In traditional building construction, the structural columns restrict the design of the buildings and the layout of furniture, so the use of specially shaped columns came into being. The finite element model of a reinforced concrete framework using specially shaped columns was established by using the ABAQUS software. The effects of concrete strength, reinforcement ratio, and axial compression ratio on the seismic performance of the building incorporating such columns were studied. The numerical analysis was performed for a tenframe structure with specially shaped columns under low reversed cyclic loading. The loaddisplacement curve, peak load, ductility coefficient, energy dissipation capacity, and stiffness degradation curve of the specially shaped column frame were obtained using the ABAQUS finite element software. The following three results were obtained from the investigation: First, when the strength of concrete in the specially shaped column frame structure was increased, the peak load increased, while the ductility and energy dissipation capacity weakened, which accelerated the stiffness degradation of the structure. Second, when the reinforcement ratio was increased in the specially shaped column frame structure, the peak load increased and the ductility and energy dissipation capacity also increased, which increased the stiffness of the structure. Third, when the axial compression ratio was increased in the structure, the peak load increased, while ductility and energy dissipation capacity reduced, which accelerated the degradation of structural stiffness.
With the increasingly improved living quality in China, popular expectation for improvements in the quality of buildings has increased. In traditional buildings, the structural columns tend to limit interior design and furniture layout and reduce the effective utilization of the indoor area [
In this study, the main input parameters were concrete strength, reinforcement ratio, and axial compression ratio. According to the requirements of
Parameters of the simulated specimens.
Specimen  Crossshaped columns  Tshaped columns  

Axial compression ratios  Reinforcement ratios (%)  Diameter (mm)  Concrete strengths  Axial compression ratios  Reinforcement ratios (%)  Diameter (mm)  Concrete strengths  
K1  0.35  1.63  22  C30  0.35  1.74  20  C30 
K2  0.35  1.63  22  C35  0.35  1.74  20  C35 
K3  0.35  1.63  22  C40  0.35  1.74  20  C40 
K4  0.35  1.63  22  C45  0.35  1.74  20  C45 
K5  0.35  1.09  18  C30  0.35  1.22  16  C30 
K6  0.35  1.35  20  C30  0.35  1.47  18  C30 
K7  0.35  2.10  25  C30  0.35  2.04  22  C30 
K8  0.2  1.63  22  C30  0.2  1.74  20  C30 
K9  0.5  1.63  22  C30  0.5  1.74  20  C30 
K10  0.65  1.63  22  C30  0.65  1.74  20  C30 
Cross sections of the specially shaped columns. (a) Crossshaped columns. (b) Tshaped columns.
In order to study effects of concrete strength, reinforcement ratio, and axial compression ratio on the seismic performance of frame structures with specially shaped columns, according to the design requirements of
Loaddisplacement hysteresis energy dissipation curve.
Ten specimens with various concrete strengths (C30, C35, C40, and C45), reinforcement ratios (1.09%, 1.35%, 1.63%, and 2.1%), and axial compression ratios (0.2, 0.35, 0.5, and 0.65) were designed for use in the present investigation. The loading models then were calculated and analyzed using the ABAQUS finite element software, from which the loaddisplacement curves, peak loads, ductility factors, energy dissipation capacities, and stiffness degradation curves of specially shaped column frames were obtained. Its calculations were made according to equation (
Finite element numerical simulation is a recognized research tool. During the process of building a finite element model, the damage plasticity model was selected to simulate concrete. Plastic damage models can simulate mechanical phenomena such as the cracking and crushing of concrete materials. The model can be combined with isotropic damage elasticity. The isotropic tensile and compressive plastic model, which represents the inelastic behavior of concrete, is a continuum damage model based on plasticity. The combined model exhibits better convergence [
In order to verify the validity of the model, a simplified mechanical model was established in ABAQUS according to the test results in the reference literature and boundary conditions and corresponding loading methods were used to restrict the bottom of the shaped columns to six degrees of freedom. That is to say, the bottom of the column can neither rotate nor move, and the node of the cushion plate on the top of the column can restrict three degrees of freedom [
Boundary condition schematic. (a) Crossshaped columns. (b) Tshaped columns.
A classic BKIN (bilinear kinematic) hardening plasticity model was applied to describe the constitutive relationship of reinforcements. The reinforcements were merged into a reinforcement cage that was built inside the whole column region. The specimen and boundary conditions were set up and simplified according to the practical situation of testing, which ensured that the simulation results would be closer to test values [
Schematic diagrams illustrating the procedure for model establishment. (a) Model of a crossshaped column. (b) Mesh of the crossshaped column. (c) Addition of constraints.
Parameters related to material properties, such as elastic modulus, Poisson’s ratio, and density, are listed in Table
Summary of material parameters.
Material  Type of material  Elastic modulus (N/mm^{2})  Poisson’s ratio  Density (kg/m^{3}) 

Reinforcement  HRB335  2.0 × 10^{5}  0.3  7.85 × 10^{3} 


Concrete  C30  3.00 × 10^{4}  0.2  2.5 × 10^{3} 
C35  3.15 × 10^{4}  
C40  3.25 × 10^{4}  
C45  3.35 × 10^{4} 
The test data from the literature [
Test parameters.
Specimen  Concrete strengths  Reinforcement type  Diameter of the reinforcement (mm)  Axial compression ratios  Thickness of the protective layer (mm) 

