Wan-xiang Chen , Li-sheng Luo , Zhi-kun Guo , Peng Yuan

a School of Civil Engineering, Sun Yat-Sen University, Zhuhai, 519082, China

b State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing, 210007, China

c Southern Marine Science and Engineering Guangdong Laboratory, Zhuhai, 519082, China

Keywords:Blast load Membrane action HFR-LWC beam Dynamic response Experimental study

A B S T R A C T The load-carrying capacities and failure patterns of reinforced concrete components can be significantly changed by membrane effects. However, limited work has been carried out to investigate the blast resistance of Hybrid Fiber Reinforced Lightweight Aggregate Concrete (HFR-LWC) members accompanying membrane action. This paper presents a theoretical approach to quantitatively depicting the membrane behavior and its contribution on the behavior of HFR-LWC beams under close-range blast loadings, and the suitability of the proposed model is validated by a series of field tests. An improved Single-Degree-of-Freedom(SDOF)model was employed to describe the dynamic responses of beam-like members under blast loadings accompanying membrane action, where the mass-load coefficient is determined according to the nonuniformly distributed load induced by close-range explosion, and the membrane action is characterized by an in-plane(longitudinal)force and a resisting moment.The elastoplastic and recovery responses of HFR-LWC beams under the combined action of blast load and membrane force were analyzed by the promoted model.A specially built end-constrain clamp was developed to provide membrane action for the beam member when they are subjected to blast load simultaneously.It is demonstrated that the analytical displacement-time histories are in good agreement with experimental results before peak deflections and that the improved SDOF model is an acceptable tool for predicting the behavior of HFR-LWC beams under blast loadings accompanying membrane action.

1. Introduction

In recent years, the increasing number of terrorist attacks and accidental explosions have attracted the attention of people engaging in disaster prevention and mitigation engineering [1].Even a relatively small explosive device delivered in a bag or backpack and placed in a close proximity to a building component or bridge pier can cause severe local damage and consequent partial collapse of the infrastructure.Therefore,it is crucial to develop new building materials or structural systems that provide an adequate level of protection against close-range blast loads.

Lightweight aggregate concrete (LWC) generally refers to concrete prepared from lightweight aggregate, ordinary sand (or lightweight sand), cement and water with a dry apparent density ranging from 1522 to 2163 kg/m3according to ACI318-19 code[2],which has been extensively used in civil construction for over 2000 years. It has been increasingly applied in high and long-span buildings due to its low density, high specific strength, good fire resistance and seismic performance in comparison with normal weight concrete [3,4]. However, its application potential is restrained by its low tensile strength, high brittleness and especially poor ductility [5]. The addition of fibers in the concrete can mitigate these negative effects and greatly improve the strength and ductility of LWCs. Some researches demonstrate that the addition of two or more kinds of fibers can produce a positive“hybrid effect” to LWC, and that the enhancement effect is obviously better than the incorporation of single fiber [6]. The tests were performed by Pan et al. [7] to investigate the mechanical properties of hybrid fiber reinforced full lightweight aggregate concrete(HFR-FLWC),in which the hybrid fibers were composed of steel fiber and polypropylene fiber, and the expanded-shale and lightweight sand was used as coarse and fine aggregates. The impact tests on hybrid fibers of polypropylene fiber and plastic steel fiber (HPP) reinforced LWC were conducted using 100 mm-SHPB equipment by Wang et al. [8]. The mechanical properties of HPP reinforced LWC under impact loading with different strain rates were systematically studied. The performance of HFR-LWC was investigated by Huo et al.[9],and results indicate that the physical and mechanical properties of LWC can be significantly improved by hybrid fibers.Libre et al.[10]attempted to improve the ductility of pumice LWC by incorporating hybrid steel and polypropylene fibers. The properties were investigated, including bulk density and workability of fresh concrete as well as compressive strength,flexural tensile strength,splitting tensile strength and toughness of hardened concrete.Results show that polypropylene fibers caused a minor change in the mechanical properties of hardened concrete,especially in the mixture of steel and polypropylene fibers.Existing researches indicate that the mechanical properties of HFR-LWC have been intensively investigated in the past decade, but limited work has been conducted on the blast resistance of HFR-LWC members.

For most beam-like members, catastrophic failures occur at large deflections accompanying membrane action[11].Therefore,it is extremely essential to estimate the load-carrying capacities of beams or slab-strip (i.e., beam-like) members under large deformation conditions.Nevertheless,past design manuals did not fully consider the membrane effect in defining resistance function but took it as a“hidden”safety factor.Commonly,an ultimate capacity of 1.5-2.0 times the yield-line value was adopted for static analysis[12]. Although membrane effects in beam-like members have caught researchers’attention for more than several decades[13,14],its application in both the numerical analysis and design has not yet matured. Regarding blast-resistant structures, the elastic and elasto-plastic models put to use from the early 1960s [12,15-17]have been the most common methods until recently. However,what has been confirmed by many experiments [13] is that these models considerably underestimated the bearing capacities of structural members. These limitations, although acceptable for structural design, are not acceptable for the response analysis of beam-like structures subjected to severe dynamic loads. The blast duration is about 1000 times shorter than that of earthquakes,and under such condition the structural behavior is governed by the inertia effect. A beam-like member that performs well under slow loading rates may collapse under faster loads. Although the capacity underestimation is a potential safety factor for statically loaded beams,under severe dynamic conditions,the load-response relationship is much more complicated.To determine the structural safety one needs to know how the loads are resisted. Obviously,analysts and designers can no longer accept“hidden”safety factors and attention has to be given to a more precise definition of resisting mechanisms.

