# Research on Dynamic Response of Concrete-Filled Steel Tube Columns Confined with FRP under Blast Loading

Authors: Jing Dong; Junhai Zhao; Dongfang Zhang; Yingping Li
Journal: Shock and Vibration
 (2019)

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## Abstract

Recently, a concrete-filled steel tube confined with fiber-reinforced polymer (FRP) has become a hot research issue as a new type of structure. These studies mainly focus on its static performance and seismic and impact behaviour, with little research on its blast resistance performance. In this study, the dynamic response of concrete-filled steel tube columns confined with FRP under blast loading was investigated. Numerical analysis was implemented using multimaterial ALE method in the finite element analysis program LS-DYNA. The proposed numerical model was validated by the SDOF result and available experimental data. And the effects of the number of FRP layers, concrete strength, and cross section were also discussed in detail based on the proposed numerical model. The results indicate that the constraints of FRP effectively enhance the blast resistance of the column, and the vulnerable parts mainly occur at the middle and two ends of the column. The blast resistance of the column can be enhanced by increasing the number of FRP layers or concrete strength. These results could provide a certain basis for blast resistance design of concrete-filled steel tubes confined with FRP.


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## Body

## 1. Introduction

In recent years, accidents caused by explosions occurred frequently. It has become an important and urgent task to prevent the structural damage and collapse of buildings under blast loading. Therefore, it is of great significance to research the dynamic response, failure modes, and damage mechanisms of building structures to improve the blast resistance performance of these structures and reduce accident losses. Concrete-filled steel tubes (CFSTs) have widely been studied and applied in civil engineering as a classical composite structure. A concrete-filled steel tube has the advantages of ease of construction, high bearing capacity, and excellent ductility [1–3]. However, in some high corrosive environments, corrosion protection of the steel tube is required, which increases the maintenance costs. Fiber-reinforced polymer (FRP) is a high-performance composite material which is composed of certain proportions of fiber material and resin matrix material. FRP has widely been used in engineering practice due to its high strength-to-weight ratios and corrosion resistance [4, 5]. Recently, a new type of structure based on CFST, CFST confined with FRP, has been proposed. CFST column confined with FRP is a composite structure of an FRP composite material and concrete-filled steel tube (Figure 1). When the concrete is confined to the steel tube, FRP is wrapped around the whole steel tube. On the one hand, the CFST column confined with FRP can be used as a new type of structure in new buildings. On the other hand, the FRP material can also be used as a repair material to reinforce existing CFST structures.Figure 1
Concrete-filled steel tube confined with FRP.Xiao et al. [6] proposed the concept of a confined concrete-filled steel tube column on the basis of concrete-filled steel tube. Taking CFST columns confined with carbon fiber-reinforced polymer (CFRP) as an example, an axial compression test and a seismic test were carried out. The test results showed that the axial bearing capacity and seismic performance of CFST columns confined with CFRP were significantly improved compared to the traditional CFST column. This new structure could be used in high-rise buildings and bridges in seismic areas. Later, many researchers focused on the ultimate bearing capacity and working mechanisms, and compression and bending experiments were implemented [7–14]. The results showed that the bearing capacity and stiffness of CFST columns confined with FRP were greatly improved compared with CFST columns. As to the studies on CFST columns confined with FRP under impulsive load, Alam et al. [15, 16] and Chen et al. [17] carried out lateral impact tests and numerical simulation. The deflection and impact force were observed, and the effect of impact velocity, vehicle mass, and FRP thickness was investigated. The results indicated that using FRP could reduce the maximum and residual displacement. Nevertheless, there is little research on the dynamic response of concrete-filled steel tube columns confined with FRP under blast loading. Although the blast load is also a type of impulsive load, the strain rate under seismic load and impact load is 10−3∼10−1 s−1 and 10−2∼101 s−1, respectively, while the strain rate under blast load is within the range from 102 to 106 s−1 [18]. Thus, the blast response of CFST columns confined with FRP still needs to be investigated.The explosion of a structure is a typical nonlinear transient dynamic issue. It is quite difficult to solve the problem of explosion mechanics using theoretical analysis methods. The experimental method not only is expensive but also cannot reflect the state of the explosion process. Numerical simulations can greatly save experimental expenses and can also observe the data and phenomena that cannot be obtained in experiments. Numerical simulation is consequently often used in the research of structural blast resistance [19]. Mutalib and Hao [20] modelled an FRP-strengthened reinforced concrete panel with additional anchorages against blast loads and analysed the dynamic response of FRP-strengthened reinforced concrete wall with different anchorage systems. Mao et al. [21] modelled an ultrahigh performance fiber-reinforced concrete panel under blast loading. The strain rate effect of the material was taken into account in the model. Zhang et al. [22] modelled concrete-filled steel tube columns under static and blast loads. The numerical results indicated that the CFST columns showed good flexural behaviour under both static and dynamic loads. In the aforementioned studies, the numerical results showed a good agreement with the corresponding experimental results, which validated the reliability of the numerical model.In this study, the dynamic response of CFST columns confined with FRP under blast loading is investigated. In Section2, a finite element (FE) model is developed using the arbitrary Lagrange–Euler (ALE) method. In Section 3, the validation of the proposed model is presented. In Section 4, the numerical results and parameter study are discussed. Conclusions are described in Section 5.


## 2. FE Model of CFST Column Confined with FRP

LS-DYNA is a well-known explicit dynamic analysis program and is one of the most commonly used software programs for numerical simulations of blast performance of structures. In this study, the g-mm-ms-MPa unit system is adopted in modelling, and a typical CFST column confined with FRP under blast loading is modelled using the commercial software ANSYS/LS-DYNA.


### 2.1. Description of Model

The column is 3000 mm high, the outer diameter of the steel tube is 500 mm, the diameter of the core concrete is 480 mm, and the thickness of the steel tube is 10 mm. The CFST column is wrapped by using a CFRP sheet with a thickness of 0.167 mm. Both ends of the column are constrained. The explosive is 300 mm × 300 mm × 300 mm TNT (mass equivalent to 44.28 kg). The scaled distance isz=L/M3=0.28m/kg1/3. The size of the air domain is 2500 mm × 2000 mm × 3600 mm. The TNT, air, steel tube, and concrete are all solid elements, and the FRP is composed of shell elements. The front view and the top view of the model are shown in Figure 2, and the completed 3D finite element model is shown in Figure 3.Figure 2
Model of the CFST column confined with FRP under blast loading. (a) Front view. (b) Top view.
(a)
(b)Figure 3
3D finite element model.


### 2.2. Material Models

#### 2.2.1. Material Model and Equation of State for Air

Material type 9 (MAT_NULL) in LS-DYNA was used to model the air, and EOS type 1 (EOS_LINEAR_POLYNOMIAL) was chosen to define the linear polynomial equation of state of air. The pressure is given by [23](1a)P=C0+C1μ+C2μ2+C3μ3+C4+C5μ+C6μ2E,(1b)μ=1V−1,where P is the pressure, C0∼C6 are the polynomial equation coefficients, V is the relative volume, and E is the internal energy per unit reference specific volume. When simulating air, the main variables and values are shown in Table 1.Table 1
Variables of air.
Variable
Density
C
0
C
1
C
2
C
3
C
4
C
5
C
6
E
0
Unit
g/mm3
—
—
—
—
—
—
—
J/mm3
Value
1.29 × 10−6
0
0
0
0
0.4
0.4
0
0.25


#### 2.2.2. Material Model and Equation of State for TNT

Material type 8 (MAT_HIGH_EXPLOSIVE_ BURN) in LS-DYNA was used to model TNT. This model is particularly suitable for modelling the detonation of a high explosive. The JWL (EOS_JWL) equation was used to define the equation of state for TNT. The JWL equation of state defines the pressure as(2)p=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωEV,where p is the explosive pressure; V is the relative volume; E0 is the initial internal energy per unit reference specific volume and ω; A, B, R1, and R2 are the material constants. When simulating TNT, the main variables and values are shown in Table 2, where D refers to the detonation velocity and PCJ represents the Chapman–Jouguet pressure.Table 2
Variables of TNT.
Variable
Density
D
PCJ
A
B
R
1
R
2
E
0
Unit
g/mm3
mm/ms
MPa
MPa
MPa
—
—
J/mm3
Value
1.64 × 10−3
6930
2.1 × 105
3.74 × 105
3230
4.15
0.95
7000


#### 2.2.3. Material Model for Steel Tube

Material type 3 (MAT_PLASTIC_KINEMATIC) in LS-DYNA was used to represent the steel tube. This model is a kinematic hardening plasticity model considering the strain rate effect, which is suitable for simulating steel material with a high strain rate. This material model is based on the Cowper–Symonds model. The yield stress is given by [24](3)σy=1+ε˙C1/pσ0+βEpεpeff,where ε˙ is the strain rate, C and p are the strain rate parameters, σ0 is the initial yield stress, β is the hardening parameter (for kinematic hardening, β=0; for isotropic hardening, β=1), Ep is the plasticity hardening modulus, and εpeff is the effective plastic strain. In this paper, the variable values of steel are listed in Table 3.Table 3
Variables of the steel material model.
Variable
RO
E
PR
SIGY
ETAN
BETA
SRC
SRP
FS
VP
Unit
g/mm3
MPa
—
MPa
MPa
—
—
—
—
—
Value
7.83 × 10−3
2.08 × 105
0.3
292.5
2100
0
40
5
0.25
RO: mass density;E: Young’s modulus; PR: Poisson’s ratio; SIGY: yield stress; ETAN: tangent modulus; BETA: hardening parameter; SRC and SRP: strain rate parameters; FS: effective plastic strain for eroding elements; VP: formulation for rate effects.


