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A finite element model (FEM) of frame-shear structure was constructed using OpenSees program based on the nonlinear flexibility theory and multi-vertical-line theory that considered bending-shear coupling, and its progressive collapse resistance under abnormal conditions was analyzed. Flexibility-based method for modeling shear wall finite element and multi-vertical-line element (SFI-MVLEM) was proposed. Method of deleting failure component elements was presented, as well as the model solving algorithm. The FEM was validated by the completed structure test. On these bases, 3 groups of typical frame-shear structure systems were designed to perform nonlinear dynamic collapse analysis under different initial failure conditions, in order to study the impact of the number of floors and earthquake resistant design on the progressive collapse resistance of frame-shear structures. Analysis results showed that, at initial failure of frame column, the residual shear wall element can well complete the internal force redistribution of structure to provide alternative force transmission path, thereby forming antiprogressive collapse force. In the case of initial failure of shear wall, C-shaped shear wall can form alternative path to diminish the vertical deformation of frame-shear structures. Final comparison shows that the structural seismic design can effectively improve their anticollapse performance.

Progressive collapse of building structures refers to the horizontal and vertical sequential damage of structures led by local failure caused by explosion, terrorist attacks, and other accidents, which results in collapse of overall structures or large-scale collapse that is disproportionate to the initial damage [

Aside from the research findings in design codes and progressive collapse resistance guidelines, scholars from various countries have also made some progressive collapse in structural testing and numerical simulation in recent years. For example, with respect to collapse testing, Yi et al. [

A typical frame-shear building structure (6-degree seismic design) is designed herein according to the

A 15-story RC frame-shear wall structure is designed according to the

Figure

Load information: floor dead load is 5.0 kN/m^{2} and live load 2.0 kN/m^{2}. Roof dead load is 7.5 kN/m^{2} and live load 0.5 kN/m^{2}, while surface roughness is set as category C.

Seismic design information: type of soil in the construction site is set as type II. Besides, seismic fortification intensity is 6 degrees, and seismic grade of shear walls and frames is grade 2. To consider the effect of seismic design on the structural antiprogressive collapse, the frame-shear structure is subjected to 8-degree seismic design, while other design information remains unchanged. The corresponding design peak ground acceleration (PGA) with a 10% probability of exceedance in 50 years equals 0.05 and 0.20 g, in which g is the acceleration of gravity. Note that because of different requirements of the maximum axial force ratios specified in the Chinese seismic design code, the steel quantity in 8-degree seismic design is large than that in 6-degree seismic design. Specifically, the maximum axial force ratio for the design intensity of 6-degree is 0.9, whereas that for the design intensity of 8-degree is 0.75. And steel dosage also results in different self-weights in these buildings. The basic dynamic properties of the frame-shear wall structures are given in Table

Parameters of structural material.

Item name | Beams | Columns | Shear walls |
---|---|---|---|

Concrete | C30 ( |
C40 ( |
C40 ( |

Longitudinal reinforcing steel | HRB335 | HRB400 | |

( |
( |
||

Hoop reinforcing steel | HPB235 ( |

Dynamic properties of the structure.

Item name | 6-degree | 8-degree | |
---|---|---|---|

Vibration periods (s) | T_{1} (1st-order translation in |
1.5 | 1.05 |

T_{2} (1st-order translation in |
1.5 | 1.05 | |

T_{3} (1st-order torsion) |
1.24 | 0.88 | |

Self-weight (t) | 3250 | 3480 |

Floor plan of the building models (unit: m).

To consider the effects of different number of stories on the progressive collapse resistance of frame-shear structure and to simplify the collapse analysis process, in this paper, 10-story and 15-story RC frame-shear models are built for separate antiprogressive collapse analyses. To consider the effect of seismic design on the structural anticollapse performance, the frame-shear structure is subjected to 8-degree seismic design and modeling for progressive collapse analysis, followed by comparative analysis of collapse models. In accordance with the provisions of GSA2010 and DoD2010, demolition analysis on key components is needed. To comprehensively analyze the antiprogressive collapse performance of frame-shear structure, the following demolition conditions are designed herein. Regarding the demolition of key columns, corner columns and the short edge column are chosen for removal. Regarding the demolition of shear walls, in accordance with the DoD2010 provisions, load-bearing walls with height greater than twice the story height should be removed in steps, while for those with height less than twice the story height, shear walls can be removed as a whole for the progressive collapse analysis. According to the design information, shear walls of this paper are lower than twice the story height, so shear walls in the center of first floor are chosen for holistic removal. Relevant demolition sites are marked in Figure