In the previous literature, there are few studies on the blast resistance of reinforced concrete members accompanying membrane action[15].In this paper,an improved SDOF model is devoted to describing the dynamic responses of fully-clamped HFR-LWC beams under the combined action of blast load and membrane force, where the mass-load coefficient is determined according to the nonuniformly distributed load induced by close-range explosion, and the membrane action is represented by a longitudinal force and a resisting moment.A specially built end-constrain clamp was developed to provide membrane action for HFR-LWC beams when subjected to blast loads, and the reliability of the proposed model was further demonstrated by a series of experimental results.

2. Description of improved SDOF model

SDOF model is a popular and robust tool in depicting the deflection response of beam-like members under blast loadings.Li and Meng [18,19] analyzed the influence of impulse shape on the structural behavior based on the maximum deflection damage criterion and elastic SDOF model. Fallah and Louca [20] depicted the structural response by analyzing SDOF systems with elasticplastic hardening and elastic-plastic softening under blast loadings. Generally, the blast loading is classified into three types according to scaled distance, including near-field blast(Z=R/Q1/3>1.2 m/kg1/3),close-range blast(Z=0.05-1.2 m/kg1/3)and contact blast(Z < 0.05 m/kg1/)) [21-23], where R is the distance between the explosive charge and the structural surface and Q is the charge weight.Response of the structural system to the blast generated by near-field explosion has been intensively studied recently. However, response of the structural component to close-range detonation has not been investigated to the level sufficient for developing engineering-level design and producing reliable design recommendation [24].

2.1. Equations of motion for improved SDOF model

Actually, most structural components cannot be treated as simply-supported members if their longitudinal motions are prevented by stiff boundaries and the edge rotations are restrained simultaneously as loading continues, such as beam-like members with fully-clamped supports[25].As shown in Fig.1,the combined action of in-plane force N and resisting moment M in both memberends raised by progressive deflection is called “membrane effect”,which has been be observed in many beam or slab structures[26].It is well known that structural members with fully-clamped supports exhibit more complex mechanical behavior and failure patterns than those with simple supports.

The structural member with continuous mass and stiffness is usually described by an equivalent undamped SDOF system[27],as shown in Fig. 2. The behavior of the beam member under blast loadings is artificially separated into two response regimes:(ⅰ) the elastic phase and (ⅱ) the plastic phase. In this model, the beam is assumed to carry a variable membrane action (i.e., the combined action of a longitudinal force N and a bending moment M as shown in Fig.2(a))throughout the analysis process.For the convenience of subsequent analysis,the clamped beam member is converted into a simply-supported beam and the membrane action is represented by a longitudinal restraint spring Snand a rotational restraint spring Smas displayed in Fig. 2(b). It is consistent with a blast scenario involving a building with a facade featuring a relatively strong cladding or glazing and subjected to an external explosion [22].

Fig.1. Membrane actions in beam-like member.

Fig. 2. Equivalent SDOF system.

As shown in Fig.2(c),the equivalent equation of motion for the beam member under blast loading accompanying membrane action can be described as:

2.1.1. Determination of close-range blast loadings

Air shock induced by close-range blast(Z=0.05-1.2 m/kg1/3)is a complex problem concerning the impingement of dense, highpressure and high-temperature detonation products [24].Measured overpressure-time histories caused by close-range blast[31-33] demonstrate that the peak pressure decreases with the increase of scaled distance and the wave arrival time increases with the scaled distance,indicating that an approximate spherical shock wave is generated. However, no significant discrepancies of wave forms and blast durations for different measured points were found. Therefore, the actual load for close-range blast acting on structural surface can be characterized as a “cap-shape” distribution [34], as shown in Fig. 3 (a).

Fig. 3. Distribution of blast load and its simplification.

Obviously,it will give rise to considerable errors if the air shock caused by close-range blast is treated as uniform distributed load in the theoretical analysis. In this analysis, the current air shock is simplified into a linearly distributed load at half-length of the beam,as shown in Fig.3(b).The overpressures on the given points,such as P1 at the mid-span section and P2 near the support of the beam,are predicted using TM5-1300 blast model[35]that derived empirically from a large number of well-designed blast experiments. It has been widely adopted to investigate the structural response under blast loads and shown a high level of accuracy with a reasonable computational cost compared to other blast models[21]. The overpressures of p（x，t） can be determined by the interpolation of p1and p2.

Supposing that the overpressures at the mid-span section and near the support of the beam are p1=P1f（t） and p2= P2f（t）respectively,wheren = P2/P1,0

The air shocks produced by explosive charge will attenuate exponentially, be it incident or reflected shock waves. For convenience, the overpressure-time history of air shock is usually simplified into an equivalent linearly decaying pressure profile[28,36].P（t）=p（x）bl·f（t）is the total lateral load applied on the top face of the beam,where b is the width of the beam,f（t）=1-t/tdis the non-dimensional time function and tdis the blast duration.

