# Damage Assessment of an RC Pier under Noncontact Blast Loading Based onP-I Curves

Authors: Lu Liu; Zhouhong Zong; Biao Tang; Minghong Li
Journal: Shock and Vibration
 (2018)

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## Abstract

Since the 9.11 attacks in New York, terrorist attacks, military strikes, and accidental explosions have caused serious damage to various infrastructures around the world. Accidental traffic explosions in bridge engineering have occurred frequently due to the increasing firework truck on the bridge and heavy fuel oil ships under the bridge. Besides, intentional car bomb explosions in terrorist attacks and military assaults are significant threats to the bridge structure. Bridge pier is a particularly important component of the overall bridge structure due to its vertical support capacity. Therefore, the damage of reinforced concrete (RC) piers under blast loading should be studied. In this paper, the pier is simplified as a single degree of freedom (SDOF) system based on energy equality, and the blast loading distribution on the surface of the pier is simplified as a uniform distribution. Then, theP-I curves of an RC pier are obtained based on the SDOF system and a uniform distribution blast loading to quickly evaluate the damage extent of an RC pier so that the P-I curves and fitting formulas for different damage extents can be used for practical engineering. After a numerical simulation verification, the P-I curves and fitting formulas based on the simplified method can supply a rapid and effective method for engineers to assess the postexplosive damage of an RC pier.


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## Body

## 1. Introduction

Since the 9.11 attacks in New York, terrorist attacks, military strikes, and accidental explosions have caused serious damage to various infrastructures around the world [1]. In recent years, accidental traffic explosions in bridge engineering have occurred frequently due to the increasing firework truck on bridges and heavy fuel oil ships under the bridge. Besides, intentional car bomb explosions in terrorist attacks and military assaults are significant threats to the bridge structure [2, 3]. Bridge pier is a particularly important component for the overall bridge structure due to its vertical support capacity; therefore, it is more vulnerable to be damaged under blast loading.There are some achievements on dynamic response, failure modes, and protective measures for structures under blast loading at home and abroad; however, there are lack of valid and fast postexplosive evaluation methods for structural damage. Therefore, it is unpredictable to assess the damage extent of the engineering structure for engineers in a simplest way. Pressure-impulse (P-I) curve has been put forward in the second world war [4] and became a current accepted method for structural postexplosive damage assessment [5], which represents an identical structural damage extent under blast loading. P-I curve is a changeless curve for each structure, so the structural postexplosive damage extent can be confirmed quickly after getting the P-I curve.P-I curve can be obtained primarily by test, numerical simulation, and theoretical analysis. Test method can reliably get the P-I curve for a specific structure by a large number of blasting experiments, in which overpressure and impulse on the structural surface at different damage extents are obtained. However, it is an economically and logistically poor choice and is hard to apply to actual engineering. Due to the large cost, few researchers use this method to get the P-I curve. Wesevich and Oswald [6] conducted a blasting experiment on a concrete block wall by a shock wave tube, in which there were 236 overpressure and impulse data points of the walls so that the P-I curve of the concrete block wall could be determined. However, there are differences between an actual explosive wave and the simulated shock wave generated by the shock wave tube.Numerical simulation can also be used to get the structuralP-I curve; however, it requires a complete understanding of the structure in order to determine numerical parameters accurately. In addition, a large amount of calculation work and better computing equipment are also important conditions to gain an exact P-I curve. Krauthammer et al. [7] have determined the P-I curves of an RC beam, a slab, columns, and other structures by simulation. Dragos et al. [8] have evaluated the damage extent of an ultrahigh-performance concrete (UHPC) slab under blast loading and determined the P-I curves in different parameters. Mutalib and Hao [9] have evaluated the damage extent of a fiber-reinforced concrete column under blast loading and determined the P-I curves by LS-DYNA according to the residual bearing capacity of columns. Column dimensions, material strength, fiber thickness, and other parameters were taken into consideration. Shim et al. [10] have assessed the blast-resistance performance of a multilayer foamed aluminum slab considering a material dynamic increase factor (DIF) at a high strain rate and obtained the P-I curve of the aluminum slab by finite element software. Domestically, Shi et al. [11] have studied RC columns under blast loading and determined the P-I curves from residual bearing capacity by ANSYS/LS-DYNA. There is no doubt that simulation is a quite accurate and low cost method to get the P-I curve under the condition of exact model parameters; however, it is hard to guarantee that the simulated structure and real structure are identical. Moreover, it is also a much more time-consuming method to evaluate structural postexplosive damage for engineers.Additionally, theoretical analytical can rapidly evaluate the structural damage extent, simplifying the structural force system. Firstly, the structural dynamic response is the most important for damage assessment, which can be got by a simplified force system. Equivalent single degree of freedom (SDOF) system has become the most extensive analytical method due to its ability to determine quickly the structuralP-I curves. Li and Meng [12] have determined the structural P-I curves by dimensional analysis and the SDOF system. Fallah and Louca [13] have conducted research on P-I curves based on elastic plastic hardening and the softening equivalent SDOF model, which can be divided into six phases, such as the elastic phase, elastic-plastic hardening phase, elastic-plastic softening phase, rigid plastic hardening phase, and rigid plastic softening phase. Ma et al. [14] have conducted an assessment on a consolidated beam at both ends based on the equivalent SDOF system considering flexural failure and shear failure. Colombo and Martinelli [15] have determined the P-I curves of RC plates and fiber RC circular plates based on the SDOF system with various diameters. Dragos and Wu [16] performed normalized processing on the structural P-I curves by the SDOF system so that the P-I curve could be applied to any impact loading. Wang et al. [17] used the SDOF system to study the P-I curves of RC slabs and beams with flexure failure and shear failure and analyze the relationship between different physical parameters and the P-I curve. This method originates from a basic mechanical model and simplifies the complex structural system; therefore, it can be used to calculate complex explosive systems rapidly and effectively due to its simple application.Currently, theP-I curve is believed to be an effective method to determine the damage assessment under blast loading in the engineering field. However, there are advantages and disadvantages in the above three methods to determine the P-I curve. Although the test method is quite accurate, it is also expensive. Furthermore, the simulation method is the most economical but is the most time-consuming [18, 19]. Last, the theoretical analytical method is the simplest and fastest method for engineers and became the most versatile method in the engineering field due to ignoring the flexural shear failure [20, 21]. Therefore, the equivalent SDOF method is used to calculate the structural dynamic response and determine the P-I curve of a bridge pier in this paper so that the damage extent can be quickly evaluated. Also, the numerical model is built to validate the SDOF model.


## 2. SDOF Systems

### 2.1. Equivalent Analysis Method

In dynamical analysis, there are three important factors: work, strain energy, and kinetic energy. The work under external loading should be calculated by structural displacement. The structural strain energy is mainly caused by tensile strain, compressive strain, shear strain, and so on. The kinetic energy is also caused by the movement and rotation of mass block [22].A structure can be replaced by an idealized system (dynamically equivalent), which shows almost the same time characteristics as the real structure. The distributed mass of a given structure is assigned to one or several concentrated masses, and the strain energy is assigned to one or several weightless and inelastic springs. In addition, the distributed loads are replaced by several concentrated loads on the concentrated mass. Therefore, this idealized equivalent system, which mainly consists of weightless springs, concentrated masses, and time-varying concentrated loads, is called a mass-spring-load dynamic equivalent system.Although a basic structure has many degrees of freedom, a basic vibration mode plays a leading role under transitory loading. Therefore, it can be simplified to study an SDOF system that has the same characteristics and vibration modes as a real structure. The SDOF system is a simplified system with only one state of motion. Figure1 shows the simplified system described using only one coordinate.Figure 1
Equivalent single degree of freedom system: (a) real structure; (b) equivalent system.
(a)
(b)


### 2.2. Equivalent System Transformation

The work under equivalent loads, strain energy, and kinetic energy in an equivalent system should be the same as the real structure and based on equal displacement and energy. Therefore, the structure can be equivalent to a mass-spring-load SDOF system, where the mass, resistance, and load are equivalent [23].


