Abstract
This paper addresses the concept of structural robustness in buildings, particularly focusing on progressive collapse, a phenomenon where localized damage leads to widespread structural failure. Resilient buildings are designed to maintain adequate performance during unexpected extraordinary events, such as explosions, impacts, or earthquakes, yet defining "adequate" performance remains complex. Design codes often refer to the "proportion" between accidental events and their consequences, highlighting the need for a deeper understanding of progressive collapse. This collapse occurs through a series of failures in structural elements, such as beams and columns, which trigger a dynamic load redistribution and result in a catastrophic domino effect. Examples of progressive collapse, such as the Ronan Point Building collapse in 1968 and the World Trade Center collapse in 2001, have driven research and regulatory efforts in this field.
While strengthening all structural elements could improve a building's damage tolerance, this approach is costly. Alternative strategies, such as redundancy and compartmentalization, are commonly used, but their effectiveness is still debated. Similarly, although anti-seismic design enhances progressive collapse resistance, it does not represent the optimal strategy for maximizing resistance. This study emphasizes the limitations of current design codes and the need for improved analytical models of progressive collapse.
The paper introduces a new simulation method based on the Discrete Element Method (DEM) to conduct large-scale parametric studies. This approach is used to identify collapse mechanisms and develop kinematic models to analyze and generalize simulation results. By extending the P-Δ method to both intact and damaged structures, simplified formulas are derived for calculating collapse loads and progressive collapse resistance. The methodology is applied to investigate the progressive collapse of 2D reinforced concrete frames subjected to the sudden removal of beams and columns. The findings highlight the importance of designing structures that can redistribute and dissipate loads, utilizing the ductility of components to avoid fragile failure behavior.
Abstract
This paper addresses the concept of structural robustness in buildings, particularly focusing on progressive collapse, a phenomenon where localized damage leads to widespread structural failure. Resilient buildings are designed to maintain adequate performance during unexpected extraordinary events, such as explosions, impacts, or earthquakes, yet defining "adequate" performance remains complex. Design codes often refer to the "proportion" between accidental events and their consequences, highlighting the need for a deeper understanding of progressive collapse. This collapse occurs through a series of failures in structural elements, such as beams and columns, which trigger a dynamic load redistribution and result in a catastrophic domino effect. Examples of progressive collapse, such as the Ronan Point Building collapse in 1968 and the World Trade Center collapse in 2001, have driven research and regulatory efforts in this field.
While strengthening all structural elements could improve a building's damage tolerance, this approach is costly. Alternative strategies, such as redundancy and compartmentalization, are commonly used, but their effectiveness is still debated. Similarly, although anti-seismic design enhances progressive collapse resistance, it does not represent the optimal strategy for maximizing resistance. This study emphasizes the limitations of current design codes and the need for improved analytical models of progressive collapse.
The paper introduces a new simulation method based on the Discrete Element Method (DEM) to conduct large-scale parametric studies. This approach is used to identify collapse mechanisms and develop kinematic models to analyze and generalize simulation results. By extending the P-Δ method to both intact and damaged structures, simplified formulas are derived for calculating collapse loads and progressive collapse resistance. The methodology is applied to investigate the progressive collapse of 2D reinforced concrete frames subjected to the sudden removal of beams and columns. The findings highlight the importance of designing structures that can redistribute and dissipate loads, utilizing the ductility of components to avoid fragile failure behavior.
Introduction
A resilient building ensures adequate performance during unexpected extraordinary events, such as explosions, impacts, earthquakes, or design and construction errors. Defining "adequate" performance is complex, and design codes, like [1], often refer to the concept of “proportion” between accidental events and their negative consequences. This highlights the importance of addressing progressive collapse in robustness-focused design, as this phenomenon can lead to a disproportionate outcome, where initially localized damage results in widespread collapse. Progressive collapse involves a series of failures of structural elements, like beams and columns, and a subsequent redistribution of dynamic loads, which can trigger additional failures and create a catastrophic "domino effect." Notable examples of progressive collapse, which have spurred research and regulatory efforts, include the 1968 partial collapse of the Ronan Point Building in London due to a gas explosion [2], the 1995 partial collapse of the Alfred P. Murrah Federal Building in Oklahoma due to a truck bomb [3], and the 2001 total collapse of the Twin Towers of the World Trade Center in New York City following aircraft impacts, explosions, and fire [4, 5].