#### 2.2.4. Material Model for Concrete

Concrete was modelled using the material type 111 (MAT_JOHNSON_HOLMQUIST_CONCRETE) in LS-DYNA. This model could be applied to simulate the concrete material under high strain, high strain rate, and high pressure. The Johnson–Holmquist–Cook model is a rate-dependent damage constitutive model proposed by Holmquist et al. [25]. The equivalent yield strength is given by(4a)σ∗=A1−D+Bp∗N1+Clnε˙ε˙0,where σ∗=σ/fc is the normalized equivalent stress; p∗=p/fc is the normalized pressure; ε˙ is the strain rate; ε˙0=1.0s−1 is the reference strain rare; and A, B, C, and N are the normalized cohesive strength, normalized pressure hardening, strain rate coefficient, and pressure hardening exponent, respectively. D represents the damage parameter and is expressed as [26](4b)D=∑Δεp+Δμpεpf+μpf=∑Δεp+ΔμpD1p∗+T∗D2,where Δεp and Δμp are the equivalent plastic strain and plastic volumetric strain, T∗=T/fc denotes the normalized maximum tensile hydrostatic pressure, and D1 and D2 are the material damage constants.The Johnson–Holmquist–Cook model takes into account the effects of large strain, high strain rate, and damage accumulation of concrete. It can describe the mechanical behaviour of concrete under explosion and high-speed impact and penetration [27]. In this paper, the variable values of the concrete are shown in Table 4.Table 4
Variables of the concrete material model.
Variable
RO
G
A
B
C
N
FC
T
EPS0
EFMIN
SFMAX
PC
UC
PL
UL
D
1
D
2
K
1
K
2
K
3
Unit
g/mm3
MPa
—
—
—
—
MPa
MPa
ms−1
—
MPa
MPa
—
MPa
—
—
—
MPa
MPa
MPa
Value
2.4 × 10−3
14860
0.79
1.6
0.007
0.61
40
4
0.001
0.01
7
13
0.001
800
0.1
0.04
1
8.5 × 104
−1.71 × 105
2.08 × 105
G: shear modulus; FC: quasi-static uniaxial compressive strength; T: maximum tensile hydrostatic pressure; ESP0: reference strain rate; EFMIN: plastic strain before fracture; SFMAX: normalized maximum strength; PC: crushing pressure; UC: crushing volumetric strain; PL: locking pressure; UL: locking volumetric strain; K1, K2, and K3: pressure constants.


#### 2.2.5. Material Model for FRP

A linear elastic material model (MAT_ELASTIC) was used to simulate FRP. FRP sheets can be considered to be in the elastic stage before breakage, so its constitutive relation can be simulated by the linear elastic stress-strain relation [28, 29]. In this paper, the variable values of FRP are listed in Table 5.Table 5
Variables of the FRP material model.
Variable
Density
Young’s modulus
Poisson’s ratio
Unit
g/mm3
MPa
—
Value
1.79 × 10−3
2.49 × 105
0.285


### 2.3. Strain Rate Effect

The dynamic behaviour of the concrete material is strain rate dependent. Both the compressive and tensile strengths vary with the strain rate. Recommended by Committee Euro-International du Béton (CEB), the dynamic increase factor (DIF) is defined as the ratio of dynamic strength to static strength. For concrete in compression, the equation of DIF is given by [30](5)CDIF=fcdfcs=ε˙ε˙s1.026α,ε˙≤30s−1,γε˙ε˙s1/3,ε˙>30s−1,where ε˙ is the strain rate, ε˙s=3×10−5s−1, α=1/5+9fcs/10, and lgγ=6.156α−2.For concrete in tension, the equation of DIF is given by(6)TDIF=ftdfts=ε˙ε˙sδ,ε˙≤1s−1,βε˙ε˙s1/3,ε˙>1s−1,where δ=1/1+8fcs/10 and lgβ=6δ−2.The DIF for yield strength of steel is adopted by [31](7)DIF=fydfys=1+ε˙C1/p,where C and p are the strain rate parameters mentioned above.Compared with steel and concrete materials, the strain rate effect of FRP is insignificant [32]. Thus, the strain rate effect of FRP could be ignored in numerical simulations [33].


### 2.4. Mesh Size

When simulating the propagation of blast wave and its interaction with the structure, the accuracy of simulation results depends largely on the element mesh size. In order to find the most proper mesh size, the model of 44.28 kg TNT in the 3200 mm × 3200 mm × 3200 mm air domain was developed and the mesh size of 20 mm, 25 mm, 30 mm, and 50 mm was adopted for simulation, respectively. The overpressure values of the four models at different distances were compared with the empirical curve from UFC 3-340-02 [34]. As shown in Figure 4, the mesh size has certain influence on the peak overpressure of the shock wave. The accuracy of the numerical simulation performs better with the smaller mesh size. The overpressure value of 25 mm is close to the results of 20 mm. However, the computation amount and analysing time increase enormously than that of 25 mm, which leads to the risk of computer memory overflow. In addition, the simulation converged well when the mesh size is 25 mm. Therefore, the mesh size of 25 mm is employed in this paper for the dual consideration of simulation accuracy and computational efficiency.Figure 4
Comparison of different mesh sizes.


### 2.5. Generation of Blast Loading

In LS-DYNA, there are three common ways to generate blast loading. The first way is to simplify the blast load, which is usually simplified to an idealized triangular ramp pressure. The load curve is defined by the keyword∗LOAD_CURVE in LS-DYNA. This method is simple in modelling, relatively high in computational efficiency, and suitable for dynamic response analysis of structural members with various distances. However, the accuracy of this method is relatively low, and it cannot simulate the propagation process of a blast wave. The second way utilizes the keyword ∗LOAD_BLAST in LS-DYNA to generate blast loading. This method requires inputting the equivalent mass of TNT, coordinates of detonation, type of blast, and other parameters in the keyword card. It has a wide application scope and the calculation results are relatively reasonable, but the shape of the explosive and the influence of obstacles on the structure cannot be taken into consideration. The third way is the fluid-structure coupling method, which requires establishing parts of air, explosive, and member. Multiple materials can be coupled by the arbitrary Lagrange–Euler (ALE) method. This method can accurately simulate the formation and propagation of the blast wave, and explosive shape and obstacles can be considered. The disadvantage of this method is that the modelling is rather complex. The computational efficiency is low due to the large computational amount. The third way, the ALE method, was applied in this study. By adding the ∗INITIAL_DETONATION keyword, the explosion is detonated at the center point of the TNT.


### 2.6. Boundaries and Erosion

With the consideration of reflection effect, a rigid plane was added to the bottom surface of the air domain using the∗RIGIDWALL_PLANAR keyword. The remaining surfaces of air domain were subjected to nonreflection boundaries. The fluid-structure interaction was processed using the ∗CONSTRAINED_LAGRANGE_IN_SOLID keyword. The nodes linked among steel tube, concrete, and FRP mesh were shared. The erosion algorithm was implemented using the ∗MAT_ADD_EROSION keyword. Under the extreme loading, the element of material is deleted when it reaches a certain value. Based on the previous studies, a principle tensile strain of 0.001 is adopted to simulate the failure of concrete as the primary criterion in this study [35].


## 2.1. Description of Model

The column is 3000 mm high, the outer diameter of the steel tube is 500 mm, the diameter of the core concrete is 480 mm, and the thickness of the steel tube is 10 mm. The CFST column is wrapped by using a CFRP sheet with a thickness of 0.167 mm. Both ends of the column are constrained. The explosive is 300 mm × 300 mm × 300 mm TNT (mass equivalent to 44.28 kg). The scaled distance isz=L/M3=0.28m/kg1/3. The size of the air domain is 2500 mm × 2000 mm × 3600 mm. The TNT, air, steel tube, and concrete are all solid elements, and the FRP is composed of shell elements. The front view and the top view of the model are shown in Figure 2, and the completed 3D finite element model is shown in Figure 3.Figure 2
Model of the CFST column confined with FRP under blast loading. (a) Front view. (b) Top view.
(a)
(b)Figure 3
3D finite element model.