#### 2.2.1. Load Coefficient

The equivalent concentrated load is the total load on the structure, multiplied by the load coefficient, as shown in the following equation [22]:(1)KL=FEF,where F is the total load on the actual structure, FE is the equivalent load, and KL is the load coefficient. FE follows the principle that work done by an equivalent load on an equivalent system is equal to that done by an actual load on an assumed deflection shape.For a structure subjected to distributed loads,(2)WD=FE⋅δmax=∫0Lpxδxdx,FE=∫0Lpx∅xdx,∅x=δxδmax,where WD is the work; δmax is the maximum structural deflection; px is the distributed loading on the structure; δx is the deflection of any position; and ∅x is the shape function, which is different in the elastic stage and plastic stage, so that KL is different as well.


#### 2.2.2. Mass Coefficient

The equivalent concentrated mass is the total distributed mass multiplied by the mass coefficient, which is shown as follows [22]:(3)KM=MEM,where M is the total mass on an actual structure, ME is the equivalent mass, and KM is the mass coefficient. ME follows the principle that kinetic energy of an equivalent system is equal to that on the deflection shape of an actual structure.For a continuous mass structure,(4)KE=12MEωδmax2=12∫0Lmxωδx2dx,ME=∫0Lmx∅2xdx,where KE is the kinetic energy, ω is a natural vibration frequency, mx is the distributed mass per unit length, and ∅x is the shape function that is caused by external loads. ∅x is different in the inelastic stage and plastic stage, so KM is also different.


#### 2.2.3. Resistance Coefficient

The equivalent resistance is the total resistance multiplied by the resistance coefficient, which is shown as follows [22]:(5)KR=RER=KE⋅δmaxK⋅δmax,where R is the total resistance of an actual structure, RE is the equivalent resistance, KR is the resistance coefficient, KE is the equivalent stiffness, and K is the total stiffness of an actual structure. RE follows the principle that strain energy of an actual structure under an assumed deflection is equal to that of an equivalent SDOF system. Because resistance of a structure is an internal force that restores the structure to its static position, KR must be equal to KL.


#### 2.2.4. Load-Mass Coefficient

The viscous damping force of a structure normally lags the inertia force by 1/4 of a period under blast loading. The durations of blast loadings are only several milliseconds, so the blast loading rapidly disappears before the damping force appears. Therefore, it is not necessary to consider the damping force in an equation of motion, which is as follows [22]:(6)MEx¨+KEx=FE,KMMx¨+KLKx=KLF,KMKLMx¨+Kx=F,KLMMx¨+Kx=F,where KLM is the load-mass coefficient and KLM=KM/KL. Therefore, the transformation of the motion equation between the actual structure and equivalent system only needs one parameter, KLM.The dynamic design factors for a cantilever support system in which one end is fixed and another end is a hinged joint are shown in Table1.Table 1
Dynamic design factors of the cantilever support system [22].
Support condition and load
Stress state
K
L
K
M
K
LM
Elastic
0.58
0.45
0.78
Elastic-plastic
0.64
0.50
0.78
Plastic
0.50
0.33
0.66


### 2.3. Equivalent SDOF Model of an RC Pier

Liu [24, 25] and Tang [26, 27] have designed a round RC pier in Figure 2, for which essential design parameters are shown in Table 2. The top of the pier is assumed a hinged constraint with two lateral displacements, and the bottom of the pier is assumed a fixed constraint. Therefore, the structural system of an RC pier is a cantilever support system.Figure 2
RC pier experiment: (a) test specimen; (b) explosive location.
(a)
(b)Table 2
Essential design parameters of the RC pier.
Specimen
Effective height
Diameter
Stirrup size
Yield strength of the stirrup
RC pier
3500 mm
400 mm
φ8 mm
300 MPa
Specimen
Longitudinal reinforcement size
Reinforcement ratio
Yield strength of longitudinal reinforcement
Concrete strength
RC pier
10φ12 mm
0.9%
400 MPa
40 MPaSeveral assumptions must be made when the RC pier is simplified as an equivalent SDOF system under blast loading. (1) The RC pier can be simplified as an ideal rigid plastic member when only bending deformation is considered, ignoring shear deformation; (2) an ideal plastic hinge appears at the bottom and midheight of the pier when the RC pier experiences bending deformation; and (3) the pier is destroyed when the displacement at midheight reachesXm [28].The load and displacement at the midheight of the pier are shown in Figure3(a) when the plastic hinge emerges at the bottom of the pier:(7)P1=8MPEL2,X1=P1L4185EEIE=8MPEL2185EEIE.Figure 3
The simplified rigid plastic model of the RC pier: (a) plastic hinge at the bottom of the pier; (b) plastic hinge at the bottom and midheight of the pier; (c) simplified rigid plastic model.
(a)
(b)
(c)The additional load and displacement at the midheight of the pier are shown in Figure3(b) when the plastic hinge emerges at the midheight of the pier:(8)P2=4MPEL2,X2=5P2L4384EEIE=20MPEL2384EEIE.The total load and displacement at the midheight of the pier are as follows when the pier becomes a geometrically unstable system:(9)P0=P1+P2=12MPEL2,X0=X1+X2=2MPEL221EEIE,where X1 is the displacement at the midheight of the pier when plastic hinge emerges at the bottom of the pier; X2 is the additional displacement at the midheight of the pier when plastic hinge emerges at the midheight of the pier; X0 is the total displacement when the pier becomes a geometrically unstable system; MPE is the plastic limit bending moment of the pier, which is the same when plastic hinge emerges at the midheight and the bottom of the pier; EE is the global equivalent elastic modulus of the pier; IE is the global equivalent moment of inertia of the pier; P1 is the horizontal pressure when plastic hinge emerges at the bottom of the pier; P2 is the additional pressure when plastic hinge emerges at the bottom and midheight of the pier; and P0 is the total pressure.The equivalent simplified SDOF model of the pier ignoring the damping force can be deduced as follows:(10)MEx¨+KEx=FEt,x<XE,MEx¨+RE=FEt,XE<x<Xm,XE=5L2MPE96EEIE,where ME=KMM, in which ME is the equivalent mass of the RC pier, M is the total mass of the RC pier, and KM=0.5; KE=KLK0, in which KE is the equivalent stiffness of the elastic stage and K0=307EEIE/L3, K0 is the equivalent initial stiffness, and KL=0.64; FEt=KLFt, where FEt is the equivalent load, Ft=PtBL, in which B is the width of the pier and L is the height of the pier; RE=KLR0, where RE is the equivalent resistance extremum and R0=12MPE/L2, in which R0 is the initial limit resistance; XE is the equivalent elastic displacement when the linear curve is equivalent to the ideal elastic-plastic curve; and Xm is the maximum plastic displacement. The resistance model of the RC pier is shown in Figure 4 [29].Figure 4
Resistance model [29].


