substantial research was accomplished by the engineering community to construct and design steel structures so that they would be more safe and reliable. But blast and impact effects on steel structures, in contrast to seismic effects, have not been adequately studied. During the lifespan of civil engineering structures, manmade hazards such as blast and impact and natural hazards like earthquakes, floods and fires may affect the structures. Structures are usually designed for probable events that may happen during their lifespan, but extreme events which they were not designed for, can result in catastrophic failure. In recent decades, some events such as 1995 Murrah Federal building bombing and 2001 attack on the World Trade Center have shown that engineering structures are vulnerable to extreme events.

A progressive collapse is a situation where local failure of a primary structural component leads to the collapse of adjoining members which, in turn, leads to additional collapse. Hence, the total collapse is disproportionate to the original cause (GSA, 2003). Increasing catastrophic events showed that the prevention or mitigation of progressive collapse must be included as a requirement in building design and analysis. Many methods have been proposed to mitigate progressive collapse and several building codes, standards, and design guidelines have discussed this issue. General Services

Linear and nonlinear analysis of progressive collapse for seismic designed steel moment frames.

M. A. Hadianfard & M. Wassegh Department of Civil and Environmental Engineering, Shiraz University of Technology, Iran

M. Soltani Mohammadi Department of Civil and Environmental Engineering, Tarbiat Modarres University , Iran

Administration (GSA, 2003) and Department of Defense (DoD, 2005) have been used more than the others for designing and analyzing of progressive collapse.

The alternate load path method is a threat independent approach that commonly is used for analysis of progressive collapse. This approach is based on removing a load-bearing element and evaluating stability of remaining structure and also its ability to bridge over the removed element.

There are different analysis procedures for the alternate path method that have been suggested in guidelines. These procedures are linear static, linear dynamic, nonlinear static and nonlinear dynamic. In recent decades, many studies have been performed to evaluate the potential of progressive collapse of buildings by computer modelling and also to evaluate the advantages and disadvantages of each four progressive collapse analysis procedures some of these studies have been performed by (Marjanishvili, 2004), (Powell, 2005), (Marjanishvili and Agnew, 2006), (McKay, 2008). A more complex nonlinear analysis is required to obtain more realistic results but it is better that the static and the dynamic analysis properly be incorporated so that the best results can be achieved for analysis of progressive collapse.

In this study, the potential of progressive collapse is evaluated for the 3-story and 6-story steel moment frames buildings that are designed for medium level and very high level of Iran seismic zones. These structures are designed per Iranian building and seismic codes. The resistance of seismic designed structures to progressive collapse is investigated using the alternate path method in accordance with both (GSA, 2003) and (UFC, 2009) in two different procedures; linear static analysis and nonlinear static analysis. For the analysis of progressive collapse the finite element SAP 2000 software is used. The effects of building height and seismic zones are evaluated and also the results of the linear analysis procedure are compared with those of nonlinear static analysis in two different progressive collapse guidelines.

2 Analysis Procedure

2.1 Linear static analysis (L.S)

The step-by-step procedure for conducting the linear elastic, static analysis in accordance with GSA is as follows: Step 1. Remove a vertical support from the location being considered and conduct a linear-static analysis with the following gravity load imposed on the bay in which the column is removed:

)25.0(2 LLDL + (1)

where DL is dead load and LL represents live load. Step 2. Check DCR (Demand-Capacity Ratio) in structural members. If the DCR of a member

violates the acceptance criteria in shear, the member is to be considered a failed member. If the flexural DCR values of a member end exceed the acceptance criteria, a hinge is placed at the member end. If hinge formation leads to failure mechanism of a member, the member is removed from the model and its loads should be redistributed to adjacent members.

Step 3. At each inserted hinge, equal-but-opposite bending moments, that its magnitude should equal the expected flexural strength of the member, are applied to the member end to each side of the hinge.

Step 4. The analysis is conducted again and the steps 1 through 3 are repeated until DCR of any member does not exceed the limit states. If moments have been redistributed throughout the entire building and DCR values are still exceeded in areas outside of the allowable collapse region defined in the guidelines, the structure will be considered to have a high potential for progressive collapse.

