Separation strain for progressive collapse analysis of reinforced concrete building using applied element method

Abstract

The basic theory of the applied element method is described in this article, and some modeling techniques for applying this analysis method to the simulation of the progressive collapse of reinforced concrete tall buildings are suggested. Separation strain, which is considered to have potential for use in realistic representation of the collapse of structures, was parametrically investigated in this study, and the appropriate value for the separation strain for use in the analysis of the progressive collapse of reinforced concrete buildings is proposed. The progressive collapse of the Murrah Federal Building in Oklahoma City in response to a bombing was analyzed by applying the suggested modeling techniques. Two alternatives for modification of the structural design of the building were developed, and the expected responses of the building with the suggested modifications were analyzed and compared with the response of the actual building to the bombing. The analysis method proposed in this article can be used to visualize the progressive collapse of tall buildings and can be effectively applied to the design of progressive-collapse-resistant structural systems for tall buildings.

Keywords

applied element method blast analysis progressive collapse separation strain tall buildings

Introduction

Progressive collapse is the collapse of all or a large part of a structure that is triggered by damage to or failure of a relatively small part of the structure. The phenomenon is of particular concern because progressive collapse is often disproportionate, that is, the collapse is out of proportion to the event that triggers it. Progressive collapse can result in huge loss of life and property damage because of its effect on the entire building (Ellingwood, 2006; Nair, 2004). Gas explosions, severe fires, impacts by vehicles or aircraft, and bomb explosions are conceivable causes of abnormal loadings that can trigger a progressive collapse.

The most widely known examples of progressive collapse may be the Ronan Point apartment collapse in 1968, which was caused by a gas explosion, the Murrah Federal Building collapse in 1998, which was caused by a terrorist bomb attack, and the World Trade Center 1 and 2 collapses on 11 September 2001, which were caused by aircraft impacts by terrorists. Over the past few decades, a considerable number of studies have been conducted on methods for the analysis and design of building structures to prevent progressive collapse (Hafez et al., 2013; Izzuddin, 2012; Kim and An, 2009; Pachenari et al., 2013; Sasani and Kropelnicki, 2008). However, many difficulties still exist in predicting and analyzing these catastrophic failures.

In the last few years, nonlinear dynamic analysis using the finite element method (FEM) has been widely applied to the analysis of progressive building collapse due to explosions and the design of building structures to prevent progressive collapse. However, this analysis method has various weaknesses, including the high rate at which the required analysis time increases with increasing complexity of the analysis model. In addition, there are technological difficulties associated with the selection of appropriate material models for use in modeling the structure with this method.

To overcome these disadvantages, the applied element method (AEM) (Lupoae et al., 2011; Salem et al., 2011; Meguro and Tagel-Din, 1998), which is another analysis method that has been applied to progressive collapse analysis, was used in this study. This method requires a relatively short amount of time to simulate a high-rise building for the purpose of a progressive collapse analysis. The analysis of the progressive collapse of and localized damage to a reinforced concrete high-rise building in response to an external explosion is performed using a program called extreme loading for structures (ELS) (Applied Science International (ASI), 2013), which is a special structural analysis program based on AEM.

In this article, the basic theory of AEM is described, with special attention given to the separation strain, which is an important function in the realistic simulation of progressive collapse. Modeling separation strain makes it possible to eliminate elements that develop excessive strain from the analysis, which results in more realistic simulation of the collapse behavior of building structures. We determined an appropriate separation strain value for use in blast analysis and progressive collapse analysis by analyzing a reinforced concrete structure and comparing its predicted behavior with experimental results.

Progressive collapse and the development of localized damage due to blast loadings were investigated by applying the proposed modeling techniques to the Murrah Federal Building (Corley et al., 1998). Two modifications to the building’s structural system that could have prevented the progressive collapse of the Murrah Federal Building were identified and were validated by performing a blast analysis of the system.

AEM

Modeling concept of AEM

The FEM has been widely used for nonlinear analysis of reinforced concrete structures. However, this method poses difficulties in the analysis of excessively deformed elements with cracking or crushing. In addition, FEM requires a substantial amount of time to perform a large-scale three-dimensional nonlinear dynamic analysis, such as a blast analysis of a tall building structure. The numerical difficulty and complexity of dealing with highly deformed elements and the time-consuming nature of finite element analysis are obstacles to its use as a practical tool in investigating the progressive collapse behavior of tall building structures. AEM, which was developed to address the faults of FEM, can be used to analyze progressive collapse in high-rise building structures. AEM permits linear and nonlinear analysis of reinforced concrete building structures and can handle highly deformed elements with relative ease. The main distinguishing feature of AEM is of the way in which element connections are represented. In FEM, inter-element continuity is represented by elements sharing nodes and elements themselves deform. On the other hand, AEM facilitates inter-element connectivity, deformation of structures, and even separation of elements using an interfacial matrix of one-dimensional springs. Figure 1 shows the basic modeling concept of AEM, with one axial spring and two shear springs used to model the one contact point of the element and form the spring matrix of the faces of an element. In ELS, which is a structural analysis program based on AEM, users can define the number of contact points on the faces of elements, and the material and structural properties of the springs are calculated automatically based on the input material properties of the structure (ASI, 2013; Meguro and Tagel-Din, 1998).