## 2.2. Material Models

### 2.2.1. Material Model and Equation of State for Air

Material type 9 (MAT_NULL) in LS-DYNA was used to model the air, and EOS type 1 (EOS_LINEAR_POLYNOMIAL) was chosen to define the linear polynomial equation of state of air. The pressure is given by [23](1a)P=C0+C1μ+C2μ2+C3μ3+C4+C5μ+C6μ2E,(1b)μ=1V−1,where P is the pressure, C0∼C6 are the polynomial equation coefficients, V is the relative volume, and E is the internal energy per unit reference specific volume. When simulating air, the main variables and values are shown in Table 1.Table 1
Variables of air.
Variable
Density
C
0
C
1
C
2
C
3
C
4
C
5
C
6
E
0
Unit
g/mm3
—
—
—
—
—
—
—
J/mm3
Value
1.29 × 10−6
0
0
0
0
0.4
0.4
0
0.25


### 2.2.2. Material Model and Equation of State for TNT

Material type 8 (MAT_HIGH_EXPLOSIVE_ BURN) in LS-DYNA was used to model TNT. This model is particularly suitable for modelling the detonation of a high explosive. The JWL (EOS_JWL) equation was used to define the equation of state for TNT. The JWL equation of state defines the pressure as(2)p=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωEV,where p is the explosive pressure; V is the relative volume; E0 is the initial internal energy per unit reference specific volume and ω; A, B, R1, and R2 are the material constants. When simulating TNT, the main variables and values are shown in Table 2, where D refers to the detonation velocity and PCJ represents the Chapman–Jouguet pressure.Table 2
Variables of TNT.
Variable
Density
D
PCJ
A
B
R
1
R
2
E
0
Unit
g/mm3
mm/ms
MPa
MPa
MPa
—
—
J/mm3
Value
1.64 × 10−3
6930
2.1 × 105
3.74 × 105
3230
4.15
0.95
7000


### 2.2.3. Material Model for Steel Tube

Material type 3 (MAT_PLASTIC_KINEMATIC) in LS-DYNA was used to represent the steel tube. This model is a kinematic hardening plasticity model considering the strain rate effect, which is suitable for simulating steel material with a high strain rate. This material model is based on the Cowper–Symonds model. The yield stress is given by [24](3)σy=1+ε˙C1/pσ0+βEpεpeff,where ε˙ is the strain rate, C and p are the strain rate parameters, σ0 is the initial yield stress, β is the hardening parameter (for kinematic hardening, β=0; for isotropic hardening, β=1), Ep is the plasticity hardening modulus, and εpeff is the effective plastic strain. In this paper, the variable values of steel are listed in Table 3.Table 3
Variables of the steel material model.
Variable
RO
E
PR
SIGY
ETAN
BETA
SRC
SRP
FS
VP
Unit
g/mm3
MPa
—
MPa
MPa
—
—
—
—
—
Value
7.83 × 10−3
2.08 × 105
0.3
292.5
2100
0
40
5
0.25
RO: mass density;E: Young’s modulus; PR: Poisson’s ratio; SIGY: yield stress; ETAN: tangent modulus; BETA: hardening parameter; SRC and SRP: strain rate parameters; FS: effective plastic strain for eroding elements; VP: formulation for rate effects.


### 2.2.4. Material Model for Concrete

Concrete was modelled using the material type 111 (MAT_JOHNSON_HOLMQUIST_CONCRETE) in LS-DYNA. This model could be applied to simulate the concrete material under high strain, high strain rate, and high pressure. The Johnson–Holmquist–Cook model is a rate-dependent damage constitutive model proposed by Holmquist et al. [25]. The equivalent yield strength is given by(4a)σ∗=A1−D+Bp∗N1+Clnε˙ε˙0,where σ∗=σ/fc is the normalized equivalent stress; p∗=p/fc is the normalized pressure; ε˙ is the strain rate; ε˙0=1.0s−1 is the reference strain rare; and A, B, C, and N are the normalized cohesive strength, normalized pressure hardening, strain rate coefficient, and pressure hardening exponent, respectively. D represents the damage parameter and is expressed as [26](4b)D=∑Δεp+Δμpεpf+μpf=∑Δεp+ΔμpD1p∗+T∗D2,where Δεp and Δμp are the equivalent plastic strain and plastic volumetric strain, T∗=T/fc denotes the normalized maximum tensile hydrostatic pressure, and D1 and D2 are the material damage constants.The Johnson–Holmquist–Cook model takes into account the effects of large strain, high strain rate, and damage accumulation of concrete. It can describe the mechanical behaviour of concrete under explosion and high-speed impact and penetration [27]. In this paper, the variable values of the concrete are shown in Table 4.Table 4
Variables of the concrete material model.
Variable
RO
G
A
B
C
N
FC
T
EPS0
EFMIN
SFMAX
PC
UC
PL
UL
D
1
D
2
K
1
K
2
K
3
Unit
g/mm3
MPa
—
—
—
—
MPa
MPa
ms−1
—
MPa
MPa
—
MPa
—
—
—
MPa
MPa
MPa
Value
2.4 × 10−3
14860
0.79
1.6
0.007
0.61
40
4
0.001
0.01
7
13
0.001
800
0.1
0.04
1
8.5 × 104
−1.71 × 105
2.08 × 105
G: shear modulus; FC: quasi-static uniaxial compressive strength; T: maximum tensile hydrostatic pressure; ESP0: reference strain rate; EFMIN: plastic strain before fracture; SFMAX: normalized maximum strength; PC: crushing pressure; UC: crushing volumetric strain; PL: locking pressure; UL: locking volumetric strain; K1, K2, and K3: pressure constants.


### 2.2.5. Material Model for FRP

A linear elastic material model (MAT_ELASTIC) was used to simulate FRP. FRP sheets can be considered to be in the elastic stage before breakage, so its constitutive relation can be simulated by the linear elastic stress-strain relation [28, 29]. In this paper, the variable values of FRP are listed in Table 5.Table 5
Variables of the FRP material model.
Variable
Density
Young’s modulus
Poisson’s ratio
Unit
g/mm3
MPa
—
Value
1.79 × 10−3
2.49 × 105
0.285


## 2.2.1. Material Model and Equation of State for Air

Material type 9 (MAT_NULL) in LS-DYNA was used to model the air, and EOS type 1 (EOS_LINEAR_POLYNOMIAL) was chosen to define the linear polynomial equation of state of air. The pressure is given by [23](1a)P=C0+C1μ+C2μ2+C3μ3+C4+C5μ+C6μ2E,(1b)μ=1V−1,where P is the pressure, C0∼C6 are the polynomial equation coefficients, V is the relative volume, and E is the internal energy per unit reference specific volume. When simulating air, the main variables and values are shown in Table 1.Table 1
Variables of air.
Variable
Density
C
0
C
1
C
2
C
3
C
4
C
5
C
6
E
0
Unit
g/mm3
—
—
—
—
—
—
—
J/mm3
Value
1.29 × 10−6
0
0
0
0
0.4
0.4
0
0.25


## 2.2.2. Material Model and Equation of State for TNT

Material type 8 (MAT_HIGH_EXPLOSIVE_ BURN) in LS-DYNA was used to model TNT. This model is particularly suitable for modelling the detonation of a high explosive. The JWL (EOS_JWL) equation was used to define the equation of state for TNT. The JWL equation of state defines the pressure as(2)p=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωEV,where p is the explosive pressure; V is the relative volume; E0 is the initial internal energy per unit reference specific volume and ω; A, B, R1, and R2 are the material constants. When simulating TNT, the main variables and values are shown in Table 2, where D refers to the detonation velocity and PCJ represents the Chapman–Jouguet pressure.Table 2
Variables of TNT.
Variable
Density
D
PCJ
A
B
R
1
R
2
E
0
Unit
g/mm3
mm/ms
MPa
MPa
MPa
—
—
J/mm3
Value
1.64 × 10−3
6930
2.1 × 105
3.74 × 105
3230
4.15
0.95
7000


## 2.2.3. Material Model for Steel Tube

Material type 3 (MAT_PLASTIC_KINEMATIC) in LS-DYNA was used to represent the steel tube. This model is a kinematic hardening plasticity model considering the strain rate effect, which is suitable for simulating steel material with a high strain rate. This material model is based on the Cowper–Symonds model. The yield stress is given by [24](3)σy=1+ε˙C1/pσ0+βEpεpeff,where ε˙ is the strain rate, C and p are the strain rate parameters, σ0 is the initial yield stress, β is the hardening parameter (for kinematic hardening, β=0; for isotropic hardening, β=1), Ep is the plasticity hardening modulus, and εpeff is the effective plastic strain. In this paper, the variable values of steel are listed in Table 3.Table 3
Variables of the steel material model.
Variable
RO
E
PR
SIGY
ETAN
BETA
SRC
SRP
FS
VP
Unit
g/mm3
MPa
—
MPa
MPa
—
—
—
—
—
Value
7.83 × 10−3
2.08 × 105
0.3
292.5
2100
0
40
5
0.25
RO: mass density;E: Young’s modulus; PR: Poisson’s ratio; SIGY: yield stress; ETAN: tangent modulus; BETA: hardening parameter; SRC and SRP: strain rate parameters; FS: effective plastic strain for eroding elements; VP: formulation for rate effects.