## 2.1. Equivalent Analysis Method

In dynamical analysis, there are three important factors: work, strain energy, and kinetic energy. The work under external loading should be calculated by structural displacement. The structural strain energy is mainly caused by tensile strain, compressive strain, shear strain, and so on. The kinetic energy is also caused by the movement and rotation of mass block [22].A structure can be replaced by an idealized system (dynamically equivalent), which shows almost the same time characteristics as the real structure. The distributed mass of a given structure is assigned to one or several concentrated masses, and the strain energy is assigned to one or several weightless and inelastic springs. In addition, the distributed loads are replaced by several concentrated loads on the concentrated mass. Therefore, this idealized equivalent system, which mainly consists of weightless springs, concentrated masses, and time-varying concentrated loads, is called a mass-spring-load dynamic equivalent system.Although a basic structure has many degrees of freedom, a basic vibration mode plays a leading role under transitory loading. Therefore, it can be simplified to study an SDOF system that has the same characteristics and vibration modes as a real structure. The SDOF system is a simplified system with only one state of motion. Figure1 shows the simplified system described using only one coordinate.Figure 1
Equivalent single degree of freedom system: (a) real structure; (b) equivalent system.
(a)
(b)


## 2.2. Equivalent System Transformation

The work under equivalent loads, strain energy, and kinetic energy in an equivalent system should be the same as the real structure and based on equal displacement and energy. Therefore, the structure can be equivalent to a mass-spring-load SDOF system, where the mass, resistance, and load are equivalent [23].


### 2.2.1. Load Coefficient

The equivalent concentrated load is the total load on the structure, multiplied by the load coefficient, as shown in the following equation [22]:(1)KL=FEF,where F is the total load on the actual structure, FE is the equivalent load, and KL is the load coefficient. FE follows the principle that work done by an equivalent load on an equivalent system is equal to that done by an actual load on an assumed deflection shape.For a structure subjected to distributed loads,(2)WD=FE⋅δmax=∫0Lpxδxdx,FE=∫0Lpx∅xdx,∅x=δxδmax,where WD is the work; δmax is the maximum structural deflection; px is the distributed loading on the structure; δx is the deflection of any position; and ∅x is the shape function, which is different in the elastic stage and plastic stage, so that KL is different as well.


### 2.2.2. Mass Coefficient

The equivalent concentrated mass is the total distributed mass multiplied by the mass coefficient, which is shown as follows [22]:(3)KM=MEM,where M is the total mass on an actual structure, ME is the equivalent mass, and KM is the mass coefficient. ME follows the principle that kinetic energy of an equivalent system is equal to that on the deflection shape of an actual structure.For a continuous mass structure,(4)KE=12MEωδmax2=12∫0Lmxωδx2dx,ME=∫0Lmx∅2xdx,where KE is the kinetic energy, ω is a natural vibration frequency, mx is the distributed mass per unit length, and ∅x is the shape function that is caused by external loads. ∅x is different in the inelastic stage and plastic stage, so KM is also different.


### 2.2.3. Resistance Coefficient

The equivalent resistance is the total resistance multiplied by the resistance coefficient, which is shown as follows [22]:(5)KR=RER=KE⋅δmaxK⋅δmax,where R is the total resistance of an actual structure, RE is the equivalent resistance, KR is the resistance coefficient, KE is the equivalent stiffness, and K is the total stiffness of an actual structure. RE follows the principle that strain energy of an actual structure under an assumed deflection is equal to that of an equivalent SDOF system. Because resistance of a structure is an internal force that restores the structure to its static position, KR must be equal to KL.


### 2.2.4. Load-Mass Coefficient

The viscous damping force of a structure normally lags the inertia force by 1/4 of a period under blast loading. The durations of blast loadings are only several milliseconds, so the blast loading rapidly disappears before the damping force appears. Therefore, it is not necessary to consider the damping force in an equation of motion, which is as follows [22]:(6)MEx¨+KEx=FE,KMMx¨+KLKx=KLF,KMKLMx¨+Kx=F,KLMMx¨+Kx=F,where KLM is the load-mass coefficient and KLM=KM/KL. Therefore, the transformation of the motion equation between the actual structure and equivalent system only needs one parameter, KLM.The dynamic design factors for a cantilever support system in which one end is fixed and another end is a hinged joint are shown in Table1.Table 1
Dynamic design factors of the cantilever support system [22].
Support condition and load
Stress state
K
L
K
M
K
LM
Elastic
0.58
0.45
0.78
Elastic-plastic
0.64
0.50
0.78
Plastic
0.50
0.33
0.66


## 2.2.1. Load Coefficient

The equivalent concentrated load is the total load on the structure, multiplied by the load coefficient, as shown in the following equation [22]:(1)KL=FEF,where F is the total load on the actual structure, FE is the equivalent load, and KL is the load coefficient. FE follows the principle that work done by an equivalent load on an equivalent system is equal to that done by an actual load on an assumed deflection shape.For a structure subjected to distributed loads,(2)WD=FE⋅δmax=∫0Lpxδxdx,FE=∫0Lpx∅xdx,∅x=δxδmax,where WD is the work; δmax is the maximum structural deflection; px is the distributed loading on the structure; δx is the deflection of any position; and ∅x is the shape function, which is different in the elastic stage and plastic stage, so that KL is different as well.


## 2.2.2. Mass Coefficient

The equivalent concentrated mass is the total distributed mass multiplied by the mass coefficient, which is shown as follows [22]:(3)KM=MEM,where M is the total mass on an actual structure, ME is the equivalent mass, and KM is the mass coefficient. ME follows the principle that kinetic energy of an equivalent system is equal to that on the deflection shape of an actual structure.For a continuous mass structure,(4)KE=12MEωδmax2=12∫0Lmxωδx2dx,ME=∫0Lmx∅2xdx,where KE is the kinetic energy, ω is a natural vibration frequency, mx is the distributed mass per unit length, and ∅x is the shape function that is caused by external loads. ∅x is different in the inelastic stage and plastic stage, so KM is also different.


## 2.2.3. Resistance Coefficient

The equivalent resistance is the total resistance multiplied by the resistance coefficient, which is shown as follows [22]:(5)KR=RER=KE⋅δmaxK⋅δmax,where R is the total resistance of an actual structure, RE is the equivalent resistance, KR is the resistance coefficient, KE is the equivalent stiffness, and K is the total stiffness of an actual structure. RE follows the principle that strain energy of an actual structure under an assumed deflection is equal to that of an equivalent SDOF system. Because resistance of a structure is an internal force that restores the structure to its static position, KR must be equal to KL.


