# Numerical Analysis on the Bending Capacity of the Concrete-Filled Micro-Steel-Tube Piles

Authors: Zaiqian Chen; Shutao Pu; Haicheng Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The finite element analysis model is developed to investigate the bending capacity of the concrete-filled micro-steel-tube (CFMST) piles. First, the classical elastic-plastic model is employed to describe the constitutive relation of the steel material, while the plastic damage model is applied to portray the concrete material. Then, the three dimensional (3D) finite element numerical model for analyzing the flexural bearing capacity of the CFMST pile is established by virtue of the ABAQUS software. The reliability and accuracy of the present numerical model is carefully validated by comparing with the experimental data. The influence of the concrete strength grade, wall thickness of the steel tube, external diameter of the steel tube, and yield strength of the steel are analyzed in detail. It is shown that increasing the wall thickness/external diameter of the steel tube and the yield strength of the steel can greatly improve the flexural bearing capacity of the CFMST pile. Finally, a concise formula for the ultimate bending moment of the CFMST pile is proposed, which could be directly applied into the future engineering design.


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## Body

## 1. Introduction

Micropile is a bored pile with a diameter typically in 150–300 mm range and internal steel-bar reinforcement, which is usually used to underpin existing structures, reduce differential settlements, and serve as the main foundation components [1–4]. As the extension of the micropile, concrete-filled micro-steel-tube (CFMST) pile, generally refers to cast-in-place pile with a pile diameter of less than 300 mm and using construction techniques such as drilling and pressure grouting. It is mainly composed of steel tube, grouting bodies, and accessories, and the corresponding strength/bearing capacity is generally higher than the micropile due to the better mechanical performance of concrete-filled steel tube. It is worthwhile noting that as the typical structural foundation elements, the conventional concrete piles are widely used to support the heavy superstructures such as buildings, bridges, and wind turbines. However, when the project is located in the unfavorable mountainous area, it is difficult to install the concrete piles into the subgrade due to the limited construction site or space. Therefore, considering the small-size construction equipment, fast construction speed, reliable material performance, and good supporting effect of the CFMST pile, it has been successfully applied into the foundation pit support, weak foundation, and slope reinforcement in the limited construction sites [5–7].In the past studies, numerous researches have been carried out to study the mechanical behavior of the concrete-filled steel tube (CFST) members under common static loads (e.g., vertical, lateral, bending, and torsional loads) and seismic loads, and a series of corresponding design theories are proposed. Han et al. [8] reviewed the development of the family of concrete-filled steel tubular structures to date and draws a research framework on CFST members. Schneider [9] presented an experimental and analytical study on the behavior of short CFST columns concentrically loaded in compression to failure. Sakino et al. [10] analyzed the interaction between the steel tube and filled concrete based on the tests of centrally loaded short columns and developed design formulas to estimate the ultimate axial compressive load capacities for CFST columns with both circular and square sections. Gajalakshmi and Helena [11] conducted an experimental study to investigate the cumulative damage of in-filled steel columns subjected to quasi-static lateral cycle load. Elchalakani et al. [12] conducted detailed experimental tests and investigated the flexural behavior of CFST under large deformation pure bending. Lu et al. [13] presented a finite element analysis model to study the flexural performance of CFST beam and proposed a strut-tie model for the load transfer mechanism of the circular composite member subjected to pure bending. Han et al. [14] established a finite element analysis (FEA) model via ABAQUS software and further studied the torsional behaviors of CFST, which provides information for the development of formulae to calculate the ultimate torsional strength. By virtue of the analytical method, Lee et al. [15] theoretically investigated the behavior of a circular CFST column under combined torsion and compression considering the confinement effect of the steel tube, softening of concrete, and spiral effects. Montejo et al. [16] and Aguirre et al. [17] studied the seismic performance of reinforced CFST pile/column via the experimental tests. Furthermore, the databases for CFST columns and beam-columns were also established [18].Most of above literature focus on the static and seismic performance of CFST members (e.g., beam and column) with common dimension. However, to be best of our knowledge, the studies on the mechanical performance of the concrete-filled micro-steel-tube (CFMST) pile are relatively limited, and the associated design is commonly based on the engineering experience and the design guidance of concrete pile. Although Wang et al. [19] carried out the experimental tests and investigated the mechanical performance of CFMST pile subjected to lateral loads, this study is limited to the lateral response. Therefore, the main objective of the present study is to develop a FEA model to analyze the flexural-bearing capacity of the CFMST pile, which is of great importance in the corresponding engineering design. Through the numerical calculation, the influence of the concrete strength grade, external diameter, and wall thickness of the steel tube and the yield strength of the steel on the flexural bearing capacity is investigated in detail. Finally, a concise calculation formula is also proposed, which provides a beneficial guidance for the design of CFMST pile.


## 2. 3D Finite Element Modeling

Considering the complexity of the flexural performance of the composite material (i.e., steel tube and inner concrete), it is hard to apply the analytical method to solve this problem. Hence, the FEM is employed to investigate the bending capacity of CFMST pile in the present study. In order to realize reliable and accurate analysis, the constitutive relation, element type, boundary condition and steel-concrete interface should be modeled reasonably that is discussed as follows.


### 2.1. Material Constitutive Models

#### 2.1.1. Steel

The classical elastic-plastic model [20] is employed to describe the constitutive relation of the steel, and the stress-strain curve of the steel material is shown in Figure 1. The entire deformation process can be divided to five stages, i.e., elastic (Oa), elastic-plastic (ab), plastic (bc), hardening (cd), and fracture (de) stages. The horizontal and vertical axes denote the strain (ε) and stress (σ) of steel, respectively. In the vertical axis, fp, fy, and fu are, respectively, the proportional limit, yield, and ultimate strength of steel. In the horizontal axis, εp = 0.8fy/Es is the maximum strain in the elastic stage; εy1 = 1.5εp is the strain at the beginning of yield stage; εy2 = 10εp is the maximum strain in yield stage; εu = 100εp is the ultimate strain. In addition, the Mises yield criterion and associated flow rule is further employed to portrait the plastic deformation behavior of the steel material. Elastic modulus and Poisson’s ratio of steel are taken as 2 × 105 MPa and 0.3.Figure 1
Typical uniaxial stress-strain curve of steel.