Figure 1.

Comparison of element connection and deformation between FEM and AEM.

Constitutive model and failure model

The springs used to connect elements in the ELS model represent the stress, distortion, and connection status between elements. Simplifying the connections using springs offers advantages in modeling partially connected elements, such as those shown in Figure 2. Although FEM cannot represent partially connected elements because elements can only be connected at predefined nodes, AEM can describe partial connection of elements using the part of the spring matrix that is not separated. This feature is particularly useful in progressive collapse analyses, which require modeling of highly deformed elements and partially connected elements.

Figure 2.

Partial element connectivity (ASI, 2013): (a) FEM and (b) AEM.

Although the use of a spring matrix simplifies the analysis of a highly deformed structure, it has drawbacks in terms of accuracy. The one-dimensional nature of springs oversimplifies the three-dimensional behavior of materials under a multi-axial stress state. A spring matrix cannot accurately represent three-dimensional material properties such as pressure hardening, strain hardening, and third invariant dependence.

In ELS, concrete is modeled using the constitutive model shown in Figure 3. Three parameters—the initial Young’s modulus, the fracture parameter representing the extent of the internal damage of the concrete, and the compressive plastic strain—are used to define the envelope for the compressive region. The stress–strain response of concrete springs subjected to tension is assumed to be linear until the cracking point is reached. After cracking, the stiffness of springs subjected to tension is set to be zero. For the constitutive model of the steel reinforcement, the uniaxial stress–strain relationship shown in Figure 4 is used. The tangent stiffness of the reinforcement is calculated based on the strain of the reinforcement spring, the loading status, and the previous loading history of the steel spring, which controls the Bauschinger effect. A set of independent springs that represent the structural properties and the locations and sizes of the steel bars in the concrete is modeled by the program, as shown in Figure 5. The steel spring set consists of one axial spring and two shear springs.

Figure 3.

Axial stress–strain curves for concrete (ASI, 2013).

Figure 4.

Axial stress–strain curves for steel (ASI, 2013).

Figure 5.

Reinforcement springs (ASI, 2013).

Blast loadings are calculated by ELS based on the user-input trinitrotoluene (TNT) charge weight, location, and detonation time. The program uses an empirical equation obtained from blast experiments. The built-in empirical equation enables the program to calculate the pressure and the duration of the pressure on the loaded surface very quickly, but it has a limitation in that the reflection and refraction of the pressure wave at the ground surface and other loaded surfaces cannot be taken into account.

Separation strain

Purpose of separation strain

In the ELS program, separation strain can be used to describe the complete failure of concrete or steel. If the strain at an element representing concrete or steel exceeds the separation strain, the spring matrix of the element is regarded as being removed from the element so that the element is separated from adjacent elements. In the case of reinforcement, the steel bars are regarded as cut if the steel stress exceeds the ultimate stress or if the concrete encircling the bars exceeds the separation strain. Therefore, the separation strain of concrete governs the collapse behavior of reinforced concrete structures, and the user can control the ductility or brittleness of the structure by changing the separation strain. The separation strain plays a very important role in the realistic simulation of the progressive collapse of reinforced concrete building structures, which was one of the objectives of this study. We identified an appropriate value for the separation strain by comparing the ELS analysis results with experimental results.

Analysis with various levels of separation strain

The analysis model used to determine the appropriate separation strain value for use in blast analysis is the same model that has been used to study the resistance of reinforced concrete members to blast loading. An experiment was performed to study the effect of blast loading on reinforced concrete members strengthened with steel-reinforced polymer (SRP) sheets (Carriere et al., 2009). From among the tested specimens, two reinforced concrete members without SRP sheets were selected for use in this study to compare the experimental results and the analysis results and thereby determine an appropriate value of the separation strain. The two specimens, A2 and B2, have almost the same properties, as shown in Table 1 and Figure 6, except for the amount of TNT used in the blasts to which they were subjected. The cross sections of the members are 150 mm × 150 mm, and the members are 2100 mm in length. The concrete cover is 20 mm, and the spacing of the stirrups is 100 mm, with two additional stirrups at each end of the supports. Ten elements are used in each direction of the cross section, and 140 elements are used in the longitudinal direction of the model. Weights of 50 and 20 kgf of TNT were exploded at a vertical distance of 2.0 m from the centers of members A2 and B2, respectively. The members were located 0.4 m above ground and were fixed by supports 300 mm from each end (Carriere et al., 2009).

