1. Introduction
In recent decades, there have been a series of building collapses due to local damaging events that were not proportional to the subsequent failures. Although the number of such progressive collapses in history is relatively small, the catastrophic consequences in terms of fatalities and other losses make protecting against progressive collapse mandatory in the structural engineering environment.
The problem of progressive collapse first began to be widely discussed after a domestic gas explosion in the 23-story Ronan Point Apartment building in 1968. Due to the very limited ability of the structural system to redistribute loads, a local failure of one panel caused the progressive collapse of upper floors (19 through 23). Subsequently, the falling debris invoked the progressive collapse of lower floors down to the ground floor. The most infamous examples include the collapse of the L’Ambiance Plaza apartment building due to construction errors in 1987, the partial collapse of the Alfred Murray Building due to explosive detonation in 1995, the complete collapse of the World Trade Center towers due to aircraft impact and fuel combustion in 2001, and some others [
1,
2,
3,
4,
5,
6,
7,
8,
9].
The tragedy data revealed the lack of a study of progressive collapse, which, in turn, intensified experimental research interests in this phenomenon. Current experiments generally focus on evaluating the capability of a structural system to bridge over a damaged area without a progressive collapse developing and formulating requirements for resistant structures. Due to the complexity of testing real-scale buildings, most experiments are conducted for scaled substructures or individual elements. Qian et al. [
10,
11] tested the scaled reinforced concrete (RC) beam–column sub-assemblages under a penultimate column removal scenario to investigate failure modes and the load redistribution capacity. Russel [
12,
13] conducted a series of static and dynamic tests of scaled RC flat slabs under various failure scenarios and examined the flexure and punching failure mechanisms. Peng et al. [
14,
15] tested the scaled RC flat slab sub-assemblages subjected to an exterior and interior column removal and analyzed the punching failure and post-punching capacity of column-slab connections. Only a few studies contain test results for complete real-scale structures. Xiao et al. [
16] performed a sequential removal of four columns of a three-story RC frame to investigate the dynamic response and load transfer mechanisms. Adam et al. [
17] carried out a test of a two-story RC building subjected to a corner column scenario and analyzed the dynamic performance of the structure and alternative load paths. Among other original studies, static [
18,
19,
20,
21,
22,
23,
24] and dynamic [
25,
26,
27] tests are also noteworthy.
Numerical simulations are also a powerful and effective way to predict the possible impact of local failures on the strength and reliability of buildings. Nonlinear dynamic analysis is often used for this purpose, considering the dynamic nature of progressive collapse events and the resulting damage of structures. In [
28], Qian et al. simulated the dynamic response of RC flat slab sub-assemblages under a two-column loss scenario and examined failure modes and force redistributions. Lui et al. [
29,
30] performed dynamic and static analyses of a multistory reinforced concrete flat-plate building under exterior and interior column removal scenarios. It has been found that the strain rate effects and compressive membrane action can significantly increase the punching resistance of a flat plate. Along with this, the use of the nonlinear static procedure estimates well the peak dynamic displacement, although overestimates slab local rotation by more than 20%. Keyvani et al. [
31] proposed the FE modeling technique to simulate punching and post-punching behavior of flat plates and validated it against test data. It has been shown that the compressive membrane forces occurring in actual flat-plate floor systems improve the slab resistance against progressive failure, and its ignoring underestimates the punching strength. Kwasniewski [
32] conducted a numerical study of the progressive collapse of an existing eight-story building and analyzed the damage resulting from various failure scenarios. Pham et al. [
33] studied the effect of blast pressure on structural resistance against progressive collapse under a column removal scenario induced by contact detonation and investigated the development of catenary action within an ultra-fast dynamic regime. Parisi et al. [
34] defined five performance limit states associated with increasing levels of damage and the corresponding load capacity for a progressive collapse design using a nonlinear dynamic analysis. Brunesi et al. [
35,
36] applied a Monte Carlo simulation to generate 2D and 3D models of low-rise RC frames. The frames were analyzed with pushdown and incremental nonlinear dynamic methods. In [
37,
38,
39], the authors compared linear and nonlinear, static and dynamic numerical procedures for a progressive collapse analysis. These procedures are outlined in the standards for determining the stress state and deformations after local failure and designing buildings resistant to progressive collapse [
40,
41,
42]. These articles demonstrate that numerical simulations following different procedures leading to different results and, therefore, require additional examination.
Current design standards describe the analysis of reinforced concrete structures using different numerical procedures, including static and dynamic methods with and without taking into account the physical nonlinearity in numerical models. In particular, the Russian design standard [
40] suggests the use of linear static, nonlinear static, and nonlinear dynamic procedures. This standard gives no restrictions on the use of any method, so it is important for civil engineers to evaluate their accuracy and predictive ability. Since the reviewed articles lack a complete comparison of numerical results following, simultaneously, for these methods and experimental data for individual elements or real-scale buildings, this paper aims to validate different numerical procedures for the analysis of RC structures. For a complete analysis, we also consider the linear dynamic procedure. Thus, the paper discusses four numerical methods common in engineering practice to simulate RC structures under local failures: linear-elastic quasistatic pulldown, nonlinear quasistatic pulldown, linear-elastic dynamic, and nonlinear dynamic procedures. The research objects are a flat RC slab and a two-story RC frame subjected to the removal of the corner support tested by Russel [
13] and Adam et al. [
17]. The goal of the research is to assess the accuracy and robustness of each of the procedures provided by the Russian standard based on a comparison with known test data.
