# Robustness of Reinforced Concrete Frames against Blast-Induced Progressive Collapse

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Global Robustness of RC Frames under Column Removal Scenario

#### 2.1. Structural Behavior Aspects

#### 2.2. Numerical Analysis for Structural Robustness

_{V}) located around the removed column reaches the value of 15–20% [19]. The latter is calculated starting from the vertical displacement of the node at the top of the removed column δ

_{V}and the length of the beam to which the node belongs L

_{b}:

_{V}= tan

^{−1}(δ

_{V}/L

_{b}),

_{d}) for the column [21]: the less the Δt

_{d}for the complete removal of the columns, the more severe the consequent structural response. In this view, the identification or setting of the column removal time interval Δt

_{d}for a certain “n” would be of value.

#### 2.3. Robustness Curves

_{L}of relevant locations along the structure have been individuated for the damage presumption (column removal), a set of damage scenarios implying an increasing damage level are defined and named “D-scenario (i,j)” (where “i” indicates the location and “j” the presumed damage level). Each D-scenario is analyzed starting from the lower damage level by implementing a nonlinear dynamic analysis (NDA in the flowchart). After each NDA, if the collapse does not occur, the pushover nonlinear analysis under lateral load is carried out to determine a point (residual capacity λu/λ%) of the robustness curve, and then the damage level is increased, and the NDA is repeated until the progressive collapse (as appropriately defined) occurs. In the proposed method, the progressive collapse of the structure is declared when the failure of a column adjacent to the columns removed or damaged due to the explosion is observed during the nonlinear dynamic analysis NDA. The procedure is carried out for different locations to obtain a set of robustness curves under blast presumed damage scenarios.

## 3. Blast-Induced Local Damage for RC Columns

#### 3.1. Blast Load

_{a}+ t

_{0}, the pressure changes in size. The absolute value of the peak pressure in the negative phase is typically smaller than the one in the positive phase. The area underpinned by the curve in the positive phase is the impulse of the blast, which has been proven to be a primary intensity measure for detonations [22]. In this study the blast intensity is determined by the equivalent TNT kilograms of explosive and the stand-off distance of the explosion from the element.

#### 3.2. Local Models and Analyses for Blast-Damage Assessment

- -
- The CAPACITY is expressed in terms of lateral displacement thresholds and associated damage levels.
- -
- The DEMAND is expressed by the peak lateral displacement and by the occurrence time of the peak lateral displacement counted from the explosion instant, called “peak response instant”. This would appear to be unusual, but our reasons will be explained in Section 4.

- (a)
- A static nonlinear (pushover) analysis to evaluate the local CAPACITY of the element under the lateral induced deformation typical of blast-loaded columns;
- (b)
- Under a certain blast load intensity, a transient dynamic nonlinear analysis to evaluate the local DEMAND.

_{r}(1 − t/t

_{d})e

^{−}

^{βt/t}

^{d},

_{r}= 2P

_{S0}(7P

_{atm}+ 4P

_{S0})/(7P

_{atm}+ P

_{S0}),

_{S0}= 1.772(1/Z

^{3}) − 0.114(1/Z

^{2}) + 0.108(1/Z),

_{S0}= 300[0.5(W)

^{1/3}], Z = R/(W)

^{1/3}, t

_{d}= 2i

_{S0}/P

_{S0},

_{atm}is the atmospheric pressure, and β is the decay coefficient, taken equal to 1.8.

^{®}commercial structural code [27], by using beam finite elements (FEs) for the column and by implementing large displacement solutions and the plastic hinges approximation for modeling material nonlinearity, the latter being modeled by considering the bending behavior at the two ends and at midspan. The elastic–plastic bilinear hardening model has been implemented for the hinges, with a rotational ductility ratio θ

_{ult}/θ

_{y}(θ

_{ult}and θ

_{y}being the ultimate and the yielding rotation, respectively) fixed to 30 and the “drop to zero” option switched on in SAP2000

^{®}. As can be seen from the figure, different blast intensities lead to different structural responses: the intensities (2 m-2 kg), (2 m-3 kg), and (2.5 m-4 kg) lead to a “damaged response” for the column, with some residual displacements after the transitory response, while the intensities (2 m-4 kg) and (3 m-20 kg) lead to the failure of the elements, something which can be recognized from the value of the maximum displacement reached (which is larger than the 30 mm limit associated with the damage 1 in Figure 4), and from the consequent decreasing of the displacements toward zero, something that is unrealistic and numerically induced. From Figure 6, it is observed that the structural response is more sensitive to the stand-off distance than to the equivalent TNT kilograms. In fact, focusing on the cases with 2 m stand-off distance, the TNT kilograms must reach the value of 4 kg to lead the collapse (2 kg and 3 kg lead to some damaged response with residual displacements), while if the stand-off distance rises to 3 m (+50%) we need to increase the TNT value to 20 kg (+400%) for the collapse.

