Investigating progressive collapse of tall structures under beam removal scenarios after earthquake is a complex subject because the earthquake damage acts as an initial condition for the subsequent scenario. An investigation is performed here on a 10 story steel moment resisting structure designed to meet the life safety level of performance when different beam removal scenarios after earthquake are considered. To this end, the structure is first subjected to the design earthquake simulated by Tabas earthquake acceleration. The beam removal scenarios are then considered at different locations assuming that both ends connections of the beam to columns are simultaneously detached from the columns; thus the removed beam falls on the underneath floor with an impact. This imposes considerable loads to the structure leading to a progressive collapse in all the scenarios considered. The results also show that the upper stories are much more vulnerable under such scenarios than the lower stories. Hence, more attention shall be paid to the beam-to-column connections of the upper stories during the process of design and construction.
Investigating the vulnerability of urban structures under a local collapse has been a worldwide interest for quite a long time. The theory behind is that, a local collapse in a limited number of structural components should not progress into the collapse of other components; so that the overall structural integrity should be kept. There are several strategies to design structures against progressive collapse; some strategies address it directly such as specific local resistance (SLR) and alternative load path method (APM), and some address it indirectly such as prescriptive design rules (PDR). Using SLR strategy, it is endeavored to provide adequate strength to the key structural components in such a way that they all can survive abnormal loads [1]. When there is a local elimination of vertical bearing components, providing an alternative load path is aimed using APM strategy. On the other hand, PDR strategy is basically a quantitative risk assessment where the scope of analysis is first defined after which the hazards are identified and ranked. The information collected in the previous steps is then employed to perform a quantitative risk analysis.
From a different perspective, although most building codes and regulations have provided adequate safety under one extreme load to a great extent, they have shown not to be supportive when there are two successive loads. The collapse of Murrah Federal Building, Oklahoma City, in 1995 and the World Trade Center (WTC) Twin Towers, New York City, in 2001 are examples of progressive collapse, when two successive loads were imposed to the buildings. Both buildings had been designed to resist earthquake loads adequately. This shows the importance of re-evaluating important structures for which unsatisfactory performance is catastrophic.
Progressive collapse analysis has received considerable attention for several decades, starting with the Ronan Point event in 1968 and continuing to receive even with more attention after the horrific event of 9/11. Progressive collapse investigations have often been concentrated on collapse under service loads [2–5]; yet, there have been limited studies on progressive collapse of structures under seismic loads. Hashemi et al. [6] scrutinized the response of concentrically braced structures under brace removal scenarios during earthquake using incremental dynamic analysis. Their investigation revealed that the loss of one or two braces resulted in a reduction in the seismic performance and an increase in the inter-story drifts. Tavakoli and Rashidi [7] investigated the possibility of progressive collapse of seismically designed steel structures under column removal scenarios. They found that structures were more susceptible to seismic progressive collapse when an external column was removed. This particularly was the case when a column was removed from the first story. Xin et al. [8] investigated the performance of super-tall mega-braced structures under seismic loads. They showed that the collapse process has a correlation with the features of the structural system, e.g., beam-to-column-connections. In a similar attempt, Zhang et al. [9] investigated the performance of a spherical shell under different peak ground accelerations (PGA) using a methodology called three-phase nonlinear simulation. Using their methodology, Zhang et al. introduced a damage index via which the performance level of the spherical shell under seismic loads could be described. Recently, Rezvani et al. [10] investigated the structural response of a generic steel frame under beam-removal scenarios after earthquake. They assumed that the gravity loads of the removed beam is applied to the beam underneath over a static process while the effect of impact was ignored for simplicity. The results showed that considering such scenarios did not result in successive collapse in other members; yet, upper stories showed to be more sensitive toward beam-removal scenarios than lower stories.