Table 1.

Material properties of concrete and steel.

	Compressive strength (MPa)	Separation strain	TNT (kg)	Yield strength (MPa)	Diameter (mm)
A2
Concrete	38.7	0.01–1.0	50
Steel				500	5.37
B2
Concrete	39.2	0.01–1.0	20
Steel				500	5.59

TNT: trinitrotoluene.

Figure 6.

Reinforced concrete specimen details.

The simulation was run for up to 10 s, and the time step for the blast load was set to $1 \times 10^{- 5} s$ . The value of the separation strain was varied from 1.0 to 0.01 to identify the value that yielded the best agreement of the experimental results and the analysis results in terms of the failure behavior of the members.

Comparison of the results

Figure 7 shows the failure shapes of the reinforced concrete members A2 and B2 after the blast experiments. In the case of A2, which was subjected to a blast of 50 kg of TNT, complete failure can be seen at three locations, in the center and at both ends. In the case of B2, which was subjected to a blast of 20 kg of TNT, complete failure can be seen at the center of the member; there was also moderate damage at both ends. The predicted failure shapes obtained from the ELS analysis for various separation strain values are shown in Figure 8. For A2, a separation strain value of 0.1 yields a failure shape very similar to that obtained experimentally. A separation strain value of 0.01 results in a prediction of more brittle behavior for A2 than that observed in the experiment. On the other hand, a separation strain value of 1.0 makes the structure too ductile; consequently, the member does not exhibit complete failure but rather excessive deformation. It can be verified that the smaller the separation strain value used, the more brittle the member’s failure behavior. The same trend was observed in the case of B2. Although a separation strain value of 0.1 results in somewhat more ductile failure being predicted than that observed in the experiment, this value produces the failure shape that most closely agrees, among the three considered, with the failure shape observed in the experiment.

Figure 7.

Failure shapes of experimental specimens due to blast loading (Carriere et al., 2009).

Figure 8.

Failure shapes indicated by analysis for various separation strains: (a) A2, separation strain 0.01; (b) A2, separation strain 0.1; (c) A2, separation strain 1.0; (d) B2, separation strain 0.01; (e) B2, separation strain 0.1; and (f) B2, separation strain 1.0.

The number of crack springs was investigated to quantitatively compare the results. Figure 9 shows the number of crack springs, which represent cracks predicted to develop in the concrete, for separation strain values of 0.01, 0.03, 0.05, 0.07, 0.1, 0.3, 0.5, 0.7, and 1.0. The analysis model for the reinforced concrete members has a cross section of 10 × 10 elements and a length of 150 elements. Every element has 25 springs on each face for normal and shear strains. The failure shape shown in Figure 7 indicates that the number of failed planes in the cross section is estimated to be 14∼17. The total number of crack springs can be roughly calculated as follows

\begin{matrix} N_{cr_sp} & = N_{p} \times N_{sp} \times 3 \times N_{el} \\ = (14 ~ 17) \times 25 \times 3 \times 100 \\ = (105, 000 ~ 127, 500) \end{matrix}

(1)

where $N_cr_s p$ is the number of crack springs, $N_{p}$ is the number of failed planes, $N_{sp}$ is the number of springs on each face of an element, and $N_{el}$ is the number of elements on a cross-sectional plane.

Figure 9.

Number of crack springs versus separation strain: (a) A2 and (b) B2.

In the case of A2, the analyses conducted with separation strain values of 0.07, 0.1, and 0.3 yielded good approximations of the number of crack springs. In the case of B2, the number of failed planes and crack springs can be estimated to be in the ranges of 9∼12 and 67,500∼90,000, respectively. Separation strain values of 0.07∼0.3 also yielded good approximations in the case of B2.

Based on the results of the qualitative and quantitative comparisons, we recommend a separation strain value of 0.1 for use in blast analysis.

Progressive collapse analysis

Murrah Federal Building in Oklahoma City

In the previous section, we suggested a separation strain value of 0.1 for use in blast analysis. In this section, this separation strain value was used in an analysis of the large-scale progressive collapse analysis of a reinforced concrete building structure. The Murrah Federal Building in Oklahoma City (American Society of Civil Engineers (ASCE), 1996; Corley et al., 1998; Tagel-Din and Rahman, 2006) was selected for use as an analysis example to verify the appropriateness of the suggested separation strain value because the failure shape and the relevant information for this building are relatively easy to obtain.