In accordance with the DoD2010 provisions, the judging mechanism for progressive collapse failure of structure in this paper is as follows: when the top displacement of removed member exceeds 1/5 of the vertical relative displacement of beam connected to it, the structure is judged as failure, which enters an irreversible collapse process. The frame-shear structure in this paper is symmetrical. Nonlinear dynamic alternate path method is used to analyze the frame-shear model, in order to ensure the accuracy of antiprogressive collapse analysis.

Multi-vertical-line shear wall element considering bending-shear coupling is developed on the basis of three-vertical-line-element theory. Initially proposed by Japanese scholars [

Element models: (a) MVLEM element; (b) SFI-MVLEM element; (c) RC panel element.

In the geometric nonlinear analysis of structure’s progressive collapse, the actual and assumed displacement fields of line elements like beams and columns differ greatly. So fine division of line elements is needed to approximate the real displacement field, in order to improve the solution accuracy. The fine division of elements will increase the number of elements, which will affect the solution efficiency. To ensure the accuracy and efficiency of solution, flexibility-based fiber line element is used in this paper [

Since the OpenSees program is different from the commercial finite element programs, if nonlinear dynamic analysis is performed by simply killing the predesigned key members applying “birth-death element” technology, the degree of freedom of the overall structure will undergo changes. Moreover, the instantaneous removal of elements leads to great dynamic effect on the structure, which is likely to result in interruption or nonconvergence of calculation process. In this paper, to achieve precise calculation, first of all, the overall calculation was conducted on the model, and the static internal force and counterforce of failure node were obtained. Then, the key components were removed and the static internal force and counterforce before the components were removed were exerted on the components at the same time so as to make the structure equal to the overall structure before the components were removed. Finally, to obtain the transient oscillation effect of the structure, the static internal force and counterforce on the key nodes were instantaneously removed, and the time was returned to zero at the same time for nonlinear dynamic calculation [

Element-deletion method.

Initial phase

Static internal force stage

Collapse stage

Regarding iterative integration and solution, this paper employs Krylov-Newton algorithm to facilitate the convergence of program computation. Transformation method is used for boundary condition processing, while SparseSYM is utilized for solving dynamic analysis equations [

The SFI-MVLEM element is used as the shear wall element to calculate the elastoplastic behaviors of RC wall corresponding zones by taking into consideration the concrete confinement effect in the confinement zone of wall-edge members and the reinforcement assignment in the nonconfined zone of wall plates. Rebar is set according to the sectional design of members. Figure

Finite element model.

Analysis model of shear wall

Computational model

Due to space constraints, four sets of shear wall specimens are selected in this paper for calculation with SFI-MVLEM shear wall elements, and the results are compared with the experimental data. The specific procedure is shown in Figure

Top force versus displacement hysteretic curve of specimens.

SW1-1

SW2-2

SW3-3

SW4-4

Ren et al. [

Reinforce details of the RC frames.

Plate fitted with reinforcing bars

Comparisons between the numerical simulations and the test results.

In order to study the differences of the full-size experiment and reduced scale experiment of frame-shear wall structure in earthquake disaster, Japan and the United States cooperated a series of structural dynamic time history experiments. The full-scale structure was tested at the Large-Scale Testing Facility at the Building Research Institute (BRI) in Tsukuba, Japan. Then, Wolfgram [

Comparisons between the numerical simulations and the test results.

