To study the progressive collapse potential of RC frames with specially shaped columns following the loss of a ground corner column, an experiment of a one-third scale, two-bay x three-bay, two-story model frame was carried out as shown in Fig. (1) [14]. The dimensions of the model frame and measuring points are displayed in Fig. (2). The height of story was 1m, and the thickness of slab was designed as 50mm. C8 steel bars were orthogonally allocated inside the slab with the interval of 150mm. The section dimensions and reinforcements of columns and beams are displayed in Fig. (3). The reinforcement details of beam-column joints are provided in Fig. (4), in which longitudinal reinforcements of beams and columns passed through joints. The grade of steel bars in beams and columns is HRB400 for the main reinforcements and HPB300 for the stirrups. The grade of steel bars in slabs is HRB400. The material properties of steel are listed in Table 1, where f_y, f_u, δ and E_s are steel yield strength, ultimate strength, ratio of elongation and elastic modulus. The average compressive strength of concrete is 42.3 MPa, which is obtained from standard 150 mm cubes. The complete stress-strain curves of concrete are displayed in Fig. (5), which are obtained from 150 mm×150 mm×300 mm prismoids. The ground column A4 designed as the failed column was substituted by a steel pipe to support the superstructure during the concrete casting process. After the maintenance of the model frame, sandbags were uniformly placed on slabs with 2.8kN/m² load applied. The vertical load in succession was applied by a 1000 kN hydraulic jack installed on top of second-floor column A4 to perform a static test. Fig. (6) depicts the details of the loading equipment in the test. The load and displacement-controlled manner was applied in the loading process. Fig. (7) shows the relationship of the force applied by the jack versus the downward displacement on top of the failed corner column. It is indicated that the collapse process consists of four distinct phases: elastic phase (OA), elasto-plastic phase (AB), plastic-hinge phase (BD) and collapse phase (DE). It is worth noting that the measured force reached its peak value of 152.8 kN at Point C.

Fig. (1). Experimental set-up.

Fig. (2). First-floor plan (dimensions in mm).

Fig. (3). Section details (dimensions in mm).

(Note: A and C stand for steel strength grade of HPB300 and HRB400 in Chinese concrete code respectively. @ stands for stirrup space. There is a decrease in spacing of beam stirrup by half of the above near beam-column joint.)

Fig. (4). Reinforcement details of beam-column joints (dimensions in mm).

Fig. (5). Stress-strain curves of concrete.

Fig. (6). Loading equipment. Note: 1. Column; 2. Load-carrying beam; 3. Girder; 4. Box-girder; 5. Hydraulic jack; 6. Inserted steel plate.

Fig. (7). Relationship between vertical load and displacement of corner column.

In Segment BD, it is noted that most of the steel bars at the beam ends had yielded along with the crush of concrete in the compression zones indicating the formation of plastic hinges, which implies that the frame began to sustain loads depending on the beam resisting mechanism. Fig. (8) displays the schematic diagram of the beam resisting mechanism, in which P is the vertical load, M_u1 and M_u2 are the bending capacity of the plastic hinges at the beam ends. The measured force decreased gradually after the peak value at Point C and could not produce a second peak force like the phenomena of some other experiments [6, 9], which means that the catenary action did not occur in this test. It is usually believed that the catenary action could enhance the progressive collapse resistance of the structure after the failure of the beam resisting mechanism, in which all the steel bars in the beams connected to the failure joint convert to be in tension after the destruction of concrete. The tensions of steel bars increase gradually with the increasing of the rotation at the beam ends. At the large deformation stage, the structure could only depend on the tensile forces of steel bars in the beams to sustain the increasing load. To activate the catenary action, the arrangement of the steel bars in the beams should meet strict requirements, in which the steel bars need to be placed throughout the failure joint and firmly anchored at other beam-column joints. Otherwise, the steel bars will fail to be fully utilized due to steel slip.

Fig. (8). Beam resisting mechanism.