#### 2.1.2. Concrete

As known to us, the behavior of concrete material is complicated due to the plastic accumulation and stiffness degradation under repeated tensile and compressive loads. CFMST piles show tensile and crushing failures when subjected to the bending load. Based on the mechanical properties of concrete, the diffuse crack model and the brittle fracture model were considered first. It is found from the numerical study that the brittle fracture model can only be used in static analysis with requiring fewer parameters and relatively poor convergence, while the diffuse crack model can only be used in dynamic analysis and works best for small loads. Hence, the plastic damage model of concrete material defined in ABAQUS is introduced to describe the mechanical behavior of concrete. The concrete plastic damage model is a continuous damage model based on isotropy in tension and compression. The concrete plastic damage model provided in ABAQUS has the following characteristics: (1) it can be applied into the finite element simulation of beams, rods, shells, and solids of concrete material or other brittle materials; (2) this model is a synthesis of uncorrelated multiaxial hardening plastic and isotropic linear damage models with being used to describe irrecoverable damage by concrete fractures; and (3) the interface model of the steel tube and the concrete is composed of the bond-slip in the tangential direction and the contact in the normal direction. The Mohr–Coulomb model is used in the tangential direction of the interface, and the contact in the normal direction is defined as “hard” contact, which specifies that the contact surfaces do not penetrate each other and do not interact when separated. This definition is consistent with the interaction at the interface of steel tube and concrete when the CFMST pile is under load. The description of the concrete plastic damage model is listed as follows:(1) Decomposition of Strain Rate. The additional assumed strain rate for the model can be decomposed as(1)ε=εel+εpl,where, ε, εel, and εpl are the total strain rate, the elastic, and the plastic parts of the strain rate, respectively.(2) Stress-Strain Relationship. (2)σ=1−dD0el:ε−εpl=Del:ε−εpl,where D0el, Del=1−dD0el, and d are the nondestructive stiffness, damaged stiffness, and stiffness degradation variable of the material, respectively. It is noted that 0 ≤ d ≤ 1. When d = 0, the material is in a nondestructive state, and when d = 1, the material is in a complete-failure state.According to the theory of continuum mechanics, the effective stressσ¯ can be written as(3)σ¯=D0el:ε−εpl.The relation between stress and effective stress can be expressed as(4)σ=1−dσ¯.For a given material at an arbitrary interface, the factor (1−d) represents the ratio of the effective area of the bearing capacity to the total area of the section. When d = 0, σ=σ¯, while when d ≠ 0, the effective stress is more representative than the stress. Therefore, it is convenient to use the effective stress to establish the plasticity correlation formula. The degradation variable is controlled by the effective stress and hardening variables; hence, we have d=dσ¯,ε¯pl.(3) Hardening Variable. The damage states in tension and compression are described by two independent hardening variables, ε¯tpl and ε¯cpl. The evolution equation of the hardening variable can be written as(5)ε¯pl=ε¯tplε¯cpl,where ε¯tpl and ε¯cplare the equivalent plastic strains in tension and compression, respectively. It is noted that micro cracks and crushing of concrete are described by an ever-increasing evolutionary variable, while the hardening variable controls the yield surface and elastic stiffness degradation.(4) Yield Function. The spatial curved surface in effective stress is described by the yield function Fσ¯,ε¯pl, which determines the state of failure and damage. For the plastic damage model, we have(6)Fσ¯,ε¯pl≤0.(5) Flow Law. The law of nonassociation is adopted and can be expressed as(7)εpl=λ∂Gσ¯∂σ¯,where λ and G are the non-negative flow factor and the plastic potential of plastic flow, respectively.(6) Damage and Stiffness Degradation. The uniaxial stress-strain relationship can be expressed in terms of the stress-plastic strain relationship as(8)σt=σtε¯tpl,T,fi,σc=σcε¯cpl,T,fi,where the subscripts t and c represent the tension and compression, respectively; T is the temperature, and fi,i=1,2,3… are other predefined variables.As shown in Figures2 and 3, when the concrete material is unloaded from a certain point in the softening stage of the stress-strain curve, the unloading response becomes weak, that is to say, the elastic stiffness of the material is damaged (reduced). The damage to elastic stiffness differs significantly in tensile and compression tests, with the damage effect becoming more pronounced as the plastic strain increases. The damage response of concrete is represented by two independent uniaxial damage variables, dt and dc, which are the functions of plastic strain, temperature, and field variables:(9)dt=dtε¯tpl,T,fi0≤dt≤1,dc=dcε¯cpl,T,fi0≤dc≤1.Figure 2
Stress-strain curve of concrete subjected to uniaxial tensile load.Figure 3
Stress-strain curve of concrete under uniaxial compressive load.The uniaxial stiffness degradation variable is a nondecreasing function of the equivalent plastic strain, and its value ranges from 0 to 1. IfE0 represents the initial elastic stiffness of the material, then the stress-strain relationship under uniaxial tension and compression is(10)σt=1−dtE0εt−ε¯tpl,σc=1−dcE0εc−ε¯cpl.In the uniaxial case, the crack development direction is perpendicular to the stress direction. The effective stress is increased due to the reduction of the effective interface area induced by the formation and propagation of cracks. In the uniaxial case, the effect of this reduction in bearing area is slightly better. The reason for this phenomenon is that the crack propagates parallel to the direction of stress at the beginning. However, when the crushing progresses more severely, the effective bearing area will significantly decrease. Hence, the effective uniaxial cohesionσ¯t and σ¯c can be written as(11)σ¯t=σt1−dt=E0εt−ε¯tpl,σ¯c=σc1−dc=E0εc−ε¯cpl.(7) Yield Condition. The yield condition of this model was established based on the yield function suggested by Lubliner et al. [21]. Then the evolution laws of tension and compression at different strengths are considered, and the yield function can be expressed in terms of effective stress as(12)Fσ¯,ε¯pl=11−αq¯−3αp¯+βε¯plσmax−γσmax−σ¯cε¯cpl≤0,in which(13)p¯=13σ¯;q¯=32S¯,where α and γ are nondimensional parameters, and S¯ is Mises equivalent stress.