An investigation is performed here on a 10 story tall steel moment resisting structure when various beam-removal scenarios after earthquake are considered. While there are a number of external reasons for the failure of connections such as explosion, here it is assumed that deficiency in design or construction is the reason of failure. The investigation is relevant to current practice considering that failure of beam-to-column-connections after earthquake has been a prevalent issue [11], though it has not received adequate attention from scholars. The investigation here is performed sequentially; when gravity and earthquake loads are first applied to the structure and it is analyzed over a nonlinear dynamic process, and then when the earthquake-damaged structure is subjected to beam-removal scenarios. The earthquake-damaged structure thus acts as an initial condition for the subsequent scenarios. It is assumed that the end connections of beams are concurrently detached from the columns’ face; hence, their gravity loads are transferred to the story underneath with an impact. While dependent on the velocity of impact, the materials nonlinearity may change significantly; it is supposed that in low velocity impact the changes can be ignored. In that case, the two following questions should be answered: 1) Do the beams in the story underneath have adequate resistance for such additional gravity loads? 2) What would be the effect of such scenarios on the response of the adjacent columns?
Impact of failed beam on the beam underneath
When a beam is removed from the structure, it falls on the beam underneath that is called hereafter receiving beam with an impact. As this is a collision between two bodies, the resulting response depends mostly on the relative velocity of the bodies at the time of impact. The impact velocities are grouped into three, low-velocity (defined as impact velocities less than 250 m/s), medium-velocity (defined as impact velocities between 250 m/s to 2000 m/s), and high-velocity (defined as impact velocities beyond 2000 m/s) [12]. As the impact velocities increase from low to high, the local behavior near the contact area may change significantly. In the low-velocity condition, the impact can significantly influence the structural response. A medium-velocity condition can cause microscopic changes in the influenced body, e.g., high-strain rate [13]. As the velocity goes to high, the changes may lead to extremely localized pressures which might be beyond the material strength [14]. For the case considered here, it is believed that the impact caused by the failed beam is rather low [15]; thus, there is no need to consider the high-strain rate as pointed out. On the other hand, when the low-velocity impact is assumed, depending on the level of flexibility of the bodies, different deformation types can be developed [16]. Here, it is assumed that the collision between two flexible bodies is a transverse impact meaning that the collision induces vibrations over some longitudinal distance from the collision point. As vibration is a function of time, a time-dependent analysis is thus required to present the bodies’ response. In frame structures, the collision of a beam to another creates a flexural response near the contact region called near field and a deformation in the far field region. Concentrating the mass of fallen beam at one point using a single degree of freedom (SDOF) model, the impact modeling can be simplified to what is described in Fig. 1(a), where it is assumed that the fallen beam with a mass of m2 and a velocity of v2 collides with the receiving beam with a mass of m1. Given that the two beams have the same final velocity immediately after the collision, the system can be shown as an equivalent SDOF system (Fig. 1(b)) where the receiving beam has a portion mass of am1 and a constant stiffness of k= 48 EI/L3. The value of a is determined considering the equivalent nodal mass of the receiving beam. There are studies have shown that a varies between of 0.375 and 0.5. For the study here, it is assumed to be 0.5 [17].
For the study planned here, both end connections of the fallen beam are supposed to detach simultaneously; the load transferred to the receiving beam will hence be linearly amplified with an impact (Fig. 2). Modeling as such implies the free fall of a body in the air, which depends on the body mass, the primary velocity, and the height (h) from which the beam falls freely. Given that the beam fallen has no primary velocity (v(t= 0)= 0), the final velocity just before the impact time is v2 =. At the impact time, the total mass mtot = am1 + m2, where am1 is a portion of the receiving beam mass, and m2 is the impacting mass. The initial velocity is thus v0 = m2v2 / mtot.
Steps of modeling
Investigating the structural response of tall structures under post-earthquake beam removal scenarios will be performed in several successive steps. As shown in Fig. 3, it starts with generation of the model and then application of the gravity loads. The structure is then analyzed through a nonlinear dynamic process in order to control whether it can meet its pre-defined performance level. If so, the beam removal scenarios are considered at various locations. The structural response under such scenarios is then monitored resulting in two situations; the structure would remain stable or it starts to experience a progressive collapse. It is worth noting that all steps are performed in a time domain fashion. For this, the gravity loads are first applied over an incremental process starting from zero to the maximum value after 2 s. The gravity loads are then kept constant throughout the analysis. The seismic loads are then applied using earthquake acceleration over a period from 2 s to 8 s. The structure is then kept stationary for 2 s after which the beam-removal scenarios are considered (Fig. 4). The reason for the 2 s rest before the beam-removal scenarios is to avoid any abruption change in the structural response after the earthquake analysis.