_{p}(called “peak response instant” in the following sections) for each scenario has a key role in the definition of the local blast demand for the column and the induced damage.

_{MAX}from Figure 6 and Figure 8 is equal to 30 mm).

## 4. Application to an Existing Structure

#### 4.1. Case Study Structure and FEM Model

^{®}structural code, by defining the nonlinear properties of the materials. The nonlinear behavior is implemented using the approximation of plastic hinges, which are obtained from the moment-rotation relationship (M-θ) evaluated from the equations provided by the Italian Standards NTC2018 [23]. All the columns of the ground floor have the cross-section already presented in Figure 5. Moreover, geometric nonlinearity is considered with large displacement and P-Δ options. As stated previously, the 2D frame is extracted from a complex structure and in order to simulate the contribution to the catenary effect and the membrane effect provided by out-of-plane beams and by the slab respectively, a dedicated nonlinear beam finite element is added, connected in parallel to each beam of the 2D frame, and named “special element” in what follows. This latter element has been modeled as an ordinary beam element provided with axial and bending plastic hinges connected to the columns of the 2D frame. The special elements have been calibrated on the basis of the membrane behavior of the floor slab as follows: (i) a dedicated fiber-based plastic model has been used in order to identify the axial and bending behavior of the 3D floor module (in-plane and out-of-plane beams plus slab); (ii) then, the plastic hinges and the elastic stiffness of the additional special elements have been calibrated in such a way that the in-plane beams of the 2D frame plus the special elements, were able to provide the same vertical elastoplastic strength and stiffness of the above mentioned floor fiber model, then taking into account for the out-of-plane membrane and catenary effects in the 2D frame model.

- -
- Evaluate the global robustness curves of the structure; and
- -
- Evaluate the blast local demand curve for the columns at the locations indicated in Figure 9.

#### 4.2. Global Robustness Results and Sensitivity Analysis

_{L}(column 3 and column 5, see Figure 9), analyses for each of the D(i,j)-scenarios (i = location, j = damage level) are developed (in order to evaluate the global robustness of the structure). A particular location is considered and starting from a certain damage level the structural response is evaluated by the nonlinear dynamic analysis. Here the damage level is intended as the number of columns that are removed in the global analysis. If failure doesn’t occur spontaneously (as defined in Section 2.2) the typical structural response is the one shown in Figure 11. Successively, the residual strength of the structure is identified using a nonlinear static analysis, then the damage level is increased (i.e., an additional element is removed together with the previous one) and, again, the structural response is evaluated. During the analysis that provides the evaluation of the residual strength of the structure (i.e., pushover analysis), the residual lateral force capacity (λu) considered is the one that corresponds to the first occurring condition between the “run-away” behavior observed in the vertical displacement time history of the nodes around the removed column or the experimentation of a vertical drift ratio (D

_{V}) bigger than 15% (see Section 2.2). Obviously, if the number of considered locations N

_{L}is different from 1, all is repeated N

_{L}times. Before starting to apply the procedure in order to identify the robustness of the structure under blast loads, various locations of hypothetical damage were assumed (Figure 9). Depending on which column is removed, the results obtained by the lateral pushover analysis are shown in Figure 12: the internal columns (No. 3, No. 4, and No. 5) are characterized by a bigger tributary area under vertical loads and the hypothetical damage to one of them could cause a bigger reduction in the capacity.

_{Y}value for the force. The slope of the linear part is identified imposing the passage for the point 0.6F

_{MAX}of the original capacity curve, while the value of F

_{Y}is obtained by imposing the equality of the areas underpinned by the bilinear curve and by the capacity curve for a fixed maximum displacement d

_{U}.

_{d}). The variation of each of these parameters influences the behavior of the structure in terms of ultimate strength and deformation; moreover, the Δt

_{d}is a very important parameter that allows the connection between the global level and local level analysis because the time interval of column removal Δt

_{d}can be considered as the time during which the damage propagates and affects the structural element under blast load effects (see Appendix A for a full report on the sensitivity analyses performed). The effect of different Δt

_{d}values on the structural robustness are shown in Figure 13, where the robustness curves obtained for Δt

_{d}= 0.01; 0.02; 0.3; 0.5 s, and for the location 2 case, are compared with each other. As expected, the less the Δt

_{d}, the lower the residual capacity obtained for a fixed damage level.