2.1.2. Equivalent coefficient for improved SDOF model

The beam member will undergo an elastic response and a plastic response successively if the applied blast load is large enough. At elastic state, the deflection equation for simply-supported beams under non-uniformly distributed load can be achieved according to the theory of material mechanics [37]. Supposing that the closerange blast load is bilinearly distributed at the span of the beam,then the deflection shape can be acquired by regarding the peak overpressure as the static pressure.It is given below:

The normalized deflection f =1 at mid-span section can be achieved by substituting x=l/2 into Eq. (3). Therefore, the deflection mode for the beam under non-uniformly distributed load is given as:

If the displacement reaches its elastic limit under uniform loading, assuming that the in-span hinge develops at beam midspan and both supports of the beam, the normalized deflected shape for inelastic deformations after plastic hinging is given as a system of rigid-link members between those plastic hinges.In this state,the beam segment between the support and the mid-span is regarded as a rigid body and the deflection shape of the beam will not be changed by the non-uniformly distributed load. Thus, the deflection mode for the beam member at plastic state can be written as:

The equivalent mass coefficient of SDOF model can be determined by assuming that the kinetic energies of the beam under blast loading are equal to that of the equivalent spring-mass system.The mass coefficients in elastic response and plastic response are given respectively:

Similarly,the equivalent load coefficient is expressed as KLorand determined respectively as:

For simplicity, the equivalent load coefficient, mass coefficient and mass-load coefficient of improved SDOF model for different load ratios n=P2/P1are summarized in Table 1. Therefore, the dynamic response of the beam-like member under nonuniformly distributed blast loading can be converted into an equivalent SDOF system based on the given coefficients.

2.1.3. Improved SDOF model based on membrane action

As shown in Fig. 2, the boundary constraint can give rise to a combined action of longitudinal force N and resisting moment M in the beam (i.e., membrane effect) with the progressive deflection,which is a self-generated additional action and accompanies the full-range response of the beam.Evidently,the membrane action is a deflection-dependent variable,and it is relevant to the structural behavior. For clamped beam-like members, the longitudinal force will be an eccentric force when the deflection exceeds certain value.η1（t）represents the equivalent lateral load (ELL) duo to the secondary moment caused by the eccentricity of the longitudinal load,and can be written as [38]:

where lis the structural length and y（t）is the current deflection at mid-span section.

The magnification method[39],which has been widely used in the design of beam-columns under static load to account for Pδ effect, is recommended by the UFC design guide (UFC 3-340-02)[40]to be applied in beam-columns subjected to axial gravity load and lateral blast load with slight modification. But the magnification factor is independent of the lateral load,and not applicable to a beam-column if the maximum lateral deflection exceeds its elastic deflection limit [22]. The P-δ effect has been successfully included in the software SBEDS[41]through the application of an equivalent uniform lateral load to the column,in which the load is obtained by equating its associated maximum moment to the equivalent moment caused by the longitudinal load with an eccentricity equal to the given deflection.

Supposing that there is a mid-span displacement y（t） for the beam at the instant t,it will induce a P-δ effect in the beam due to the initial deflection caused by longitudinal force N. Nevertheless,there will also generate an additional reverse moment M at both beam-ends due to the rotational constraint of the supports [38].Therefore, the total moment caused by the longitudinal force and the bending moment is given by Ny（t）- M. As shown in Fig. 4,supposing that the total equivalent lateral load-induced moment atthe mid-span section of the beam is equal to the secondary moment caused by the longitudinal load and the resisting moment,the following equation gives:

Table 1 Summarizations of equivalent coefficient for improved SDOF model.

Fig. 4. Equivalent lateral load-induced moment.

The resisting moment can be acquired by the rotational restraint stiffness of the support and the deflection of the beam as [42]:

Substituting Eq. (10) into Eq. (9) yields:

The full-range responses of the beams under blast loadings accompanying membrane action can be depicted by following equations:

Fig. 5. Linearly decaying blast load.

2.2. Dynamic responses of clamped beams under close-range blast loadings

In this analysis, the response of clamped beams under blast loadings are divided into three phases (including elastic response,plastic response and recovery response) according to the relationship between the maximum displacement and the elastic limit of the structural member.An equivalent linearly decaying blast load is applied on the beam P（t）=Pm×（1-t/td）[36],as shown in Fig.5,where Pmis the peak overpressure.

2.2.1. Elastic response

The elastic response Eq. (12a) can be simplified into the following equation by supposing thatand K’ = K-

where ωNis the membrane-dependent natural frequency of the beam, which is similar to the vibration frequency proposed by Bazant et al. [44]. The general solution of Eq. (13) can be obtained as:

where the arrival time for the maximum displacement is given byrespectively.