## 2.2.4. Load-Mass Coefficient

The viscous damping force of a structure normally lags the inertia force by 1/4 of a period under blast loading. The durations of blast loadings are only several milliseconds, so the blast loading rapidly disappears before the damping force appears. Therefore, it is not necessary to consider the damping force in an equation of motion, which is as follows [22]:(6)MEx¨+KEx=FE,KMMx¨+KLKx=KLF,KMKLMx¨+Kx=F,KLMMx¨+Kx=F,where KLM is the load-mass coefficient and KLM=KM/KL. Therefore, the transformation of the motion equation between the actual structure and equivalent system only needs one parameter, KLM.The dynamic design factors for a cantilever support system in which one end is fixed and another end is a hinged joint are shown in Table1.Table 1
Dynamic design factors of the cantilever support system [22].
Support condition and load
Stress state
K
L
K
M
K
LM
Elastic
0.58
0.45
0.78
Elastic-plastic
0.64
0.50
0.78
Plastic
0.50
0.33
0.66


## 2.3. Equivalent SDOF Model of an RC Pier

Liu [24, 25] and Tang [26, 27] have designed a round RC pier in Figure 2, for which essential design parameters are shown in Table 2. The top of the pier is assumed a hinged constraint with two lateral displacements, and the bottom of the pier is assumed a fixed constraint. Therefore, the structural system of an RC pier is a cantilever support system.Figure 2
RC pier experiment: (a) test specimen; (b) explosive location.
(a)
(b)Table 2
Essential design parameters of the RC pier.
Specimen
Effective height
Diameter
Stirrup size
Yield strength of the stirrup
RC pier
3500 mm
400 mm
φ8 mm
300 MPa
Specimen
Longitudinal reinforcement size
Reinforcement ratio
Yield strength of longitudinal reinforcement
Concrete strength
RC pier
10φ12 mm
0.9%
400 MPa
40 MPaSeveral assumptions must be made when the RC pier is simplified as an equivalent SDOF system under blast loading. (1) The RC pier can be simplified as an ideal rigid plastic member when only bending deformation is considered, ignoring shear deformation; (2) an ideal plastic hinge appears at the bottom and midheight of the pier when the RC pier experiences bending deformation; and (3) the pier is destroyed when the displacement at midheight reachesXm [28].The load and displacement at the midheight of the pier are shown in Figure3(a) when the plastic hinge emerges at the bottom of the pier:(7)P1=8MPEL2,X1=P1L4185EEIE=8MPEL2185EEIE.Figure 3
The simplified rigid plastic model of the RC pier: (a) plastic hinge at the bottom of the pier; (b) plastic hinge at the bottom and midheight of the pier; (c) simplified rigid plastic model.
(a)
(b)
(c)The additional load and displacement at the midheight of the pier are shown in Figure3(b) when the plastic hinge emerges at the midheight of the pier:(8)P2=4MPEL2,X2=5P2L4384EEIE=20MPEL2384EEIE.The total load and displacement at the midheight of the pier are as follows when the pier becomes a geometrically unstable system:(9)P0=P1+P2=12MPEL2,X0=X1+X2=2MPEL221EEIE,where X1 is the displacement at the midheight of the pier when plastic hinge emerges at the bottom of the pier; X2 is the additional displacement at the midheight of the pier when plastic hinge emerges at the midheight of the pier; X0 is the total displacement when the pier becomes a geometrically unstable system; MPE is the plastic limit bending moment of the pier, which is the same when plastic hinge emerges at the midheight and the bottom of the pier; EE is the global equivalent elastic modulus of the pier; IE is the global equivalent moment of inertia of the pier; P1 is the horizontal pressure when plastic hinge emerges at the bottom of the pier; P2 is the additional pressure when plastic hinge emerges at the bottom and midheight of the pier; and P0 is the total pressure.The equivalent simplified SDOF model of the pier ignoring the damping force can be deduced as follows:(10)MEx¨+KEx=FEt,x<XE,MEx¨+RE=FEt,XE<x<Xm,XE=5L2MPE96EEIE,where ME=KMM, in which ME is the equivalent mass of the RC pier, M is the total mass of the RC pier, and KM=0.5; KE=KLK0, in which KE is the equivalent stiffness of the elastic stage and K0=307EEIE/L3, K0 is the equivalent initial stiffness, and KL=0.64; FEt=KLFt, where FEt is the equivalent load, Ft=PtBL, in which B is the width of the pier and L is the height of the pier; RE=KLR0, where RE is the equivalent resistance extremum and R0=12MPE/L2, in which R0 is the initial limit resistance; XE is the equivalent elastic displacement when the linear curve is equivalent to the ideal elastic-plastic curve; and Xm is the maximum plastic displacement. The resistance model of the RC pier is shown in Figure 4 [29].Figure 4
Resistance model [29].


## 3. Blast Loading and Damage Failure Criterion

### 3.1. Equivalent Blast Loading of an RC Pier

Blast loading on a structure is a complex loading and is related to the explosive position, explosive mass, structural shape, and surrounding environment. The blast type can be divided into a near-field explosion and a far-field explosion according to the explosive position and scale distance when explosive is located on the ground. The blast loading along the height of the pier can be assumed to be a trapezoidal distribution in the near-field explosion, which is examined in this paper. A simplified pressure distribution is shown in Figure5, where the overpressure times at any height of the pier are considered to be equal, and the negative pressure of the blast shock wave is ignored, as it is negligible when compared to the positive pressure.Figure 5
Simplified pressure distribution: (a) trapezoidal load distribution; (b) real pressure history; (c) simplified pressure history.
(a)
(b)
(c)The blast loading can be equivalent in accordance with the impulse equivalence on the structure. First, the peak pressure of equivalent uniform loading can be obtained from the following equation:(11)P¯=12P1max+P2max.Then, the equivalent duration of blast loading can be deduced from the duration of blast loading in the center of the pier, so the impulse of an equivalent uniform load can be calculated as(12)td=2IP¯.The blast loading history in the real pressure-time function can be simplified to the triangular blast loading in Figure5(c), as shown in the following equation:(13)Pt=Pmax1−ttd,0<t<td,Pt=0,td<t,where Pmax is the reflected overpressure peak of the blast shock wave and td is the duration of the simplified triangular blast loading.


### 3.2. Damage Failure Criteria of an RC Pier

It is assumed that the RC pier only experiences flexural failure under blast loading, as the maximum ductile deformation usually occurs at the midheight of the pier. Then, the damage extent of the RC pier can be evaluated according to the TM5-1300 manual [17]. The bearing angle of the pier and the ratio of maximum deflection at midheight to half height are usually an indicator of the flexural failure criterion of the pier. The flexural failure criterion of an RC pier is shown in Table 3.Table 3
The failure criterion of an RC pier.
Index
Slight damage
Moderate damage
Severe damage
Collapse
Bearing angle (°)
0∼2
2∼5
5∼12
>12
Maximum lateral displacement (mm)
0∼61
61∼153
153∼364
>364It can be concluded from the above criterion that the reinforcement has yielded, but the concrete in the compression zone has not yet been destroyed when the bearing angle is 0°∼2°. The concrete outside the compression reinforcement is crushed when the bearing angle is greater than 2°, leading to the compressive stress of concrete being transferred to the reinforced bar, causing the strength of the pier to decease. The concrete on both sides spalls when the bearing angle is greater than 5°, causing the overall bearing capacity to decrease. The reinforcement in the tension zone yields, or the concrete in the compression zone is crushed completely when the bearing angle is greater than 12°, meaning the pier collapses.


## 3.1. Equivalent Blast Loading of an RC Pier

Blast loading on a structure is a complex loading and is related to the explosive position, explosive mass, structural shape, and surrounding environment. The blast type can be divided into a near-field explosion and a far-field explosion according to the explosive position and scale distance when explosive is located on the ground. The blast loading along the height of the pier can be assumed to be a trapezoidal distribution in the near-field explosion, which is examined in this paper. A simplified pressure distribution is shown in Figure5, where the overpressure times at any height of the pier are considered to be equal, and the negative pressure of the blast shock wave is ignored, as it is negligible when compared to the positive pressure.Figure 5
Simplified pressure distribution: (a) trapezoidal load distribution; (b) real pressure history; (c) simplified pressure history.
(a)
(b)
(c)The blast loading can be equivalent in accordance with the impulse equivalence on the structure. First, the peak pressure of equivalent uniform loading can be obtained from the following equation:(11)P¯=12P1max+P2max.Then, the equivalent duration of blast loading can be deduced from the duration of blast loading in the center of the pier, so the impulse of an equivalent uniform load can be calculated as(12)td=2IP¯.The blast loading history in the real pressure-time function can be simplified to the triangular blast loading in Figure5(c), as shown in the following equation:(13)Pt=Pmax1−ttd,0<t<td,Pt=0,td<t,where Pmax is the reflected overpressure peak of the blast shock wave and td is the duration of the simplified triangular blast loading.