### 2.2. Establishment of the Finite Element Numerical Analysis Model

Figure4 shows the simply supported beam subjected to two symmetrically concentrated loads. It should be pointed out that the corresponding midspan of the beam is under the purely bending action (i.e., no shear force). Based on this simply supported member structure, the finite element numerical analysis model (see Figure 5) is established via the ABAQUS software. During the laboratory test, the material parameters of the poured concrete are strength grade C30, material density ρ = 2400 kg/m3, design value of compressive strength fc = 14.3 MPa, design value of tensile strength ft = 1.43 MPa, elastic modulus E = 30 GPa, Poisson’s ratio ν = 0.2, and length of the member L = 1.5 m. In the numerical analysis, unless otherwise stated, the material parameters of concrete adopt the above values. Besides, the plastic parameters of concrete are defined as the expansion angle, eccentricity, ratio of biaxial to uniaxial compressive strength, and viscosity parameter are selected as 20°, 0.1, 1.16, and 0.0001, respectively; the morphological parameter K that affects the yield surface is fixed at 2/3. Finally, based on the basic material parameters and plastic parameters of the concrete, the stress and strain relationships and damage parameters are directly generated in the ABQUAS software.Figure 4
Simply supported beam under two symmetrically vertical loads.Figure 5
Diagram of the mesh division.Nodes on the slave surface cannot invade the master surface, but nodes on the master surface invade the slave surface between nodes on the slave surface in ABAQUS software. Generally, slave surface is with finer meshes. When the mesh density is the same, the salve surface is the surface of softer material. The concrete is the master surface due to the larger grid, and the steel tube is slave surface.The interaction between contact surfaces involves two parts, one is the normal action between contact surfaces, and the other is the tangential action between contact surfaces. The “hard contact” is selected to describe the normal action and the tangential action in ABAQUS software. The steel tube and solid concrete adopt hexahedral elements with the type being C3D8R. In the numerical analysis, the displacement controlled loading method is adopted, and the loading is ceased as the maximum bending moment that the CFMST pile can bear is achieved. In general engineering practice, the displacement of the control pile is 20 mm, and the bending moment corresponding to the displacement of the pile 20 mm is selected as the ultimate flexural bearing capacity.


## 2.1. Material Constitutive Models

### 2.1.1. Steel

The classical elastic-plastic model [20] is employed to describe the constitutive relation of the steel, and the stress-strain curve of the steel material is shown in Figure 1. The entire deformation process can be divided to five stages, i.e., elastic (Oa), elastic-plastic (ab), plastic (bc), hardening (cd), and fracture (de) stages. The horizontal and vertical axes denote the strain (ε) and stress (σ) of steel, respectively. In the vertical axis, fp, fy, and fu are, respectively, the proportional limit, yield, and ultimate strength of steel. In the horizontal axis, εp = 0.8fy/Es is the maximum strain in the elastic stage; εy1 = 1.5εp is the strain at the beginning of yield stage; εy2 = 10εp is the maximum strain in yield stage; εu = 100εp is the ultimate strain. In addition, the Mises yield criterion and associated flow rule is further employed to portrait the plastic deformation behavior of the steel material. Elastic modulus and Poisson’s ratio of steel are taken as 2 × 105 MPa and 0.3.Figure 1
Typical uniaxial stress-strain curve of steel.