A straightforward but prohibitively expensive method to enhance a building's damage tolerance is to reinforce all its structural elements. Alternative approaches involve redundancy and compartmentalization (see, e.g., [6, 7]), and design codes integrate measures to support these concepts, such as adding ties, using highly ductile structural elements, and recommending moment-resistant connections (see [8] for a review on measures to improve structural robustness). While these solutions generally improve resistance to progressive collapse, their effectiveness is still debated in certain contexts [9]. Similarly, although anti-seismic design typically enhances progressive collapse resistance (see, e.g., [10]), the conclusions of this work demonstrate that it does not provide the most effective strategy for optimizing collapse resistance.
The limitations of current design codes for structural robustness emphasize the need to better understand the physics of progressive collapse. Simple analytical models can only be developed for specific cases, such as tower collapses (see, e.g., [4]). For more complex structures, like 2D models, fully nonlinear dynamic simulations are required, typically within the framework of the Alternate Load Path Method (ALPM) [11]. In ALPM, structural elements are suddenly removed from the model to simulate an initial damage event. Modern simulation algorithms incorporate ALPM and allow for detailed analyses (e.g., [12]). However, the goal is to derive general results and examine the impact of various design parameters, such as the strength and ductility of beams and columns or the structural hierarchy, rather than focusing on the collapse of a specific structure. For this purpose, a new simulation code based on the Discrete Element Method (DEM) has been used to simulate as numerical experiments to derive simplified analytical models that offer convenient formulas for measuring and designing progressive collapse resistance. This paper, starting from an extension of the P-Δ method method applied to both intact and damaged structures, proposes a strategy to derive formulas for collapse loads and progressive collapse resistance. The approach involves an initial exploration stage, in which DEM simulations identify possible collapse mechanisms. Subsequently, simple kinematic models are developed to replicate these mechanisms. These kinematic models allow for an analytical interpretation and generalization of the simulation results. As an application of the methodology, the progressive collapse of regular 2D reinforced concrete frames subjected to the sudden removal of beams and columns from various parts of the structure is investigated.
The concept of structural robustness, already introduced in various calculation codes, is a fundamental requirement of structures for their ultimate resistance in the event of damage, even minimal, without manifesting consequences or collapses disproportionate to the cause/action.
In other words, it is desirable to ensure, starting from the design stages, that the structure is able to absorb a certain amount of extra load redistributing it or dissipating it in such a way as to exploit the ductility characteristics of the components and materials as much as possible, without inducing trends with fragile behavior.
Framed structures
This section analyzes the methods of calculating the limit load or the multiplier of the collapsing loads defining the concept of structural Robustness, which implies the search for the variation or reduction of the ultimate load multiplier starting from the intact structure and then moving on to the structure in case of damage.
Intact structures
To analyze the case of structures subject to compression and evaluate their critical multiplier of axial loads, the use of the P-Δ method applied to frames with movable nodes is used in structural engineering, which constitutes a simplified method but which takes into account all the unstable effects and allows to evaluate the overall response of the structure [20].
Therefore, considering a frame made up of horizontal beams supported by vertical pillars, it is possible to apply to this structural system a set of vertical loads Vi and horizontal Qi acting at the various floors; in this case the overall equilibrium equations can be written in matrix form, where the matrix of horizontal actions Qi acting on the various floors must correspond to an equivalent load Qeq which considers the action expressed by the equilibrium matrix E of the unstable effects P-Δ due to the vertical loads, considering the multiplier of the vertical loads λ and the contribution
of the vector D of the displacements of the structure at the various floors.
The above can therefore be expressed as follows:
- (1) Q eq = λED
- (2) Q = ( κ t + λE ) D
Where κt is the translational stiffness matrix of the considered frame. Proceeding with the search for the critical collapse multiplier λi is obtained by setting:
- (3) det ( κ t + λE ) = 0
Damaged structures
Now considering the case of the same frame structure seen in the previous chapter in which damage is introduced
due to an unforeseen event to one of the floors (assuming for example to remove a certain number of beams and pillars).