_{L}= 2): location 1 implies that the first element removed is column 3, while for location 2 the first element removed is column 5. In both cases, damage level 2 corresponds to the progressive collapse of the structure due to the spontaneous failure of a column adjacent to the removed or damaged one as a result of the explosion. Damage level 0.5, instead, corresponds to the loss of 50% of the transversal section; this means that the explosion results in a loss of element stiffness and capacity, but not in a collapse. The damaged element continues to carry the axial load but there are no dynamic effects due to the loss of the column. It should be noted that for damage level 1.5, considerations are similar to those reported for damage level 0.5: in this case, one column is completely removed, and the loss of 50% of the second column’s section is considered. All the pushover curves are reported in Appendix A, where the effect of different values of the investigated parameters (damping ζ and removal time of the column Δt

_{d}) is discussed.

#### 4.3. Local Demand

_{d}to the different intensity measures of the explosion (stand-off distance from the ignition and equivalent kilograms of TNT). In this view, the second step of the procedure contemplates the local analyses from which the results already presented are obtained. In particular, the focus is on the evaluation of the capacity curve of the column (Figure 4) and on the assessment of the blast intensities, stand-off distance, and equivalent kilograms of TNT, which define the blast local demand (Figure 8). It is then essential to link together these two analysis levels: from the curve in Figure 4 it is possible to evaluate the capacity of the element and the value of displacement corresponding to a certain damage level (i.e., damage level 1, failure of the element, occurs in correspondence with the ultimate displacement of the midspan node; damage level 0.5, the partial failure of the element, occurs in correspondence with the first drop of the element capacity from its maximum value, see Figure 4); from the demand relationship between the intensity measures of the explosion and the peak response instant Δt

_{p}shown in Figure 8, it is possible to associate the specific value of the column removal time Δt

_{d}with a certain blast intensity. This can be done by interpreting the peak response instant Δt

_{p}obtained from the local analysis, as the column removal time Δt

_{d}used in the global analysis (i.e., Δt

_{d}= Δt

_{p}).

#### 4.4. Blast Scenario-Dependent Robustness

- (a)
- LOCAL DAMAGE PRESUMPTION. First, by using the local capacity curve defined in Figure 4, a certain presumed local damage level is associated with a certain peak lateral displacement d
_{peak}; - (b)
- BLAST SCENARIO DEFINITION. Second, by means of the blast local demand curve in Figure 8, it is possible to associate the d
_{peak}value previously identified with a particular blast scenario characterized by a certain blast intensity (a stand-off distance and a certain value of equivalent kilograms of TNT), and correspond it with a peak response instant Δt_{d}; - (c)
- ROBUSTNESS SELECTION. Finally, the appropriate robustness for the presumed local damage above can be selected among the robustness curves evaluated in Figure 13 as the one obtained by the column removal time interval equal to the peak response instant Δt
_{d}and then associated with the above-identified blast scenario.

_{d}with the considered displacement at midspan node (Figure 15b), the values of λ

_{u}/λ connected to the corresponding removal time interval of the column are selected from Figure 15c. In this case, for example, a blast scenario with a stand-off distance equal to 2 m and 3 equivalent kg of TNT, which is characterized by a peak response instant of about 0.02 s, is associated to the local damage 1; the value corresponding to damage level 1 of the robustness curve with Δt

_{d}= 0.02 s is considered from Figure 15c and used for the construction of the BSR curve of Figure 15d at the same damage level. A local damage level of 0.5 occurs for a blast scenario characterized by a stand-off distance equal to 2.5 m and 4 equivalent kg of TNT (peak response instant equal to 0.01 s): the value of robustness belonging to curve Δt

_{d}= 0.01 s is used in order to identify the point on the BSR curve of Figure 15d at a damage level of 0.5.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{d}). Considering the first parameter, Figure A1 shows the effects of the variation of the damping ratio ζ for location 1 when the removal interval Δt

_{d}is set equal to 0.02 s. As the damping index increases, the maximum vertical displacement and the time necessary to dampen the free oscillations of the removed column node decreases. For successive analyses, the damping ratio is set equal to 4% since smaller values (e.g., 1% in the figure) do not determine a significative difference in terms of “damping” time. For the sake of completeness, it has to show that a very large damping index, such as 0.5, leads to an almost absence of oscillation. The bi-linear pushover curves in Figure A2 show the case of damage 1, with the instantaneous removal of column 3, and depict the influence of the damping index parameter at the same removal time interval Δt

_{d}, set equal to 0.02 s. As the damping index decreases, the above described dynamic amplification effect leads to a decreasing in both the stiffness and strength of the damaged frame (i.e., after the removal of the column) under lateral load.