### 2.1.2. Concrete

As known to us, the behavior of concrete material is complicated due to the plastic accumulation and stiffness degradation under repeated tensile and compressive loads. CFMST piles show tensile and crushing failures when subjected to the bending load. Based on the mechanical properties of concrete, the diffuse crack model and the brittle fracture model were considered first. It is found from the numerical study that the brittle fracture model can only be used in static analysis with requiring fewer parameters and relatively poor convergence, while the diffuse crack model can only be used in dynamic analysis and works best for small loads. Hence, the plastic damage model of concrete material defined in ABAQUS is introduced to describe the mechanical behavior of concrete. The concrete plastic damage model is a continuous damage model based on isotropy in tension and compression. The concrete plastic damage model provided in ABAQUS has the following characteristics: (1) it can be applied into the finite element simulation of beams, rods, shells, and solids of concrete material or other brittle materials; (2) this model is a synthesis of uncorrelated multiaxial hardening plastic and isotropic linear damage models with being used to describe irrecoverable damage by concrete fractures; and (3) the interface model of the steel tube and the concrete is composed of the bond-slip in the tangential direction and the contact in the normal direction. The Mohr–Coulomb model is used in the tangential direction of the interface, and the contact in the normal direction is defined as “hard” contact, which specifies that the contact surfaces do not penetrate each other and do not interact when separated. This definition is consistent with the interaction at the interface of steel tube and concrete when the CFMST pile is under load. The description of the concrete plastic damage model is listed as follows:(1) Decomposition of Strain Rate. The additional assumed strain rate for the model can be decomposed as(1)ε=εel+εpl,where, ε, εel, and εpl are the total strain rate, the elastic, and the plastic parts of the strain rate, respectively.(2) Stress-Strain Relationship. (2)σ=1−dD0el:ε−εpl=Del:ε−εpl,where D0el, Del=1−dD0el, and d are the nondestructive stiffness, damaged stiffness, and stiffness degradation variable of the material, respectively. It is noted that 0 ≤ d ≤ 1. When d = 0, the material is in a nondestructive state, and when d = 1, the material is in a complete-failure state.According to the theory of continuum mechanics, the effective stressσ¯ can be written as(3)σ¯=D0el:ε−εpl.The relation between stress and effective stress can be expressed as(4)σ=1−dσ¯.For a given material at an arbitrary interface, the factor (1−d) represents the ratio of the effective area of the bearing capacity to the total area of the section. When d = 0, σ=σ¯, while when d ≠ 0, the effective stress is more representative than the stress. Therefore, it is convenient to use the effective stress to establish the plasticity correlation formula. The degradation variable is controlled by the effective stress and hardening variables; hence, we have d=dσ¯,ε¯pl.(3) Hardening Variable. The damage states in tension and compression are described by two independent hardening variables, ε¯tpl and ε¯cpl. The evolution equation of the hardening variable can be written as(5)ε¯pl=ε¯tplε¯cpl,where ε¯tpl and ε¯cplare the equivalent plastic strains in tension and compression, respectively. It is noted that micro cracks and crushing of concrete are described by an ever-increasing evolutionary variable, while the hardening variable controls the yield surface and elastic stiffness degradation.(4) Yield Function. The spatial curved surface in effective stress is described by the yield function Fσ¯,ε¯pl, which determines the state of failure and damage. For the plastic damage model, we have(6)Fσ¯,ε¯pl≤0.(5) Flow Law. The law of nonassociation is adopted and can be expressed as(7)εpl=λ∂Gσ¯∂σ¯,where λ and G are the non-negative flow factor and the plastic potential of plastic flow, respectively.(6) Damage and Stiffness Degradation. The uniaxial stress-strain relationship can be expressed in terms of the stress-plastic strain relationship as(8)σt=σtε¯tpl,T,fi,σc=σcε¯cpl,T,fi,where the subscripts t and c represent the tension and compression, respectively; T is the temperature, and fi,i=1,2,3… are other predefined variables.As shown in Figures2 and 3, when the concrete material is unloaded from a certain point in the softening stage of the stress-strain curve, the unloading response becomes weak, that is to say, the elastic stiffness of the material is damaged (reduced). The damage to elastic stiffness differs significantly in tensile and compression tests, with the damage effect becoming more pronounced as the plastic strain increases. The damage response of concrete is represented by two independent uniaxial damage variables, dt and dc, which are the functions of plastic strain, temperature, and field variables:(9)dt=dtε¯tpl,T,fi0≤dt≤1,dc=dcε¯cpl,T,fi0≤dc≤1.Figure 2
Stress-strain curve of concrete subjected to uniaxial tensile load.Figure 3
Stress-strain curve of concrete under uniaxial compressive load.The uniaxial stiffness degradation variable is a nondecreasing function of the equivalent plastic strain, and its value ranges from 0 to 1. IfE0 represents the initial elastic stiffness of the material, then the stress-strain relationship under uniaxial tension and compression is(10)σt=1−dtE0εt−ε¯tpl,σc=1−dcE0εc−ε¯cpl.In the uniaxial case, the crack development direction is perpendicular to the stress direction. The effective stress is increased due to the reduction of the effective interface area induced by the formation and propagation of cracks. In the uniaxial case, the effect of this reduction in bearing area is slightly better. The reason for this phenomenon is that the crack propagates parallel to the direction of stress at the beginning. However, when the crushing progresses more severely, the effective bearing area will significantly decrease. Hence, the effective uniaxial cohesionσ¯t and σ¯c can be written as(11)σ¯t=σt1−dt=E0εt−ε¯tpl,σ¯c=σc1−dc=E0εc−ε¯cpl.(7) Yield Condition. The yield condition of this model was established based on the yield function suggested by Lubliner et al. [21]. Then the evolution laws of tension and compression at different strengths are considered, and the yield function can be expressed in terms of effective stress as(12)Fσ¯,ε¯pl=11−αq¯−3αp¯+βε¯plσmax−γσmax−σ¯cε¯cpl≤0,in which(13)p¯=13σ¯;q¯=32S¯,where α and γ are nondimensional parameters, and S¯ is Mises equivalent stress.


## 2.1.1. Steel

The classical elastic-plastic model [20] is employed to describe the constitutive relation of the steel, and the stress-strain curve of the steel material is shown in Figure 1. The entire deformation process can be divided to five stages, i.e., elastic (Oa), elastic-plastic (ab), plastic (bc), hardening (cd), and fracture (de) stages. The horizontal and vertical axes denote the strain (ε) and stress (σ) of steel, respectively. In the vertical axis, fp, fy, and fu are, respectively, the proportional limit, yield, and ultimate strength of steel. In the horizontal axis, εp = 0.8fy/Es is the maximum strain in the elastic stage; εy1 = 1.5εp is the strain at the beginning of yield stage; εy2 = 10εp is the maximum strain in yield stage; εu = 100εp is the ultimate strain. In addition, the Mises yield criterion and associated flow rule is further employed to portrait the plastic deformation behavior of the steel material. Elastic modulus and Poisson’s ratio of steel are taken as 2 × 105 MPa and 0.3.Figure 1
Typical uniaxial stress-strain curve of steel.