## 2.2.4. Material Model for Concrete

Concrete was modelled using the material type 111 (MAT_JOHNSON_HOLMQUIST_CONCRETE) in LS-DYNA. This model could be applied to simulate the concrete material under high strain, high strain rate, and high pressure. The Johnson–Holmquist–Cook model is a rate-dependent damage constitutive model proposed by Holmquist et al. [25]. The equivalent yield strength is given by(4a)σ∗=A1−D+Bp∗N1+Clnε˙ε˙0,where σ∗=σ/fc is the normalized equivalent stress; p∗=p/fc is the normalized pressure; ε˙ is the strain rate; ε˙0=1.0s−1 is the reference strain rare; and A, B, C, and N are the normalized cohesive strength, normalized pressure hardening, strain rate coefficient, and pressure hardening exponent, respectively. D represents the damage parameter and is expressed as [26](4b)D=∑Δεp+Δμpεpf+μpf=∑Δεp+ΔμpD1p∗+T∗D2,where Δεp and Δμp are the equivalent plastic strain and plastic volumetric strain, T∗=T/fc denotes the normalized maximum tensile hydrostatic pressure, and D1 and D2 are the material damage constants.The Johnson–Holmquist–Cook model takes into account the effects of large strain, high strain rate, and damage accumulation of concrete. It can describe the mechanical behaviour of concrete under explosion and high-speed impact and penetration [27]. In this paper, the variable values of the concrete are shown in Table 4.Table 4
Variables of the concrete material model.
Variable
RO
G
A
B
C
N
FC
T
EPS0
EFMIN
SFMAX
PC
UC
PL
UL
D
1
D
2
K
1
K
2
K
3
Unit
g/mm3
MPa
—
—
—
—
MPa
MPa
ms−1
—
MPa
MPa
—
MPa
—
—
—
MPa
MPa
MPa
Value
2.4 × 10−3
14860
0.79
1.6
0.007
0.61
40
4
0.001
0.01
7
13
0.001
800
0.1
0.04
1
8.5 × 104
−1.71 × 105
2.08 × 105
G: shear modulus; FC: quasi-static uniaxial compressive strength; T: maximum tensile hydrostatic pressure; ESP0: reference strain rate; EFMIN: plastic strain before fracture; SFMAX: normalized maximum strength; PC: crushing pressure; UC: crushing volumetric strain; PL: locking pressure; UL: locking volumetric strain; K1, K2, and K3: pressure constants.


## 2.2.5. Material Model for FRP

A linear elastic material model (MAT_ELASTIC) was used to simulate FRP. FRP sheets can be considered to be in the elastic stage before breakage, so its constitutive relation can be simulated by the linear elastic stress-strain relation [28, 29]. In this paper, the variable values of FRP are listed in Table 5.Table 5
Variables of the FRP material model.
Variable
Density
Young’s modulus
Poisson’s ratio
Unit
g/mm3
MPa
—
Value
1.79 × 10−3
2.49 × 105
0.285


## 2.3. Strain Rate Effect

The dynamic behaviour of the concrete material is strain rate dependent. Both the compressive and tensile strengths vary with the strain rate. Recommended by Committee Euro-International du Béton (CEB), the dynamic increase factor (DIF) is defined as the ratio of dynamic strength to static strength. For concrete in compression, the equation of DIF is given by [30](5)CDIF=fcdfcs=ε˙ε˙s1.026α,ε˙≤30s−1,γε˙ε˙s1/3,ε˙>30s−1,where ε˙ is the strain rate, ε˙s=3×10−5s−1, α=1/5+9fcs/10, and lgγ=6.156α−2.For concrete in tension, the equation of DIF is given by(6)TDIF=ftdfts=ε˙ε˙sδ,ε˙≤1s−1,βε˙ε˙s1/3,ε˙>1s−1,where δ=1/1+8fcs/10 and lgβ=6δ−2.The DIF for yield strength of steel is adopted by [31](7)DIF=fydfys=1+ε˙C1/p,where C and p are the strain rate parameters mentioned above.Compared with steel and concrete materials, the strain rate effect of FRP is insignificant [32]. Thus, the strain rate effect of FRP could be ignored in numerical simulations [33].


## 2.4. Mesh Size

When simulating the propagation of blast wave and its interaction with the structure, the accuracy of simulation results depends largely on the element mesh size. In order to find the most proper mesh size, the model of 44.28 kg TNT in the 3200 mm × 3200 mm × 3200 mm air domain was developed and the mesh size of 20 mm, 25 mm, 30 mm, and 50 mm was adopted for simulation, respectively. The overpressure values of the four models at different distances were compared with the empirical curve from UFC 3-340-02 [34]. As shown in Figure 4, the mesh size has certain influence on the peak overpressure of the shock wave. The accuracy of the numerical simulation performs better with the smaller mesh size. The overpressure value of 25 mm is close to the results of 20 mm. However, the computation amount and analysing time increase enormously than that of 25 mm, which leads to the risk of computer memory overflow. In addition, the simulation converged well when the mesh size is 25 mm. Therefore, the mesh size of 25 mm is employed in this paper for the dual consideration of simulation accuracy and computational efficiency.Figure 4
Comparison of different mesh sizes.


## 2.5. Generation of Blast Loading

In LS-DYNA, there are three common ways to generate blast loading. The first way is to simplify the blast load, which is usually simplified to an idealized triangular ramp pressure. The load curve is defined by the keyword∗LOAD_CURVE in LS-DYNA. This method is simple in modelling, relatively high in computational efficiency, and suitable for dynamic response analysis of structural members with various distances. However, the accuracy of this method is relatively low, and it cannot simulate the propagation process of a blast wave. The second way utilizes the keyword ∗LOAD_BLAST in LS-DYNA to generate blast loading. This method requires inputting the equivalent mass of TNT, coordinates of detonation, type of blast, and other parameters in the keyword card. It has a wide application scope and the calculation results are relatively reasonable, but the shape of the explosive and the influence of obstacles on the structure cannot be taken into consideration. The third way is the fluid-structure coupling method, which requires establishing parts of air, explosive, and member. Multiple materials can be coupled by the arbitrary Lagrange–Euler (ALE) method. This method can accurately simulate the formation and propagation of the blast wave, and explosive shape and obstacles can be considered. The disadvantage of this method is that the modelling is rather complex. The computational efficiency is low due to the large computational amount. The third way, the ALE method, was applied in this study. By adding the ∗INITIAL_DETONATION keyword, the explosion is detonated at the center point of the TNT.


## 2.6. Boundaries and Erosion

With the consideration of reflection effect, a rigid plane was added to the bottom surface of the air domain using the∗RIGIDWALL_PLANAR keyword. The remaining surfaces of air domain were subjected to nonreflection boundaries. The fluid-structure interaction was processed using the ∗CONSTRAINED_LAGRANGE_IN_SOLID keyword. The nodes linked among steel tube, concrete, and FRP mesh were shared. The erosion algorithm was implemented using the ∗MAT_ADD_EROSION keyword. Under the extreme loading, the element of material is deleted when it reaches a certain value. Based on the previous studies, a principle tensile strain of 0.001 is adopted to simulate the failure of concrete as the primary criterion in this study [35].


## 3. Verification of FE Model

The CFST column confined with FRP is a rapidly developed new type of composite material component in buildings recently. The references of blast tests for CFST column confined with FRP cannot be searched currently at home and abroad. Therefore, the finite element model of CFST column confined with FRP is not able to be verified directly by the relevant experiments. The accuracy and reliability of the abovementioned model was verified from three aspects, including the blast field tests of CFST column and reinforced concrete (RC) column with FRP and equivalent single-degree-of-freedom model, respectively. (1) The proposed FE model was verified against the blast tests of CFST columns in order to validate the developed model and the material models for air, TNT, steel tube, and concrete. And this situation was regarded as a special case when the FRP layer was 0. (2) To further verify the material model for FRP and ALE methods, the developed model was validated against the blast tests of RC columns confined with FRP. (3) A theoretical analysis on CFST column confined with FRP under blast loading was investigated to verify the dynamic response of numerical simulation. The FE model of the CFST column confined with FRP under blast loading precisely developed in this study was verified by the three above aspects. Based on the validated FE model, some exploratory researches were investigated and discussed.


### 3.1. CFST Column Blast Test

The blast field tests of three CFST columns and one concrete-filled double-skin steel tube column were carried out [36]. The steel tube was made of hot-rolled seamless steel and filled with C40 fine stone concrete. The specific specimen parameters are listed in Table 6, where H is the height of the column, D is the outside diameter of the steel tube, t is the thickness of the steel tube, M is the mass of TNT, H′ is the height of TNT, and L is the distance between the TNT and the specimen.Table 6
Parameters of the CFST column blast test.
Specimen
H (mm)
D (mm)
t (mm)
M (kg)
H′ (m)
L (m)
SC-1
1800
273
7
3
0.9
1.6
SC-2
1800
273
7
50
0.9
0.7
SC-3
1800
273
7
50
0.9
8This study performed numerical simulation verification on the test results of SC-1 and SC-2. A cube with dimensions of 125 mm × 125 mm × 125 mm was used to simulate 3 kg TNT explosive, and the air domain size was 2500 mm × 1000 mm × 2500 mm. Both ends of the CFST column were fixed, and displacement and rotation constraints were added.Since the SC-1 test is a small-charge explosion test, there is no obvious damage phenomenon or displacement on SC-1. The experimental data of six measure points on the column SC-1 are compared with the numerical simulation results. As shown in Figure5, the measure points #1, #2, #3, #4, #5, and #6 are set at the top, middle, and bottom of the front side and the back side of the column, respectively. Table 7 shows the comparison between the experimental and numerical data. It is found that all the numerical results are within 16% of the experimental value. The discrepancies at #3 and #6 are mainly caused by the site environment. The site of blast tests is an open and wide field, but the ground in simulation is modelled by using the limited rigid plane. This may affect the enhancement of overpressure reflection effect at the top of the column. Moreover, the results of blast tests are also influenced by incidental factors such as ideal air condition and climate. Even so, the distribution of simulation corresponds with experimental results and the overall discrepancy is within rational range.Figure 5
Measure points of SC-1.Table 7
Comparison of experimental results and numerical simulation results (SC-1).
Measure point
Arrival time of blast wave (ms)
Peak value of blast wave (MPa)
Experiment
Simulation
Discrepancy (%)
Experiment
Simulation
Discrepancy (%)
#1
0.689
0.699
1.45
1.364
1.384
1.47
#2
0.515
0.499
3.11
6.202
5.726
7.67
#3
0.616
0.699
13.47
1.350
1.408
4.30
#4
1.169
1.099
5.99
0.361
0.365
1.11
#5
0.894
0.899
0.56
0.448
0.485
8.26
#6
1.296
1.099
15.2
0.401
0.366
8.73The model of column SC-2 was generated using the same method as that of SC-1. A 350 mm × 300 mm × 300 mm cube was used to simulate 50 kg of TNT. Since this case was a close-in blast field, obvious damage of the specimen occurred.Figure6 displays the failure mode of column SC-2 under the close-in blast load of 50 kg TNT explosive. Figure 6(a) shows the experimental results, and Figure 6(b) shows the numerical results. By comparison, it can be seen that the numerical results are close to the experimental results, and obvious deformations in the middle of the column were observed in both test and numerical results. Figure 7 illustrates the displacement-time history curve of midheight point of column SC-2 in the numerical simulation. The maximum displacement of the middle-height point is 91.276 mm according to the figure, while the test result is 93 mm. The deviation rate is only 1.85%.Figure 6
Damage mode of SC-2. (a) Experimental results. (b) Numerical results.
(a)
(b)Figure 7
Displacement-time history curve of the midspan of SC-2.From the comparisons on parameters of the blast wave, it shows that the numerical results are in good agreement with the experimental results. As a result, the developed FE model can rationally reflect the dynamic response of CFST columns under blast loading. In other words, the generation of blast loading, material models, and state of equations for air and explosive in the proposed FE model are reasonable.