As is seen, failure criteria are controlled twice; when the seismic analysis is performed and then when the beam-removal scenarios are considered. Dependent on for which performance level the structure is designed, failure criteria may vary. As shown in Fig. 5, there are mainly four performance levels which are operational (O), immediate occupancy (IO), life safety (LS), and collapse prevention (CP). These performance levels are often distinguished by drift values, and rotation of beams as explained hereafter. For the study here, the structure is designed to meet the LS level of performance, meaning that the two following criteria shall be met: controlling rotation of beams and controlling actions of columns. As for the rotation control, the force-deformation diagram shown schematically in Fig. 5 is used.
The rotation (q) is defined as the ratio of deflection (∆) experienced by the structural member to the length of the member (L). The yield rotation (qy) is then defined as the ratio of yield deflection (∆y) to the member length. The yield rotations of beams and columns are determined using Eqs. (1)–(3), respectively.
As for controlling the columns actions, the axial load has a dominant role to consider the column as deformation-controlled or force-controlled. This difference is considered to check the potential of failure. When the compressive axial loads are smaller than 50% of PCL, i.e., the axial compression capacity, the column is considered deformation-controlled. If the axial compression capacity goes beyond 50%, the column is considered force-controlled. The moment-force interaction of the column elements is determined using Eqs. (4)–(6). In this case, a column is considered as to have failed if the demand over capacity ratio (DCR) exceeds unity.
Only the deformation-controlled action is required for beams. This checks rotations of beams in single separate step of the analysis; thus the performance level is determined. When the performance level of a beam goes beyond the LS level of performance, the beam is considered as to have failed. This is worth mentioning that at the LS level of performance, the plastic rotation in seismically compact beams is between 6qy and 8qy. When the plastic rotation goes beyond 8qy, it corresponds to the CP level of performance.
Case study
Figure 6 shows the plan view of a 10-story steel moment resisting structure designed for a PGA of 0.3g and for meeting the LS level of performance. As seen, there are five span lengths of 6000 mm over the X-X direction and four span lengths of 7000 mm over the Y-Y direction. The height of each story is 3200 mm. The building is dimensioned for the load combination of 6.0 kPa for the dead load and 2.0 kPa for the live load in stories 1 to 9, and 5.0 kPa for the dead load and 1.5 kPa for the live load in story 10. A combination of 100% dead load and 20% live load is used to find the required mass for calculating the earthquake load [18]. The flooring system is a 100 mm one-way concrete slab with a compressive strength of 25 MPa. The yield and ultimate stress of steel profiles are 240 MPa and 370 MPa, respectively. Poisson’s ratios of steel and concrete are respectively 0.2 and 0.23. The specifications of the beams and columns are shown in Tables 1 and 2. In this table, the plates added on the flange and web of beam elements are of 1/3 of the beam’s length. A strain hardening modulus of 2%E is used for consideration of the nonlinear behavior of the steel as shown in Fig. 7.
To monitor the structural response under beam-removal scenarios, eight scenarios at different locations and stories are considered as introduced in Table 3. As is seen, there are six scenarios over the X-X direction, and two more scenarios over the Y-Y direction (Fig. 6). The last two scenarios are considered to further investigate the effect of span length on the structural response under the beam-removal scenarios. To do that, the structure is first subjected to the Tabas earthquake acceleration. The results of the nonlinear dynamic analysis are then controlled to understand whether the structure can meet the LS performance level criteria. If so, the beam-removal scenarios are considered where applicable. All steps of the analysis are performed using ABAQUS [19] finite element package.