**Figure A1.**Displacement of the node at the top of column 3 (location 1): effect of variation of damping ratio (Δt

_{d}values are expressed in seconds).

**Figure A2.**Pushover curve of 2D RC frame structure—damage level 1: effect of variation of damping ratio.

_{d}for location 3. As it appears in Figure A3, which shows the results of a column removal analysis that captures the effects of amplification in terms of displacement and in terms of geometric and material nonlinearities, the displacement of the node at the top of the removed column increases as Δt

_{d}decreases. There is a small difference between the cases with Δt

_{d}= 0.5–0.3 and Δt

_{d}= 0.01–0.02. It is also possible to note that the value of Δt

_{d}has a certain influence on the amplitude of the oscillations around the residual displacement and on the damping shown in the time histories.

**Figure A3.**Displacement of the node at the top of column 3: effect of variation of Δt

_{d}(removal time of the column). Δt

_{d}values are expressed in seconds.

_{d}affects the capacity of the damaged structure. Although there are just slight differences between the curves for the cases considered, if the Δt

_{d}value also decreases the overall capacity decreases. It is important to understand the effect that different values of Δt

_{d}have in terms of decreasing the capacity of the structure because this parameter can be used to simulate the damage induced by different blast scenarios.

**Figure A4.**Pushover curve of 2D RC frame structure—damage level 1: effect of variation of Δt

_{d}(removal time of the column). Δt

_{d}values are expressed in seconds.

_{d}. Coherently with Figure A4, where the pushover curves for damage level 1 are reported, the decrease of Δt

_{d}determines a decrease in the residual capacity for both locations. Damage level 2 always determines the progressive collapse of the structure, while damage level 1 causes a drop in initial capacity of about 20% for L1 and about 30% for L2. Similarly, Figure A7 and Figure A8 show the effect of the variation of the other parameter investigated, the damping ratio, with a defined removal time of the column (Δt

_{d}= 0.02 s). The trend related to capacity losses remains the same as previously discussed: if the damping index increases, there is an increase in the capacity of the RC frame compared to cases with a smaller damping value. Even in this case, damage level 2 causes the collapse of the structure and damage level 1 determines a reduction of the structure’s capacity.

**Figure A5.**Robustness curves for 2D RC frame structure for location 1: effect of variation of Δt

_{d}(removal time of the column). Δt

_{d}values are expressed in seconds.

**Figure A6.**Robustness curves for 2D RC frame structure for location 2: effect of variation of Δt

_{d}(removal time of the column). Δt

_{d}values are expressed in seconds.

**Figure A7.**Robustness curves for 2D RC frame structure for location 1: effect of variation of damping ratio. Δt

_{d}values are expressed in seconds.

**Figure A8.**Robustness curves for 2D RC frame structure for location 2: effect of variation of damping ratio. Δt

_{d}values are expressed in seconds.

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**Figure 1.**Typical robustness curves; different markers represent different locations for the presumed damage along the structure.

**Figure 2.**Flowchart of the procedure to evaluate the structural robustness against blast damage [17].

**Figure 4.**Capacity curve of the local element (RC column shown in Figure 5).

**Figure 5.**Cross section of the RC column analyzed; structural scheme considered and acting load. Sizes are in mm if not specified otherwise.

**Figure 6.**Typical results of a local numerical analysis for demand assessment: time histories of midspan node lateral displacement due to lateral blast load.

**Figure 13.**Robustness curves for 2D RC frame structure for location 2: effect of variation of Δt

_{d}(removal time of the column). Δt

_{d}values are expressed in seconds.

**Figure 15.**BSR curves evaluation summary. Local damage presumption (

**a**); blast scenario definition (

**b**); robustness performance selection (

**c**); and BSR curve (

**d**).

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**MDPI and ACS Style**

Francioli, M.; Petrini, F.; Olmati, P.; Bontempi, F.
Robustness of Reinforced Concrete Frames against Blast-Induced Progressive Collapse. *Vibration* **2021**, *4*, 722-742.
https://doi.org/10.3390/vibration4030040

**AMA Style**

Francioli M, Petrini F, Olmati P, Bontempi F.
Robustness of Reinforced Concrete Frames against Blast-Induced Progressive Collapse. *Vibration*. 2021; 4(3):722-742.
https://doi.org/10.3390/vibration4030040

**Chicago/Turabian Style**

Francioli, Mattia, Francesco Petrini, Pierluigi Olmati, and Franco Bontempi.
2021. "Robustness of Reinforced Concrete Frames against Blast-Induced Progressive Collapse" *Vibration* 4, no. 3: 722-742.
https://doi.org/10.3390/vibration4030040