2.2.2. Plastic response

Transition from elastic response into plastic response is noted to occur if the maximum displacement of the beam exceeds its elastic limit.The peak value of the blast load Pmis taken as the static load uniformly acting on the beam, as shown in Fig. 6. The deflection mode X（x）satisfies the following equation according to Li et al.[36]:

Fig. 6. Schematic diagram of clamped beam under uniform load.

and

where Muis the ultimate moment according to literature[45].It is given by:

where Asis the area of tensile bar and h0is the effective height of section. The equation of motion for the beam at y ≥yecan be rewritten as:

where Rm=12Mu/l [30] and

If the maximum elastic displacement of the beam arrives during the stage of blast action,but the peak plastic displacement occurs at the stage of free vibration, then the initial conditions for plastic response can be determined by:

where tpis the initial time of plastic response and the termination of elastic response is determined bySimilarly, the plastic displacement and corresponding arrival time can be obtained respectively by:

On the other hand, if the maximum elastic and plastic displacements both arrive at the stage of free vibration, the initial conditions can be expressed as:

2.2.3. Response of elastic recovery

The maximum displacement of the beam will be followed by a response of elastic recovery owing to the structural resilience. In this stage,the elastic deformation is resumed rapidly,and the beam member will theoretically stay at the position corresponding to permanent deformation as the kinetic energy inputted by explosive charge is entirely dissipated. The equation of motion for recovery response can be written as:

The general solution of Eq.(24)based on the initial condition ofis derived by:

where this the initial time of recovery response; ypsis the permanent displacement and it should be determined by the maximum displacement and the elastic response.

2.2.4. Dynamic Increased Factor of HFR-LWC

Concrete(including HFR-LWC)is a strain-rate sensitive material.The material properties and failure mechanism will be changed if a rapid load acts on the structure.The common phenomenon is that the material strength increases with loading rate, i.e. strain rate effect.It is usually scaled by Dynamic Increased Factor(DIF),a ratio of dynamic strength to static strength.

Concrete sensitivity to strain rate is accounted for in SDOF blast analysis by using the DIF.For close-range response,the strain rates are in the range between 102and 104s-1[40]. The dynamic strengths of HPP reinforced LWC under different strain rates were experimentally studied.After investigation by the author[8],it was found that the DIF varied between 1.18 and 1.65 for strain rates in the range between 61 and 110 s-1,which is slightly larger than that of normal weight concrete. It is difficult to capture the accurate strain rate at the specific region of the structural member in the theoretical analysis. Based on the interpolation of experimental data provided in literature [8], a DIF = 1.55 corresponding to the minimum strain rate (i.e., 10 s-1recommended by UFC 3-340-01[40]) is adopted in the improved SDOF model.

3. Experimental programs

In order to investigate the effect of membrane action on the blast resistance of HFR-LWC beams, a series of blast tests were performed at the test site of Army Engineering University of PLA in China. In the tests, membrane forces (N and M) are introduced at element supports through specifically designed clamps, which are formed by two steel plates tensioned with a variable number of tie rods. Finally, the experimental results were employed to validate the rationality of proposed model.

3.1. Loading device for membrane action

Membrane action is simulated by a specially manufactured endconstraint clamp. As shown in Fig. 7(a), the tested beam is horizontally supported on two steel rollers that are pre-located at the rigid abutments, and thus membrane action will be generated by the end-constrain clamp. Two steel plates with large stiffness are parallelly installed on the end-surfaces of the beam, and they are tensioned by several longitudinal steel rods with nuts. The steel rods are symmetrically arranged at the front and rear faces of the beam with a vertical space of 40 mm between two rods in vertical direction,keeping the top steel rods within the neutral plane of the beam.A steel pad covering the end-surface below the neutral plane of the beam is used to prevent the transversal steel bars from immersing into the concrete. Three transversal steel bars corresponding to the vertically positions of the steel rods are horizontally arranged between the steel plate and steel pad, where the grooves in both sides are designed to avoid the slides of the bars.The end-constrain clamp is pre-manufactured in the factory using Q345 steel. The thickness of the steel plate and steel pad are both 30 mm. The steel rod has a measured length of 1800 mm and diameter of D = 20 mm, and the steel bar has a diameter of D=20 mm.As the beam deflects under transversal blast loadings,changes in geometry tend to cause its edges to move outward and react against the end-constrain clamp. The outward movement of the beam is restrained by the steel plate, and the steel rods will provide a longitudinal in-plane compression N below the neutral plane. Meantime, the progressive deflection drives the beam-end section to rotate around the neutral plane, which will give rise to different tensile stress of steel rods in the vertical direction.Generally, the stress increases with the distance from the neutral plane, and a rotation-induced moment M is generated at both beam-ends. Accordingly, the longitudinal force N and resisting moment M at both beam-ends are self-generated by the endconstrain clamp as deflection progresses, and the beam will behave as a clamped member. As referenced samples, a simplysupported HFR-LWC beam and a plain HFR-LWC beam with the same dimensions and reinforcement ratios are also prepared for blast-resistance tests.

Fig. 7. Loading-device of membrane action.

Fig. 8. Schematic diagram for calculation of constraint stiffness.

As shown in Fig. 7(b), the rod ③is passed through the original neutral plane of the beam, various boundary constraints can be achieved by changing the configurations of steel rod. In order to describe the membrane effects quantitatively, the longitudinal constraint stiffness Snand the rotational constraint stiffness Smare introduced according to literature [46]. As displayed in Fig. 8, the end-surface of the beam will rotate around its neutral plane as deflection progresses, and thus the constraint stiffness of the endconstrain clamp can be theoretically obtained according to the plane section assumption.

