# A Parametric Computational Study of RC Building Structures under Corner-Column Removal Situations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Study

#### 2.1. Description of the Building and Test

^{2}columns and flat-slabs that were 0.20 m thick. The distance between column edges was 5 m and the floor-to-ceiling height was 2.8 m. B500S reinforcement steel bars (yield stress of 500 MPa) were used for the reinforcement of slabs and columns. The design of the building followed the requirements of Eurocodes [38,39,40]. A more detailed description of the reinforcement bars of the concrete elements can be found in Adam et al. [28]. Table 1 gives the concrete mechanical properties of slabs and columns obtained on the day of the test, which have been used in the FE model validation.

^{2}and applied to a tributary area of 5 × 5 m

^{2}on the corner bay where the column was to be removed. At the edges of this corner bay, a façade load of 0.56 kN/m was also applied on the first floor simulating the load of a hypothetical outer wall. The resulting load configuration represents a design load case under accidental scenarios [28,36]. Figure 1 contains a sketch and a photograph of the loads (in red in the sketch) and the building prepared for the test.

#### 2.2. FE Model

_{ct}(Table 1). This bilinear response was defined by an intermediate point with a strain of 5.5 × 10

^{−4}, stress equal to 0.33 f

_{ct}and a final point with a strain of 25 × 10

^{−4}and a stress equal to f

_{ct}/100 [46]. Damage was also introduced in the model in terms of tension and compression damage [41] by defining two independent parameters, named dt and dc, respectively. Both parameters are used to identify the inelastic behaviour of concrete when subjected to tensile and compressive stresses in the presence of tensile cracking and compressive crushing, respectively. In both cases, multilinear relationships are assumed as it is considered in ABAQUS by default [41]. The rest of the parameters of the damage plasticity model, such as the dilatation angle, eccentricity, fb0/fc0 and K, were considered by default [41]. The steel reinforcement bars (yield stress of 500 MPa, Young modulus of 200,000 MPa and Poisson ratio of 0.3) were introduced in the slabs using perfect bond smeared layers [28]. The measured strain rate in the tests were lower than 1/s, as discussed in [13]. Strain rate values registered during the test with Digital Image Correlation (DIC) showed a peak strain rate value of 0.32/s, which in concrete structures is normally representative of small dynamic events such as earthquakes or heavy traffic. Micallef et al. [47] showed that, for values less than 10/s, this effect can be ignored as it was adopted in the FE analysis. Figure 2 shows the FE model that was developed.

^{3}and 78.5 kN/m

^{3}for concrete and steel, respectively). The superimposed experimental load was also modelled as uniform and line loads in the FE model. Fully-fixed boundary conditions were adopted for the bottom nodes of the ground columns, while nodes in the slab belonging to the intersections with the columns were coupled as plate constraints to ensure that all the nodes in these regions had rigid body displacements.

_{1}= 0.2785 s; based on the test results [28]). A preliminary step with static-linear analysis (SLA) was carried out to obtain the load of column P3 before its sudden removal. This force (F3) was then applied to the model prepared for the DNLA. The removed column (P3) was not modelled in this analysis (Figure 2) but was substituted by the equivalent force F3 obtained from the SLA. During DNLA, the concentrated load, F3, and the gravity loads (i.e., self-weight and superimposed loads) were increased gradually in the first step (from t = 0 s to t = 1 s) to obtain a stable response of the structure and zero vertical displacement in the position of the concentrated force, F3 (column P3 before column removal). In the second step, the concentrated force, F3, was gradually deactivated, as defined by the column removal time parameter. For the FE validation, this column removal time parameter was given a value of 0.1 s, i.e., the value obtained in the test [28]. The third step was defined until t = 2 s to analyse the dynamic non-linear structure response after the corner-column removal.

#### 2.3. FE Model Validation

#### 2.4. Parametric Study

_{ct}), (ii) flexural reinforcement ratio around the P2/P6 column-slab joint (ρ), (iii) number of floors (nf), (iv) column removal time (crt) and (v) the applied superimposed load (q). Ten different models were processed, establishing the model described in Section 2.3 as the reference model but considering the same tensile strength (2.0 MPa) for both slabs. Models built for the parametric study vary by only one parameter at a time. Three different f

