# A Simplified Method for Assessing the Response of RC Frame Structures to Sudden Column Removal

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## Abstract

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## 1. Introduction

## 2. Modelling Sudden Column Removal on Frame structures

#### 2.1. Maximum Dynamic Response of the System

- a linear elastic phase with tangent stiffness ${k}_{e}$. This phase ends at a yielding force ${F}_{y}$, corresponding to a displacement ${u}_{y}={F}_{y}/{k}_{e}$;
- a hardening phase with plasticity. The tangent stiffness of this phase has tangent stiffness ${k}_{p}$. The system increases its displacement up to ${u}_{+}$, when its velocity is null;
- the unloading follows a linear elastic law with tangent stiffness equal to ${k}_{e}$.

#### 2.2. Dimensional Analysis for the Dynamic Phase of the Sudden Damage

#### 2.3. Dynamic Amplification Factor (DAF)

## 3. Proposed Method for the Simplified Analysis

- 1.
- the capacity and the ductility properties of the beams are assessed and a bending moment-curvature (BM-$\chi $) relationship is obtained. The bending moment at yielding, ${M}_{y}$, and the ultimate bending moment, ${M}_{u}$, and the corresponding kinematic quantities, ${\chi}_{y}$ and ${\chi}_{u}$, are considered as representative for the behaviour of the concrete cross-section;
- 2.
- the axial force, $\mathrm{N}$, acting in the removed column is computed considering the loads applied on its tributary area, that is, half of the dead and live loads each beam plus the weight of the upper columns.
- 3.
- the mechanism that originates following column removal is studied and the upper bound theorem of the plastic analysis is applied. With reference to Figure 7b, the replacement axial force $\mathrm{N}$ previously computed is inserted in the mechanism as an upward force located in A; bending moments m${}^{*}$ are applied in the plastic hinges. A downward variable force, that is, column removal force, $\lambda 1$ is inserted in A. The value of $\lambda $ for which the virtual work principle (VWP) holds defines an upper bound collapse load, $\lambda \left(\right)open="("\; close=")">{m}^{*}$ [37];
- 4.
- the chord rotation of beams ends, ${\theta}^{*}$, is computed and an approximate downward displacement of node A is obtained as ${\delta}^{*}={\theta}^{*}\ell $;
- 5.
- the compliance curve is built repeating points 3. and 4. considering the yielding and ultimate capacities of the beams. The yielding force, ${F}_{y}$, is computed through the load multiplier $\lambda \left(\right)open="("\; close=")">{M}_{y}$ and the corresponding displacement, ${\delta}_{y}$, through the chord rotation at yielding, ${\theta}_{y}$. A similar approach is required for the ultimate point, that is, for ${F}_{u}$ and ${\delta}_{u}$. The tangent stiffnesses in the elastic and in the plastic with hardening part of the curve are evaluated as$${k}_{e}=\frac{{F}_{y}}{{\delta}_{y}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{k}_{p}=\frac{{F}_{u}-{F}_{y}}{{\delta}_{u}-{\delta}_{y}};$$
- 6.
- the displacement due to a quasi-static column removal process is evaluated as$${u}_{st}=\frac{\mathrm{N}}{{k}_{e}}.$$The dimensionless quantities ${\mathrm{\Pi}}_{1}$, ${\mathrm{\Pi}}_{2}$ and ${\mathrm{\Pi}}_{3}$ are computed and the dynamic amplification factor is evaluated following the steps reported in Appendix A;
- 7.
- the maximum dynamic displacement is computed as ${u}_{dyn}=\mathrm{D}\mathrm{A}\mathrm{F}\phantom{\rule{4pt}{0ex}}{u}_{st}$.

