Progressive Collapse Resistance of RC Frames under a Side Column Removal Scenario: The Mechanism Explained

2.1 Experimental Program

One of the most critical internal column loss scenarios is loss of a penultimate-external column on the ground floor of a structure. First, with lack of external lateral supports, the development of catenary action in beams and tensile membrane action in slabs relies solely on the perimeter columns and the perimeter compressive ring formed within the deflected slabs, as illustrated in Fig. 1a. Secondly, with large deformations, it is possible that the sum of catenary forces and tensile membrane forces may pull the perimeter columns at the ground floor inwards, triggering a progressive collapse, as shown in Fig. 1b. Therefore, this research focused on the behavior of a two-span, two-bay, single-story RC frame subjected to side column loss.

A four-span, eight-story and four-bay RC frame structure was designed considering the concrete design code and seismic design code of China (GB50010-2010 2010; GB50011-2010 2010). It should be noted that the Chinese code is generally similar to Eurocode 2 although the load and resistance factors are slightly different. It is expected that a building designed in accordance with Eurocode 2 will possess a marginally higher factor of safety given the differences in load and resistance factors between the two codes. Table 1 summarizes the details of the prototype frame. A one-third scale model of a segment of the ground story of the original frame was made to be used in the collapse experiments. The floor height of the model frame was 1100 mm. The floor plan details of the reinforcement and cross-sectional dimensions that were used in the model frame are shown in Fig. 2. The middle column on the A-axis, namely, the side column of the model frame was removed when the model was built in the laboratory, but the corresponding frame joint was intact. In fact, the support of the removed side column were considered, namely, the prototype frame was intact when it was designed according to design codes.

Items	Floor height	Bay span	Depth span	Beam size		Column size
				Depth		Width	Depth	Width
	3300 mm	3900 mm	5400 mm	Bay direction	Depth direction
				350 mm	450 mm	200 mm	400 mm	400 mm
Items	Floor loads		Seismic fortification intensity			Materials
	Live (other/roof)	Constant (other/roof)	7		Concrete	Longitudinal steel bars		Stirrups
	2.5/0.5 kN/m2	6.0/7.5 kN/m2			C30	HRB335		HPB300

C30 of concrete strength grade denote the characteristic value of compression strength for cube dimensions of 150 × 150 × 150 mm is 30 MPa (1 MPa = 0.145 ksi). HRB335 of steel bar types denote the characteristic value of strength for hot-rolled ribbed steel bar is 335 MPa; HPB300 of steel bar types denote the characteristic value of strength for hot-rolled plain steel bar is 300 MPa.

Yi et al. (2008) constructed a middle column of a planar frame by stacking two mechanical jacks and a load cell, and investigated the structural response before and after the middle column was removed. In the initial stage of the experiment, we applied a constant vertical load to the top of the removed column area and the model load was achieved by the unloading of the mechanical jacks. In our experiment, no constant load was placed on the top of the area where the column was removed at the beginning of the experiment. The step-by-step loading process was initiated by a MTS servo actuator on the top of the removed side column. In fact, the results of the two kinds of loading mode are almost equivalent, but the latter is less complicated. Therefore, the second loading mode was adopted in the literature (Sadek et al. 2011) and our experiment. Actually, a static experimental progressive collapse evaluation is presented in the paper. However, a typical building structure exhibits a highly nonlinear dynamic response under a sudden column loss scenario. Based on the energy conservation principle, Izzuddin et al. (2008) proposed a simplified method by which the simplified dynamic response and progressive collapse resistance can be derived from the static progressive collapse response.

Figure 3 depicts the details of the instrumentation layout and test setup. The load was placed on the top of the area of the removed side column with a MTS servo actuator, as shown in Fig. 3. A rate of 3 mm/min was used to apply the load under displacement control. Vertical and horizontal displacement transducers and steel stain gauges were used in the model frame test. Downward displacements of the top of the removed side column were imposed until failure. Frame collapse is defined in this study as the rupture of tension steel bars in the floor beams; in fact, the progressive collapse-resistance capacity reached the peak at the same time.

2.2 Finite Element Modeling

Due to the limitations of experimental conditions and cost, the structural response information obtained in the experiment was not enough. A computational study of the response of the structural progressive collapse described in this paper was carried out using explicit time integration in LS-DYNA. Both geometrical and material nonlinearities were accounted for in the analysis and this included fracture with element erosion. Concrete was represented in the model by finely meshed solid elements, and beam elements represented reinforcing bars. The reinforcing bar steel properties were modeled with a piecewise-linear plasticity model (Material 24 in LS-DYNA) and stress–strain curves were based on the tensile test data. The measured values of mechanical properties of steel bars and concrete are shown in Table 2. The engineering stress–strain curves, however, were transformed into true stress–strain curves. A continuous surface cap model was used for the concrete material (Material 159 in LS-DYNA). Our model captured the dominant characteristics of concrete responses, including softening caused by damage accumulation and confinement effects. Element erosion was used to model fractures, with the values for erosion strain for an element size calibrated to the failure strain values from the tensile tests. Solid elements were not eroded in this study to avoid excessive loss of stiffness. Since this study focus on the resistance of the whole structure rather than the simulation of cracking, cracks are not modeled explicitly in order to simplify the model and reduce the computational cost. But concrete cracking can be reflected by contours of the damage index computed by the concrete material model. The bottom of all columns except the removal side column was assumed to be fixed. In the computational study, the loading mode was the same as the experiment. Due to the symmetry, only one-half of the whole model was established in the computation, and the overview of the model is shown in Fig. 4.