The total longitudinal force can be expressed as N=∑Ni,where the subscript i represents the layer number of steel rods and i=1,2,3, respectively. Considering that the steel rods are symmetrically arranged at the front and rear sides of the beam,the tensile force of steel rod in ith layer can be expressed as:

Table 2 Restraint grades of end-constraint clamp.

As a general example shown in Fig. 8, the total tensile force is obtained by:

The average longitudinal movement Δ of the steel plate at either beam-end is given as:

and thus the longitudinal constraint stiffness can be achieved by substituting Eq. (27) and Eq. (28) into following formulas:

where, Esis the elastic modulus of the steel rod;is the average tensile strain of the steel rod in ith layer; A is the cross-sectional area of the steel rod; L is the half length of the steel rod and d is the central space between two steel rods in vertical direction.

The resultant force provided by the steel rod is an eccentric load regarding to the neutral axis of the beam. Then, the resisting moment generated by the end-constrain clamp can be derived by:

Based on the geometric relationshipit can be further obtained that:

In addition, the relationship of rotation angle and rotational constraint stiffness can be written as:

Finally,the rotational constraint stiffness Smcan be determined by substituting Eq. (30) and Eq. (31) into Eq. (32), and this gives:

Table 3 Mix proportion of HFR-LWC.

Fig. 9. Raw materials.

Table 4 Properties of shale ceramsite.

Table 5 Material properties of fiber.

The longitudinal constraint stiffness Snand rotational constraint stiffness Smbased on other rod-arrangements can also be theoretically achieved similarly. For convenience, the restraint-grades ranging from I to Ⅳare introduced to represent the restraint strengths of end-constraint clamps and they are listed in Table 2.

3.2. HFR-LWC mixture and material properties

3.2.1. HFR-LWC preparation

All specimens were prepared at Army Engineering University of PLA in China using locally available materials.The mix proportions are provided in Table 3 and the raw materials are shown in Fig. 9.Crushed shale ceramsite with irregular surface was selected as coarse aggregate to provide better interlocking effect and increase the contact area between the aggregate and the cement paste[47].In this paper, P·O42.5 Portland cement (Chinese Standard GB175-2007 [48]) was used. As shown in Table 4, the continuously graded class 900 shale ceramsite with a diameter between 5 and 15 mm was used as lightweight coarse aggregate. It has a bulk density of 882 kg/m3that’s in consistent with the Chinese standards[49] and the water absorption for 1 h is 5%. River sand with fineness modulus of 2.6 and bulk density of 2570 kg/m3was used as fine aggregate.Hybrid fibers of polypropylene fiber and plastic steel fiber were incorporated to improve the ductility of LWC.As shown in Table 5, the tensile strengths of polypropylene fiber and plastic steel fiber are over 280 MPa and 500 MPa,respectively. They have elastic modulus over 3.8 GPa and 7.0 GPa, diameters 0.0342 mm and 0.6-0.9 mm,and lengths 6 mm and 35 mm,respectively.Silica fume of average particle diameter between 0.1 and 0.3 μm with SiO2content of 98% was used as the mineral admixture. To ensure adequate workability of fresh mixture and reduce the balling (or clumping) effect of fibers [50], a poly-carboxylic type superplasticizer (SP) with a water reducing ratio of 20% was used in all mixture.

3.2.2. Material properties of HFR-LWC

The mixing procedure was performed according to Ref. [8].During the mixing,hybrid fibers were pre-mixed and gradually fed into the cement mixture by hand to ensure excellent distribution.The fresh concrete was cast in molds and demolded after 24 h,and then kept in the casting site for 28 days until testing time. Six 100 mm × 100 mm × 100 mm cubic HFR-LWC samples were prepared and nine beam specimens were simultaneously fabricated in this test. The compressive strengths of HFR-LWC and plain LWC were measured by using hydraulic pressure testing machine with a maximum load capacity of 2500 kN, and the average cubic compressive strengths of HFR-LWC and plain LWC according to Chinese Standard GB/T50081-2002 [51] are 47.97 MPa and 34.80 MPa respectively, as shown in Table 6. In addition, the dry apparent density of the HFR-LWC sample measured before strengthtest is 1910 kg/m3, which is smaller than that of normal concrete,about 2600 kg/m3, and has a better specific strength.

Table 6 Material properties of LWC.

Fig.10. Reinforcement arrangements and specimen fabrications (unit: mm).

3.3. Specimen preparation

The main purpose of this experiment is to investigate the influence of restraint-grades and scaled distances on the dynamic behavior and the failure patterns of HFR-LWC beams.A total of nine beam specimens with dimensions of h×b=200 mm×100 mm in cross-section and l = 1500 mm in length are manufactured based on the similarity principle.As displayed in Fig.10,the longitudinal reinforcements are constructed with four or six 6 mm-diameter HRB400 steel rebars, and they are symmetrically arranged in the cross-section of the beam.The corresponding reinforcement ratios are 0.28%and 0.42%,respectively.Fifteen stirrups spaced uniformly at 100 mm in full span are arranged to enhanced the shear capacity of the beam. All reinforcements have a clear concrete cover of 10 mm.The reinforcement has a yield strength of 653 MPa,Young’s modulus of 200 GPa and an elongation of 8%.