UFC 2009 recommends similar approach for the alternate path method except some characters such as applied load, and acceptance criteria. In UFC 2009, moreover gravity loads also lateral loads

are involved in analysis. In this guideline there are two different loading cases; one apply for deformation-controlled actions and another for force-controlled actions. The loading case for deformation-controlled actions is as follows:

[ ]LDG LDLD5.02.1 +Ω= (2)

where GLD represents increased gravity loads for deformation-controlled actions for linear static analysis. D and L represent dead load and live load respectively and ΩLD represents load increase factor that for steel framed structures is calculated as follows:

1.19.0 +=Ω LIFLD m (3)

where mLIF is the smallest m of any primary beam, girder, spandrel or wall element that is directly connected to the columns directly above the column removal location. Loading case for force-controlled actions is as follows:

[ ]LDG LFLF 5.02.1 +Ω= (4)

where GLF represents increased gravity loads for force-controlled actions for linear static analysis and ΩLF represents load increase factor that is equal 2 for steel framed structures. For any cases of loading, deformation or force-controlled actions, a lateral load that is calculated as follows must be added to the gravity loads:

∑= PLLAT 002.0 (5)

where ∑ P is sum of the gravity loads acting on only that floor. The GSA 2003 and UFC 2009 proposed the use of the DCR as a criterion to determine the failure

of main structural members by the linear analysis procedure. In the GSA 2003 the strength reduction factor is not applied and the inherent strength is obtained by multiplying the nominal strength with the overstrength factor of 1.1. Depending on the width/thickness ratio of the members, the acceptance criteria (DCR) vary from 1.25 to 3. In the UFC 2009 the limit state of DCRs is equal to 1.0 and strength reduction factors are used for calculating member strength.

2.2 Nonlinear static analysis (N.S)

For linear static and nonlinear static analysis the GSA 2003 uses dynamic amplification factor of 2 in load combination. The UFC 2009 uses the following combination of gravity loads to calculate the deformation-controlled and force-controlled actions:

[ ]LDG NN 5.02.1 +Ω= (6)

where GN is increased gravity loads for nonlinear static analysis and ΩN represents dynamic increase factor that is calculated as follows for steel framed structures:

+

+=Ω83.0

76.008.1

y

pra

N

θθ

(7)

where θpra is the plastic rotation angle and θy is the yield rotation. For the nonlinear analysis procedures, the guidelines specify maximum plastic hinge rotation and

ductility as acceptance criteria for progressive collapse.

3 Case study for progressive collapse The structures are considered in this study are the three-story and six-story intermediate steel moment frames structures that have been designed in accordance with building and seismic codes of Iran. It is assumed that the type of used steel is St-37 and the strain hardening is 2% for these structures. Height of stories is 3.2 m and spans of the structures are 5, 6 and 7 m. The structures have block-joists floors that theirs pattern of loading has shown in Figure 1. For ignoring effects of connections it is assumed that, connections have been designed based on sections capacity. The structures are designed in two different seismic zones of Iran; medium level of risk and very high level of risk. We tried to choose specifications of the structures with due considerations of current constructions of steel structures in Iran. Applied gravity loads to the structures are assumed to be as shown in Table 1.

Table 1 Gravity Loads

Figure 1. Structure plan, joist direction, numbering of columns.

4 Analysis of structural models for progressive collapse

4.1 Linear static analysis (L.S)

4.1.1 GSA 2003, L.S According to width/thickness ratio of the beam sections, flexure acceptance criteria (DCR) for all of the beams are 3. The acceptance criteria for columns (DCR) are also according to their width/thickness ratio and for all columns are 2.

For the 3-story structure that designed in the zone of moderate seismic risk, in the cases that columns 1 and 6 were removed in the first story, the results of alternate path analysis showed that DCR for axial, flexure and shear forces, in all elements did not exceed the limit states. But when the column 3 is removed, shear DCR of a beam in the second story that is located in 1-2 bay of frame C exceeds the limit states. Therefore the beam is removed from the model and second iteration of alternate path analysis is carried out after redistribution of loads. The results of second iteration show that DCR of shear forces also exceed the limit states of beams in 1-2 bay of frame C in both first and third stories (Fig. 2). As a result, the occurrence of progressive collapse is possible for the mentioned bay of the structure when the column 3 is removed.