Material		Diameter measurement, mm	Young’s modulus, MPa	Yield strength, MPa	Ultimate strength, MPa	Rupture strain, δ 10
						Necking zone	Outside necking zone
Steel bars	Ф 8	8.0	2.0 × 105	347.0	488.6	25.3 %	16.1 %
	Ф 6	6.4	2.0 × 105	283.2	443.8	31.0 %	20.6 %
	Ф 3	3.5	/	331.0	387.6	/	/
	Young’s modulus, MPa			Compression strength of concrete prism, MPa
Concrete	3.05 × 105			36.8

Defining a one-dimensional contact interface (Contact_1d in LS-DYNA) was used to model bond-slip behavior between the solid elements that represented concrete as well as the beam elements that represented reinforcing bars. Two sets of nodes were required to define the contact, with the concrete nodes specified as master nodes and the reinforcement nodes specified as slave nodes. The parameters of the bond-slip model were selected with reference (Shi and Li 2009). Bond slip was not considered for column longitudinal and transverse reinforcement. The beam elements that represented the reinforcing bars were constrained to be within the solid elements with the CONSTRAINED_LAGRANCE_IN_SOLID card.

3.1 Elastic Stage

As show in Fig. 5, Section OA can be considered as the elastic stage with cracking of frame beams observed at State A, and the displacement of the removed column was less than 5 mm in this stage. It can be seen from Fig. 7 that the bottom reinforcement was in tension and the stress was almost linearly increased with increases in the vertical displacement of the removed column in the longitudinal and transverse beams at this stage. However, the values of the stress were lower than the corresponding yield values. The top reinforcement was in compression and the stress was very small in the longitudinal and transverse beams at this stage. As shown in Fig. 8, the stress of the slab bottom reinforcement was very small at this stage. The above analysis indicated that the frame beams and slabs were almost in the elastic state.

3.2 Elastoplastic Stage

In Fig. 5, Section AB is the elastoplastic stage, and the displacement of the removed column was about 28 mm at State B. In this stage, the vertical load no longer increased linearly with increasing vertical displacement of the removed column. As shown in Fig. 7, the bottom reinforcement had entered the yield state in this stage. From the stress of the slab bottom reinforcement, it was found that the stress of a part of the reinforcement near the removed column obviously increased with increases in the vertical displacement of the removed column, but the stress of the others far from the removed column was still very small at this stage. Based on the experimental observation, it was found that the concrete of frame slabs had obviously cracked, and plastic hinges in the frame beam ends near the removed column had formed in this stage.

3.3 Plastic Stage

In Fig. 5, Section BC is the plastic stage, and the displacement of the removed column was about 68 mm at State C. displacement of the column that was removed was approximately 68 mm in State C. The increasing rate of the vertical load in this stage with increasing vertical displacement of the removed column decreased significantly, and the deformations were dominated by plastic rotations of the frame beams. From the longitudinal reinforcement stress at the frame beam ends near the removed column, it was observed that the top reinforcement changed to tension from compression in the longitudinal beams in this stage. However, due to the lack of lateral support or constraint, the top reinforcement in the transverse beam was still under the compression state at this stage. As shown in Fig. 8, the stress of a part of the reinforcement near the removed column significantly increased with increases in the vertical displacement of the removed column, and the maximum of the stress had exceeded 200 MPa. However, the greater the distance from the removed column, the smaller the stress became.

3.4 Composite Stage of Catenary and Tensile Membrane

In Fig. 5, Section CD is the composite stage of thecatenary and tensile membrane. After the plastic stage Section BC, it can be observed that the tension cracks in concrete penetrated through the compression zones in frame beams and slabs in the experiment, which indicated the moments of resistance at the plastic hinges in the frame beams and the plastic hinge lines in the frame slabs can be ignored at this stage. Figure 7 shows that the bottom longitudinal reinforcement in the longitudinal frame beams had entered the strain-hardening range, and the top longitudinal reinforcement in the longitudinal frame beams was fully in tension at this stage and had entered the yield state at State D. However, in the transverse frame beam and the transverse direction of the frame slabs, due to the lack of lateral support or constraint, the beam and slabs worked approximately as a cantilever beam and slabs. Therefore, at this stage, the transverse frame beam and transverse direction of the frame slabs almost entirely failed, as shown in Fig. 9. Figure 7b shows that the top reinforcement in the transverse beam was still under the compression state and the stress fluctuated obviously at this stage. Also, after the displacement of the removed column more than 100 mm, the tensile stress of the bottom reinforcement in the transverse beam was not reliable. A similar phenomenon was found in the transverse direction of the frame slabs. The above analysis indicated that the catenary mechanism and the tensile membrane mechanism can only be formed in the longitudinal frame beams and the longitudinal direction of frame slabs, respectively. Figure 8 shows that a part of the longitudinal reinforcement near the removed column had entered the yield state at State D. At State D, the structure attained a maximum vertical load of 60.7 kN at a vertical displacement of 345 mm, at which point one of the bottom steel bars of frame beams adjacent to the removed column on A-axis ruptured, as shown in Fig. 10. The experimental observation was consistent with the numerical analysis results, as shown in Fig. 7.