## 3.2. Damage Failure Criteria of an RC Pier

It is assumed that the RC pier only experiences flexural failure under blast loading, as the maximum ductile deformation usually occurs at the midheight of the pier. Then, the damage extent of the RC pier can be evaluated according to the TM5-1300 manual [17]. The bearing angle of the pier and the ratio of maximum deflection at midheight to half height are usually an indicator of the flexural failure criterion of the pier. The flexural failure criterion of an RC pier is shown in Table 3.Table 3
The failure criterion of an RC pier.
Index
Slight damage
Moderate damage
Severe damage
Collapse
Bearing angle (°)
0∼2
2∼5
5∼12
>12
Maximum lateral displacement (mm)
0∼61
61∼153
153∼364
>364It can be concluded from the above criterion that the reinforcement has yielded, but the concrete in the compression zone has not yet been destroyed when the bearing angle is 0°∼2°. The concrete outside the compression reinforcement is crushed when the bearing angle is greater than 2°, leading to the compressive stress of concrete being transferred to the reinforced bar, causing the strength of the pier to decease. The concrete on both sides spalls when the bearing angle is greater than 5°, causing the overall bearing capacity to decrease. The reinforcement in the tension zone yields, or the concrete in the compression zone is crushed completely when the bearing angle is greater than 12°, meaning the pier collapses.


## 4.P-I Curves

### 4.1. Relations between Loads and Resistance

The structure itself has a natural vibration period. This vibration is related to the load and structure. The blast loading can be divided into impulse loading, pressure-impulse loading, and pressure loading according to theP-I curve, so that there are three different relationships between blast loading and resistance, as shown in Table 4. From the first curve, the time of maximum deflection is longer than the loading time (tm/t0>3), so the impulse loading causes structural destruction. From the third curve, the time of maximum deflection is shorter than the loading time (0.1>tm/t0), so the pressure loading causes structural destruction. However, when 3>tm/t0>0.1, both the pressure loading and impulse loading cause structural destruction.Table 4
Load-resistance history.
Load-resistance history
Pressure type
High pressure
Medium pressure
Low pressure
Load type
Impulse
Pressure impulse
Pressure
Duration
Short
Medium
Long
Reaction time
Long
Medium
Short
t
m
/
t
0
t
m
/
t
0
>
3
3
>
t
m
/
t
0
>
0.1
0.1
>
t
m
/
t
0
Note. tm is the time of maximum deflection and t0 is the loading time.


### 4.2. The Characteristics ofP-I Curves

Figure6 shows the classic P-I curve diagram, in which each curve is representative of the same damage extent of the structure. There are two asymptotes: the pressure asymptote and the impulse asymptote, which define the critical values of overpressure and impulse, respectively. The impulse asymptote physically shows the minimum impulse when the pier experiences some damage extent under blast loading. Moreover, the pressure asymptote physically shows the minimum pressure when the pier experiences some damage extent under blast loading.Figure 6
TypicalP-I curve diagram.The diagram is divided into two parts according to theP-I curve: zone A and zone B. When the blast loading is in zone A, the actual structural damage extent is more significant than what is stated by the P-I curve. When the blast loading is in zone B, the actual structural damage extent is less than what is stated by the P-I curve. There are a series of P-I curves in a structural damage extent diagram, where each curve corresponds to a different damage extent. However, the damage extent of a structure can be distinguished between different curves, which are defined as slight damage, moderate damage, severe damage, and failure.


### 4.3.P-I Curves of an RC Pier

To determine theP-I curve of the pier, the blast loading on the structure must follow the above assumptions. Then, the P-I curves of an experimental RC pier can be drawn according to the above failure criteria when the top of the pier produces 2°, 5°, and 12° angular displacements, which are shown in Figure 7.Figure 7
Calculated data and fittingP-I curves.From Figure7, the coordinate system is divided into four parts by three damage curves. The pier experiences slight damage when the value is below the θ=2° curve, and the pier experiences moderate damage when the value is between the θ=2° curve and the θ=5° curve. The pier experiences severe damage when the value is between the θ=5° curve and the θ=12° curve, and finally, the pier collapses and fails when the value is above the θ=12° curve. Three equal damage lines are fitted and shown in the following equations:(14)P−0.66I−2.351.5=0.08616,P−1.70I−5.951.5=0.08870,P−3.20I−14.351.5=0.20841.From the abovementioned equations, the corresponding pressure asymptotes are 0.66 MPa, 1.70 MPa, and 3.2 MPa when the top of the pier produces 2°, 5°, and 12° angular displacements, respectively. In other words, the minimum overpressures are 0.66 MPa, 1.70 MPa, and 3.2 MPa when the RC pier experience slight damage, moderate damage, and severe damage, respectively.The corresponding impulse asymptotes are 2.35 MPa·ms, 5.95 MPa·ms, and 14.35 MPa·ms when the top of the pier produces 2°, 5°, and 12° angular displacements, respectively. In other words, the minimum impulses are 2.35 MPa·ms, 5.95 MPa·ms, and 14.35 MPa·ms when the RC pier experiences slight damage, moderate damage, and severe damage, respectively.From the diagram, the curved section of theP-I curve is relatively short, implying that it is more difficult to produce flexural shear failure than flexural failure and shear failure.Once theP-I curves of the RC pier are determined, the damage extent of the RC pier can be quickly evaluated by the overpressure and impulse.


## 4.1. Relations between Loads and Resistance

The structure itself has a natural vibration period. This vibration is related to the load and structure. The blast loading can be divided into impulse loading, pressure-impulse loading, and pressure loading according to theP-I curve, so that there are three different relationships between blast loading and resistance, as shown in Table 4. From the first curve, the time of maximum deflection is longer than the loading time (tm/t0>3), so the impulse loading causes structural destruction. From the third curve, the time of maximum deflection is shorter than the loading time (0.1>tm/t0), so the pressure loading causes structural destruction. However, when 3>tm/t0>0.1, both the pressure loading and impulse loading cause structural destruction.Table 4
Load-resistance history.
Load-resistance history
Pressure type
High pressure
Medium pressure
Low pressure
Load type
Impulse
Pressure impulse
Pressure
Duration
Short
Medium
Long
Reaction time
Long
Medium
Short
t
m
/
t
0
t
m
/
t
0
>
3
3
>
t
m
/
t
0
>
0.1
0.1
>
t
m
/
t
0
Note. tm is the time of maximum deflection and t0 is the loading time.