Figure

Removal of the corner columns in building: (a) 10-story model; (b) 15-story model; (c) 10-story model of the 8-degree seismic design; (a)–(c) vertical displacement of the joint at the top of the removed column on the

Figure

Removal of central shear wall in building: (a) 10-story model; (b) 15-story model; (c) 10-story model of the 8-degree seismic design; (a)–(c) vertical displacement of the joint at the top of the removed column on the

Figure

Removal of short-side middle column in building: (a) 10-story model; (b) 15-story model; (c) 10-story model of the 8-degree seismic design; (a)–(c) vertical displacement of the joint at the top of the removed column on the

With the development of economy, high-rise structures continue to spring up; progressive collapse resistance design of high-rise structures has been gaining increasing attention. Anticollapse designs are made abroad mainly in accordance with the US GSA2010 and DoD2010 design guidelines. In China, the primary references are the

In recent years, the engineering community’s understanding of the design of structural antiprogressive collapse lies in the mere anticollapse design of important buildings in accordance with the codes, while ignoring the influences of anticollapse design on the earthquake, fire, and other disaster resistance of structures. This does not meet the requirements on the establishment of multihazard prevention system. For example, addition of intrabeam longitudinal reinforcement layout after the anticollapse design may result in “strong beam weak column” damage of the structure during the earthquake. In addition, with the increase of structural rebar, the impact on rebar increases under fire scenario, which may pose threats to the stability of high-rise structures. Therefore, for safety consideration, a secondary seismic or fire-resistance design may be required after the anticollapse design. This means repeated design, which may result in increased material consumption and aggravated designer’s responsibility. This paper attempts to discuss and analyze the structural progressive collapse using influencing factors such as seismic design and story number. We find that the direct seismic design of structure without collapse design can effectively improve its progressive collapse resistance. However, further in-depth study is needed, and design method suitable for seismic, fire, and progressive collapse resistance of high-rise structures needs to be put forward.

Existing methods for anticollapse analysis of building structures mainly include the linear static alternate path method, nonlinear static alternate path method, linear dynamic alternate path method, and nonlinear dynamic alternate path method, of which the nonlinear dynamic alternate path method is the most accurate method for antiprogressive collapse analysis. Its analytical procedure is as follows: firstly, nonlinear numerical model of structure is constructed; key members are removed from top to bottom story-by-story, and nonlinear dynamic analysis is performed; if the structure undergoes progressive collapse and rebar in the collapse subarea is increased, then the demolition analysis cycle is continued until the structure no longer collapses. The method is computationally difficult to be widely used. Lu et al. [

Linear static alternate path method in GSA2010 mostly uses the internal force reduction coefficient and dynamic magnification factor to consider the dynamic response problem of structures. However, in practical engineering, nonlinear and dynamic factors will simultaneously produce effects on structure, which may lead to large computational error. Thus, progressive collapse specifications need to be further perfected and supplemented [

Parameter settings in the codes for progressive collapse resistance design are mostly based on the previous collapse disaster data, which are highly occasional. Besides, testing of large-scale structures is costly, demanding in terms of test platform, data collection, and testing personnel, and makes verification of design parameters difficult. In additions, the current structural theory and building material performance are all significantly better than the past, which will thus lead to certain errors in the calculation of parameters.

Through the above nonlinear progressive collapse analysis of high-rise frame-shear structures, we can draw the following conclusions:

Nonlinear finite element model of high-rise frame-shear structure is built by using the novel shear wall element SFI-MVLEM and the flexible line fiber element. The feasibility of simulating progressive structural collapse with SFI-MVLEM and line flexibility elements is verified via collapse tests on substructures comprising four shear wall members and two frame sets as an example. Meanwhile, the use of progressive collapse analysis method and the integral iteration and solution methods proposed in this paper can well obtain the progressive collapse analysis data of structures. This indicates that the finite element program OpenSees can be applied to the seismic and progressive collapse analyses of large-scale high-rise structures.

Considering the effects of story number and seismic design on the structural progressive collapse resistance in the high-rise frame-shear structures, three typical cases are designed for the nonlinear collapse analysis of high-rise structures. The results show that the frame-shear structure has a fairly complete transmission path upon initial failure of the corner column to prevent collapse behavior. When the shear wall produced initial damage, shear wall edge frame beams formed anticollapse strength at end bending moment; the reinforcing steel bar in the beam provided catenary mechanism in the form of tension, formed anticollapse strength and connected with the wall around the ones where shear failed, and could effectively alter force transferring path, so it could effectively resist continuity of collapse, and frame-shear wall structure had small deformation. In the case of short-side middle column failure, the structural vertical deformation is smaller than the corner column failure. Finally, anticollapse performance of high-rise structures is well improved after 8-degree seismic design.

The authors declare that they have no conflicts of interest.