4.1 Catenary Mechanism of Beams

Figure 6 shows the failure pattern of the frame model. Due to the entire failure of the transverse frame beam and the transverse direction of the frame slabs in the failure zone, the effect of the transverse beam and the transverse direction of the slabs can be ignored, while calculating the progressive collapse resistance of the whole structure at a collapse limit state.

Based on the experimental observation, it can be found that the axes of the longitudinal frame beams are almost still straight at the collapse limit state. Figure 12 shows the vertical displacement of different positions on the longitudinal frame beams at the collapse limit state, which is obtained from the numerical results and is consistent with the experimental observation. Therefore, the model of the progressive collapse resistance of RC beams can be proposed as shown in Fig. 13. The progressive collapse resistance of frame beams was obtained as (Hou and Yang 2014)

$$ P_{\text{ub}} = \frac{{\left( {L_{1} + L_{2} } \right)v_{\text{u}} }}{{L_{1} L_{2} }} \cdot \left( {A_{\text{th}} } \right)f_{\text{y}} $$

(1)

where \( L_{1} \) and \( L_{2} \) are the spans of beam 1 and beam 2, respectively, \( v_{u} \) is the vertical displacement of the removed column, A _th is the area of steel bars through whole span, and \( f_{y} \) is the yield stress of the steel bars in frame beams. The model was originally proposed by Li et al. (2011), and the reliability was verified by Hou and Yang (2014).

4.2 Tensile Membrane of Slabs

Based on the failure phenomenon as shown in Fig. 6, the internal area of the frame slabs surrounded by the negative moment yield lines is viewed as an analysis object, and its boundaries are assumed to be rectangular. Thus, the model of the progressive collapse resistance of frame slabs is proposed as shown in Fig. 14. Based on the analysis of the progressive collapse process, just the effect of the longitudinal direction of the frame slabs on the progressive collapse resistance of the whole structure was considered, and the moments of resistance at the plastic hinge lines in the frame slabs can be ignored at the collapse limit state. Therefore, the boundaries of the analysis object can only bear a pulling force, as shown in Fig. 14. Let \( l_{y} \) (the side lengths of slabs ① and ② in the direction of the Y axis) be equal to the projection lengths of the positive moment yield lines in the corresponding position, it can be noted that the load-carrying capacity of the curved boundaries in Fig. 6 and the rectangular boundaries in Fig. 14 are equivalent.

In Fig. 6, for slab BOK, based on the vertical displacement of different positions on the longitudinal frame beams at the collapse limit state (Fig. 12) and the compatibility of deformation between frame beams and slabs, it can be found that the slab edge BK is still straight at the collapse limit state. Figure 15 displays the vertical displacement of different positions in the transverse span centers of slabs at the collapse limit state, which was obtained from the numerical results. It was observed that there was relatively little change in the vertical displacement in the Section OA1 range compared to Section OA2. Therefore, the line section OA1 can approximately be assumed to be straight. Thus, we assumed that the slabs GJK and HIK (Fig. 14), surrounded by positive and negative moment yield lines and outer edges of slabs, were still in two different planes at the collapse limit state, respectively.

4.3 Validation

Based on analysis and experimental results, it is noted that the limit vertical displacement of the frame structure is controlled by frame beams on the A-axis. Based on the literature (Hou and Yang 2014), the calculated value of the limit vertical displacement of the removed column (\( v_{u} \)) is 356.7 mm. Substituting the value of \( v_{u} \), the geometric dimensions and the properties of steel bars of the frame beams on the A-axis into Eq. (1), the progressive collapse resistance of frame beams (\( P_{\text{ub}} \)) can be calculated 41.5 kN.

Based on the deformation compatibility condition of the frame beams and slabs, the limit vertical displacement of Point K (\( v \)) is equal to the limit vertical displacement of the removed column (\( v_{u} \)). Therefore, the value of \( v \) should be 356.7 mm. Substituting the value of \( v \), the geometric dimensions and the properties of steel bars of the frame slabs into Eqs. (2), (3) and (4), the progressive collapse resistance of frame slabs is 14.9 kN.

By using the principle of superposition, the progressive collapse resistance of the whole frame structure is 56.4 kN. The calculated value is 7.1 % smaller than the experimental result. Portions of the steel bars have entered the hardening stage in the collapse limit state, but steel hardening is not considered in the model. Thus, the progressive collapse resistance obtained from the model is somewhat conservative. Moreover, it can be found that the progressive collapse resistance of frame slabs is 26.4 % of the progressive collapse resistance of the whole frame structure.