Dynamic increase factor for RC frame with specially shaped columns against progressive collapse

3.1. Element type

The finite-element model of the RC frame with specially shaped columns is established by OpenSees. The beam and column elements are simulated using the Force-Based Beam-Column Element from the OpenSees library [12]. By determining the resistance and stiffness matrix of control sections, the resistance and tangent stiffness matrix of the entire element is integrated along the length in the light of Gauss integration. Slabs are not simulated in the modeling on which loads are applied to the adjoining beams by the rule of the two-way slab. To consider the enhancement of slabs on beams, the beam elements with T-shaped or L-shaped sections are employed in the numerical analysis [13]. It should be noted that the width of the effective flange is evaluated according to Chinese code for concrete structures [14].

3.2. Material properties and constitutive models

The element is composed of fiber sections in which material properties of concrete and steel bars are respectively assigned to the fibers. As displayed in Fig. 2, Hysteretic Material from OpenSees is employed herein to model steel bars. Concrete02 Material derived from the modified Kent-Park model [15] is used to describe the stress-strain relationship of concrete, as shown in Fig. 3. The section of beams and columns is divided into the protective layer concrete and core 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. 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:

For descending part of the curve:

For horizontal part of the curve:

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:

where $\sigma$ and $\epsilon$ are the concrete stress and strain; $f_c^{\text{'}}$ is the compressive strength of concrete cylinder; $f_{yh}$ is the yield strength of stirrup; ${\rho }_s$ is the volume-stirrup ratio of beams or columns; $h\mathrm{\text{'}}$ is the width of core concrete; and $s_h$ is the stirrup spacing. The ultimate compressive strain of the core concrete is derived as:

Fig. 2Rebar constitutive model

Fig. 3Concrete constitutive model

3.3. Model validation

To investigate the progressive collapse potential of RC frames with specially shaped columns subjected to 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 (Fig. 4) [16]. The dimensions of the model frame and measuring points are displayed in Fig. 5. The height of story was 1m, and the thickness of slab was designed as 50mm. The section dimensions and reinforcements of columns and beams are displayed in Fig. 6. The material properties of steel are listed in Table 1, where $f_y$, $f_u$, $\delta$ 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. Sandbags were uniformly placed on slabs with 2.8 kN/m² load applied. The ground Column A4 designed as the failed column was substituted by a steel pipe to support the superstructure during concrete casting process and was instantaneously removed to simulate the loss of the corner column. After the dynamic test, vertical load was applied by a 1000 kN hydraulic jack installed on top of second-floor Column A4 to perform a static test. Fig. 7 depicts the details of the loading equipment in the static test. Based on OpenSees, a finite-element model completely replicates the test, in which the properties and dimensions are identical to those in the test. The comparison between numerical and experimental results is shown in Fig. 8. It can be seen that the finite element model is accurate enough to conduct the progressive collapse analysis for RC frame with specially shaped columns, which could provide theoretical basis for the following analyses.

Fig. 4Experimental set-up

Fig. 5First-floor plan (dimensions in mm)

Fig. 6Section 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.)

a) L-shaped column

b) T-shaped column

c) Cross-shaped column

d) Beam

Fig. 7Loading equipment. (Note: 1. Column; 2. Load-carrying beam; 3. Girder; 4. Box-girder; 5. Hydraulic jack; 6. Steel inserted plate)

4.1. Building characteristics

A seven-bay $x$ three-bay, six-story RC frame structure with specially shaped columns is designed in accordance with the requirements of Chinese code for seismic design of buildings [17] and specification for concrete structures with specially shaped columns [18]. The layout of the frame is displayed in Fig. 9. The height of story is 3.3 m, and the thickness of slab is 120 mm. The concrete strength grade of beams, floors and columns is C30, which is estimated at approximately 24 MPa for a standard concrete cylinder. The concrete protective layer thickness is 30 mm. The yield strength is 400 MPa for the main reinforcements and 300 MPa for the stirrups. The yield strength is 300 MPa for the reinforcements in slabs. The dead loads are 7.5 kN/m² for roof and 5 kN/m² for other floors. The live loads are 0.5 kN/m² for roof and 2 kN/m² for other floors. The reinforcements of beams and columns derived from PKPM, which is a widely used architectural engineering design CAD software, are presented in Fig. 10.