The columns are introduced with an initial sinusoidal imperfection amplitude of H/300 (H is the story height) so that the probable buckling and post-buckling strength can be captured during the analysis. This imperfection is applied over the unfavorable direction of the columns. Co-rotational transformation of the geometric stiffness matrix is also considered; hence, it is possible to involve large displacement, P-D effects, and the residual deformation effects into the analysis. Plastification of beams and columns over the member’s length and the cross-sections is considered using fiber elements. In addition, to ensure a smooth transaction between the elastic and plastic parts, a transaction curve is introduced at the intersection of the first and second tangents.
Analysis results
The structural sections and properties were introduced in the previous section. The analysis starts with the application of gravity loads to the structure followed by earthquake loads simulated using Tabas earthquake acceleration. The acceptance criteria at the LS performance level are then controlled using Eqs. (1)–(6). The results shown in Table 4 confirm that the structure meets the defined criteria. The last step of the analysis is to apply the beam-removal scenarios introduced in Table 3.
The application of beam-removal scenarios to the earthquake-damaged structure is carried out using the procedure explained in Section 2 and Fig. 3. When a beam is removed from a structure, its gravity loads are transferred to the lower story with an impact. In that case, the structure should be considered as to have failed if one of the two following criteria is not met: the rotation of the receiving beams goes beyond the CP criterion, and the DCR of the columns goes beyond 1.0.
Beams’ reactions under the scenarios
Figure 8(a) shows the structural response under Scenario 1 just after the collision of beam 2-5-BC with the corresponding beam on the first floor, i.e. , the receiving beam. To provide a better understating of this scenario, Figs. 8(b)–8(c) shows the beam 2-5-BC while falling and then when it has collided with the receiving beam.
The variation of rotation over time in the left connection of the receiving beam is shown in Fig. 9. As is seen, the maximum rotation immediately after the collision is 0.077 rad, which is beyond the allowable rotation at the CP level of performance that is 0.072 rad. This means the receiving beam is no longer able to carry its allocated loads. The structure would thus experience a progressive collapse.
Figure 10 shows the structural response under Scenario 2 where beam 6-5-BC is removed from the structure and collides with the receiving beam. The results show that immediately after the collision, the maximum rotation at the left side of the receiving beam exceeds 0.104 rad. Given that the beam should be considered as to have failed if the rotation goes beyond 0.082 rad, the beam here has thus failed, resulting in the structural progressive collapse.
Figure 11 shows rotation variation over time in Scenario 3 where beam 9-5-BC is removed from the structure. While the maximum allowable rotation in this scenario at the CP level of performance is 0.112 rad, the results show that the rotation of the left connection of the receiving beam after the collision is 0.133 rad, meaning that the receiving beam has failed.
The structural response under Scenarios 4 to 6, where the beam-removal scenarios are applied to the inner frame of 3, is shown in Fig. 12. As is seen, the structure collapses in all scenarios just after the collision of the removed beam with the corresponding receiving beam. In Scenario 4 (Fig. 12(a)), the maximum allowable rotation in the receiving beam at the CP level of performance is 0.072 rad while the rotation just after the collision is 0.078 rad. Similarly, in Scenario 5 (Fig. 12(b)), the rotation of the receiving beam at the CP level of performance should not exceed 0.08 rad while it is increases to 0.091 rad after the collision. In Scenario 6 (Fig. 12(c)), the maximum rotation of the receiving beam at the CP level should be limited to 0.114 rad while it goes beyond 0.133 rad after the collision. These show that the inner frame does not have adequate robustness under such scenarios.
Scenarios 7 and 8 are to consider the effect of span length on the structural response under beam-removal scenarios. To this end, in Scenario 7, the external span 4-5 on the 9th story of Frame F is removed from the structure. Figure 13(a) shows the receiving beam just after the collision, and Fig. 13(b) shows the rotation of the left connection of the beam versus time. As seen, the maximum rotation goes beyond 0.166 rad after the impact. The maximum allowable rotation of the receiving beam at the CP level of performance is 0.112 rad meaning that the receiving beam should be considered as to have failed after the impact.