In this way equation (3) can be transformed, including two new contributions that take into account the balancing effects Ed and the displacements Dd induced by the damaged structure which add to the stabilizing Ei and displacement Di effects of the intact structure, as follows:
- (4) Q = [ κ t + λ (Ei + Ed )](Di + Dd )
Starting from the equation obtained (4) it is therefore possible to derive the critical collapse multiplier λd:
- (5) det [κt + λ (Ei + Ed )] = 0
Overall response
At this point, the research approach for structural strength can be identified by defining the percentage of reduction of the collapse multiplier between the intact structure and the damaged structure, or by introducing the Robustness ratio R, defined as:
- (6) R = λd / λi
Progressive Collapse of 2D Framed Structures: Simulations with DEM
As an application of the proposed methodology, 2D framed structures shown in Figure 1 are considered. These structures are made of reinforced concrete with high plastic strain and rotation capacity. They all have the same overall size but differ in the number of structural cells, denoted by n. The frames with n = 2 and n = 5 can be seen as hierarchical reorganizations of the frame with n = 11, achieved by using a primary structure composed of fewer but larger structural elements. To accomplish this, the cross-sectional area and reinforcement of the beams and columns are made proportional to their respective lengths, L and H. This ensures that the slenderness of the structural elements remains constant across different values of n. The hierarchical level of a frame is defined as the ratio 1/n. More details on the material parameters, cross-sections, and design strategy can be found in [6, 15].
The frames are subjected to the sudden removal of beams and columns within a designated damage area, as shown in Figure 1. The damage areas have the same size across frames with different n values, representing accidental events with the same destructive energy or spatial extent, such as explosions or impacts. The damage areas are sized such that one-third of the columns on a given floor are removed. Four different initial damage positions are considered (see Figure 1). The case involving the damage position CB is discussed in detail in [6, 15, 19], so we will summarize the main findings for this case only. For the other damage positions, we present the simulation results and provide an analytical interpretation.
Figure 1 Studied frames and damage positions (dashed area). B = bottom, T = top, C = central, E = external.
Bending Collapse
Bending collapse occurs before damage when a static triple hinge mechanism forms in the beams (see Figure 2-a), reflecting their high plastic capacity. After damage at position CB, simulations in [6, 15] reveal that a dynamic triple hinge mechanism develops for frames with n = 2 (see Figure 2-b), while four hinges form when n = 5 or n = 11 (see Figure 2-c). For frames where n > 2, bending collapse typically affects only the elements directly above the damage area, leading to a partial collapse that helps compartmentalize the damage and prevents it from spreading horizontally. However, the combination of high beam plastic capacity and high loads (near qIu) can induce a significant catastrophic inertial effect, where the central portion of the structure collapses and drags the rest of the structure down (see [6, 15]).
For damage at position EB, a double-hinge mechanism is observed, which also holds true for position EM when n = 5 or n = 11 (see Figure 2-d,e). Conversely, when the damage is at position EM and n = 2, the beam above the damage area becomes a cantilever, failing in bending after the formation of a single plastic hinge (see Figure 2-f). For lateral damage, dynamic dragging effects were not observed. Additional simulations showed that damage positions intermediate between CB and EB are equivalent to position CB, as long as the external column remains intact. This suggests that a single intact column at the edge of a beam, under dynamic conditions, is sufficient to fully constrain rotations.
The collapse loads obtained from the simulations are summarized in Figure 3. qIu = L remains independent of the damage position or hierarchical level, which is a result of the design rules used (see [6, 15]). In contrast, qc = L decreases with n because the concentration of bending moments after damage increases with the number of columns removed from one storey. qc = L also decreases as the damage shifts from position CB to EB, since the double hinge collapse mechanism triggered by damage at the external position corresponds to a four-hinge mechanism with double the span. Consistent with the previously described behavior, the cases of damage at positions EM and EB are similar, except when n = 2, due to the cantilever rupture mechanism (Figure 2-f). Since qIu = L remains constant, R1 follows the same trend as qc = L, indicating that hierarchical structures with smaller n values are more resistant to bending collapse after damage.
Figure 2 (a) Static bending collapse before damage. Dynamic bending collapse after damage
in position (b) CB: n = 2, (c) CB: n = 11, (d) EB: n = 5, (e) EM: n = 5, and (f) EM: n = 2.
(g) Pancake collapse with damage in position CB: n = 5 (similar for the intact structure). (h)
Impact driven collapse after damage in ET position: n = 5.