_{ct}values were considered: 2.0, 2.5 and 3.0 MPa, the lowest value being similar to the experimental one and adopting two higher levels. The flexural reinforcement ratio around the P2/P6 column-slab joint was approximately equal to 0.8%, which could be considered relatively high, so a value of 0.5% was also considered. The number of floors was also taken into account due to its possible influence in the Vierendeel action and the dynamic response, adding two and five extra levels to those of the test (i.e., four and seven floors). As the column removal time clearly affects the dynamic response of the structure, for this parameter, three extra levels were considered: 0.01 s, 0.05 s (both smaller than that in the test) and 0.2 s (higher than that in the test). The lower value is acceptable according to the limit of T/10 [36], where T (vibration period) was 0.28 s according to the experimental results. The applied superimposed load was also increased from 5.3 kN/m

^{2}to 9 kN/m

^{2}, which represents the maximum design load for a Ultimate Limit State (ULS) in persistent and transient design situations and was also near the flexural capacity of the slab. Table 3 shows the parameter values in each model.

## 3. Results

#### 3.1. Vertical Displacement

#### 3.2. Drift

#### 3.3. Vertical Reactions in the Adjacent Columns

#### 3.4. Computed Tensile Damage Maps

## 4. Discussion

#### 4.1. Activation of Alternative Load Paths (ALPs)

^{2}/m [40], while in accidental situations during Vierendeel-type action, the reinforcement required is considerably higher (see comments below).

^{2}/m [40], being around 5.9 cm

^{2}/m, 9.1 cm

^{2}/m and 12.4 cm

^{2}/m for bending moments of 40 kN*m/m, 60 kN*m/m and 80 kN*m/m, respectively. The differences between the parameters, as shown in Figure 14 and Figure 15, are significant for the reinforcement ratio, number of floors and column removal time parameters. However, concrete tensile strength has no influence on the distribution of the analysed bending moments. The results of model M_q9 confirm that the structure yielded, because this model was that of the Ultimate Limit State (ULS) of the structure.

#### 4.2. Dynamic Amplification and Load Increase Factors (DAFs and LIFs)

_{LD}and LIF

_{LD}were computed as the difference in the maximum displacement on the top of the removed column (P3) between a static-nonlinear or static-linear and a dynamic-nonlinear FE model, respectively. In the forces, the DAF

_{LF}and LIF

_{LF}were computed as the difference in the axial force of columns P2/P6 between a static-nonlinear or static-linear and a dynamic-nonlinear FE model, respectively. Table 7 shows the results obtained.

_{LD}values in general were between 1.80 and 2.70, which can be considered consistent with test results, theoretical considerations and codes [27,28,37]. The lower values obtained within this range are related to stiffer models where the dynamic amplification was lower and where damping was higher; for a single degree of freedom system with 7.3% damping (test value), the theoretical DAF

_{LD}is 1.79, which is similar to that obtained in M_Ref. The low value of DAF

_{LD}in M_crt0.2 was due to the slow removal of the column (i.e., low dynamic amplification). The higher values obtained for DAF

_{LD}were predicted in models with parameter leading to higher deformations, accelerations and lower damping effects. In such cases, DAF

_{LD}was larger than 2, which could be justified on (a) undesired overly stiff numerical response predicted by the static nonlinear models and (b) valid theoretical considerations, i.e., as shown in [13], for a single degree of freedom system with nonlinear parabolic response and no damping, the DAF

_{LD}is larger than 2 (e.g., 10% or higher depending on the curvature of the parabola). The uncommonly high value of DAF

_{LD}in M_q9 was due to the formation of a yield line mechanism in the slab leading to development of plastic deflections captured in the dynamic nonlinear analysis. In this case, the values of DAFs and LIFs are strongly influenced by the flexural capacity of the slab, as shown by the yield lines in Figure 11e. It should be noted that values of LIF

_{LD}are always higher because material non-linear effects are included in the factor besides the dynamic effects.

_{LF}and LIF

_{LF}values are not significantly different, and they are also relatively constant with changes in the parameters in the study. All the values are around 1.10, slightly inferior to the 1.20–1.30 which the authors obtained in previous studies [13,28] and which can be justified by (i) the fact that the present study considered experimental damping in the FE models and (ii) the small differences found in the reactions peak values between experimental and computational results. Model M_q9 again obtained a value of 1.00 because the load was exactly the yielding load. Model M_crt0.2 had a slightly lower value (1.04), again due to the slow removal of the column.