## 4.2. The Characteristics ofP-I Curves

Figure6 shows the classic P-I curve diagram, in which each curve is representative of the same damage extent of the structure. There are two asymptotes: the pressure asymptote and the impulse asymptote, which define the critical values of overpressure and impulse, respectively. The impulse asymptote physically shows the minimum impulse when the pier experiences some damage extent under blast loading. Moreover, the pressure asymptote physically shows the minimum pressure when the pier experiences some damage extent under blast loading.Figure 6
TypicalP-I curve diagram.The diagram is divided into two parts according to theP-I curve: zone A and zone B. When the blast loading is in zone A, the actual structural damage extent is more significant than what is stated by the P-I curve. When the blast loading is in zone B, the actual structural damage extent is less than what is stated by the P-I curve. There are a series of P-I curves in a structural damage extent diagram, where each curve corresponds to a different damage extent. However, the damage extent of a structure can be distinguished between different curves, which are defined as slight damage, moderate damage, severe damage, and failure.


## 4.3.P-I Curves of an RC Pier

To determine theP-I curve of the pier, the blast loading on the structure must follow the above assumptions. Then, the P-I curves of an experimental RC pier can be drawn according to the above failure criteria when the top of the pier produces 2°, 5°, and 12° angular displacements, which are shown in Figure 7.Figure 7
Calculated data and fittingP-I curves.From Figure7, the coordinate system is divided into four parts by three damage curves. The pier experiences slight damage when the value is below the θ=2° curve, and the pier experiences moderate damage when the value is between the θ=2° curve and the θ=5° curve. The pier experiences severe damage when the value is between the θ=5° curve and the θ=12° curve, and finally, the pier collapses and fails when the value is above the θ=12° curve. Three equal damage lines are fitted and shown in the following equations:(14)P−0.66I−2.351.5=0.08616,P−1.70I−5.951.5=0.08870,P−3.20I−14.351.5=0.20841.From the abovementioned equations, the corresponding pressure asymptotes are 0.66 MPa, 1.70 MPa, and 3.2 MPa when the top of the pier produces 2°, 5°, and 12° angular displacements, respectively. In other words, the minimum overpressures are 0.66 MPa, 1.70 MPa, and 3.2 MPa when the RC pier experience slight damage, moderate damage, and severe damage, respectively.The corresponding impulse asymptotes are 2.35 MPa·ms, 5.95 MPa·ms, and 14.35 MPa·ms when the top of the pier produces 2°, 5°, and 12° angular displacements, respectively. In other words, the minimum impulses are 2.35 MPa·ms, 5.95 MPa·ms, and 14.35 MPa·ms when the RC pier experiences slight damage, moderate damage, and severe damage, respectively.From the diagram, the curved section of theP-I curve is relatively short, implying that it is more difficult to produce flexural shear failure than flexural failure and shear failure.Once theP-I curves of the RC pier are determined, the damage extent of the RC pier can be quickly evaluated by the overpressure and impulse.


## 5. Numerical Verification

### 5.1. Geometric Model

To study the validity of above computational method and the accuracy ofP-I curves, numerical simulation is the most common and effective method [30]. In this paper, ANSYS/LS-DYNA, dynamic explicit analysis software, is used to build a finite element model of an RC pier, which is shown in Figure 8. The diameter of circular cross section is 400 mm, and the effective height of the pier is 3500 mm. Other geometric parameters are set according to the experimental pier. In this model, concrete is simulated by 8-noded solid164 element, and steel bars are simulated by 2-noded beam161 element, whose mesh size is 20 mm. The number of concrete elements is 55500, and the number of steel bar elements is 3030. The bottom of the pier is set as a fixed constraint, and the top of the pier is set as a hinged constraint. Different elements are coupled with common nodes ignoring relative slippage due to a fairly short explosion time.Figure 8
Finite element model of an RC pier.


### 5.2. Material Model

The concrete uses the material model∗MAT_CONCRETE_DAMAGE_REL3, called the K&C model, which mainly analyzes the dynamical response of the concrete structure under blast loading [31]. The model is the modified pseudotensor concrete material, including three independent failure surfaces, damage effects, and strain rate effects to control complex mechanical properties [32]. The failure criterion of this model is determined by initial yield failure surface, maximum strength failure surface, and residual strength failure surface, whose three invalid compression meridians consist of eight independent parameters. The compression meridian formulas are as follows:(15)Δσy=aoy+Pa1y+a2yP,Δσm=ao+Pa1+a2P,Δσr=Pa1f+a2fP,where Δσy is the equivalent yield strength; Δσm is the equivalent failure strength; Δσr is the equivalent residual strength; aoy, a1y, a2y, ao, a1, a2, a1f, and a2f are the material constant determined by the unconfined compression test and three axial compression test; P is the hydrostatic pressure. The specific function relation of failure surface under different stress states are as follows:(16)Δσr=ηΔσm+1−ηΔσyλ≤λn,Δσh=ηΔσm+1−ηΔσyλ≥λn,where λ is the damage variable that is function of the equivalent plastic strain, λn is the damage turning point between strengthening phase and softening phase, and η is the function of the damage value that ranges from zero to one.K&C model also considers the damage effects including shear deformation damage and volume damage under three-direction stretching. The damage variables are as follows:(17)λ=∫0εP¯dεP¯rf1+P/rfftb1,P≥0,λ=∫0εP¯dεP¯rf1+P/rfftb2,P≥0,λ=b3fdkdεV−εV,yield,where rf is the coefficient of the strain rate effect, b1 is the damage effect under compression, b2 is the damage effect under tension, b3 is the damage effect under volume stretching, dεP¯ is the equivalent plastic strain increment, fd is the stress limiting factor, kd is the internal variable factor, εV is the volumetric strain, and εV,yield is the volumetric strain at the yield point.This model is a strain rate-related materials whose strength is increased by the strain rate versus increasing factor of strain rate. Therefore, the increasing function is as follows:(18)Δσe=rfΔPrf,where rf is the coefficient of the strain rate effect and P is the hydrostatic pressure.However, this complicated concrete constitutive has been embedded into LS-DYNA software; therefore, only density, passion ratio, uniaxial compressive strength, and dynamic increase factor are input into software, and other parameters are generated automatically. In this paper, the strain rate effect of concrete modes comes from the modified models. The DIFs [33] of concrete tension strength are seen in the following equations:(19)TDIF=ftdfts=0.26lgε˙d+2.06,ε˙d≤1/s,TDIF=ftdfts=2lgε˙d+2.06,1/s<ε˙d≤2/s,TDIF=ftdfts=1.44331lgε˙d+2.22766,2/s<ε˙d≤150/s,where ftd is the dynamic tension strength of concrete at the strain rate of ε˙d and fts is the static tension strength of concrete at the strain rate of ε˙ts (ε˙ts=10−6/s).The DIFs of concrete compressive strength [31] are seen in the following equations:(20)CDIF=fcdfcs=0.0419lgε˙d+1.2165,ε˙d≤30/s,CDIF=fcdfcs=0.8988lgε˙d2−2.8255lgε˙d+3.4907,ε˙d>30/s.The steel bar uses material model∗MAT_PLASTIC_KINEMATIC, called the plasticity kinematic model, which uses the Cowper–Symonds model to consider the dynamic constitutive relationship [34]. This model consists of the isotropy and kinematic hardening model related with strain rate and can consider material failure. Cowper–Symonds model can be defined as following equation:(21)σY=1+ε˙C1/pσ0+βEpεpeff,where σ0 is the initial yield stress, ε˙ is the strain rate, C and p are the strain rate parameter of the C-S model, εpeff is the effective plastic strain, and β is the hardening parameter consisting of 0 (only kinematic hardening) and 1 (only isotropic hardening). Ep is the plastic hardening modulus and Etan is the tangent modulus, which are related as follows:(22)Ep=EtanEE−Etan.In this paper, the DIF [34] of the steel bar is as follows:(23)DIF=ε˙10−4α,α=0.074−0.04fy414,where ε˙ is the strain rate of reinforcement, fy is the yield strength, and the application ranges are 10−4s−1≤ε˙≤255s−1 and 270MPa≤fy≤710MPa.The DIFs of concrete and the steel bar are shown in Figure9. The material parameters are mainly in Table 5.Figure 9
DIF of concrete (a) and steel bar (b).
(a)
(b)Table 5
Material parameters.
∗MAT_CONCRETE_DAMAGE_REL3
Concrete
Material number
1
Density
2.5 × 103 kg/m3
Poisson ratio
0.2
Unconfined compressive strength
4.0 × 107 Pa
∗MAT_PIECEWISE_LINEAR_PLASTICITY
Longitudinal bar
Stirrup
Material number
2
3
Yield strength
4.0 × 108 Pa
3.0 × 108 Pa
Density
7.85 × 103 kg/m3
7.85 × 103 kg/m3
Elastic modulus
2.0 × 1011 Pa
2.0 × 1011 Pa
Poisson ratio
0.3
0.3
Tangent modulus
2.0 × 109 Pa
2.0 × 109 Pa
Failure strain
0.15
0.15