In this test, the steel bars in the beams connected to the failure column were not placed throughout the joint on top of the failure corner column. Then the joint dimensions of the frame with specially shaped columns are much smaller than those of rectangular column frame. As shown in Fig. (9), the concrete at beam-column joints experienced crushing and spalling with the exposure of steel bars resulting in weak anchorage effect for steel bars. In the combined function of the above two factors, the catenary action was not generated in the test due to the weak tie force in the steel bars of beams. The variations of steel strain in beams observed in the test reveal that the compressive steel bars in the beam resisting mechanism could not completely convert to be tensile with increasing of the rotation at the beam ends, which also indicates that there would not be any catenary action to bear greater loads in latter period. Therefore, the frame sustained loads mainly depending on the beam resisting mechanism after the loss of a ground corner column and the experimental parameters related to the beam resisting mechanism have significant effects on collapse-resistance performance of the frame.

Fig. (9). Experimental details.

OpenSees is employed to simulate the above test. The finite element model (Fig. 10) completely replicates the test, in which the properties and dimensions are identical to those in the test. The frame beam and column elements are simulated using the Force-Based Beam-Column Element from the OpenSees library [16]. Slabs are not simulated in the modeling, on which loads are applied to the adjoining beams by the rule of the two-way slab. For instance, the loads of the hatched areas are submitted to Beam A1-A2 and B2-C2 as uniformly distributed loads respectively in Fig. (11). As Sasani [17] adopted the beam elements with T-shaped or L-shaped sections to consider the enhancement of slabs on beams and achieved good agreement with test results, this kind of element section (Fig. 12) is also employed in the numerical analysis. It should be noted that the width of the effective flange is evaluated according to Chinese code for concrete structures [18].

(1)

where b_f' is the width of the effective flange, l₀ is the span of beams, b is the width of beams, s_n is the net distance between beams, and h_f' is the height of the flange. Substituting test parameters into Eqn. (1), b_f' = 400 is taken as the width of the effective flange.

Fig. (10). Finite-element model.

Fig. (11). Rule of the two-way slab (dimensions in mm).

Fig. (12). Beam section considering effect of slabs (dimensions in mm).

The element is composed of the fiber sections in which material properties of concrete and steel bars are respectively assigned to the fibers. As displayed in Fig. (13), Hysteretic Material from OpenSees is employed herein to model steel bars. Related steel parameters in the model are summarized in Table 2. Concrete02 Material derived from the modified Kent-Park model [19] is used to describe the stress-strain relationship of concrete as shown in Fig. (14). The section of beams and columns is divided into the protective layer concrete (Unconfined Concrete) and core concrete (Confined Concrete). The ultimate compressive strain of the protective layer concrete is taken as 0.004, while the ultimate compressive strength is taken as 0 in consideration of the spalling of concrete. Related concrete parameters in the model are summarized in Table 3. The concrete strength enhancement coefficient K needs to be taken into account in the core concrete. Its constitutive equations and related parameters are defined as follows:

For rising part of the curve:

(2)

For descending part of the curve:

(3)

For horizontal part of the curve:

(4)

where K and Z_m are, respectively, the concrete strength enhancement coefficient due to stirrup constraints and the strain softening slope. They are mathematically expressed as:

(5)

(6)

where f '_c is the compressive strength of concrete cylinder; f_yh is the yield strength of stirrup; ρ_s is the volume-stirrup ratio of beams or columns; h' is the width of core concrete; and s_h is the stirrup spacing. The ultimate compressive strain of concrete is derived as:

(7)

Fig. (13). Reinforcement steel constitutive model.

Fig. (14). Concrete constitutive model.

The comparison between numerical and experimental results is shown in Figs. (15 and 16). As shown in Fig. (15), good agreement is achieved in the initial stiffness and behavior in the later period of the collapse stage. At the plastic-hinge stage, the simulated stiffness degrades obviously resulting in a lower peak load. It can be seen that the simulated stiffness at the early period of the collapse stage almost remains unchanged, whereas the test stiffness decreases visibly. The simulated horizontal displacements of the frame are roughly consistent with experimental results, which are compared in Fig. (16). In general, the finite element model is accurate enough to simulate the collapse process, which could provide advices for further parametric analyses.

Fig. (15). Relationship between vertical load and displacement of corner column.