Fig. 8Comparison between numerical and experimental results

a) Strain for Steel 1 in dynamic test

b) Strain for Steel 2 in dynamic test

c) Vertical load vs. displacement of corner column curve in static test

4.2. Analysis method

In this paper, the corner column A1, inner column B3 and side column A3 are respectively designed as the initial failure column on the basis of the alternate path analysis method to perform the ND and NS analysis by employing OpenSess. The corresponding element type and constitutive models have been discussed in Section 3, which is verified to be accurate enough. The load combination in the ND analysis can be described as:

where $DL$ and $LL$ are dead load and live load, respectively [8]. The steps required in performing the ND analysis are: 1) build the finite element model which is shown in Fig. 9; 2) perform static analysis to estimate the internal force $P_0$ at the top of the removed column; 3) perform modal analysis on the residual structure; 4) as shown in Fig. 11, impose the internal force $P_0$ on the residual structure in the opposite direction to make the analysis model static equivalent to the primary model, apply the downward load $P$ depicted in Fig. 12 at the top of the removed column and perform the ND analysis. In Fig. 12, $t_f$ is the failure time which is as 0.1 times as the fundamental period of the residual structure [4].

Fig. 9Model layout (units: mm)

Fig. 10Reinforcement details of beams and columns (units: mm)

a) Beam B1

b) Beam B2

c) Beam B3

d) L-shaped column

e) T-shaped column

In the NS analysis, the failure column element should be removed from the model firstly and a dynamic multiplier $\alpha$ should be used to amplify the load combination on the affected bays adjacent to the failure column. The affected bays include the floor enclosed by axes 1, 2, A and B for the corner column removal scenario, axes 2, 4, A and C for the inner column removal scenario (Fig. 13) and axes 2, 4, A and B for the side column removal scenario. The load combination applied on the affected bays above the failure column can be expressed as:

While the load combination applied on other bays can be obtained according to Eq. (9), in which the dynamic multiplier is not considered.

Fig. 11Schematic diagram in ND analysis

a) Equivalent to primary model

b) Simulate to remove failure column

Fig. 12Loading function

Fig. 13Affected bays for inner column removal scenario

4.3. Analysis parameters

The Rayleigh damping of the frame should be taken into account in the ND analysis. The first two frequencies of the residual structure derived from modal analysis can be employed to calculate the mass damping coefficient ${\alpha }_1$ and the stiffness damping coefficient ${\beta }_1$ as follows [19]:

where ${\omega }_1$ and ${\omega }_2$ are the first two natural frequencies of the residual structure derived from modal analysis; and the damping ratio of concrete $\xi$ is 0.05.

As the two-way shear and torsion are not coupled in the fiber section of OpenSees, the uniaxial elastic constitutive for shear and torsion need to be defined as a supplement. The shear and torsional stiffness of beam and column sections are displayed in Table 2, in which $A_{sy}$ and $A_{sz}$ are the shear area along $y$- and $z$-axis direction in the local coordinate of sections, $T$ is the torsion constant, $k_{sy}$ and $k_{sz}$ which are evaluated according to Eqs. (13-14) are the shear stiffness along $y$- and $z$-axis direction, and $k_t$ derived from Eq. (15) is the torsional stiffness [20]. The shear modulus of concrete $G_c$ is 12413 MPa.

5.1. Influence of the location of failure column

The displacement time histories on top of the failure column for Column A1 removal scenario are depicted in Fig. 14, in which the dashed lines denote the vertical displacements for different $\alpha$ in the NS analysis. It can be seen that the vertical displacement increases gradually with the increasing of the situation of the failure column and surges to the peak value at the top floor. It is observed that the displacement in the NS analysis could approximate the maximum value in the ND analysis when $\alpha$ reaches 1.62-1.70. The relationships between axial forces of adjacent columns and $\alpha$ are presented in Fig. 15, in which the dashed lines denote the peak axial forces in the ND analysis and corresponding $\alpha$. The axial forces in the NS analysis increase linearly with the increasing of $\alpha$. It is noted that the $\alpha$ corresponding to peak axial forces in the ND analysis are all less than 1.60, which verifies the safety of the $\alpha$ evaluated previously. It is suggested that the NS analysis for the frame with specially shaped columns subjected to corner column loss has enough safety reserve at a DIF of 1.70.