3.4. Blast-resistance test

3.4.1. Experimental setup

Blast-resistant tests were carried out by a self-designed setup(including a specially built end-constraint clamp and an excavated test pit)at the field test site while HFR-LWC beams were cured for 28 days. As shown in Fig. 11, a “L” shape excavated test pit with dimensions of 2020 mm in length,710 mm in width and 800 mm in height was cast using C60 reinforced concrete. The tested beam is located horizontally on the steel-roller supports and kept its top face parallel to the ground. An effective span of 1300 mm for the HFR-LWC beam is provided by this constraint, to ensure that the specimen can bend downward within its vertical plane in the test pit.The top of the test pit is covered by 20 mm-thickness steel plate to prevent the wires and the recording apparatus from being damaged by shock waves. A hole with the same dimension of 100 mm×1500 mm is reserved on the steel plate so that the blast load can act on the beam.Two angle steels anchored to the test pit by steel bolts are used to provide an upward restraint against the HFR-LWC beam’s rebound caused by tensile waves.

Blast load generated by cubic TNT charge of 1 kg,2 kg and 3 kg(corresponding scaled distance changing from 1 to 0.69 m/kg1/3)is applied on the tested beam, respectively. They are artificially divided into three groups according to the restraint grade, TNT charge weight and reinforcement ratio in the following discussion and the testing details are listed in Table 7.

3.4.2. Data acquisition

The acquisitive data includes the tensile strains of steel rods,the reflected overpressure-time histories of shock waves and the deflection-time histories of specimen. All sensors and the endconstrain clamps are pre-installed before the detonation of explosive charges. As displayed in Fig. 12, a cubic TNT charge is suspended over the center of the beam by keeping the standoff distance of 1000 mm from the top surface of the beam to the explosive charge. TNT charge is centrally initiated by an electric detonator, and a close-range air blast (Z = R/Q1/3≤1.2 [24]) is generated. Three strain gauges are installed at each steel rod, and their average value is taken as the representative strain. Owing to the structural symmetries and blast loads, three pressure transducers and three LVDTs spaced at about 220 mm are installed at half span of the beam and the covered steel plate,respectively. All dynamic signals are captured using DH8302-data acquisition instrument. Before formal detonation, the reliability of the measurement system is checked by a pre-loading test with 0.5 kg-TNT charge.The overpressure-time histories produced by 1 kg,2 kg and 3 kg-TNT charges are displayed in Fig.13,respectively.Fig.14 shows the distributions of peak overpressure versus the span of the tested beam. It is apparent that an approximate spherical shock wave is generated in this test,and that the descending rate of overpressure increases with the decrease of scaled distance, indicating that a closer range detonation will give rise to severe local pressure,thus resulting in unavoidable errors in the theoretical analysis by assuming that the shock wave is uniformly distributed in full length.

Fig.11. Blast-resistant test setup (unit: mm).

Table 7 Blast-resistant testing cases.

Fig.12. Measured arrangements for blast-resistant test (unit: mm).

4. Experimental results and model validations

4.1. Failure patterns of HFR-LWC beams

Generally,the failure modes of RC members under blast loading are classified into three types: bending failure, bending-shear failure and direct shear failure [31]. A flexural failure pertains to the combination of local and global deformation, a bending-shear failure occurs near the support sections due to the large shear gradient, and a direct failure in the member body is caused by the high-frequency stress waves.If the magnitude of the stress waves is large enough,a massive failure may occur suddenly in the member body.The failure in the second phase is governed by the shear near the support regions and it greatly depends on the blast impulse.If a member can survive into the third phase, it will exhibit a quite typical response characteristic and the failure is governed by large global as well as local deformations. As shown in Fig.15, all HFRLWC beams appear flexural failures in ductile manners. Some uniformly distributed cracks were found at the full span of HFRLWC beams, but no spalled concrete were observed even in the case of 3 kg-TNT blast load,demonstrating that the ductility of the tested beam is significantly improved by hybrid incorporation of polypropylene fibers and plastic steel fibers. There were no noticeable shear cracks close to the supports, unlike plain LWC members which failed abruptly with penetrating cracks and spalled local concrete[52].