### 5.3. Blast Loading

There are three methods to apply blast loading on the bridge pier. The first one is fluid-solid coupling [35] that mainly builds air and explosive element to simulate propagation of the shock wave and interaction between the shock wave and structure by some keywords, which is a direct research method; however, its computational efficiency is very low. The second one is ConWep methods [36] that applies blast loading from a conventional weapon on the structure, which only considers explosive location, explosive equivalent, detonating time, and explosive type; however, it only supply two shock waves with incident angle, namely, free-field spherical shock wave and hemispherical shock wave. The third one used in this paper is simplified blast loading (Figure 10) that simplifies traditional pressure history to triangle loading in time domain. In dynamics, the loading can be equivalent to impulse loading when the natural vibration period of the structure is more than five times as long as the duration of loading. The duration of blast loading is normally millisecond and far below the natural vibration period of the structure; therefore, this model can be used to simulate blast loading reasonably. This method has high computational efficiency, simple model, and clear physical concepts.Figure 10
Blast loading model on the bridge pier.According to the above research and hypothesis, blast loading distributed on the vertical pier is also triangle distribution on the front face due to explosive detonating at the bottom of the pier. Then, blast loading distributed on the lateral pier for circular cross section can be simplified as uniform loading shown in Figure10, which proves to be effective. This FE model has been verified according to the explosion test in [20]; therefore, it can be used to calculate other explosive cases.


### 5.4. Displacement Analysis

LS-DYNA software is a program package for nonlinear explicit analysis, including geometric nonlinearity, material nonlinearity, and interfacial state nonlinearity such as friction and contact. Although the calculation of blast loading has been simplified, it will still take a large number of time to compute each case. In addition, the postexplosive assessment of the structure by numerical simulation is still accurate. But numerical simulation is too convenient to apply for engineers. Therefore, it is necessary to validate the above theoretical analysis by numerical simulation in this paper to evaluate the postexplosive damage rapidly by theP-I curve for engineers.There are three points selected separately on differentP-I curves, so there are nine overpressures and nine impulses. Each set of overpressure and impulse are loaded on the simulative model, and displacements in the middle of the pier are calculated in Table 6. All numerical displacements are less than the theoretical displacements, but all the errors are less than 20%. Therefore, theoretical displacements and numerical displacements are in good agreement. A simplified theoretical analysis can effectively evaluate the damage of the pier, so that it can supply a rapid and effective assessment method for engineers.Table 6
Displacement comparison.
Theoretical displacement (mm)
P (MPa)
I (MPa·ms)
Numerical displacement (mm)
Error (%)
2°
61.0
10.00
2.39
59.8
1.97
0.66
30.00
50.5
17.21
0.82
3.00
52.5
13.93
5°
153.0
5.00
6.04
141.0
7.84
1.70
30.00
136.0
11.11
1.73
8.00
123.0
19.60
12°
364.0
10.00
14.45
329.0
9.62
3.20
30.00
319.0
12.36
3.22
20.00
311.0
14.56
Error = (theoretical displacement − numerical displacement)/theoretical displacement × 100%.


## 5.1. Geometric Model

To study the validity of above computational method and the accuracy ofP-I curves, numerical simulation is the most common and effective method [30]. In this paper, ANSYS/LS-DYNA, dynamic explicit analysis software, is used to build a finite element model of an RC pier, which is shown in Figure 8. The diameter of circular cross section is 400 mm, and the effective height of the pier is 3500 mm. Other geometric parameters are set according to the experimental pier. In this model, concrete is simulated by 8-noded solid164 element, and steel bars are simulated by 2-noded beam161 element, whose mesh size is 20 mm. The number of concrete elements is 55500, and the number of steel bar elements is 3030. The bottom of the pier is set as a fixed constraint, and the top of the pier is set as a hinged constraint. Different elements are coupled with common nodes ignoring relative slippage due to a fairly short explosion time.Figure 8
Finite element model of an RC pier.