### 3.2. RC Columns Confined with FRP Blast Test

The field blast tests on eight RC pier columns with different protection were carried out by Liu [37]. Among the eight pier columns, specimen JS5 is an RC column wrapped with one layer of CFRP sheet. Blast tests were carried out under four different cases. The specific parameters of the four cases are listed in Table 8, where H is the effective height of the column, D is the diameter of the column, tf is the thickness of CFRP, M is the mass of TNT, H′ is the height of TNT, and L is the distance between the column and TNT. The effective height of the circular column is 3500 mm, and the diameter is 400 mm. To facilitate the constraints, the top of the column is increased by 200 mm, while the actual height of the column is 3700 mm. The size of the RC base is 1000 mm × 1000 mm × 500 mm. The concrete strength grade is C40, and the reinforcement ratio is 0.9% (as shown in Figure 8).Table 8
Four different cases of specimen JS5.
Case no.
H (mm)
D (mm)
t
f (mm)
M (kg)
H′ (m)
L (m)
Case 24
3500
400
0.167
0.4
0.33
1.5
Case 25
3500
400
0.167
1
0.33
1.5
Case 26
3500
400
0.167
2
0.33
1.5
Case 27
3500
400
0.167
1
0.33
0Figure 8
Geometry and reinforcement details of specimen JS5.In this test, four pressure transducers were installed on the front face of the specimens at different positions along the height of the column. The height of the four measure points were 330 mm, 1330 mm, 2330 mm, and 3300 mm, respectively. Three acceleration sensors were arranged on the back surface and their heights were 300 mm, 1750 mm, and 3300 mm, respectively. The sensor arrangements are shown in Figure9.Figure 9
Sensor arrangements of specimen JS5. (a) Pressure sensor. (b) Acceleration sensor.Numerical simulation was based on Cases 25 and 27. The model was also generated according to the modelling methods mentioned in Section2. Because the pressure sensor was damaged in the blast test, a small number of effective data were measured in the experiment. The numerical results were compared with those effective data. In Case 25, the arrival time of peak pressure measured from #2 pressure transducer was 2.56 ms and the peak pressure was 0.32 MPa. The results obtained from numerical simulation were 2.14 ms and 0.317 MPa, respectively. Additionally, Figure 10 shows the comparison between the experimental results and numerical analysis on #3 acceleration sensor. It can be learned from the above comparisons that the numerical analysis results were essentially in agreement with the experimental results. Thus, the linear elastic model for FRP and the multimaterial ALE method are applicable to predict the propagation of the blast wave and the blast response of reinforced concrete confined with FRP composite column.Figure 10
Acceleration-time history curve of #3 acceleration sensor.


### 3.3. Equivalent Single-Degree-of-Freedom Model

An equivalent single-degree-of-freedom model is a simplified method to study the dynamic deflection of structure [38]. Under blast loading, numerous vibration modes of the structural member will be generated. The domination mode of vibration can be considered as the deformation state of the member. The equivalent single-degree-of-freedom model (SDOF) was initially proposed by Norris et al. [39] and Biggs [40]. It replaces the motion state of the actual structure with the motion state of a concentrated mass spring. As illustrated in Figure 11, the structure is simplified to only one degree of freedom; that is, the mass of the structure is concentrated at a certain point, and then the point is limited to vibrate in a certain direction under dynamic loads. The equivalent mass (Me), equivalent load (Fe), and equivalent stiffness (Ke) can be obtained using the principle of energy conservation. The maximum dynamic displacement of the structure can be solved using the motion equation of the SDOF system. The motion equation of the ideal SDOF system ignoring the damping force is(8a)Mey¨+Key=Fetelastic stage,(8b)Mey¨+Re=Fetplastic stage,where y is the displacement of the equivalent system and Re is the equivalent resistance. Adopting the load coefficient KL = Fe/F, mass coefficient KM = Me/M, resistance coefficient KR = Re/R, and load-mass coefficient KLM = KM/KL, the above motion equations can be rewritten as(9a)KLMMy¨+K0y=Ftelastic stage,(9b)KLMMy¨+R0=Ftplastic stage,where K0 and R0 are the stiffness and resistance of actual structure, respectively.Figure 11
Equivalent single-degree-of-freedom model. (a) Actual structure. (b) Equivalent system.
(a)
(b)As to this study, the dynamic displacement of the proposed model in Section2 was also calculated by the SDOF method to further verify the numerical model. Since the predicated column has two ends fixed, the coefficients of the system are shown in Table 9.Table 9
Conversion coefficients of the system.
Load and support condition
Stage
K
L
K
M
K
LM
Elastic
0.53
0.41
0.77
Elastic-plastic
0.64
0.50
0.78
Plastic
0.50
0.33
0.66The resistance-deflection function for the SDOF system is illustrated in Figure12 [41]. The maximum displacement in the elastic stage can be deduced as follows:(10)y0=ReKe=KRR0KLK0=R0K0,where R0 = 16Mpe/L, Mpe is the plastic limit bending moment, L is the length of the column, and K0 = 307EI/L3, where E and I are the elastic modulus and inertia moment, respectively.Figure 12
Resistance-deflection function.According to blast resistance criteria of US Army, the maximum deflection of elastic-plastic SDOF system is given by [34](11)ym=μy0,where μ is the factor which can be found in the empirical curve given by UFC-3-340-02. The maximum displacement of the predicated column is 23.36 mm after calculating by the SDOF method. The numerical result is 24.13 mm, which is in close agreement with the result of SDOF. The maximum displacement of numerical simulation is verified by the theoretical result.In summary, the reliability of the finite element model is verifie1d against the blast tests of CFST columns and reinforced concrete columns confined with FRP. The accuracy of numerical analysis in dynamic response under blast loading is further confirmed by the SDOF result. In consequence, it is believed that the FE model developed in this paper can effectively predict the dynamic response of CFST columns confined with FRP under blast loading.