In a similar vein, Scenario 8 is where the internal beam of 3-4 in the same story and frame is removed from the structure. The structural response under such scenario is shown in Fig. 14. As is shown, maximum rotation of the receiving beam goes beyond the allowable one at the CP level (0.112 rad) meaning that the beam has failed after the collision. A comparison between the results of Scenarios 7 and 8 shows that the external beam is more vulnerable to collapse than the internal beam. In addition, the results of the last two scenarios confirm that beams with larger span lengths are more susceptible to collapse than beams with shorter span lengths. As well, the studied scenarios confirm that upper stories are more vulnerable to collapse than lower stories.
Columns reactions under the scenarios
Here, performance of the columns under the above-mentioned scenarios is controlled. In Scenario 1, Fig. 15 shows variations of the axial force and bending moment over time in the column 5B of the first story. As seen, while the earthquake loads are applied to the structure, the DCR remains almost constant. However, the figure shows that both the axial force and the bending moment change notably just after the collision.
To understand whether the column has failed under Scenario 1, Eqs. (4)–(6) are controlled. If the value of DCR in these equations goes beyond 1.0, the column shall be considered to have failed. Substituting the properties of column C1 (from Table 1), and using Fye = 1.10Fy [20] into the equations, the DCR is accounted for. The results show that the maximum value of the DCR is 0.18; thus, the impact has no considerable effect on the column’s performance.
Similarly, Fig. 16 shows the variation of axial force and bending moment versus time in the column 5B when Scenario 2 is applied. As seen, there are considerable fluctuations just after the collision of the removed beam with the receiving beam. Controlling the DCR of the column based on Eqs. (4)–(6), however, shows that the DCR is 0.089 meaning that the column has not collapsed due to the impact.
In a similar manner, Fig. 17 shows the variations of axial forces and bending moments over time in columns B-8, C-1, C-5, and C-8 under the Scenarios 3 to 6, respectively. As is seen, in all scenarios, although forces and moments changes significantly after the collision, no collapse occurs because the DCR remains below 1.0. This confirms that the beam-removal scenarios mainly affect the response of beams.
Concluding remarks
Investigating progressive collapse of tall structures has been a worldwide interest for decades. There is however, rare studies to address the seismic progressive collapse of a structure, where it is first subjected to earthquake before a structural member is removed from it. The reason for not considering such scenarios might be due to the fact that most building codes provide adequate safety for only one extreme load and not for when there are two successive extreme loadings. Yet, there are reports showing that progressive collapse of structures after earthquake has been a prevalent issue; indicating the necessity of studying seismic progressive collapse of structures. This paper was to investigate the vulnerability of a 10 story tall steel moment resisting structure designed to meet the life safety level of performance where various beam removal scenarios after earthquake were considered. To this end, the structure was first subjected to the Tabas earthquake acceleration. The life safety performance level of the structure was then controlled. Eight beam-removal scenarios in various locations were then considered. The scenarios covered both internal and external spans from lower and upper stories, as well as different span lengths. It was assumed that when a beam is removed from the structure, the beam removed could collide with the beam underneath with an impact. The forces created from such impact are highly dependent on the story height and the mass of the removed beams. Two failure criteria were considered: 1) the rotation of beams exceeding 8qy (corresponding to the collapse prevention performance level); 2) the demand over capacity ratio (DCR) in columns exceeding 1.0. Checking the first criterion showed that the structure collapses progressively in all the scenarios. The beam removal scenarios, however, had no significant effect on the DCR of the columns. The results also indicted that beams located at the upper stories are much more susceptible to failure than those located on the lower stories.
It can therefore be mentioned that, failure of beam-to-column connections after earthquake can be catastrophic as it can potentially result in a progressive collapse of other members. Meantime, as upper stories are more vulnerable to such failure, more attention shall be paid to upper stories in the process of design and construction. It is also worth noting that the analyses conducted here were based on the assumption that the strain rate effect can be ignored. While this might be the case for structures with similar configurations, the results here cannot simply be generalized to all types of structures.
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