## 5.2. Material Model

The concrete uses the material model∗MAT_CONCRETE_DAMAGE_REL3, called the K&C model, which mainly analyzes the dynamical response of the concrete structure under blast loading [31]. The model is the modified pseudotensor concrete material, including three independent failure surfaces, damage effects, and strain rate effects to control complex mechanical properties [32]. The failure criterion of this model is determined by initial yield failure surface, maximum strength failure surface, and residual strength failure surface, whose three invalid compression meridians consist of eight independent parameters. The compression meridian formulas are as follows:(15)Δσy=aoy+Pa1y+a2yP,Δσm=ao+Pa1+a2P,Δσr=Pa1f+a2fP,where Δσy is the equivalent yield strength; Δσm is the equivalent failure strength; Δσr is the equivalent residual strength; aoy, a1y, a2y, ao, a1, a2, a1f, and a2f are the material constant determined by the unconfined compression test and three axial compression test; P is the hydrostatic pressure. The specific function relation of failure surface under different stress states are as follows:(16)Δσr=ηΔσm+1−ηΔσyλ≤λn,Δσh=ηΔσm+1−ηΔσyλ≥λn,where λ is the damage variable that is function of the equivalent plastic strain, λn is the damage turning point between strengthening phase and softening phase, and η is the function of the damage value that ranges from zero to one.K&C model also considers the damage effects including shear deformation damage and volume damage under three-direction stretching. The damage variables are as follows:(17)λ=∫0εP¯dεP¯rf1+P/rfftb1,P≥0,λ=∫0εP¯dεP¯rf1+P/rfftb2,P≥0,λ=b3fdkdεV−εV,yield,where rf is the coefficient of the strain rate effect, b1 is the damage effect under compression, b2 is the damage effect under tension, b3 is the damage effect under volume stretching, dεP¯ is the equivalent plastic strain increment, fd is the stress limiting factor, kd is the internal variable factor, εV is the volumetric strain, and εV,yield is the volumetric strain at the yield point.This model is a strain rate-related materials whose strength is increased by the strain rate versus increasing factor of strain rate. Therefore, the increasing function is as follows:(18)Δσe=rfΔPrf,where rf is the coefficient of the strain rate effect and P is the hydrostatic pressure.However, this complicated concrete constitutive has been embedded into LS-DYNA software; therefore, only density, passion ratio, uniaxial compressive strength, and dynamic increase factor are input into software, and other parameters are generated automatically. In this paper, the strain rate effect of concrete modes comes from the modified models. The DIFs [33] of concrete tension strength are seen in the following equations:(19)TDIF=ftdfts=0.26lgε˙d+2.06,ε˙d≤1/s,TDIF=ftdfts=2lgε˙d+2.06,1/s<ε˙d≤2/s,TDIF=ftdfts=1.44331lgε˙d+2.22766,2/s<ε˙d≤150/s,where ftd is the dynamic tension strength of concrete at the strain rate of ε˙d and fts is the static tension strength of concrete at the strain rate of ε˙ts (ε˙ts=10−6/s).The DIFs of concrete compressive strength [31] are seen in the following equations:(20)CDIF=fcdfcs=0.0419lgε˙d+1.2165,ε˙d≤30/s,CDIF=fcdfcs=0.8988lgε˙d2−2.8255lgε˙d+3.4907,ε˙d>30/s.The steel bar uses material model∗MAT_PLASTIC_KINEMATIC, called the plasticity kinematic model, which uses the Cowper–Symonds model to consider the dynamic constitutive relationship [34]. This model consists of the isotropy and kinematic hardening model related with strain rate and can consider material failure. Cowper–Symonds model can be defined as following equation:(21)σY=1+ε˙C1/pσ0+βEpεpeff,where σ0 is the initial yield stress, ε˙ is the strain rate, C and p are the strain rate parameter of the C-S model, εpeff is the effective plastic strain, and β is the hardening parameter consisting of 0 (only kinematic hardening) and 1 (only isotropic hardening). Ep is the plastic hardening modulus and Etan is the tangent modulus, which are related as follows:(22)Ep=EtanEE−Etan.In this paper, the DIF [34] of the steel bar is as follows:(23)DIF=ε˙10−4α,α=0.074−0.04fy414,where ε˙ is the strain rate of reinforcement, fy is the yield strength, and the application ranges are 10−4s−1≤ε˙≤255s−1 and 270MPa≤fy≤710MPa.The DIFs of concrete and the steel bar are shown in Figure9. The material parameters are mainly in Table 5.Figure 9
DIF of concrete (a) and steel bar (b).
(a)
(b)Table 5
Material parameters.
∗MAT_CONCRETE_DAMAGE_REL3
Concrete
Material number
1
Density
2.5 × 103 kg/m3
Poisson ratio
0.2
Unconfined compressive strength
4.0 × 107 Pa
∗MAT_PIECEWISE_LINEAR_PLASTICITY
Longitudinal bar
Stirrup
Material number
2
3
Yield strength
4.0 × 108 Pa
3.0 × 108 Pa
Density
7.85 × 103 kg/m3
7.85 × 103 kg/m3
Elastic modulus
2.0 × 1011 Pa
2.0 × 1011 Pa
Poisson ratio
0.3
0.3
Tangent modulus
2.0 × 109 Pa
2.0 × 109 Pa
Failure strain
0.15
0.15


## 5.3. Blast Loading

There are three methods to apply blast loading on the bridge pier. The first one is fluid-solid coupling [35] that mainly builds air and explosive element to simulate propagation of the shock wave and interaction between the shock wave and structure by some keywords, which is a direct research method; however, its computational efficiency is very low. The second one is ConWep methods [36] that applies blast loading from a conventional weapon on the structure, which only considers explosive location, explosive equivalent, detonating time, and explosive type; however, it only supply two shock waves with incident angle, namely, free-field spherical shock wave and hemispherical shock wave. The third one used in this paper is simplified blast loading (Figure 10) that simplifies traditional pressure history to triangle loading in time domain. In dynamics, the loading can be equivalent to impulse loading when the natural vibration period of the structure is more than five times as long as the duration of loading. The duration of blast loading is normally millisecond and far below the natural vibration period of the structure; therefore, this model can be used to simulate blast loading reasonably. This method has high computational efficiency, simple model, and clear physical concepts.Figure 10
Blast loading model on the bridge pier.According to the above research and hypothesis, blast loading distributed on the vertical pier is also triangle distribution on the front face due to explosive detonating at the bottom of the pier. Then, blast loading distributed on the lateral pier for circular cross section can be simplified as uniform loading shown in Figure10, which proves to be effective. This FE model has been verified according to the explosion test in [20]; therefore, it can be used to calculate other explosive cases.


## 5.4. Displacement Analysis

LS-DYNA software is a program package for nonlinear explicit analysis, including geometric nonlinearity, material nonlinearity, and interfacial state nonlinearity such as friction and contact. Although the calculation of blast loading has been simplified, it will still take a large number of time to compute each case. In addition, the postexplosive assessment of the structure by numerical simulation is still accurate. But numerical simulation is too convenient to apply for engineers. Therefore, it is necessary to validate the above theoretical analysis by numerical simulation in this paper to evaluate the postexplosive damage rapidly by theP-I curve for engineers.There are three points selected separately on differentP-I curves, so there are nine overpressures and nine impulses. Each set of overpressure and impulse are loaded on the simulative model, and displacements in the middle of the pier are calculated in Table 6. All numerical displacements are less than the theoretical displacements, but all the errors are less than 20%. Therefore, theoretical displacements and numerical displacements are in good agreement. A simplified theoretical analysis can effectively evaluate the damage of the pier, so that it can supply a rapid and effective assessment method for engineers.Table 6
Displacement comparison.
Theoretical displacement (mm)
P (MPa)
I (MPa·ms)
Numerical displacement (mm)
Error (%)
2°
61.0
10.00
2.39
59.8
1.97
0.66
30.00
50.5
17.21
0.82
3.00
52.5
13.93
5°
153.0
5.00
6.04
141.0
7.84
1.70
30.00
136.0
11.11
1.73
8.00
123.0
19.60
12°
364.0
10.00
14.45
329.0
9.62
3.20
30.00
319.0
12.36
3.22
20.00
311.0
14.56
Error = (theoretical displacement − numerical displacement)/theoretical displacement × 100%.


## 6. Conclusions

Based on the equivalent SDOF method in this paper, the damage extent of an RC pier is studied under blast loading, and the main conclusions are as follows:(a)
The RC pier can be simplified as the equivalent SDOF system based on energy equality, and the blast loading can be simplified as a uniform distribution so that the computation time is greatly reduced. According to this method, the dynamic response and damage extent of the pier can be evaluated quickly. Therefore, this method can be applied to practical engineering.(b)
Based on the SDOF systems and the uniform explosive load, theP-I curves and corresponding fitting formulas of the RC pier are obtained. The minimum overpressures are 0.66 MPa, 1.70 MPa, and 3.20 MPa when the RC pier experiences slight damage, moderate damage, and severe damage, respectively, and the minimum impulses are 2.35 MPa·ms, 5.95 MPa·ms, and 14.35 MPa·ms when the RC pier experiences slight damage, moderate damage, and severe damage, respectively. The damage extent of the RC pier in the case of any overpressure and impulse can be quickly evaluated through the P-I curves.(c)
Through the numerical simulation verification, the midspan displacements calculated by the simplified method and the midspan numerical displacements are in good agreement. The damage extent obtained by the simplified method and the damage extent obtained by numerical simulation are also in good agreement. Therefore,P-I curves based on the simplified method can used to effectively evaluate the damage extent of an RC pier, which can supply a rapid and effective assessment for engineers.



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*Source: 9204036-2018-08-16.xml*