For the 3-story structure that designed in the zone of very high level of seismic risk, in all cases that columns 1, 6, and even column 3 were removed, the results of alternate path analysis showed that DCR for shear, flexure and axial forces in all elements did not exceed the limit states.

For the 6-story structure, for two seismic zones when columns 1, 6 and even column 3 are removed, the DCR of axial, flexure and shear forces, in all members do not exceed the limit states.

Therefore, as the height of the structure increases and the structure is designed for higher seismicity, the potential of progressive collapse decreases.

Load type Story Roof Dead load (kg/m2) 600 650

Live load (kg/m2) 200 150

Perimeter walls

(kg/m)

700 300

a. First iteration b. Second iteration

Figure 2. Location of failed beams in the case of removing column 3, moderate seismic zone, (GSA 2003).

4.1.2 UFC 2009, L.S Because of existence of different methods for calculating of deformation-controlled and force-controlled actions, the UFC 2009 produces two different cases of loading. Therefore, for evaluating of progressive collapse of the structure, two analysis models are evaluated. In these structures, the models related to deformation-controlled actions, for bays that are directly affected by removed columns, are considered under the gravity load of GLD=4.07 (1.2 D + 0.5 L). This load for force-controlled actions is GLF =2 (1.2 D + 0.5L). It is remarkable that analysis of progressive collapse in each models, must be done under gravity loads and lateral loads. The lateral loads are calculated from equation 5. For these structures the lateral load for level of the roof is 6822.4865 (N) and for the other floors are 7300.0704 (N).

For the 3-story structure designed in moderate level of seismic risk, when the column 3 is removed, the beams in 1-2 bay of frame C in all three stories violate shear limit states and failed in the first iteration of alternate path analysis. When the column 6 is removed, in frame 2, shear DCR in beams of A-B bay in all three stories exceed from limit states. In the case of removing column 1, the beams of 1-2 bay of frame A after three iterations of alternate path analysis violate the shear DCR and so progressive collapse for the structure will be probable.

For the 3-story structure designed in the zone of very high level of seismic risk, as it does for structure in the moderate seismic zone, in the cases of removing column 3 and column 6 the shear DCRs exceeded from limit state but the amount of the DCR were less than the corresponding DCR of the structure in the moderate zone. But when the column 1 is removed, all shear DCRs do not violate the limit states, whereas, for the structure in the moderate zone, after three iteration progressive collapse occurred in the structure.

For the 6-sotry structure in the moderate level of seismic risk, when the column 1 and column 6 are removed, the DCR of axial, flexure and shear forces, in all members do not exceed the limit states. But in the case of removing column 3, the shear DCR of beams located in the 1-2 bay of the frame C, in all six stories violate the limit states and fail. This can be resulted from the direction of joist and gravity loading pattern that apply heavy loads on this bay.

For the 6-story structure in the very high seismic zone, when columns 1, 6 and even column 3 is removed, the DCR of axial, flexure and shear forces, in all members do not exceed the limit states.

In spite of possibility of occurrence of progressive collapse for 6-story structure designed for moderate seismicity in the case of removing column 3, the DCRs are less than the three stories structure and it can be concluded, as the same of according to GSA, that probability of occurrence of progressive collapse decreases with increasing the height of the structure and with designing for higher seismicity.

4.2 Nonlinear static analysis (N.S)

The vertical push-over or push-down analysis for progressive collapse was carried out by gradually increasing the gravity loads, then the relationship between the vertical displacement at the location of the removed column and the increasing loads was evaluated in each cases of column removing. The details of this analysis are as follows:

4.2.1 GSA 2003, N.S In accordance with GSA 2003 guideline, for push-down analysis, the gravity load factor is 2 and the lateral loads do not involve in the load combinations. Fig. 3a and Fig. 3b show the push-down curves of the 3-story structure designed in the moderate and very high seismic zones respectively. Fig. 3c and Fig. 3d show the push-down curves of the 6-story structure that were designed in moderate seismic zone and very high seismic zone respectively. In the figures the load factor of 1.0 shows the state that the vertical load reached the full gravity load for nonlinear static progressive collapse analysis in accordance with the GSA guideline. It is observed that the yield strength is higher in the 6-story structure than the 3-story structure and also the yield strength is higher in the structure that located in the zone of very high seismic risk. It is also observed that for all cases, when the column 3 is removed, because of loading pattern, the structure has the least capacity. In all cases of 3-story structure the load factors at yield are less than 1.0, which implies that the structures have the potential for possible progressive collapse.