A progressive collapse is characterized by great a uncertainty of the initiating event. Such actions can be both dynamic (earthquake, internal gas explosion, external blast, vehicle impact, buckling, extreme fire action, demolition [
33,
43,
44,
45,
46,
47]) and quasi-static (soil changes due to changes in the groundwater level, karst processes, etc. [
48]). Since the initial local failure duration is almost impossible to determine reliably, it is often assumed equal to zero. In this paper, it is also assumed that local failure occurs instantaneously.
It is known that the initial geometrical imperfections mainly reduce the rigidity and strength of compressive members and can affect failure modes and alternative load paths [
49,
50]. However, the presence of imperfections is not described in the reference experiments, so their effects are not considered in the numerical models.
4. Discussion
This study developed three-dimensional FE models of an RC flat slab and two-story RC frame for the corner support failure scenario. We compared the vertical displacements known from experiments [
12,
17] and using numerical procedures recommended by the Russian design code [
40] (linear static, nonlinear static, and nonlinear dynamic) and linear dynamic procedure, also quite popular in engineering practice.
A comparison of the results following the nonlinear static procedure (NSP) showed that loading by the internal force of the removed element with the opposite sign led to an overestimation of displacements. In the first problem, the displacements were higher than the experimental ones by only 10%, which resulted in engineering accuracy. Moreover, the nonlinear static analysis demonstrated that the slab was resistant to a progressive collapse, which was also consistent with the experiment. However, because the nonlinear static calculation did not consider the real inertial forces and the increase in the dynamic tensile strength for concrete, the nature of the damage on the top face of the slab was somewhat inconsistent with the experiment. In the experiment, the main crack region (yield line) was located between the central supports, while, in the nonlinear static analysis, it was between the central and the corner supports.
In contrast, the mismatch with the test data in the second problem was over 400%. The displacement values obtained from nonlinear static analysis led to a conclusion about the progressive collapse of the RC frame, which did not correspond to the test data. Thus, the proposed Russian standard nonlinear static analysis of different structures with the same dynamic increase factor equal to 2.0 was incorrect and too conservative. Using the improved dynamic increase factor for a nonlinear static analysis, depending on the structure parameters, would achieve more reliable results and an economical design.
The linear static procedure (LSP) did not consider nonlinear behavior and led to underestimated results. The mismatch with the test data was about 70% in the first problem and 50% in the second one. It was clear that in this procedure, the load increase factor must take into account both the effects of the forces of inertia and nonlinear effects.
The peak vertical displacements obtained following linear static and linear dynamic procedures were relatively close in both problems. We can conclude that the DIF = 2 used in the linear static procedure was reasonable, but only when there were no effects due to physical nonlinearity. However, considered RC structures presented strongly nonlinear behavior when one of the load-bearing elements failed. A comparison of dynamic solutions showed that consideration of physical nonlinearity was necessary.
Thus, the Russian standard suggests to apply the same load multiplier of 2.0 both for the linear static and nonlinear static analysis. This coefficient is not appropriate for cases where nonlinear response is expected. Since, for reinforced concrete structures, the response in the nonlinear range was typical due to the low tensile strength of concrete, the use of a load multiplier equal to 2.0 was very limited.
Correct results in both problems were obtained using only the nonlinear dynamic procedure. The mismatch with the test data in the first problem was in the range of 1.5–3.2%, and in the second problem did not exceed 7%. Damage fields from the nonlinear dynamic simulations also corresponded to the cracking patterns from experiments and could also be used to analyze the resistance to a progressive collapse.
Current American guidelines [
41,
42] allow to determine the load multipliers more accurately depending on the type of analysis and structural parameters of building. We calculated it for two-story RC frame to analyze the structure following these guidelines and compared results with the Russian standard.
The load increase factor (LIF) for the linear static procedure was calculated as follows:
where m
LIF is the smallest m factor determined for each structural element directly connected or located above the removal column. For m
LIF = 6, we obtained LIF = 8.
The dynamic increase factor (DIF) for nonlinear static procedure was calculated as follows:
where θ
pra is the plastic rotation angle and θ
y is the yield rotation. Given that θ
pra = 0.05 and θ
y = 0.032, we obtained DIF = 1.26. Compared to the uniform multiplier of 2.0 proposed by the Russian standard [
40], the calculated LIF and DIF following the DoD guideline [
41] looked more reasonable. Using these values as multipliers to the masses loaded on the slabs above the removal column, we obtained the peak displacements equal to 61.8 mm and 54.7 mm from LSP and NSP, respectively (see
Figure 13).
As can be seen, the linear static and nonlinear static pushdown procedures with different load multipliers according to the DoD guideline fit the experimental results much better. For the LSP, the error was 28.5%; for the NSP it was about 14%. It is also important to note that in both cases the results were conservative.
Therefore, static pushdown procedures outlined in [
41,
42] allowed for more accuracy, although the linear static procedure still resulted in overestimated peak displacements. A perspective direction for the numerical investigation of the progressive collapse of RC buildings is to develop and clarify the linear static procedure based on the results obtained from the nonlinear dynamic method. These researches will allow us to obtain accurate stresses and deformations of structures using a linear analysis that will be much less time-consuming than nonlinear dynamic simulations.
Due to the high accuracy and consistency with experiments, the nonlinear dynamic high-fidelity models of RC structures could also be used in damage identification methods for structural health monitoring under different operational and environmental conditions [
48]. It can help determine the presence of damage in a structure, identify the geometric location of damage, quantify damage severity, and even predict the remaining service life of a structure.