+Z1  C50  HRB500  16  0.39  25 
TZ2  C40  HRB335  12  0.18  20 
Based on the experimental test data, ABAQUS was used to establish a finite element model with the same control parameters, and the hysteretic performance of the resulting structure was analyzed. The hysteresis curve, skeleton curve, and test curve can be compared, as shown in Figure
Hysteresis curves of +Z1 and TZ2. (a) +Z1 test hysteresis curve. (b) +Z1 simulated hysteresis curve. (c) +Z1 skeleton curve comparison. (d) TZ2 test hysteresis curve. (e) TZ2 simulated hysteresis curve. (f) TZ2 skeleton curve comparison.
As shown in Figure
The “hysteresis curve” and “skeleton curve” were defined as follows: The “hysteresis curve” refers to the deformation response of a structure under repeated loading and unloading. It reflects the deformation characteristics, stiffness degradation, and energy consumption of the structure. This response is the basis for determining the restoring force model and carrying out nonlinear seismic response analysis. It is also known as the “restoring force curve.” The “skeleton curve” refers to the curve obtained by translating a section of a stressstrain curve loaded in the same direction (tension or compression) beyond the maximum stress of the previous load [
The hysteresis curves of the reinforced concrete frame structure with shaped columns, with various concrete strength levels, under low reversed cyclic loading, are shown in Figure
Hysteresis curves of frames with various concrete strength levels. (a) K1. (b) K2. (c) K3. (d) K4.
Skeleton curves of frames with various concrete strength levels.
As shown in Figure
According to Figure
Curves of coefficients of energy dissipation with various concrete strength levels.
According to Figure
According to Figure
Relationship between various concrete strength levels and ductility coefficients.
By contrast, from curves shown in Figure
According to equation (
Stiffness degradation curves of frames with various concrete strengths. (a) K1. (b) K2. (c) K3. (d) K4.
The probability density function can be stated as follows [
As shown in Figure
The hysteresis curves of a reinforced concrete frame with shaped columns, with various reinforcement ratios, under low reversed cyclic loading, are shown in Figure
Hysteresis curves of frames with various reinforcement ratios. (a) K5. (b) K6. (c) K1. (d) K7.
Skeleton curves of frames with various reinforcement ratios.
As shown in Figure
According to Figure
Curves of coefficients of energy dissipation for various reinforcement ratios.
According to Figure
According to Figure
Relationship between various reinforcement ratios and ductility coefficients.
From the curve shown in Figure
According to equation (
Stiffness degradation curves of frames with various reinforcement ratios. (a) K5. (b) K6. (c) K1. (d) K7.
As shown in Figure
The hysteresis curves of frames with various axial compression ratios under low reversed cyclic loading are shown in Figure
Hysteresis curves of frames with various axial compression ratios. (a) K8. (b) K1. (c) K9. (d) K10.
Skeleton curves of frames with various axial compression ratios.
The reinforced concrete frame structures with shaped columns exhibited relatively strong plastic deformation capacities, as shown in Figure
According to Figure
Curves of coefficients of energy dissipation for various axial compression ratios.
The relationship between the energy dissipation curves of frame structures with shaped columns can be derived from Figure
According to Figure
Relationship between various axial compression ratios and ductility coefficients.
Figure
According to equation (
Stiffness degradation curves of frames with various axial compression ratios. (a) K8. (b) K1. (c) K9. (d) K10.
As shown in Figure
Numerical simulation analysis was performed on a tenframe structure with specially shaped columns under low reversed cyclic loading, according to which the following conclusions were drawn:
In reinforced concrete frame structures with shaped columns, with an increase in the strength of concrete, the peak load gradually increased, presenting arced hysteresis curves. Energy dissipation capacity and deformation capacity were gradually decreased, and there was an increase of 15–20% over the initial stiffness value. The stiffness degradation curves of frames with smaller concrete strengths were flatter, with a longer descent stage, and they exhibited better seismic performance.
In reinforced concrete frame structures with specially shaped columns, increasing the reinforcement ratio gradually increased the peak load, resulting in arced hysteresis curves with a relatively broad hysteresis loop. The energy dissipation capacity and deformation capacity were gradually increased, and initial stiffness was increased by approximately 10%. The slopes of the slow descent stage and descent stage were relatively smaller, which delayed the failure of the specimens.
In reinforced concrete frame structures with specially shaped columns, with an increase in the axial compression ratio, the peak load was increased significantly, and arced hysteresis curves were evident with fuller hysteresis loops when the axial compression ratio was smaller. The energy dissipation capacity and deformation capacity of the structure were gradually decreased, and the initial stiffness value was decreased by approximately 28%. A smaller axial compression ratio can limit the development of damage and delayed the failure of the specimens.
Representation of the mechanical properties and size specification of the screw steel with a tensile yield strength of 335 MPa
Axial compression design value of columns/total section area multiplied by the design value of the axial compressive strength of the concrete
Area of longitudinal reinforcement of a reinforced concrete member/effective total area of the component
Maximum earthquake intensity
Peak strength of the curve
Earthquake intensity (for determination of seismic intensity) providing criteria for the evaluation of seismic fortifications
Shape parameter
Displacement applied to the structure
Ductility coefficient which is the ratio of maximum deformation to yield deformation
Energy dissipation coefficient.
The research data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this article.
The authors gratefully acknowledge the project support from the National Natural Science Foundation of China (No. 51578120) and the Graduate Science and Technology Innovation Projects in Northeast Petroleum University (YJSCX2017025NEPU).