## 3.1. CFST Column Blast Test

The blast field tests of three CFST columns and one concrete-filled double-skin steel tube column were carried out [36]. The steel tube was made of hot-rolled seamless steel and filled with C40 fine stone concrete. The specific specimen parameters are listed in Table 6, where H is the height of the column, D is the outside diameter of the steel tube, t is the thickness of the steel tube, M is the mass of TNT, H′ is the height of TNT, and L is the distance between the TNT and the specimen.Table 6
Parameters of the CFST column blast test.
Specimen
H (mm)
D (mm)
t (mm)
M (kg)
H′ (m)
L (m)
SC-1
1800
273
7
3
0.9
1.6
SC-2
1800
273
7
50
0.9
0.7
SC-3
1800
273
7
50
0.9
8This study performed numerical simulation verification on the test results of SC-1 and SC-2. A cube with dimensions of 125 mm × 125 mm × 125 mm was used to simulate 3 kg TNT explosive, and the air domain size was 2500 mm × 1000 mm × 2500 mm. Both ends of the CFST column were fixed, and displacement and rotation constraints were added.Since the SC-1 test is a small-charge explosion test, there is no obvious damage phenomenon or displacement on SC-1. The experimental data of six measure points on the column SC-1 are compared with the numerical simulation results. As shown in Figure5, the measure points #1, #2, #3, #4, #5, and #6 are set at the top, middle, and bottom of the front side and the back side of the column, respectively. Table 7 shows the comparison between the experimental and numerical data. It is found that all the numerical results are within 16% of the experimental value. The discrepancies at #3 and #6 are mainly caused by the site environment. The site of blast tests is an open and wide field, but the ground in simulation is modelled by using the limited rigid plane. This may affect the enhancement of overpressure reflection effect at the top of the column. Moreover, the results of blast tests are also influenced by incidental factors such as ideal air condition and climate. Even so, the distribution of simulation corresponds with experimental results and the overall discrepancy is within rational range.Figure 5
Measure points of SC-1.Table 7
Comparison of experimental results and numerical simulation results (SC-1).
Measure point
Arrival time of blast wave (ms)
Peak value of blast wave (MPa)
Experiment
Simulation
Discrepancy (%)
Experiment
Simulation
Discrepancy (%)
#1
0.689
0.699
1.45
1.364
1.384
1.47
#2
0.515
0.499
3.11
6.202
5.726
7.67
#3
0.616
0.699
13.47
1.350
1.408
4.30
#4
1.169
1.099
5.99
0.361
0.365
1.11
#5
0.894
0.899
0.56
0.448
0.485
8.26
#6
1.296
1.099
15.2
0.401
0.366
8.73The model of column SC-2 was generated using the same method as that of SC-1. A 350 mm × 300 mm × 300 mm cube was used to simulate 50 kg of TNT. Since this case was a close-in blast field, obvious damage of the specimen occurred.Figure6 displays the failure mode of column SC-2 under the close-in blast load of 50 kg TNT explosive. Figure 6(a) shows the experimental results, and Figure 6(b) shows the numerical results. By comparison, it can be seen that the numerical results are close to the experimental results, and obvious deformations in the middle of the column were observed in both test and numerical results. Figure 7 illustrates the displacement-time history curve of midheight point of column SC-2 in the numerical simulation. The maximum displacement of the middle-height point is 91.276 mm according to the figure, while the test result is 93 mm. The deviation rate is only 1.85%.Figure 6
Damage mode of SC-2. (a) Experimental results. (b) Numerical results.
(a)
(b)Figure 7
Displacement-time history curve of the midspan of SC-2.From the comparisons on parameters of the blast wave, it shows that the numerical results are in good agreement with the experimental results. As a result, the developed FE model can rationally reflect the dynamic response of CFST columns under blast loading. In other words, the generation of blast loading, material models, and state of equations for air and explosive in the proposed FE model are reasonable.


## 3.2. RC Columns Confined with FRP Blast Test

The field blast tests on eight RC pier columns with different protection were carried out by Liu [37]. Among the eight pier columns, specimen JS5 is an RC column wrapped with one layer of CFRP sheet. Blast tests were carried out under four different cases. The specific parameters of the four cases are listed in Table 8, where H is the effective height of the column, D is the diameter of the column, tf is the thickness of CFRP, M is the mass of TNT, H′ is the height of TNT, and L is the distance between the column and TNT. The effective height of the circular column is 3500 mm, and the diameter is 400 mm. To facilitate the constraints, the top of the column is increased by 200 mm, while the actual height of the column is 3700 mm. The size of the RC base is 1000 mm × 1000 mm × 500 mm. The concrete strength grade is C40, and the reinforcement ratio is 0.9% (as shown in Figure 8).Table 8
Four different cases of specimen JS5.
Case no.
H (mm)
D (mm)
t
f (mm)
M (kg)
H′ (m)
L (m)
Case 24
3500
400
0.167
0.4
0.33
1.5
Case 25
3500
400
0.167
1
0.33
1.5
Case 26
3500
400
0.167
2
0.33
1.5
Case 27
3500
400
0.167
1
0.33
0Figure 8
Geometry and reinforcement details of specimen JS5.In this test, four pressure transducers were installed on the front face of the specimens at different positions along the height of the column. The height of the four measure points were 330 mm, 1330 mm, 2330 mm, and 3300 mm, respectively. Three acceleration sensors were arranged on the back surface and their heights were 300 mm, 1750 mm, and 3300 mm, respectively. The sensor arrangements are shown in Figure9.Figure 9
Sensor arrangements of specimen JS5. (a) Pressure sensor. (b) Acceleration sensor.Numerical simulation was based on Cases 25 and 27. The model was also generated according to the modelling methods mentioned in Section2. Because the pressure sensor was damaged in the blast test, a small number of effective data were measured in the experiment. The numerical results were compared with those effective data. In Case 25, the arrival time of peak pressure measured from #2 pressure transducer was 2.56 ms and the peak pressure was 0.32 MPa. The results obtained from numerical simulation were 2.14 ms and 0.317 MPa, respectively. Additionally, Figure 10 shows the comparison between the experimental results and numerical analysis on #3 acceleration sensor. It can be learned from the above comparisons that the numerical analysis results were essentially in agreement with the experimental results. Thus, the linear elastic model for FRP and the multimaterial ALE method are applicable to predict the propagation of the blast wave and the blast response of reinforced concrete confined with FRP composite column.Figure 10
Acceleration-time history curve of #3 acceleration sensor.


## 3.3. Equivalent Single-Degree-of-Freedom Model

An equivalent single-degree-of-freedom model is a simplified method to study the dynamic deflection of structure [38]. Under blast loading, numerous vibration modes of the structural member will be generated. The domination mode of vibration can be considered as the deformation state of the member. The equivalent single-degree-of-freedom model (SDOF) was initially proposed by Norris et al. [39] and Biggs [40]. It replaces the motion state of the actual structure with the motion state of a concentrated mass spring. As illustrated in Figure 11, the structure is simplified to only one degree of freedom; that is, the mass of the structure is concentrated at a certain point, and then the point is limited to vibrate in a certain direction under dynamic loads. The equivalent mass (Me), equivalent load (Fe), and equivalent stiffness (Ke) can be obtained using the principle of energy conservation. The maximum dynamic displacement of the structure can be solved using the motion equation of the SDOF system. The motion equation of the ideal SDOF system ignoring the damping force is(8a)Mey¨+Key=Fetelastic stage,(8b)Mey¨+Re=Fetplastic stage,where y is the displacement of the equivalent system and Re is the equivalent resistance. Adopting the load coefficient KL = Fe/F, mass coefficient KM = Me/M, resistance coefficient KR = Re/R, and load-mass coefficient KLM = KM/KL, the above motion equations can be rewritten as(9a)KLMMy¨+K0y=Ftelastic stage,(9b)KLMMy¨+R0=Ftplastic stage,where K0 and R0 are the stiffness and resistance of actual structure, respectively.Figure 11
Equivalent single-degree-of-freedom model. (a) Actual structure. (b) Equivalent system.
(a)
(b)As to this study, the dynamic displacement of the proposed model in Section2 was also calculated by the SDOF method to further verify the numerical model. Since the predicated column has two ends fixed, the coefficients of the system are shown in Table 9.Table 9
Conversion coefficients of the system.
Load and support condition
Stage
K
L
K
M
K
LM
Elastic
0.53
0.41
0.77
Elastic-plastic
0.64
0.50
0.78
Plastic
0.50
0.33
0.66The resistance-deflection function for the SDOF system is illustrated in Figure12 [41]. The maximum displacement in the elastic stage can be deduced as follows:(10)y0=ReKe=KRR0KLK0=R0K0,where R0 = 16Mpe/L, Mpe is the plastic limit bending moment, L is the length of the column, and K0 = 307EI/L3, where E and I are the elastic modulus and inertia moment, respectively.Figure 12
Resistance-deflection function.According to blast resistance criteria of US Army, the maximum deflection of elastic-plastic SDOF system is given by [34](11)ym=μy0,where μ is the factor which can be found in the empirical curve given by UFC-3-340-02. The maximum displacement of the predicated column is 23.36 mm after calculating by the SDOF method. The numerical result is 24.13 mm, which is in close agreement with the result of SDOF. The maximum displacement of numerical simulation is verified by the theoretical result.In summary, the reliability of the finite element model is verifie1d against the blast tests of CFST columns and reinforced concrete columns confined with FRP. The accuracy of numerical analysis in dynamic response under blast loading is further confirmed by the SDOF result. In consequence, it is believed that the FE model developed in this paper can effectively predict the dynamic response of CFST columns confined with FRP under blast loading.


## 4. Results and Discussion

The proposed FE analysis model is rational from the above verification. After analysis and postprocessing, the dynamic response of a typical CFST column confined with FRP mentioned in Section2 is discussed as follows.


### 4.1. Displacement

To study the displacement variation of the CFST column confined with FRP under blast loading, displacement contours in the direction of theX-axis (i.e., the propagation direction of the blast wave) at different times are shown in Figures 13 and 14, where Figure 13 is the front view and Figure 14 is the left view.Figure 13
Contours ofX-displacement at different time periods (front view): (a) t = 0.4 ms, (b) t = 0.9 ms and (c) t = 2 ms.
(a)
(b)
(c)Figure 14
Contours ofX-displacement at different time periods (left view): (a) t = 0.4 ms, (b) t = 0.9 ms, and (c) t = 2 ms.
(a)
(b)
(c)As seen from Figures13 and 14 that when t = 0.4 ms, the blast wave arrives at the column. The displacement at the middle of the column is relatively obvious, reaching 9.92 mm. As time goes on, the displacement of the column increases. When t = 0.9 ms, the displacement of the column reaches the maximum value of 24.13 mm. When t = 2 ms, the displacement in the middle of the column decreases and the displacements at both ends of the column increases. The displacement presents a symmetrical distribution with large middle and small ends for the whole.Figure15 illustrates the displacement-time history curve of the midheight point (i.e., point A in Figure 2) of the CFST column confined with FRP under blast loading. It can be seen from the figure that no transverse displacement occurs within 0–0.2 ms of the explosion. During this period, the blast wave propagates in the air after the detonation. After 0.2 ms, the displacement increases rapidly along the propagation direction. The maximum lateral displacement is 24.13 mm at 0.9 ms. The displacement of the column subsequently decreases and vibrates approximately 12 mm.Figure 15
Displacement-time histories of the midheight point of CFSTs confined with FRP.