## 5. Conclusions

_{ct}), (ii) flexural reinforcement ratio (ρ), (iii) number of floors (nf), (iv) column removal time (crt) and (v) the applied superimposed load (q). The influence of each parameter was studied on deflections, drifts, forces, damage, Alternative Load Paths (ALPs) and Dynamic Amplification and Load Increase Factors (DAFs and LIFs), for both displacements (DAF

_{LD}, LIF

_{LD}) and loads (DAF

_{LF}, LIF

_{LF}). From the results obtained, the following conclusions can be drawn:

- The tensile concrete strength has a strong influence on the numerical predictions of displacements and structural damage. This affects the computed DAF
_{LD}, with low values of f_{ct}generally giving more realistic values of DAF_{LD}. This parameter did not have a significant effect on the reactions/forces. - The same conclusions can be drawn for the influence of the flexural reinforcement ratio, although in this case it also had a significant influence on the member reaction/force values. However, this influence is not transferred to the computed DAF
_{LF}, which is not significantly influenced by this parameter. - The number of floors also has a clear influence on both vertical displacements and member reaction/forces. However, DAF
_{LD}and DAF_{LF}are not influenced by this parameter. - Column removal time has a strong influence on vertical displacements and bending moments, although it barely affects the reaction forces’ peak and residual force values. Due to the wide variation found, it is always recommended to assume the worst possible case at the limit of the GSA recommendations [36] to establish a column removal time equal to a tenth of the structure’s fundamental period without the missing column (T/10).
- The applied superimposed load had a strong influence on all the aspects studied. A comparison was made between a load combination for ULS and a common design accidental load. Results showed that considering a more aggressive scenario than a design accidental situation could result in making different and erroneous recommendations on the safe side, in some cases (e.g., excessive deflections or forces can be considered in a design accidental situation but this is a persistent and transient situation), but these recommendations would still be unsafe for the DAF
_{LF}(i.e., resulting in a lower value than in a design accidental situation). - The test and FE models in this paper showed that Vierendeel action was the main ALP of the structure for the particular case investigated, i.e., without infill walls and subjected to a design compliant corner-column removal situation. The rest of the column above the removed one acts as a key element in effectively activating this ALP. However, for this ALP to be safe, a series of reinforcement details should be in the design phase of the structure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Numerical reaction in P1 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate compression.

**Figure A2.**Numerical reaction in P4 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate compression.

**Figure A3.**Numerical reaction in P5 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time, and (

**e**) superimposed load. Positive values indicate compression.

**Figure A4.**Numerical reaction in P7 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate compression.

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**Figure 2.**Finite Element (FE) model without the ground floor column P3 (column removed). Colours represent sections with different amounts of reinforcement according to the building details [28].

**Figure 3.**Comparison of the residual vertical deformed shape of the first floor between P2 and P3 in the experimental test [28] and FE model.

**Figure 4.**Comparison of the residual horizontal displacement/drift of the structure between the test [28] and the FE model. Positive values indicate horizontal displacement towards the removed column.

**Figure 5.**Comparison of the test [28] and FE model for the axial force increments in columns P1, P2/P6, P5 and P7. Positive values indicate compression.

**Figure 6.**Comparison of the tensile damage (DAMAGET) between the FE model and test [28].

**Figure 7.**Numerical residual deformed shapes along 1st floor corner-bay slab between P2 and P3 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate downward displacements.

**Figure 8.**Numerical residual horizontal drift obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load.

**Figure 9.**Numerical reaction in P2 obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate compression.

**Figure 10.**Numerical axial force in P3 (first floor) obtained by varying (

**a**) concrete tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time and (

**e**) superimposed load. Positive values indicate compression.

**Figure 11.**Computed tensile damage at t = 2 s obtained for (

**a**) M_fct2.5, (

**b**) M_ρ0.5, (

**c**) M_nf7, (

**d**) M_crt0.2 and (

**e**) M_q9.

**Figure 12.**Conceptualisation of static column removal effects (deformed shape, axial and shear forces and bending moments) of the building under study in design situations before (initial state) and after (cantilever and Vierendeel) column removal.

**Figure 13.**Shear and axial force of first floor column P3 before and after column removal for the different models. Positive axial force values indicate compression.

**Figure 14.**Influence of different parameters on bending moments from P2 to P3 in the first slab: (

**a**) tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time, (

**e**) superimposed load.

**Figure 15.**Influence of different parameters on bending moments from P2 to P3 in the second slab: (

**a**) tensile strength, (

**b**) flexural reinforcement ratio, (

**c**) number of floors, (

**d**) column removal time, (

**e**) superimposed load.

Mechanical Property | Element | Result [MPa] |
---|---|---|

Compressive strength(f_{c}) | 1st slab | 30.5 |

2nd slab | 31.1 | |

Tensile strength(f_{ct}) | 1st slab | 2.44 |

2nd slab | 1.83 | |

Elastic Modulus(E) | Ground floor columns | 29,275 |

1st slab | 28,810 | |

1st floor columns | 29,403 | |

2nd slab | 33,119 |

**Table 2.**Comparison of the peak and residual values of maximum vertical displacement and drift between the test [28] and the FE model.