## 2.1.2. Concrete

As known to us, the behavior of concrete material is complicated due to the plastic accumulation and stiffness degradation under repeated tensile and compressive loads. CFMST piles show tensile and crushing failures when subjected to the bending load. Based on the mechanical properties of concrete, the diffuse crack model and the brittle fracture model were considered first. It is found from the numerical study that the brittle fracture model can only be used in static analysis with requiring fewer parameters and relatively poor convergence, while the diffuse crack model can only be used in dynamic analysis and works best for small loads. Hence, the plastic damage model of concrete material defined in ABAQUS is introduced to describe the mechanical behavior of concrete. The concrete plastic damage model is a continuous damage model based on isotropy in tension and compression. The concrete plastic damage model provided in ABAQUS has the following characteristics: (1) it can be applied into the finite element simulation of beams, rods, shells, and solids of concrete material or other brittle materials; (2) this model is a synthesis of uncorrelated multiaxial hardening plastic and isotropic linear damage models with being used to describe irrecoverable damage by concrete fractures; and (3) the interface model of the steel tube and the concrete is composed of the bond-slip in the tangential direction and the contact in the normal direction. The Mohr–Coulomb model is used in the tangential direction of the interface, and the contact in the normal direction is defined as “hard” contact, which specifies that the contact surfaces do not penetrate each other and do not interact when separated. This definition is consistent with the interaction at the interface of steel tube and concrete when the CFMST pile is under load. The description of the concrete plastic damage model is listed as follows:(1) Decomposition of Strain Rate. The additional assumed strain rate for the model can be decomposed as(1)ε=εel+εpl,where, ε, εel, and εpl are the total strain rate, the elastic, and the plastic parts of the strain rate, respectively.(2) Stress-Strain Relationship. (2)σ=1−dD0el:ε−εpl=Del:ε−εpl,where D0el, Del=1−dD0el, and d are the nondestructive stiffness, damaged stiffness, and stiffness degradation variable of the material, respectively. It is noted that 0 ≤ d ≤ 1. When d = 0, the material is in a nondestructive state, and when d = 1, the material is in a complete-failure state.According to the theory of continuum mechanics, the effective stressσ¯ can be written as(3)σ¯=D0el:ε−εpl.The relation between stress and effective stress can be expressed as(4)σ=1−dσ¯.For a given material at an arbitrary interface, the factor (1−d) represents the ratio of the effective area of the bearing capacity to the total area of the section. When d = 0, σ=σ¯, while when d ≠ 0, the effective stress is more representative than the stress. Therefore, it is convenient to use the effective stress to establish the plasticity correlation formula. The degradation variable is controlled by the effective stress and hardening variables; hence, we have d=dσ¯,ε¯pl.(3) Hardening Variable. The damage states in tension and compression are described by two independent hardening variables, ε¯tpl and ε¯cpl. The evolution equation of the hardening variable can be written as(5)ε¯pl=ε¯tplε¯cpl,where ε¯tpl and ε¯cplare the equivalent plastic strains in tension and compression, respectively. It is noted that micro cracks and crushing of concrete are described by an ever-increasing evolutionary variable, while the hardening variable controls the yield surface and elastic stiffness degradation.(4) Yield Function. The spatial curved surface in effective stress is described by the yield function Fσ¯,ε¯pl, which determines the state of failure and damage. For the plastic damage model, we have(6)Fσ¯,ε¯pl≤0.(5) Flow Law. The law of nonassociation is adopted and can be expressed as(7)εpl=λ∂Gσ¯∂σ¯,where λ and G are the non-negative flow factor and the plastic potential of plastic flow, respectively.(6) Damage and Stiffness Degradation. The uniaxial stress-strain relationship can be expressed in terms of the stress-plastic strain relationship as(8)σt=σtε¯tpl,T,fi,σc=σcε¯cpl,T,fi,where the subscripts t and c represent the tension and compression, respectively; T is the temperature, and fi,i=1,2,3… are other predefined variables.As shown in Figures2 and 3, when the concrete material is unloaded from a certain point in the softening stage of the stress-strain curve, the unloading response becomes weak, that is to say, the elastic stiffness of the material is damaged (reduced). The damage to elastic stiffness differs significantly in tensile and compression tests, with the damage effect becoming more pronounced as the plastic strain increases. The damage response of concrete is represented by two independent uniaxial damage variables, dt and dc, which are the functions of plastic strain, temperature, and field variables:(9)dt=dtε¯tpl,T,fi0≤dt≤1,dc=dcε¯cpl,T,fi0≤dc≤1.Figure 2
Stress-strain curve of concrete subjected to uniaxial tensile load.Figure 3
Stress-strain curve of concrete under uniaxial compressive load.The uniaxial stiffness degradation variable is a nondecreasing function of the equivalent plastic strain, and its value ranges from 0 to 1. IfE0 represents the initial elastic stiffness of the material, then the stress-strain relationship under uniaxial tension and compression is(10)σt=1−dtE0εt−ε¯tpl,σc=1−dcE0εc−ε¯cpl.In the uniaxial case, the crack development direction is perpendicular to the stress direction. The effective stress is increased due to the reduction of the effective interface area induced by the formation and propagation of cracks. In the uniaxial case, the effect of this reduction in bearing area is slightly better. The reason for this phenomenon is that the crack propagates parallel to the direction of stress at the beginning. However, when the crushing progresses more severely, the effective bearing area will significantly decrease. Hence, the effective uniaxial cohesionσ¯t and σ¯c can be written as(11)σ¯t=σt1−dt=E0εt−ε¯tpl,σ¯c=σc1−dc=E0εc−ε¯cpl.(7) Yield Condition. The yield condition of this model was established based on the yield function suggested by Lubliner et al. [21]. Then the evolution laws of tension and compression at different strengths are considered, and the yield function can be expressed in terms of effective stress as(12)Fσ¯,ε¯pl=11−αq¯−3αp¯+βε¯plσmax−γσmax−σ¯cε¯cpl≤0,in which(13)p¯=13σ¯;q¯=32S¯,where α and γ are nondimensional parameters, and S¯ is Mises equivalent stress.


## 2.2. Establishment of the Finite Element Numerical Analysis Model

Figure4 shows the simply supported beam subjected to two symmetrically concentrated loads. It should be pointed out that the corresponding midspan of the beam is under the purely bending action (i.e., no shear force). Based on this simply supported member structure, the finite element numerical analysis model (see Figure 5) is established via the ABAQUS software. During the laboratory test, the material parameters of the poured concrete are strength grade C30, material density ρ = 2400 kg/m3, design value of compressive strength fc = 14.3 MPa, design value of tensile strength ft = 1.43 MPa, elastic modulus E = 30 GPa, Poisson’s ratio ν = 0.2, and length of the member L = 1.5 m. In the numerical analysis, unless otherwise stated, the material parameters of concrete adopt the above values. Besides, the plastic parameters of concrete are defined as the expansion angle, eccentricity, ratio of biaxial to uniaxial compressive strength, and viscosity parameter are selected as 20°, 0.1, 1.16, and 0.0001, respectively; the morphological parameter K that affects the yield surface is fixed at 2/3. Finally, based on the basic material parameters and plastic parameters of the concrete, the stress and strain relationships and damage parameters are directly generated in the ABQUAS software.Figure 4
Simply supported beam under two symmetrically vertical loads.Figure 5
Diagram of the mesh division.Nodes on the slave surface cannot invade the master surface, but nodes on the master surface invade the slave surface between nodes on the slave surface in ABAQUS software. Generally, slave surface is with finer meshes. When the mesh density is the same, the salve surface is the surface of softer material. The concrete is the master surface due to the larger grid, and the steel tube is slave surface.The interaction between contact surfaces involves two parts, one is the normal action between contact surfaces, and the other is the tangential action between contact surfaces. The “hard contact” is selected to describe the normal action and the tangential action in ABAQUS software. The steel tube and solid concrete adopt hexahedral elements with the type being C3D8R. In the numerical analysis, the displacement controlled loading method is adopted, and the loading is ceased as the maximum bending moment that the CFMST pile can bear is achieved. In general engineering practice, the displacement of the control pile is 20 mm, and the bending moment corresponding to the displacement of the pile 20 mm is selected as the ultimate flexural bearing capacity.