The bending moment diagram illustrated in Figure 4-a and the application of the kinematic theorem to the mechanism shown in Figure 5-a provide analytical formulas for the static collapse loads in bending (B) before any structural damage occurs. These formulas are based on the assumptions of perfectly brittle rupture in the linear elastic regime (el) or ideal plasticity (pl).
(7)
Figure 3 (a) Collapse loads in bending from the simulations. (b) Residual strength.
(8)
Equation 7 can be employed also for damage position EB (and EM if n > 2), after substituting nr;c with 2 nrc - 1, where nrc is the actual number of removed columns. This is justified by the fact that a structure with damage in EB position and undergoing bending collapse can be regarded as the left half of a twice as big symmetric structure with damage in position CB. This leads to a substantial reduction of the critical loads.
Finally, in case of damage in EM positionand n = 2, the cantilever schemes in Figure 4-b and 5-b provide the critical loads:
(9)
Figure 4 Simplified static schemes and bending moment diagram for perfectly brittle collapse
in bending mode. (a) Generic beam of the intact structure, (b) cantilever scheme for n = 2 and
damage in EM position, (c) damage in CB position. The latter scheme is also valid for damage
in EB position (and EM position if n > 2).
Figure 5 Simplified static schemes for ideally plastic collapse in bending mode
The outcomes of the simulations confirm that the collapse load qc decreases as the number of columns removed from each story (nr,c) increases. This behavior is in line with the physical expectation that removing more columns reduces the overall structural capacity, concentrating the load on fewer remaining elements. The expressions for the robustness factor R1, which can be calculated directly from an extension of equation 6, show that bending collapse results in the most severe consequences in terms of R1. Specifically, when one-third of the columns are removed from a story, the simulations indicate that R1 is approximately 25%, which aligns with the expected outcomes based on the geometry and material characteristics used in the analysis.
Additionally, these analytical expressions serve as a valuable tool for understanding the structural behavior under progressive collapse conditions. They allow engineers and designers to predict the structural response to different damage scenarios and, ultimately, to improve the design of resilient structures that can withstand extreme events without leading to disproportionate or catastrophic failures. The ability to calculate the collapse load and robustness factor helps in determining the level of safety and damage tolerance that can be incorporated into buildings, offering more effective strategies for progressive collapse prevention and mitigation.
Conclusions
This study provides an in-depth analysis of the structural response to progressive collapse under various damage scenarios, focusing on bending collapse mechanisms in reinforced concrete frames. Using a combination of numerical simulations and analytical models, the research highlights several key findings and contributions to the understanding of structural robustness.
First, the study demonstrates that bending collapse is a critical failure mode in the event of damage, especially when a significant portion of the structural elements (such as columns) are removed. The findings reveal that structural elements above the damage area play a major role in the collapse mechanism, with partial collapse often confined to the immediate vicinity of the damage, as seen in frames with higher hierarchical levels. However, when subjected to large dynamic loads, the collapse may propagate further, creating a domino effect that could lead to more widespread structural failure.
Second, the application of the Discrete Element Method (DEM) simulations alongside traditional analysis methods (such as the P-Δ method) has allowed for a more comprehensive understanding of the collapse load and the factors influencing structural robustness. The simulations reveal that critical loads decrease as more columns are removed from a given story, confirming that the structure's ability to withstand collapse is highly dependent on the number of remaining intact structural elements. Moreover, the analytical expressions derived in this study provide a solid foundation for calculating collapse resistance and can be used as a tool for designing buildings with improved robustness.
Lastly, the results indicate that hierarchical structures with fewer structural elements (lower n) are better suited to resist bending collapse after damage, as they are more efficient in redistributing the applied loads. These findings underscore the importance of considering both redundancy and structural hierarchy in the design of resilient buildings.
In conclusion, the study provides valuable insights into the complex behavior of structures under progressive collapse, emphasizing the need for advanced analytical tools to accurately predict collapse mechanisms and ensure the design of buildings capable of withstanding extraordinary events. The proposed methodology offers a practical approach for designing more robust structures that can absorb and redistribute loads in the event of unforeseen damage, ultimately contributing to the safety and resilience of the built environment. Further investigations into other damage scenarios and load distributions will help refine the models and improve the predictive capability of progressive collapse analysis.
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