Component | Value | Test | FE Model | Error [mm] | |
---|---|---|---|---|---|

Maximum vertical displacement [mm] | Peak | −48.1 | −41.9 | 6.2 | |

Residual | −42.8 | −40.5 | 2.3 | ||

Drift [mm] | Floor 1 | Peak | 1.40 | 0.84 | 0.56 |

Residual | 0.40 | 0.38 | 0.02 | ||

Floor 2 | Peak | 4.80 | 2.34 | 2.46 | |

Residual | 2.40 | 1.53 | 0.87 |

Model | Parameters | ||||
---|---|---|---|---|---|

f_{ct} [MPa] ^{1} | ρ^{1} | nf | crt [s] | q [kN/m^{2}] ^{1} | |

Reference (M_Ref) | 2.0 | 0.8% | 2 | 0.1 | 5.3 |

M_fct2.5 | 2.5 | 0.8% | 2 | 0.1 | 5.3 |

M_fct3.0 | 3.0 | 0.8% | 2 | 0.1 | 5.3 |

M_ρ0.5 | 2.0 | 0.5% | 2 | 0.1 | 5.3 |

M_nf4 | 2.0 | 0.8% | 4 | 0.1 | 5.3 |

M_nf7 | 2.0 | 0.8% | 7 | 0.1 | 5.3 |

M_crt0.01 | 2.0 | 0.8% | 2 | 0.01 | 5.3 |

M_crt0.05 | 2.0 | 0.8% | 2 | 0.05 | 5.3 |

M_crt0.2 | 2.0 | 0.8% | 2 | 0.2 | 5.3 |

M_q9 | 2.0 | 0.8% | 2 | 0.1 | 9.0 |

^{1}Considered equal for slabs 1 and 2.

**Table 4.**Vertical displacement peak and residual values obtained at five points on the 1st floor slab from P2 edge to P3 edge. Units in mm.

Model | Distance [m] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.62 | 1.67 | 3.33 | 4.35 | 5 | ||||||

Peak | Residual | Peak | Residual | Peak | Residual | Peak | Residual | Peak | Residual | |

M_Ref | 4.47 | 4.39 | 17.61 | 17.38 | 36.60 | 36.09 | 44.46 | 43.84 | 46.88 | 46.18 |

M_fct2.5 | 3.14 | 2.97 | 11.84 | 11.17 | 24.10 | 22.51 | 29.33 | 27.33 | 31.32 | 29.02 |

M_fct3 | 2.74 | 2.54 | 9.99 | 9.16 | 20.15 | 18.24 | 24.45 | 22.06 | 26.20 | 23.63 |

M_ρ0.5 | 7.54 | 7.48 | 27.78 | 27.64 | 57.35 | 57.20 | 69.83 | 69.46 | 72.85 | 72.48 |

M_nf4 | 7.209 | 7.17 | 26.96 | 26.78 | 56.09 | 55.53 | 68.09 | 67.53 | 71.23 | 70.41 |

M_nf7 | 8.37 | 8.26 | 29.94 | 29.63 | 61.63 | 60.82 | 74.54 | 73.55 | 77.71 | 76.52 |

M_crt0.01 | 6.05 | 6.00 | 23.70 | 23.60 | 50.14 | 49.72 | 60.45 | 60.13 | 63.42 | 63.32 |

M_crt0.05 | 5.48 | 5.39 | 21.62 | 21.38 | 45.28 | 44.73 | 55.06 | 54.33 | 57.83 | 56.99 |

M_crt0.2 | 3.44 | 3.39 | 13.44 | 13.27 | 27.61 | 27.23 | 33.50 | 33.03 | 35.62 | 35.11 |

M_q9 | 5.32 | 5.32 | 230.7 | 230.7 | 446.80 | 446.80 | 533.54 | 533.54 | 556.30 | 556.30 |