## 3. Laboratory Test and Model Verification

In order to verify the reliability of the developed finite element analysis model, the laboratory test was carried out. The details of the laboratory test, including the specimen size, test devices, and test process is discussed as follows.


### 3.1. Specimen Dimension

Concrete strength grade, thickness of steel tube, diameter of the piles, and yield strength of steel are the main factors affecting the bending strength of CFMST pile. The size of seamless steel tube commonly used in engineering is selected as listed in Table1. Furthermore, the specimen preparation process mainly includes concrete mixing, pouring, vibration, curing, and demolding. The concrete should be evenly distributed during vibration and maintained for 28 days.Table 1
Dimension of CFMST piles.
No.L (m)Concrete strength gradeD (mm)t (mm)fy (MPa)CFMST pile 11.5C301404235CFMST pile 21.5C301685235Note. L and D denote the length and diameter of the CFMST pile, respectively; t and fy are the thickness and yield strength of steel tube, respectively.


### 3.2. Test Devices and Process

The devices used in the test contain the YAS-2000 pressure tester (see Figure6), dial indicator, and DH3818-2 static strain tester. During the test, the CFMST pile should be placed under the test bench of YAS-2000 pressure tester, and the vertical pressure generated by the pressure pump is at L/3 location of the CFMST pile (see Figure 3). The test pressure, displacement of the pressure pump, and the test time can be recorded by the computer control system. The longitudinal strain of CFMST pile under the action of vertical forces can be collected by the DH3818-2 static strain tester. The strain is tested using a quarter bridge strain gauge with the strain gauges being installed on the top, right, and bottom of the cross section (see Figure 7).Figure 6
Diagram of YAS-2000 pressure tester.Figure 7
Schematic diagram of the layout of strain gauges.The whole test contains the following seven steps: (1) Stick the strain gauges on the surface of the specimen according to the connection method of 1/4 bridge, and record the position of the strain gauge; (2) lift the test specimen onto the test bench of YAS-2000 testing machine and adjust the position of the test piece; (3) connect the pasted strain gauge lead with DH3818-2 static strain gauge, and then number each group of strain gauges to facilitate later reading; (4) install and adjust the dial indicator; (5) check whether the above types of work are completed correctly; (6) set the test specimen parameters and load parameters on the computer control system of YAS-2000 pressure testing machine; and (7) start the test and collect the corresponding data till the specimen fails. It is noted that in the initial stage of the test, there exists a process from loose contact to perfect bonding between the loading device and specimen which may cause some noisy data and minor unreal plastic region. Hence, these data should be carefully checked and treated, and we should also minimize the gap between the loading device and specimen. Moreover, the final failure status of one specimen is shown in Figure8.Figure 8
Specimen after failure.


### 3.3. Verification of the Developed Finite Element Analysis Model

Figure9 shows the comparison of the deflection in the central section between numerical results and laboratory test data. It is stated again that in general engineering practice, the bending moment corresponding to the deflection of the pile 20 mm is taken as the ultimate flexural bearing capacity (Mu). For CFMST pile 1, Mu from the experiment and the numerical calculation are 54.7 kN·m and 54.4 kN·m, respectively. For CFMST pile 2, the corresponding Mu are 81.0 kN·m and 81.3 kN·m, respectively. It can be found that the numerical results have good agreement with the experimental data, which indicates that the developed finite element analysis model can be applied into the engineering practice. Moreover, for CFMST piles 1 and 2, the slope of the hardened section of the experimental results is greater than that of the numerical results, and the corresponding moment of the experimental results is also higher than that of the numerical results.Figure 9
Comparison of the deflection in the central section between experimental data and numerical results.
(a)(b)


## 3.1. Specimen Dimension

Concrete strength grade, thickness of steel tube, diameter of the piles, and yield strength of steel are the main factors affecting the bending strength of CFMST pile. The size of seamless steel tube commonly used in engineering is selected as listed in Table1. Furthermore, the specimen preparation process mainly includes concrete mixing, pouring, vibration, curing, and demolding. The concrete should be evenly distributed during vibration and maintained for 28 days.Table 1
Dimension of CFMST piles.
No.L (m)Concrete strength gradeD (mm)t (mm)fy (MPa)CFMST pile 11.5C301404235CFMST pile 21.5C301685235Note. L and D denote the length and diameter of the CFMST pile, respectively; t and fy are the thickness and yield strength of steel tube, respectively.


## 3.2. Test Devices and Process

The devices used in the test contain the YAS-2000 pressure tester (see Figure6), dial indicator, and DH3818-2 static strain tester. During the test, the CFMST pile should be placed under the test bench of YAS-2000 pressure tester, and the vertical pressure generated by the pressure pump is at L/3 location of the CFMST pile (see Figure 3). The test pressure, displacement of the pressure pump, and the test time can be recorded by the computer control system. The longitudinal strain of CFMST pile under the action of vertical forces can be collected by the DH3818-2 static strain tester. The strain is tested using a quarter bridge strain gauge with the strain gauges being installed on the top, right, and bottom of the cross section (see Figure 7).Figure 6
Diagram of YAS-2000 pressure tester.Figure 7
Schematic diagram of the layout of strain gauges.The whole test contains the following seven steps: (1) Stick the strain gauges on the surface of the specimen according to the connection method of 1/4 bridge, and record the position of the strain gauge; (2) lift the test specimen onto the test bench of YAS-2000 testing machine and adjust the position of the test piece; (3) connect the pasted strain gauge lead with DH3818-2 static strain gauge, and then number each group of strain gauges to facilitate later reading; (4) install and adjust the dial indicator; (5) check whether the above types of work are completed correctly; (6) set the test specimen parameters and load parameters on the computer control system of YAS-2000 pressure testing machine; and (7) start the test and collect the corresponding data till the specimen fails. It is noted that in the initial stage of the test, there exists a process from loose contact to perfect bonding between the loading device and specimen which may cause some noisy data and minor unreal plastic region. Hence, these data should be carefully checked and treated, and we should also minimize the gap between the loading device and specimen. Moreover, the final failure status of one specimen is shown in Figure8.Figure 8
Specimen after failure.