The dimensions and the steel ratio of the main structural members were determined from published information (ASCE, 1996). The compressive strength of the concrete and the yield strength of the reinforcement were 35 and 500 MPa, respectively. The steel ratio of the secondary members was assumed to be 1.5%. The amount of TNT (1750 kg) and the spot at which it was exploded were the same in the simulation as in the real event. Figure 10 shows the damage to and the extent of the progressive collapse of the Murrah Federal Building after the 1995 terrorist bombing.

Figure 10.

The Murrah Federal Building damaged after a terrorist bombing (ASCE, 1996).

Three loading phases were used to define the loading sequence. The first loading phase encompassed static loads such as the self-weight and live loads. The second phase encompassed the blast loading. The pressure due to a blast at locations close to the explosion is normally very high and decreases rapidly with distance from the source. The duration of the pressure wave is very short close to the blast source and increases with distance from the source. A very short time step is required to accurately describe the pressure history. This very short time step should be sustained until the pressure at the far end of the building decays. A time step of 0.0001 s and duration of 0.2 s were used for the second loading phase in this analysis.

The third loading phase encompasses the progressive collapse of the building. A longer time step can be used for the third loading phase to reduce the overall analysis time, but it should be shorter than the shortest natural frequency of the structure considered in the analysis. A time step of 0.001 s was used for the third phase in this analysis, and the simulation was run for 20 s to investigate the complete collapse of the building.

Figure 11 shows the results of the analysis of the progressive collapse of the Murrah Federal Building. The pictures in Figure 11 represent the collapse shape at 0.2, 1.0, 3.0, and 10.0 s after detonation of the bomb. The colors in the pictures represent vertical displacements in the range of 0.0–3.0 m. Upward and downward vertical displacements greater than 3.0 m are indicated in white. Figure 11(a) shows that within 0.2 s after detonation, the blast pressure due to the explosion had already been applied to the columns and beams close to the detonation spot. A few of the columns damaged by the direct effect of the blast wave and the beams and slabs on the third and fourth level show some damage caused by the loss of supporting columns. Within 3.0 s after detonation, three of the front columns had been completely ruptured and spandrel beams on the third floor had been severely damaged. Within 10.0 s, almost all of the front columns, beams, and slabs were falling down. With the exception of the mega columns on the left side and the rightmost bay, all of the front part of the building had totally collapsed within 10.0 s. A comparison of the simulated collapsed shape at 10.0 s with Figure 9, which shows the actual collapsed shape of the building after the bombing, shows that the simulated and actual collapsed shapes are almost identical. This comparison demonstrates that the analysis method described in this article can be used to predict the progressive collapse of reinforced concrete buildings. In addition, this example demonstrates that the separation strain value suggested in this article is appropriate for use in the analysis of progressive collapse as well as in blast analysis.

Figure 11.

Progressive collapse analysis of the Murrah Federal Building: (a) 0.2 s, (b) 1.0 s, (c) 3.0 s, and (d) 10.0 s.

Design alternatives

There are a few methods available for designing structural systems to prevent progressive collapse. ASCE 7 (2005) defines two categories of approaches for reducing the possibility of progressive collapse: direct design approaches and indirect design approaches. Direct design approaches include the alternative path method, which requires the structure to provide an alternative load path so that the damage is absorbed and major collapse is averted, and the specific load resistance method, which requires the structure to provide sufficient strength to resist failure. Indirect design approaches are based on the implicit consideration of resistance to progressive collapse through the structural integrity that can be achieved through the provision of minimum levels of strength, continuity, and ductility.

The alternative path method was used in this study to design structural systems that might have prevented the progressive collapse of the Murrah Federal Building. When we look at the progressive collapse sequence shown in Figure 10, we see that the rupture of the transfer girder at the third level appears to be the trigger for the progressive collapse of the upper floors. The first design alternative, shown in Figure 12(a), involves constructing columns at the ground level to reduce the span of the girders by half. The additional five columns have the same dimensions and reinforcement details as the perimeter columns at the ground level. The girders supported by the additional columns are expected to provide the needed alternative load path. The second alternative, shown in Figure 12(b), involves the use of a secondary transfer girder at the top of the building. This additional transfer girder has the same size and properties as the transfer girder at the third level. The additional transfer girder acts as a structural system on which the columns below hang. The second alternative may be less effective than the first alternative in resisting the progressive collapse of the building; however, it does permit a wide entrance span at the ground level.

Figure 12.