## 4. An Example

- 1.
- The analytical model proposed by Park and Paulay [39] was adopted for the evaluation of the bending moments and the corresponding curvatures. It results a bending moment at yielding equal to ${M}_{y}=94.5$ kNm and a curvature equal to ${\chi}_{y}=1.18\times {10}^{-2}$ m${}^{-1}$. Referring to the ultimate capacity, it follows that at the ultimate concrete compressive strain, ${\epsilon}_{cu}=0.0035$, the compression steel is not yielding. The ultimate bending moment and curvature are ${M}_{u}=102.8$ kNm and ${\chi}_{u}=8.16\times {10}^{-2}$ m${}^{-1}$, respectively.
- 2.
- The axial force in the selected column was evaluated considering the tributary area of the column, that is, half length of the beam. The force results from the following expression$$\mathrm{N}={n}_{f}\left(\right)open="("\; close=")">\mathrm{D}\mathrm{L}+\mathrm{L}\mathrm{L}{n}_{f}+{g}_{c}\left(\right)open="("\; close=")">{n}_{f}-1$$
- 3.
- The upper bound theorem of the plastic analysis was applied and the load multiplier that defines the collapse load $\lambda \left(\right)open="("\; close=")">{m}^{*}$ is determined as$$\lambda \left(\right)open="("\; close=")">{m}^{*}\ell +{g}_{b}\left(\right)open="("\; close=")">\ell -{w}_{c}$$Substituting the values of the yielding and the ultimate bending moments it results $\lambda \left(\right)open="("\; close=")">{M}_{y}$ and $\lambda \left(\right)open="("\; close=")">{M}_{u}$, respectively.
- 4.
- The chord rotation of beams end were computed through the analytical models proposed in the Italian Building Code [40,41] for an equivalent cantilever: the elastic rotation is due to an elastic curvature increasing from zero (at the free end of the cantilever) to ${\chi}_{y}$ (at the opposite end); the plastic curvature, that is, ${\chi}_{u}-{\chi}_{y}$, is kept constant on the plastic hinge length [42]. According to the aforementioned approach, chord rotation at yielding is ${\theta}_{y}=0.01112$ radians, while ultimate chord rotation is equal to ${\theta}_{u}=0.03117$ radians. The corresponding downward displacements are ${\delta}_{y}=44$ mm and ${\delta}_{u}=125$ mm.
- 5.
- The bilinear compliance curve is derived from the data previously obtained. It results ${F}_{y}=283.5$ kN and ${F}_{u}=308.5$ kN. Thus, the tangent stiffnesses in the elastic and the hardening phase are computed through Equations (7), resulting in ${k}_{e}=$6,443,181 N/m and ${k}_{p}=$311,482 N/m.
- 6.
- the displacement due to a quasi-static column removal process was computed through Equation (8) as ${u}_{st}=28.4$ mm. The displacement at yielding to be inserted in the dimensionless quantities computed at the following point is ${u}_{y}={\delta}_{y}=44$ mm.
- 7.
- The dimensionless quantities were computed according to Section 2.2 as$${\mathrm{\Pi}}_{2}=\frac{N}{{F}_{y}}=0.640\phantom{\rule{2.em}{0ex}}{\mathrm{\Pi}}_{3}=\frac{{k}_{e}}{{k}_{p}}=0.049$$
- 8.
- The dynamic amplification factor is$$\mathrm{D}\mathrm{A}\mathrm{F}=\frac{1.379}{0.640}=2.155,$$

#### Special Case: Side Column Removal

## 5. Comparison with Numerical Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Solution of the Nonlinear SDOF System

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**Figure 1.**Scheme for the evaluation of the damage on a frame. The central bottom column is the removed element. (

**a**) First, the internal forces in the element are evaluated. (

**b**) Then, the element is removed and a nodal force, equal but opposite to the internal force is applied to the top node. (

**c**) Finally, an opposite force is suddenly applied to the system to simulate sudden column removal. In (

**d**) a sketch of the equivalent SDOF system is reported.

**Figure 2.**Sketch of the single degree of freedom (SDOF) system. The nonlinear device is denoted with letter (d). On the right-hand side the dynamic forces acting on the mass are reported: $m\ddot{u}$ is the inertia force, ${F}_{R}$ is the restoring force of the device, which depends on its elongation u, f is the applied force.

**Figure 3.**Elastic-plastic law of the connecting device. Explanations of the letters in the plot are reported in the text.