## 3.3. Verification of the Developed Finite Element Analysis Model

Figure9 shows the comparison of the deflection in the central section between numerical results and laboratory test data. It is stated again that in general engineering practice, the bending moment corresponding to the deflection of the pile 20 mm is taken as the ultimate flexural bearing capacity (Mu). For CFMST pile 1, Mu from the experiment and the numerical calculation are 54.7 kN·m and 54.4 kN·m, respectively. For CFMST pile 2, the corresponding Mu are 81.0 kN·m and 81.3 kN·m, respectively. It can be found that the numerical results have good agreement with the experimental data, which indicates that the developed finite element analysis model can be applied into the engineering practice. Moreover, for CFMST piles 1 and 2, the slope of the hardened section of the experimental results is greater than that of the numerical results, and the corresponding moment of the experimental results is also higher than that of the numerical results.Figure 9
Comparison of the deflection in the central section between experimental data and numerical results.
(a)(b)


## 4. Numerical Results and Discussions

The main factors affecting the flexural strength of CFMST pile are concrete strength grade, wall thickness of steel tube, external diameter of pile, and yield strength of the steel. Meanwhile, considering the common size specification of seamless steel tube, the detailed numerical simulation scheme is carefully planned as shown in Table2.Table 2
Numerical simulation scheme.
Influencing factorMember numberConcrete strength gradeD (mm)t (mm)fy (MPa)Concrete strength gradeSC30C301687235SC50C50SC70C70Wall thickness of steel tubeST1C301685235ST26ST37External diameter of steel tubePD1C301407235PD2168PD3219Yield strength of steelQ235C301687235Q345345Q390390Based on the aforementioned numerical analysis model via ABAQUS software, the influence of four affecting factors on the flexural behavior of the concrete-filled micro-steel-tube pile is discussed.


### 4.1. Influence of Concrete Strength Grade

Figure10 shows the influence of concrete grade on the bending moment in the central section vs. deflection. It can be seen from Figure 10 that the concrete grade has negligible influence on the bending moment-deflection curve when the deflection is small, while it has obvious influence on the ultimate flexural strength. It can be also observed that the ultimate flexural strength gradually increases with increasing concrete strength grade.Figure 10
Effect of concrete grade on the bending moment in the central section vs. deflection.To further present the contribution of the concrete and steel tube in the bending capacity, we consider the following three cases with same length and external diameter, i.e., Case 1 CFMST pile, Case 2 hollow steel tube, and Case 3 solid concrete. The used calculation parameters areL = 1500 mm, D = 168 mm, t = 7.0 mm, fy = 235 MPa, and degree of concrete C30. It can be seen from Figure 11 that the ultimate bending moment value of Case 1, Case 2, and Case 3 are, respectively, 102.91 kN·m, 82.74 kN·m, and 20.17 kN·m. That is to say, the bending strength of the composite material pile is the summation of that of the hollow steel tube and solid concrete. Besides, the bending strength ratios of steel tube and solid concrete to the composite material pile is about 80% and 20%, respectively. This means that for the composite material pile, the steel tube makes the most contributions on the flexural strength, and the solid concrete can improve the flexural strength in certain degree.Figure 11
Variation of bending moment with deflection at midspan of the member for three cases.


### 4.2. Influence of the Wall Thickness of the Steel Tube

Figure12 shows the influence of wall thickness of steel tube on the bending moment vs. deflection and ultimate flexural strength. It can be seen from Figure 12 that wall thickness has significant influence on the development of bending moment vs. deflection. The bending moment and ultimate flexural strength increase greatly with increasing wall thickness for the fixed deflection. Particularly, the ultimate flexural strength increases linearly with increasing wall thickness, which could provide useful guidance for the engineering design.Figure 12
Effect of wall thickness of steel tube on the bending moment at midspan vs. deflection.


### 4.3. Influence of the External Diameter of the Steel Tube

Figure13 shows the influence of the external diameter of steel tube on the bending moment at midspan vs. deflection and the ultimate flexural strength. It can be seen from Figure 13 that the bending moment and the ultimate flexural strength increase remarkably with the increasing external diameter. The main reason for this phenomenon is that flexural rigidity increases greatly with increasing external diameter of the steel tube, resulting in the improved flexural bearing capacity.Figure 13
Effect of external diameter of steel tube on the bending moment at midspan vs. deflection.


### 4.4. Influence of Yield Strength of Steel

Figure14 shows the influence of the external diameter of steel tube on the bending moment vs. deflection. It is found in Figure 14 that in the elastic stage (i.e., the deflection is small), the yield strength of steel has negligible influence on the bending moment. However, when it is in the elastoplastic stage, the bending moment increases with increasing yield strength of the steel. It can be concluded that increasing the yield strength of the steel can effectively improve the flexural bearing capacity of the CFMST pile.Figure 14
Effect of yield strength of steel on the bending moment at midspan vs. deflection.


## 4.1. Influence of Concrete Strength Grade

Figure10 shows the influence of concrete grade on the bending moment in the central section vs. deflection. It can be seen from Figure 10 that the concrete grade has negligible influence on the bending moment-deflection curve when the deflection is small, while it has obvious influence on the ultimate flexural strength. It can be also observed that the ultimate flexural strength gradually increases with increasing concrete strength grade.Figure 10
Effect of concrete grade on the bending moment in the central section vs. deflection.To further present the contribution of the concrete and steel tube in the bending capacity, we consider the following three cases with same length and external diameter, i.e., Case 1 CFMST pile, Case 2 hollow steel tube, and Case 3 solid concrete. The used calculation parameters areL = 1500 mm, D = 168 mm, t = 7.0 mm, fy = 235 MPa, and degree of concrete C30. It can be seen from Figure 11 that the ultimate bending moment value of Case 1, Case 2, and Case 3 are, respectively, 102.91 kN·m, 82.74 kN·m, and 20.17 kN·m. That is to say, the bending strength of the composite material pile is the summation of that of the hollow steel tube and solid concrete. Besides, the bending strength ratios of steel tube and solid concrete to the composite material pile is about 80% and 20%, respectively. This means that for the composite material pile, the steel tube makes the most contributions on the flexural strength, and the solid concrete can improve the flexural strength in certain degree.Figure 11
Variation of bending moment with deflection at midspan of the member for three cases.