The height of beam section is set as 120, 180 and 240 mm (180 mm is chosen in the test) to conduct the analysis respectively. The influence of the height of beam section on collapse-resistance behavior is shown in Fig. (17). It is seen that the effect of the height of beam section on the collapse-resistance behavior of the frame is significant. The stiffness and collapse-resistance capacity remarkably increase with the increasing of the height of beam section. However, the plastic-hinge stage of the collapse process becomes shorter, which implies weaker rotation capacity of beam-column joints and ductility of the frame. The variations of horizontal displacements on top of the first-floor frame columns as a function of the vertical displacement of the failure joint are depicted in Fig. (18). With the increase of the height of beam section, the horizontal displacement increases at the measuring point 1-2 and decreases distinctly at the measuring point 1-3.

Fig. (17). Influence of height of beam section on collapse-resistance behavior.

Fig. (18). Influence of height of beam section on horizontal displacement.

The rebar ratio of the upper or lower steel bars of beam is set as 0.70%, 1.09% and 1.57% (1.09% is chosen in the test) respectively. As shown in Fig. (19a), the stiffness of the frame begins to increase in the elasto-plastic stage resulting in better collapse-resistance capacity with the increasing of the upper steel bars which serve as tensile reinforcements at the ends of beams. The lower steel bars adjacent to the failure column play an important role in the beam resisting mechanism as main tensile reinforcements. It is noted from Fig. (19b) that the growth of the lower steel bars enhances the collapse-resistance capacity significantly and lengthens the plastic-hinge stage which indicates better rotation capacity and ductility. The influences of the rebar ratio of beam on horizontal displacements of the frame are displayed in Figs. (20 and 21). Compared with the upper steel bars, the lower steel bars of beams play an important role to submit the vertical loads to other components efficiently, which improves the overall deformability of the frame. Thus, the horizontal displacements increase with the increasing of the rebar ratio of lower steel bars.

Fig. (19). Influence of rebar ratio of beam on collapse-resistance behavior.

Fig. (20). Influence of rebar ratio of upper steel bars on horizontal displacement.

Fig. (21). Influence of rebar ratio of lower steel bars on horizontal displacement.

The rebar ratio of slab is set as 0.38%, 0.67% and 1.05% (0.67% is chosen in the test) to conduct the analysis respectively. The steel reinforcements of slab in the range of the effective flange width are beneficial to the structural integrity and effectively participate in the beam resisting mechanism with the upper steel bars of beam at the beam ends. It can be seen from Fig. (22) that the effect of the rebar ratio of slab is similar with that of upper steel bars of beam, in which the stiffness and collapse-resistance capacity appear to increase with the increasing of the rebar ratio. The variations of horizontal displacements on top of the first-floor frame columns as a function of the vertical displacement of the failure joint are shown in Fig. (23). It is illustrated that the rebar ratio of slab has little impact on horizontal displacements of the frame, which is also similar with the results of the rebar ratio of upper steel bars of beam.

Fig. (22). Influence of rebar ratio of slab on collapse-resistance behavior.

Fig. (23). Influence of rebar ratio of slab on horizontal displacement.

The limb length of specially shaped column is set as 160, 240 and 320 mm (240 mm is chosen in the test) respectively. As displayed in Fig. (24), the lengthening of the limb of specially shaped column enhances the horizontal restraining stiffness of the structure, which increases the initial stiffness but has no influence on the collapse-resistance capacity. It could be concluded that the collapse-resistance capacity of the frame with specially shaped columns following the removal of a ground corner column is mainly related to the characteristics of beams and slabs. Specially shaped columns which serve as constraint elements at the beam ends only have influence on the lateral stiffness of the frame. The influence of the limb length of specially shaped column on horizontal displacements of the frame is depicted in Fig. (25). The lengthening of the limb of specially shaped column makes shorter beam spans and stronger restraints at beam-column joints, which results in reduced out-of-plane torsion of beam sections in the collapse process of the frame. Thus, the horizontal displacements at the two measuring points reduce significantly with the increasing of the limb length of specially shaped column. In this study, the thickness of slab and rebar ratio of specially shaped column are also analyzed respectively. It is indicated that they have little influence on the collapse-resistance behavior of the frame in rational value domain.

Fig. (24). Influence of limb length of specially shaped column on collapse-resistance behavior.

Fig. (25). Influence of limb length of specially shaped column on horizontal displacement.