Model | Vertical Displacement in P3 [mm] | Increase/Decrease [%] |
---|---|---|

M_Ref | 46.18 | - |

M_fct2.5 | 29.02 | −37.2% |

M_fct3.0 | 23.63 | −48.8% |

M_ρ0.5 | 72.48 | 57.0% |

M_nf4 | 70.41 | 52.5% |

M_nf7 | 76.52 | 65.7% |

M_crt0.01 | 63.32 | 37.1% |

M_crt0.05 | 56.99 | 23.4% |

M_crt0.2 | 35.11 | −24.0% |

M_q9 | 556.30 | 1104.7% |

**Table 6.**Residual and peak horizontal drift values obtained in each floor for each model. Units in mm.

Model | Floor | |||
---|---|---|---|---|

1st Floor | 2nd Floor | |||

Peak | Residual | Peak | Residual | |

M_Ref | 0.61 | 0.21 | 1.99 | 1.36 |

M_fct2.5 | 0.73 | 0.28 | 1.85 | 1.16 |

M_fct3 | 0.71 | 0.27 | 1.76 | 1.11 |

M_ρ0.5 | 0.45 | 0.06 | 1.95 | 1.30 |

M_nf4 | 0.41 | 0.07 | 2.05 | 1.37 |

M_nf7 | 0.16 | 0.01 | 1.58 | 1.35 |

M_crt0.01 | 0.63 | 0.14 | 2.14 | 1.46 |

M_crt0.05 | 0.62 | 0.16 | 2.06 | 1.38 |

M_crt0.2 | 0.53 | 0.22 | 1.70 | 1.30 |

M_q9 | −0.74 | −0.46 | 1.57 | 1.05 |

**Table 7.**Dynamic Amplification Factors (DAFs) and Load Increase Factors (LIFs) for displacements (DAF

_{LD}and LIF

_{LD}) and forces (DAF

_{LF}and LIF

_{LF}).

Model | Static Nonlinear | Static Linear | Dynamic Nonlinear | DAF_{LD} | LIF_{LD} | DAF_{LF} | LIF_{LF} | |||
---|---|---|---|---|---|---|---|---|---|---|

Max. Disp. [mm] | Axial Force P2/P6 [kN] | Max. Disp. [mm] | Axial Force P2/P6 [kN] | Max. Disp. [mm] | Axial Force P2/P6 [kN] | |||||

M_Ref | 25.6 | 317.4 | 14.1 | 317.3 | 46.9 | 340.1 | 1.83 | 3.33 | 1.07 | 1.07 |

M_fct2.5 | 20.6 | 316.0 | 14.1 | 317.3 | 31.3 | 346.4 | 1.52 | 2.22 | 1.10 | 1.09 |

M_fct3.0 | 18.4 | 316.1 | 14.1 | 317.3 | 26.2 | 351.4 | 1.42 | 1.86 | 1.11 | 1.11 |

M_ρ0.5 | 26.9 | 317.9 | 14.1 | 317.2 | 72.7 | 350.6 | 2.70 | 5.16 | 1.10 | 1.11 |

M_nf4 | 27.8 | 645.0 | 15.2 | 649.6 | 70.3 | 720.7 | 2.53 | 4.63 | 1.12 | 1.11 |

M_nf7 | 30.7 | 1138.8 | 18.1 | 1152.1 | 77.7 | 1267.4 | 2.53 | 4.29 | 1.11 | 1.10 |

M_crt0.01 | 25.6 | 317.4 | 14.1 | 317.3 | 63.4 | 352.6 | 2.48 | 4.56 | 1.11 | 1.11 |

M_crt0.05 | 25.6 | 317.4 | 14.1 | 317.3 | 57.8 | 343.6 | 2.26 | 4.10 | 1.08 | 1.08 |

M_crt0.2 | 25.6 | 317.4 | 14.1 | 317.3 | 35.6 | 330.5 | 1.39 | 2.52 | 1.04 | 1.04 |

M_q9 | 76.3 | 411.4 | 18.7 | 399.3 | 556.8 | 398.6 | 7.30 | 29.78 | 0.97 | 1.00 |

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**MDPI and ACS Style**

Buitrago, M.; Bertolesi, E.; Garzón-Roca, J.; Sagaseta, J.; Adam, J.M.
A Parametric Computational Study of RC Building Structures under Corner-Column Removal Situations. *Appl. Sci.* **2020**, *10*, 8911.
https://doi.org/10.3390/app10248911

**AMA Style**

Buitrago M, Bertolesi E, Garzón-Roca J, Sagaseta J, Adam JM.
A Parametric Computational Study of RC Building Structures under Corner-Column Removal Situations. *Applied Sciences*. 2020; 10(24):8911.
https://doi.org/10.3390/app10248911

**Chicago/Turabian Style**

Buitrago, Manuel, Elisa Bertolesi, Julio Garzón-Roca, Juan Sagaseta, and José M. Adam.
2020. "A Parametric Computational Study of RC Building Structures under Corner-Column Removal Situations" *Applied Sciences* 10, no. 24: 8911.
https://doi.org/10.3390/app10248911