### 4.2. Effective Stress of Steel Tube

The stress state of CFST column confined with FRP under blast loading is very complicated. The equivalent stress, or von Mises stress, is usually used to represent its stress characteristics. Figure16 displays the effective stress contours of the CFST column confined with FRP at different times.Figure 16
Effective stress contours of the steel tube at different time periods (front view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)As seen from Figure16, the stress of the column increases rapidly at 0.3 ms. The peak stress in middle part reaches 410.93 MPa, exceeding the static yield strength of the steel tube. As time goes on, the areas of greater stress extend from middle of the column to the two ends of the column gradually. At 1 ms and 2 ms, the stress in most of the elements of the middle and two end parts is among 303–377 MPa, which exceeds the static yield strength of the steel tube. However, the stress in the transition areas between the middle and two ends is relatively small and does not exceed the yield strength. At 3 ms, the stress in the middle and two end parts of the column decreases. The effective stress is generally distributed symmetrically above and below. Therefore, the vulnerable parts of a CFST confined with FRP are the middle and two ends (Figure 17).Figure 17
Vulnerable parts of the steel tube.


### 4.3. Effective Stress of Concrete

Figure18 shows the effective stress contours of concrete at different times. Compared with the effective stress of the steel tube, the effective stress of the concrete is distributed evenly along the longitudinal member. In the initial stage, the effective stress of the concrete in the middle of the column increases rapidly, exceeding the static ultimate stress of concrete. The effective stress in other areas is relatively small and does not exceed the ultimate stress. Due to the EROSION keywords, elements will be deleted if the strain value exceeds the failure strain. As time elapses, some elements in the middle section stop working first. Then, stress increases at the top and bottom of the column. These elements stop working, and shear failure occurs in the column. Stress in the middle part of the backside of the column increases gradually, and stress of most of the elements in this area is in the range of 44.06 MPa to 52.62 MPa, exceeding the static ultimate stress of the concrete. Thus, the vulnerable parts of the concrete are still the middle and two end parts of the column.Figure 18
Effective stress contours of concrete at different time periods (left view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)


### 4.4. Effective Stress of FRP

The effective stress contours of FRP at different times are displayed in Figure19. The stress distribution of FRP is similar to that of a steel tube. In the beginning, the area with higher stress is still the middle part of the column and later the stress at the two ends increases. The stress is higher in the middle part and two ends, while lower in the transition areas.Figure 19
Effective stress contours of a steel tube at different time periods (front view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)


### 4.5. Parametric Study

#### 4.5.1. Number of FRP Layers

To study the effect of number of FRP layers on the blast resistance of the CFST column confined with FRP, the dynamic response of the CFST columns confined with FRP with different layers under blast loading was simulated by changing the thickness of the FRP while keeping other parameters unchanged.Figure20 illustrates the displacement-time history curves of the CFST columns confined with FRP when the FRP is 0, 1, 2, 3, and 4 layers (the thicknesses of the FRP are 0, 0.167 mm, 0.334 mm, 0.501 mm, and 0.668 mm, respectively). When the number of FRP layers is 0, namely, a traditional CFST without the FRP constraint, the maximum displacement of the member under blast loading is 30.17 mm. When the number of FRP layers is 1, the maximum displacement of the column is 24.13 mm, a reduction of 6.04 mm compared with a nonretrofitted CFST column. It is found that the constraint of FRP could effectively improve the blast resistance of CFST columns. When the number of FRP layers is 2, 3, and 4, the maximum displacements are 22.11 mm, 19.52 mm, and 19.37 mm, respectively. This indicates that with an increase in the number of FRP layers, the blast resistance performance of the member is further improved, but its effect is gradually weakened. As seen from the figure, the displacement of a three-FRP layer column and a four-FRP layer column is relatively close, but it is obvious that the three-FRP layer is more economical than the four-FRP layer.Figure 20
Comparison of midheight displacement with different thicknesses of FRP.


#### 4.5.2. Concrete Strength

The dynamic response of the CFST columns confined with FRP with four different concrete strength grades under the same blast loading is analysed while keeping the other parameters unchanged. The displacement-time history curves of the midheight point of the four cases are shown in Figure21.Figure 21
Comparison of midheight displacement with different comprehensive strengths of concrete.When the strength grades of the concrete are C35, C40, C45, and C50, the maximum displacements of the CFST columns confined with FRP are 28.32 mm, 24.13 mm, 21.65 mm, and 18.90 mm, respectively. The maximum displacement decreases gradually with an increase in the concrete strength grade. This indicates that the blast resistance of a CFST confined with FRP can be enhanced by improving the concrete strength.


#### 4.5.3. Cross Section

In the engineering application, the CFST columns with the square section are more widely used because of its convenient joint connection. To study the influence of the cross section on the blast resistance performance, two square-section column models are generated. One column is the same size with the circular one (500 mm × 500 mm × 3000 mm), and the other column is the same material consumptions with the circular one (namely, the section size is 443 mm × 443 mm and the thickness of steel tube is 9 mm). The numerical results of the square section cases are compared with those of the circular section case. The displacement-time history curves of the two cases are shown in Figure22.Figure 22
Comparison of midheight displacement with different cross sections. (a) Same section size. (b) Same material consumptions.
(a)
(b)Under the condition of the same section size, the maximum displacement of the square section column with the same section size is 21.99 mm, which is 8.87% less than that of the circular section member. The blast resistance performance of the square section member is better than that of the circular section member, but the square section member has 27.39% higher amounts of FRP, steel, and concrete than the circular member. Under the condition of the same material consumptions, the maximum displacement of the column with the square section is 30.27 mm. The column with circular section shows better blast performance than the one with square section. This is mainly because the circular section member has an excellent aerodynamic profile. Parts of the blast wave detours, and the impact energy reduces.


## 4.1. Displacement

To study the displacement variation of the CFST column confined with FRP under blast loading, displacement contours in the direction of theX-axis (i.e., the propagation direction of the blast wave) at different times are shown in Figures 13 and 14, where Figure 13 is the front view and Figure 14 is the left view.Figure 13
Contours ofX-displacement at different time periods (front view): (a) t = 0.4 ms, (b) t = 0.9 ms and (c) t = 2 ms.
(a)
(b)
(c)Figure 14
Contours ofX-displacement at different time periods (left view): (a) t = 0.4 ms, (b) t = 0.9 ms, and (c) t = 2 ms.
(a)
(b)
(c)As seen from Figures13 and 14 that when t = 0.4 ms, the blast wave arrives at the column. The displacement at the middle of the column is relatively obvious, reaching 9.92 mm. As time goes on, the displacement of the column increases. When t = 0.9 ms, the displacement of the column reaches the maximum value of 24.13 mm. When t = 2 ms, the displacement in the middle of the column decreases and the displacements at both ends of the column increases. The displacement presents a symmetrical distribution with large middle and small ends for the whole.Figure15 illustrates the displacement-time history curve of the midheight point (i.e., point A in Figure 2) of the CFST column confined with FRP under blast loading. It can be seen from the figure that no transverse displacement occurs within 0–0.2 ms of the explosion. During this period, the blast wave propagates in the air after the detonation. After 0.2 ms, the displacement increases rapidly along the propagation direction. The maximum lateral displacement is 24.13 mm at 0.9 ms. The displacement of the column subsequently decreases and vibrates approximately 12 mm.Figure 15
Displacement-time histories of the midheight point of CFSTs confined with FRP.


## 4.2. Effective Stress of Steel Tube

The stress state of CFST column confined with FRP under blast loading is very complicated. The equivalent stress, or von Mises stress, is usually used to represent its stress characteristics. Figure16 displays the effective stress contours of the CFST column confined with FRP at different times.Figure 16
Effective stress contours of the steel tube at different time periods (front view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)As seen from Figure16, the stress of the column increases rapidly at 0.3 ms. The peak stress in middle part reaches 410.93 MPa, exceeding the static yield strength of the steel tube. As time goes on, the areas of greater stress extend from middle of the column to the two ends of the column gradually. At 1 ms and 2 ms, the stress in most of the elements of the middle and two end parts is among 303–377 MPa, which exceeds the static yield strength of the steel tube. However, the stress in the transition areas between the middle and two ends is relatively small and does not exceed the yield strength. At 3 ms, the stress in the middle and two end parts of the column decreases. The effective stress is generally distributed symmetrically above and below. Therefore, the vulnerable parts of a CFST confined with FRP are the middle and two ends (Figure 17).Figure 17
Vulnerable parts of the steel tube.