Two structural systems for preventing progressive collapse: (a) design alternative 1 and (b) design alternative 2.

Blast analyses of design alternatives

Blast analyses of the two design alternatives were carried out under the same conditions as the blast analysis of the original building design, in terms of the applied loads, the number of elements, and the material properties assumed, except for the addition of either additional columns at the ground level or a secondary transfer girder at the top level. Figure 13 shows the behavior of the first design alternative in the simulation. Within 0.2 s of the detonation, the columns at the ground level suffered significant damage, to almost the same degree as the ground-level columns in the original model, as shown in Figure 10(a). However, Figure 13(b) shows that after 10.0 s, the collapse was limited to the third floor and the upper floors and the right part of the building remained intact. The surviving beams attached to the mega columns on the left side and the remaining right part of the transfer girder on the third floor greatly limited the extent of the collapse. Therefore, the additional columns at the ground level are considered an effective way to prevent the progressive collapse of the building. The analysis results for the second design alternative, which involves the use of an additional transfer girder at the top level, are shown in Figure 14. Figure 14(a), which shows the response of the building at 0.2 s, indicates that the columns at the ground level and the transfer girder at the third level were damaged to approximately the same degree as in the original structure. Figure 14(b), which shows the response of the building at 10.0 s, shows that the additional transfer girder at the top level provided an alternative load path, in spite of the complete failure of the transfer girder at the third level, and thus prevented the progressive collapse that occurred in the original structure. The transfer girder at the top level was capable of bridging the missing columns at the ground level and thereby limiting the extent of the damage.

Figure 13.

Progressive collapse analysis of design alternative 1: (a) 0.2 s and (b) 10.0 s.

Figure 14.

Progressive collapse analysis of design alternative 2: (a) 0.2 s and (b) 10.0 s.

Figure 15 presents the ratios of the collapsed floor area to the total floor area for each of the three scenarios. While the original structure suffered damage to 27% of the total floor area, the first and second design alternatives reduced the damage to 9% and 12% of the total floor area, respectively.

Figure 15.

Comparison of collapsed floor areas.

A main characteristic of progressive collapse is the disproportionate nature of the damage with respect to the input loading. In the case of the Murrah Federal Building, the amount of damage considered proportionate to a blast loading of 1750 kg of TNT is estimated to be approximately 10%. The disproportionate amount of damage (27%) that resulted from the progressive collapse could have been prevented by providing an alternative load path.

Conclusion

Numerical simulation of the progressive collapse of a building structure requires special nonlinear dynamic analysis techniques that are capable of realistically simulating large deformations, material nonlinearity, material properties at high strain rate, and blast loadings. In this study, the AEM was used to perform numerical simulations of progressive collapse. The authors found that the separation stain value used in AEM plays a very important role in reproducing the realistic collapse behavior of a reinforced concrete building structure. An appropriate separation strain value for use in progressive collapse analysis was identified by comparing numerical simulation results and experimental results for reinforced concrete members subjected to blast loadings. The suggested separation strain value was used in an analysis of the progressive collapse of the Murrah Federal Building in Oklahoma City and in analyses of two design alternatives that could have reduced the extent of the collapse. The following conclusions can be drawn from the results of the study:

AEM can be used effectively to analyze the progressive collapse of reinforced concrete building structures. Although AEM is based on the overly simplified premise that all of the complicated material behaviors involved can be represented by springs, it can nonetheless yield reasonable results with limited computing requirements.

The separation strain value used in the ELS program controls the ductility of the analysis model. Because steel bars in concrete members are considered to be cut if the strain in the surrounding concrete exceeds the separation strain, the ductility of the steel bars should be taken into account in determining the separation strain of the concrete. Based on a comparison of simulation results and experimental results, a separation strain value of 0.1 is recommended for use in blast analysis of reinforced concrete structures.

Based on a progressive collapse analysis of the Murrah Federal Building in Oklahoma City, the recommended separation strain value is also considered appropriate for use in progressive collapse analysis. The results of the analysis of the Murrah Federal Building demonstrate that the ELS program can realistically simulate the collapse of building structures.

Two design alternatives for preventing the progressive collapse of the Murrah Federal Building were identified and analyzed. The results indicate that the extent of the damage to the original building (27% of the total floor area) could have been reduced to 9% and 12% by the first and second design alternatives, respectively. The results of this study show that nonlinear dynamic programs such as ELS can be powerful tools in designing building structures to resist progressive collapse.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by High-tech City Development Works through the Korea Institute of Construction & Technology Evaluation and Planning (KICTEP) and was funded by the Ministry of Land, Transport, and Maritime Affairs (13CHUD-B053109-04-000000).

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