For 6-story building in the moderate seismic zone, when column 3 is removed, the structure fails before reaching the full loads. When columns 1 and 6 are removed, the full load can be reached and the acceptance criteria are satisfied for all beams and columns. In the very high seismic zone in all cases, the full load can be reached and the acceptance criteria are satisfied for all beams and columns.

Fig. 5 shows the deformed shape of frame A in the case of removing column 1, it can be observed that, in all beams of 1-2 bay of frame A, plastic hinges formed but plastic rotation did not exceed 6 % radian, which is less than the acceptance criterion of 21 % in accordance with GSA 2003.

4.2.2 UFC 2009, N.S In the UFC 2009 guideline, the gravity load factor for push-down analysis is calculated from Eq.7. Fig. 4a and Fig. 4b show the push-down curves of the 3-story structure designed for moderate seismicity and very high seismicity respectively and Fig. 4c and Fig. 4d show the push-down curves of the 6-story structure that were designed in moderate seismic zone and very high seismic zone respectively. In the figures the load factor of 1.0 corresponds to the state that the loads reached the gravity load specified in the UFC guideline.

It is observed that the yield strength is higher in the structure designed for very high seismicity than the structure designed for moderate seismicity and also the yield strength is higher in the 6-story structure than the 3-story structure. It is also observed that when the column 3 is removed, because of gravity loading pattern, the structure has the least capacity. In all cases of 3-story building the load factors at yield are less than 1.0, which implies that the structures may have the potential for possible progressive collapse.

For 6-story building in the moderate seismicity, as can be shown from Fig. 4c, when columns 1 and 6 are removed, the full load can be reached and the acceptance criteria are satisfied for all beams and columns. When column 3 is removed, even though the full loads according to UFC 2009 can be reached, the acceptance criteria can’t be satisfied and the structure collapses progressively.

Generally, it can be observed that, in accordance with the UFC push-down analysis, the capacity of the structures for progressive collapse is higher than the corresponding capacity in GSA and the effects of lateral loads, increase with increasing the height of structures.

a. Moderate seismic zone , 3-story b. Very high seismic zone, 3-story

c. Moderate seismic zone, 6-story d. Very high seismic zone, 6-story Figure 3. Push-down curves in accordance with GSA 2003.

a. Moderate seismic zone, 3-story b. Very high seismic zone, 3-story

c. Moderate seismic zone, 6-story d. Very high seismic zone, 6-story Figure 4. Push-down curves in accordance with UFC 2009.

Figure 5. Rotation of plastic hinges in radian, 6-story and moderate seismic zone, GSA 2003.

5 Conclusions This study investigated the progressive collapse resisting capacity of 3-story and 6-story steel moment frames structures designed in moderate and very high seismic zones of Iran. For analysis of progressive collapse, linear static and nonlinear push-down analysis methods were used in accordance with GSA 2003 and UFC 2009 guidelines. The findings of this study are summarized as follows:

• In general, the potential of progressive collapse decreases with increasing the height of the structures and in short steel structures, there is not enough redundancy to redistribute loads of the failed elements. Also for steel structures designed for higher seismicity, there is less possibility of occurrence of progressive collapse.

• In linear static analysis, because of more gravity loads and existence of lateral load in load combinations of UFC 2009, the resisting-capacity of progressive collapse is less than the GSA 2003. But for nonlinear static analysis, the load factor in UFC for bays that directly affected by removed column, is less than of the GSA and the capacity of the steel structures for progressive collapse is higher than GSA.

• It is suggested that for mitigating progressive collapse, the gravity loads and direction of joist, should not have one-way patterns, so that gravity loads will not be concentrated in some elements and the potential of progressive collapse can be decreased in the structure.

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