## 4.3. Effective Stress of Concrete

Figure18 shows the effective stress contours of concrete at different times. Compared with the effective stress of the steel tube, the effective stress of the concrete is distributed evenly along the longitudinal member. In the initial stage, the effective stress of the concrete in the middle of the column increases rapidly, exceeding the static ultimate stress of concrete. The effective stress in other areas is relatively small and does not exceed the ultimate stress. Due to the EROSION keywords, elements will be deleted if the strain value exceeds the failure strain. As time elapses, some elements in the middle section stop working first. Then, stress increases at the top and bottom of the column. These elements stop working, and shear failure occurs in the column. Stress in the middle part of the backside of the column increases gradually, and stress of most of the elements in this area is in the range of 44.06 MPa to 52.62 MPa, exceeding the static ultimate stress of the concrete. Thus, the vulnerable parts of the concrete are still the middle and two end parts of the column.Figure 18
Effective stress contours of concrete at different time periods (left view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)


## 4.4. Effective Stress of FRP

The effective stress contours of FRP at different times are displayed in Figure19. The stress distribution of FRP is similar to that of a steel tube. In the beginning, the area with higher stress is still the middle part of the column and later the stress at the two ends increases. The stress is higher in the middle part and two ends, while lower in the transition areas.Figure 19
Effective stress contours of a steel tube at different time periods (front view): (a)t = 0.3 ms, (b) t = 1 ms, (c) t = 2 ms, and (d) t = 3 ms.
(a)
(b)
(c)
(d)


## 4.5. Parametric Study

### 4.5.1. Number of FRP Layers

To study the effect of number of FRP layers on the blast resistance of the CFST column confined with FRP, the dynamic response of the CFST columns confined with FRP with different layers under blast loading was simulated by changing the thickness of the FRP while keeping other parameters unchanged.Figure20 illustrates the displacement-time history curves of the CFST columns confined with FRP when the FRP is 0, 1, 2, 3, and 4 layers (the thicknesses of the FRP are 0, 0.167 mm, 0.334 mm, 0.501 mm, and 0.668 mm, respectively). When the number of FRP layers is 0, namely, a traditional CFST without the FRP constraint, the maximum displacement of the member under blast loading is 30.17 mm. When the number of FRP layers is 1, the maximum displacement of the column is 24.13 mm, a reduction of 6.04 mm compared with a nonretrofitted CFST column. It is found that the constraint of FRP could effectively improve the blast resistance of CFST columns. When the number of FRP layers is 2, 3, and 4, the maximum displacements are 22.11 mm, 19.52 mm, and 19.37 mm, respectively. This indicates that with an increase in the number of FRP layers, the blast resistance performance of the member is further improved, but its effect is gradually weakened. As seen from the figure, the displacement of a three-FRP layer column and a four-FRP layer column is relatively close, but it is obvious that the three-FRP layer is more economical than the four-FRP layer.Figure 20
Comparison of midheight displacement with different thicknesses of FRP.


### 4.5.2. Concrete Strength

The dynamic response of the CFST columns confined with FRP with four different concrete strength grades under the same blast loading is analysed while keeping the other parameters unchanged. The displacement-time history curves of the midheight point of the four cases are shown in Figure21.Figure 21
Comparison of midheight displacement with different comprehensive strengths of concrete.When the strength grades of the concrete are C35, C40, C45, and C50, the maximum displacements of the CFST columns confined with FRP are 28.32 mm, 24.13 mm, 21.65 mm, and 18.90 mm, respectively. The maximum displacement decreases gradually with an increase in the concrete strength grade. This indicates that the blast resistance of a CFST confined with FRP can be enhanced by improving the concrete strength.


### 4.5.3. Cross Section

In the engineering application, the CFST columns with the square section are more widely used because of its convenient joint connection. To study the influence of the cross section on the blast resistance performance, two square-section column models are generated. One column is the same size with the circular one (500 mm × 500 mm × 3000 mm), and the other column is the same material consumptions with the circular one (namely, the section size is 443 mm × 443 mm and the thickness of steel tube is 9 mm). The numerical results of the square section cases are compared with those of the circular section case. The displacement-time history curves of the two cases are shown in Figure22.Figure 22
Comparison of midheight displacement with different cross sections. (a) Same section size. (b) Same material consumptions.
(a)
(b)Under the condition of the same section size, the maximum displacement of the square section column with the same section size is 21.99 mm, which is 8.87% less than that of the circular section member. The blast resistance performance of the square section member is better than that of the circular section member, but the square section member has 27.39% higher amounts of FRP, steel, and concrete than the circular member. Under the condition of the same material consumptions, the maximum displacement of the column with the square section is 30.27 mm. The column with circular section shows better blast performance than the one with square section. This is mainly because the circular section member has an excellent aerodynamic profile. Parts of the blast wave detours, and the impact energy reduces.


## 4.5.1. Number of FRP Layers

To study the effect of number of FRP layers on the blast resistance of the CFST column confined with FRP, the dynamic response of the CFST columns confined with FRP with different layers under blast loading was simulated by changing the thickness of the FRP while keeping other parameters unchanged.Figure20 illustrates the displacement-time history curves of the CFST columns confined with FRP when the FRP is 0, 1, 2, 3, and 4 layers (the thicknesses of the FRP are 0, 0.167 mm, 0.334 mm, 0.501 mm, and 0.668 mm, respectively). When the number of FRP layers is 0, namely, a traditional CFST without the FRP constraint, the maximum displacement of the member under blast loading is 30.17 mm. When the number of FRP layers is 1, the maximum displacement of the column is 24.13 mm, a reduction of 6.04 mm compared with a nonretrofitted CFST column. It is found that the constraint of FRP could effectively improve the blast resistance of CFST columns. When the number of FRP layers is 2, 3, and 4, the maximum displacements are 22.11 mm, 19.52 mm, and 19.37 mm, respectively. This indicates that with an increase in the number of FRP layers, the blast resistance performance of the member is further improved, but its effect is gradually weakened. As seen from the figure, the displacement of a three-FRP layer column and a four-FRP layer column is relatively close, but it is obvious that the three-FRP layer is more economical than the four-FRP layer.Figure 20
Comparison of midheight displacement with different thicknesses of FRP.


## 4.5.2. Concrete Strength

The dynamic response of the CFST columns confined with FRP with four different concrete strength grades under the same blast loading is analysed while keeping the other parameters unchanged. The displacement-time history curves of the midheight point of the four cases are shown in Figure21.Figure 21
Comparison of midheight displacement with different comprehensive strengths of concrete.When the strength grades of the concrete are C35, C40, C45, and C50, the maximum displacements of the CFST columns confined with FRP are 28.32 mm, 24.13 mm, 21.65 mm, and 18.90 mm, respectively. The maximum displacement decreases gradually with an increase in the concrete strength grade. This indicates that the blast resistance of a CFST confined with FRP can be enhanced by improving the concrete strength.


## 4.5.3. Cross Section

In the engineering application, the CFST columns with the square section are more widely used because of its convenient joint connection. To study the influence of the cross section on the blast resistance performance, two square-section column models are generated. One column is the same size with the circular one (500 mm × 500 mm × 3000 mm), and the other column is the same material consumptions with the circular one (namely, the section size is 443 mm × 443 mm and the thickness of steel tube is 9 mm). The numerical results of the square section cases are compared with those of the circular section case. The displacement-time history curves of the two cases are shown in Figure22.Figure 22
Comparison of midheight displacement with different cross sections. (a) Same section size. (b) Same material consumptions.
(a)
(b)Under the condition of the same section size, the maximum displacement of the square section column with the same section size is 21.99 mm, which is 8.87% less than that of the circular section member. The blast resistance performance of the square section member is better than that of the circular section member, but the square section member has 27.39% higher amounts of FRP, steel, and concrete than the circular member. Under the condition of the same material consumptions, the maximum displacement of the column with the square section is 30.27 mm. The column with circular section shows better blast performance than the one with square section. This is mainly because the circular section member has an excellent aerodynamic profile. Parts of the blast wave detours, and the impact energy reduces.


## 5. Conclusion

In this paper, the dynamic response of the CFST columns confined by FRP under blast loading was numerically investigated. Displacement, effective stress, and design parameters of a CFST confined with FRP were discussed in detail by using the developed finite element model, and the innovative conclusions based on these studies were summarized as follows:(1)
Verified by the blast tests of CFST columns and RC columns confined with FRP and SDOF results, the FE model of the CFST column confined with FRP under blast loading using the multimaterial ALE method developed in this study is reliable to simulate the dynamic response. Using this method, the CFST column confined with FRP under blast loading with various scaled distances can further be modelled and analysed.(2)
Under blast loading, the displacement of the column reaches the peak value in a short time and then drops to a certain range of vibration. When the scaled distancez is 0.28 m/kg1/3, the maximum displacement is 24.13 mm and the deformation-span ratio is 0.8%. The CFST column confined with FRP performs a good blast resistance.(3)
The stresses of the steel tube, concrete, and FRP generally show a symmetrical distribution. The stresses in the middle part and two ends of the column are higher, exceeding the static ultimate stress. The numerical results can provide a certain basis for blast protection of the CFST columns and a blast-resistance design of the CFST columns confined with FRP.(4)
The stiffness and bearing capacity of the columns can be enhanced by increasing the number of FRP layers or concrete strength. However, as the number of layers increases, the blast resistance enhancement effect decreases. When the number of layers is greater than 3 layers, it is of little significance to increase the layers for improving the blast resistance performance of the columns.(5)
With the same cross section size, a column with a square cross section shows better blast resistance performance than the one with a circular cross section, but the materials consumptions are also greater. With the same materials, a column with a circular-section shows better blast performance than the one with a square cross section due to the wave detouring effect.Further experimental study of CFST columns confined with FRP under blast loading is necessary to implement, and it will be one of our future works.



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*Source: 8692310-2019-07-10.xml*