The failure patterns for beams A-34 without the end-constrain clamp and beam A-42, A-71 and A-36 with different restraint grades are displayed in Fig.15(a)-(d). It is found that beam A-34 exhibits the most serious damage compared with other HFR-LWC beams with end-constrain clamps under the blast loadings induced by 3 kg-TNT charge. Many major cracks opened up approximately at mid-span and propagate the entire depth of the beam.Major cracks were followed by a few visible diagonal cracks close to the supports. Both penetrating cracks of shale ceramsite and broken fibers due to crack opening were also observed, suggesting that coarse aggregate shows the interlocking effect to some extent. This is largely contributed by the fiber bridging effect of macro-cracks, allowing greater deformation of the beam before failure. However, for beam A-42 and A-71 with restraint grades I and II respectively, several major tensile cracks developed at the bottom of mid-span,followed by an obvious shear crack close to the supports. The widths and number of cracks decreased greatly compared with beam A-34. For beam A-36 with a higher restraint level, multiple fine cracks were seen at the mid-span and propagated up to two-thirds depth of the beam.They slightly opened up illustrating that the beam has reached its plastic state. But there were no noticeable shear cracks close to the supports. The propagation velocity of the cracks for HFR-LWC beams under blast loadings can be effectively prevented by end-constrain clamps.The presence of hybrid fibers facilitates widening of cracks through the action of fibers in stitching the macro-cracks, which allows progressive failure of structure with pulling out of the fibers.Fig.15(c),(e)and(f)shows HFR-LWC beams with restraint grade II subjected to explosion of 3 kg, 2 kg and 1 kg-TNT charges, respectively. It is apparent that there were no visible cracks on the bottom face along the full lengths of the beams and that obvious residual deformations were observed in beam A-75 when subjected to explosion of 1 kg-TNT charge, demonstrating that the HFR-LWC beams were in the elastic state. For beam A-11, under blast loading induced by 2 kg-TNT charge,it was seen that some uniform tensile cracks developed on the bottom face along entire length of the beam. The cracks at the mid-span region spread upward and penetrated the entire section, indicating plastic response of the beam.But no noticeable shear cracks were observed.The concrete in compression region was not crushed and fell off due to the bridging effect of hybrid fibers. While scaled distance further decreased to 0.69 m/kg1/3, such as for beam A-71 shown in Fig. 15(c), obvious residual deformation was found and multiple tensile cracks were observed along the entire length of the beam.Above failure patterns illustrate that the HFR-LWC beams will be damaged severely as scaled distance decreases from 1 m/kg1/3to 0.69 m/kg1/3,but no direct shear failures can be seen although it is subjected to the blast load of 3 kg-TNT charge.These ductile failures are attributed to the membrane effect and the enhancement effect of hybrid fibers. The number and widths of cracks in beam A-76 with a reinforcement ratio of 0.42% obviously reduce compared with that in beam A-71. However, the discrepancies of failure patterns between beam A-36 and beam A-73 are inapparent although the restraint levels upgrade from restraint grade II to restraint grade I. It is indicated that membrane action is a deflectiondependent behavior and will not be fully activated for small deflections as the reinforcement ratio increases.In addition,the blast resistance and ductility of HFR-LWC beams can be greatly enhanced by the incorporation of hybrid fibers.The fiber pulling-out failures enable the structural members to absorb more blast energies before collapsing. However, a deep inspection into the crack pattern of beam A-13 displayed in Fig.15(i) reveals that more than ten shear cracks were observed at full span besides two main cracks penetrating the entire height of the beam.It is illustrated that the plain LWC beam failed in a more brittle manner compared with the HFRLWC beam.

Fig.13. Overpressure-time histories for different TNT charges.

Fig.14. Distributions of overpressure in full-span.

4.2. Deflection response of HFR-LWC beams

For the cases considered herein, the blast duration is much shorter than the response period of structure.Therefore,the energy imparted to the HFR-LWC beam by blast loading is considered as impulsive loading.The maximum deflection response to impulsive loads and the vibration around permanent displacement until energy is consumed entirely can be captured by the LVDTs installed on the beam specimen. Deflection-time histories of several typical tested beams listed in Table 7 are given in Fig.16. In the test, the deflection-time histories of several recorded points were not captured because the LVDTs were damaged by spalled concrete.It is evident that all beam specimens reached their maximum displacements abruptly when exposed to close-range blast, and then experienced obvious rebound processes. The downward incident shock wave was converted into tensile wave while reaching the bottom of tested beam,which resulted in a drastic upward rebound of the member after its maximum deflection response. Post-blast inspection reveals that the angle steels at the two supports which had been pre-anchored to the test pit by steel bolts were severely deformed and that the equilibrium position of the beam were changed.Hence,some measured points were unable to capture the actual residual displacement accurately. It should be noted that more attention should be paid to the peak displacements of protective structures.Thus,the deflection response before reaching the first peak value was employed to validate the reliability of improved SDOF model in the subsequent analysis.

Fig.15. Failure patterns of LWC beam.

4.3. Tensile stress of steel rods

The strain-time histories of steel rods are displayed in Fig. 17,where the symbol rod-i represents the average strain of steel rods at ith-layer as shown in Fig.8.A notable phenomenon from Fig.17 is the remarkable divergence of strain-time histories between the steel rod under blast loading and that under static loading reported by Cheng et el.[53],illustrating that the self-generated membrane action shows evident load dependence. The steel rods are elongated with progressive deflection of the HFR-LWC beam, and the tensile strain reaches its peak value simultaneously with the arrival of maximum deflection. After that, the tensile strain is dropped suddenly and followed by a continuous vibration with small amplitude, until the kinetic energy of the HFR-LWC beam is dissipated entirely.The four-point bending tests[53]of HFR-LWC beams manufactured in the same batch indicate that the in-plane force generated by the steel rod increased with the deflection of the beam and kept a constant value until the HFR-LWC beam collapsed.The development of membrane action exhibits a slow climbing process, and the slope of membrane action-deflection curve depends on the loading rate. Contrary to that observed in static loading tests, it is anticipated that membrane force reaches its maximum in a short time, which is almost synchronous with the arrival of peak deflection. Therefore, blast resistance of the HFRLWC beam is mainly enhanced by the end-constrain clamp at the stage before membrane force reaching its peak value.However,the subsequent tensile force of the steel rod has little contribution on the reserve capacity of the beam and thus can be neglected because the following deflection of the HFR-LWC beam will never exceed its previous peak value.