## 4.2. Influence of the Wall Thickness of the Steel Tube

Figure12 shows the influence of wall thickness of steel tube on the bending moment vs. deflection and ultimate flexural strength. It can be seen from Figure 12 that wall thickness has significant influence on the development of bending moment vs. deflection. The bending moment and ultimate flexural strength increase greatly with increasing wall thickness for the fixed deflection. Particularly, the ultimate flexural strength increases linearly with increasing wall thickness, which could provide useful guidance for the engineering design.Figure 12
Effect of wall thickness of steel tube on the bending moment at midspan vs. deflection.


## 4.3. Influence of the External Diameter of the Steel Tube

Figure13 shows the influence of the external diameter of steel tube on the bending moment at midspan vs. deflection and the ultimate flexural strength. It can be seen from Figure 13 that the bending moment and the ultimate flexural strength increase remarkably with the increasing external diameter. The main reason for this phenomenon is that flexural rigidity increases greatly with increasing external diameter of the steel tube, resulting in the improved flexural bearing capacity.Figure 13
Effect of external diameter of steel tube on the bending moment at midspan vs. deflection.


## 4.4. Influence of Yield Strength of Steel

Figure14 shows the influence of the external diameter of steel tube on the bending moment vs. deflection. It is found in Figure 14 that in the elastic stage (i.e., the deflection is small), the yield strength of steel has negligible influence on the bending moment. However, when it is in the elastoplastic stage, the bending moment increases with increasing yield strength of the steel. It can be concluded that increasing the yield strength of the steel can effectively improve the flexural bearing capacity of the CFMST pile.Figure 14
Effect of yield strength of steel on the bending moment at midspan vs. deflection.


## 5. Proposed Calculation Method for Flexural Bearing Capacity

At present, the determination of the ultimate flexural capacity of the CFMST pile is mostly based on the unified theory of concrete-filled steel tube. However, in the unified theory, the determination of the ultimate flexural bearing capacity does not distinguish the function of the steel tube and concrete [22]. Hence, in the present study, we further distinguish the steel tube from the concrete based on the results of finite element numerical analysis, and the expression of the flexural bearing capacity of CFMST pile can be written as(14)Mu=KsMs+KcMc,where Mu is the ultimate flexural capacity of CFMST pile; Ks and Kc are the influencing coefficient due to hoop effect; Ms and Mc are, respectively, the flexural bearing capacity characteristic value of steel tube and concrete, which can be expressed as (i = s, c)(15)Mi=fiWi,where fs and fc are the yield strength value of steel and the standard value of the cube compressive strength of concrete, respectively; Ws and Wc are moment of resistance of steel and concrete.Making use of the fundamental calculation formula proposed by the technical code for concrete-filled steel tubular structures [22], and the finite element numerical results corresponding to Table 2, the fitted ultimate bending strength formula for the CFMST pile can be defined as(16)Mu=3.62Ms+0.561.15+αsθ+αcθ2Mc,in which(17)θ=fAsfcAc;αs=0.175f212+0.973;αc=−0.102fc14.3+0.032,where θ is the hooping coefficient; As and Ac are the cross-sectional areas of steel tube and concrete, respectively; αs, αc are the material influencing coefficient; and f is design value of the compressive strength of steel tube.Table3 shows the comparison of the calculation results by fitted equation (16) with the numerical results corresponding to the numerical scheme given in Table 2. The symbol “| |” denotes the absolute value of the relative error. It can be seen from Table 3 that the results from the developed formula agree well with the numerical results (e.g., the absolute value of relative error is less than 5%), which indicates that the fitted equation (16) is reliable. Then, we further compare the result from equation (16) with that from experiment and unified theory for member ST1. Mu gained from fitted equation (20), experiment (i.e., CFMST pile 2), and unified theory [22] are equal to 80.55 kN·m, 81.0 kN·m, and 89.52 kN·m, respectively. The above comparison shows that the result from fitted equation (16) is very close to that from the experiment, and both two values are lower than the result from the unified theory, indicating that the proposed calculation method is reliable and could be directly applied into engineering design.Table 3
Comparison of the numerical results with those from proposed formula.
Member numberResults from equation (16) (kN·m)Numerical results (kN·m)|Relative error| (%)SC30105.88102.912.89SC50110.33110.380.05SC70114.87118.983.45ST180.5581.340.97ST293.4590.353.43ST3105.88102.912.89PD170.9167.574.94PD2105.88102.912.89PD3189.12197.964.47Q235105.88102.912.89Q345151.72146.163.81Q390170.39177.053.76


## 6. Conclusions

In this paper, we establish the 3D numerical analysis model via the ABAQUS software to analyze the flexural behavior of the CFMST pile. In the numerical model, the plastic deformation of the steel material and the plastic damage of the concrete are taken in consideration. After verifying the reliability and accuracy of the 3D numerical model, we have directly applied it to study the influence of the concrete strength grade, wall thickness of the pile, external diameter of the pile, and yield strength of the steel on the flexural behavior of the composite material pile. Finally, the concise calculation formula for the ultimate flexural bearing capacity is also proposed and validated through comparison with the results from numerical analysis model, laboratory test, and unified theory. The main conclusions can be drawn as follows:(1)
The ultimate flexural strength gradually increases with increasing concrete strength grade: Wall thickness and external diameter of the steel tube have significant influence on the development of bending moment vs. deflection. The bending moment and ultimate flexural strength increase greatly with increasing wall thickness or external diameter.(2)
The yield strength of steel has negligible influence on the bending moment when the deflection is small: However, the bending moment increases greatly with increasing yield strength of the steel when the deflection is large.(3)
Based on the results of the finite element numerical calculations, the expression of the ultimate flexural bearing capacity of the concrete-filled micro-steel-tube pile is proposed. After careful comparison, it is found that this newly developed calculation formula could be directly applied into the flexural bearing capacity design of the concrete-filled micro-steel-tube pile.



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*Source: 2317594-2022-10-17.xml*