Table 3 summarizes the results for Column B3 removal scenario in detail, in which the displacement of the NS analysis could approximate the maximum value of the ND analysis when $\alpha$ approaches 1.17-1.21. The peak axial forces of adjacent columns in the ND analysis are all less than those in the NS analysis, except for Column C3 with a slight deviation of 2 %. In general, the DIF of 1.21 is adequate to be used in the NS analysis for the frame with specially shaped columns subjected to inner column loss. The results for Column A3 removal scenario are displayed in Table 4. It can be seen that the displacement in the NS analysis approximates the maximum value in the ND analysis at $\alpha$ of 1.26-1.32. It is clear that the peak axial forces of adjacent columns in the ND analysis are all less than those in the NS analysis. Consequently, the DIF of 1.32 should be adopted in the NS analysis for side column removal scenario.

Fig. 14Displacement time histories for Column A1 removal scenario

a) First floor

b) Second floor

c) Third floor

d) Fourth floor

e) Fifth floor

f) Sixth floor

The relationship between $\alpha$ and the story on which the failure column locates is exhibited in Fig. 16. The superstructure above the failure corner column tends to be a cantilever structure resulting in more remarkable dynamic effect with the increase of the story. As a result, the $\alpha$ for the corner column removal scenario increases gradually and reaches its peak value at the top floor. Whereas the $\alpha$ for the inner or side column removal scenario decreases slightly resulting from the less loads on the superstructure. It should be noted that the DIF could be obtained at the top floor for the corner column removal scenario and at the first floor for the inner or side column removal scenario generally. Although sustained more loads than the corner column, the $\alpha$ for the inner or side column removal scenario at each floor is much less as the joint on top of the failure column is firmly constrained by more connected beams. It is concluded that the $\alpha$ reduces with the increment of the constraint at the failure joint. In other words, the DIF for the corner, side or inner column removal scenario decreases in turn.

Fig. 15Relationship between axial forces of adjacent columns and α

a) First floor

b) Second floor

c) Third floor

d) Fourth floor

e) Fifth floor

f) Sixth floor

Fig. 16Relationship between story and α

5.2. Influence of the reinforcement of frame

In order to look into the influence of the reinforcements on the DIF, the structure is designed for the seismic intensities of 7 and 8 degrees respectively, in which the reinforcements increase approximately 17 % and 87 % compared with the frame designed of 6 degree. The corresponding PGA values of the design earthquake are 100 cm/s² and 200 cm/s². It can be seen from Fig. 17 that the variations of $\alpha$ for frames designed of 7 and 8 degrees are consist with that of 6 degree. As displayed in Table 5, the DIF for the frame designed of 7 degree are completely identical with that of 6 degree and the DIF for the frame designed of 8 degree in the corner, inner and side column removal scenarios are 1.74, 1.20 and 1.33, respectively. It is indicated that the reinforcements of the frame with specially shaped columns has relatively little effect on the DIF. The values of 1.8, 1.2 and 1.4 are suggested as the DIF in NS alternate path analysis of the frame with specially shaped columns subjected to the corner, inner and side column removal scenarios, respectively.

Fig. 17Relationship between story and α for frames of 7 and 8 degrees

a) 7 degree

b) 8 degree

5.3. Influence of the span of beam

In this study, the span of Beam B1 is chosen as 3.9 m, 4.2 m, 4.5 m (the value of 4.5 m is adopted in previous study) and 4.8 m to investigate the influence of the plane layout of the frame on the DIF. As the DIF is obtained at the top floor for the corner column removal scenario and at the first floor for the inner and side column removal scenarios, the study herein are conducted for failure columns on the first, third and sixth floors, as shown in Fig. 18. It can be seen that the loading area of the affected bays above the failure column increases with the elongation of the beam span, which results in linear growth of $\alpha$ for the inner and side column removal scenarios. It is noted that the variation of $\alpha$ for the corner column removal scenario is not obvious resulting from relatively small loading area of the affected bays and constraints at the failure joint from connected beams, which increases slightly on the first and third floors, but decreases on the top floor with the increase of the span of Beam B1.

Fig. 18Relationship between span of Beam B1 and α

a) First floor

b) Third floor

c) Sixth floor