**Figure 4.**Loading force - displacement curves for various values of ${\mathrm{\Pi}}_{3}$. Continuous lines relates to the dynamic analysis, that is, ${\mathrm{\Pi}}_{1}^{dyn}$, while the dashed lines relate to the quasi-static loading, that is, ${\mathrm{\Pi}}_{1}^{st}$.

**Figure 5.**Dynamic amplification factor for various normalised loading and stiffness ratios. Squares represents the dynamic amplification factor (DAF) obtained through the numerical simulations.

**Figure 6.**Dynamic amplification factor for various stiffness ratios ${\mathrm{\Pi}}_{3}$ in the range $\left(\right)$ for ${\mathrm{\Pi}}_{2}=1$. The grey line corresponds to DAF=2. Squares represents the DAF obtained through the numerical simulations.

**Figure 7.**A sketch of the plastic hinges that originates when the column is removed is sketched in (

**a**). The mechanism can be described with a unique coordinate, for example, the rotation $\phi $. The forces applied on the structure onto which the virtual work principle (VWP) is applied is depicted in (

**b**).

**Figure 8.**A sketch of the test frame onto which the simplified method is applied. The removed middle column is reported in red, the removed side column in violet. The corresponding control nodes are marked with letters A${}_{1}$ and A${}_{2}$, respectively. The geometry and the reinforcement layouts of beams and columns are depicted on the right-hand side.

**Figure 9.**Compliance curves for the four reinforcement amounts in the beams and column removal position. Solid lines (–) correspond to the FE simulations, while dashed lines (- -) to the simplified bilinear compliance law. The downward displacement is limited to 200 mm in main plot, while the secondary axes (that have the same labels as the primary ones) depict the full pushdown analysis up to 1400 mm.

**Figure 10.**Time-histories of the downward displacement of the control point under the sudden column removal process (for both middle and side elements). Column removal starts at 0.1 s and lasts 0.01 s. The dashed lines illustrate the dynamic displacements obtained with the simplified method herein presented.

**Table 1.**Comparison between numerical and simplified analysis by varying the amount of reinforcement in the beams.

Beams | Removed | ${\mathit{u}}_{\mathit{st},\mathit{num}}$ | ${\mathit{u}}_{\mathit{dyn},\mathit{num}}$ | DAF${}_{\mathit{num}}$ | ${\mathit{u}}_{\mathit{dyn},\mathit{simpl}}$ | Error |
---|---|---|---|---|---|---|

Reinforcement | Column | (mm) | (mm) | (mm) | ||

3 + 3$\varphi $16 | Middle | 32.4 | 124.4 | 3.84 | 122.5 | −1.5% |

3 + 3$\varphi $20 | Middle | 23.3 | 47.7 | 2.04 | 49.8 | +4.4% |

4 + 4$\varphi $16 | Middle | 25.8 | 57.6 | 2.23 | 61.3 | +6.4% |

3 + 3$\varphi $20 | Side | 39.2 | 63.2 | 1.61 | 58.7 | −7.7% |

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**MDPI and ACS Style**

Biagi, V.D.; Kiakojouri, F.; Chiaia, B.; Sheidaii, M.R.
A Simplified Method for Assessing the Response of RC Frame Structures to Sudden Column Removal. *Appl. Sci.* **2020**, *10*, 3081.
https://doi.org/10.3390/app10093081

**AMA Style**

Biagi VD, Kiakojouri F, Chiaia B, Sheidaii MR.
A Simplified Method for Assessing the Response of RC Frame Structures to Sudden Column Removal. *Applied Sciences*. 2020; 10(9):3081.
https://doi.org/10.3390/app10093081

**Chicago/Turabian Style**

Biagi, Valerio De, Foad Kiakojouri, Bernardino Chiaia, and Mohammad Reza Sheidaii.
2020. "A Simplified Method for Assessing the Response of RC Frame Structures to Sudden Column Removal" *Applied Sciences* 10, no. 9: 3081.
https://doi.org/10.3390/app10093081