Crack propagations obviously lagged behind blast action. For simplicity,it is assumed that the tensile stress of the steel rods has fully developed in the elastic response of the beam.The section area and elastic modulus of the steel rod are pre-determined in the test.The tensile force is derived according to Hooke’s law,and the total membrane force can be further achieved by adding all tensile forces of the steel rods together. As mentioned earlier, the membrane contribution can be characterized by the peak strain of the steel rod. The peak membrane forces corresponding to the maximum strains of steel rods taken from Fig.17 are summarized in Table 8.It is clear that the values of membrane action are sensitive to deflections and restraint grades. Generally, larger deflections and restraint grades will give rise to higher membrane forces. The membrane force generated by the end-constrain clamp with restraint grade III increases by 44.8%-184.3%, compared with that generated by the end-constrain clamp with restraint grade I when the HFR-LWC beam is subjected to explosion of 3 kg-TNT charge.On the other hand,the membrane force will increase by 95.6%-148.6%for the HFR-LWC beam with restraint grade II if the scaled distance decreases from 1 m/kg1/3to 0.69 m/kg1/3.

Fig.16. Displacement-time histories of LWC beam.

4.4. Validation of improved SDOF model

The reliability of improved SDOF model presented in this paper is verified by the available experimental results. In the theoretical analysis, the dimensions of the HFR-LWC beam is taken as 200 mm × 100 mm × 1300 mm, with a total mass of 57.3 kg. The compressive strength of HFR-LWC is 47.97 MPa and the elastic modulus is 25.7 GPa. Longitudinal reinforcement has a yield strength of 653 MPa and Young’s modulus of 201 GPa. The massload coefficient of SDOF model is determined according to the nonlinearly distributed load of close-range blast, in which the overpressure and blast duration are predicted using TM5-1300[35], and the peak overpressure ratio n = P2/P1is derived by the interpolation based on Table 1. In the improved model, the deflection-dependent membrane action is considered by Eq. (11)and the shock wave is simplified into linearly decaying blast load.

The characteristic time of shock wave generated by different TNT charges based on the promoted SDOF model is listed in Table 9.It is observed from Figs. 16 and 17 that the arrival time of peak membrane force is extremely close to that of the maximum displacement of the HFR-LWC beam, illustrating that membrane action on the blast resistance of the member has been fully applied.Therefore,the membrane forces generated by the peak stress of the steel rods can be acceptably employed to represent the blastresistant contribution of end-constrain clamps. The displacementtime histories of the mid-span section for HFR-LWC beams under blast loadings accompanying membrane action based on improved SDOF model are displayed in Fig. 18, in which the recorded time history curves for the testing cases listed in Table 7 are also given.

Fig.17. Strain-time histories of steel rods.

Table 8 Tensile strains of steel rod and longitudinal forces.

Table 9 Characteristic times of air shock.

Analytical results show that the deflection of the HFR-LWC beam reached its peak value abruptly, followed by an elastic vibration branch. Fig.18(f) indicates that the deflection vibrated around its equilibrium position and no permanent deformations were found during the full dynamic response,showing that the HFR-LWC beam behaves in elastic state when subjected to explosion of 1 kg-TNT charge,which can be confirmed by Fig.15(f)that almost no visible cracks and permanent deformations were observed. The similar phenomenon was also seen in the predicted displacement-time histories of beam A-11 as shown in Fig. 18(e), because the beam with restraint grade II suffered the blast load induced by a smaller TNT charge (2 kg-TNT) than other specimens with the same reinforcement ratio of 0.28%. However, displacement-time curves are comprised of a remarkable branch of plastic response and a branch of elastic response when the HFR-LWC beam is subjected to explosion of 3 kg-TNT charge.In the elasto-plastic response,the midspan displacement of the HFR-LWC beam exceeded its elastic limit suddenly and the membrane action was activated subsequently with the progression of plastic deflection. In addition, the maximum mid-span displacement is a noteworthy parameter for the component of protective construction. The displacement-time histories in Fig. 18 and the characteristic time listed in Table 10 show that the analytical values are in good agreement with the measured results for pre-peak displacements, suggesting that the improved SDOF model is an acceptable tool to predict the peak deflections of HFR-LWC beams under blast loadings accompanying membrane action. It should be emphasized that in most cases,plastic behavior predominates the response in the analytical model,while most experimental results show elastic behavior.The obvious rebound branches can be observed in Fig.18(a),(b),(c),(d),(g)and(h), which lead to a considerable difference between the two methods.During regular detonations, the tested beams are simply supported on the steel rollers,and two angle steels at the supports are used to prevented the beams from bouncing up caused by reflected tensile waves. However, the post-blast inspections reveal that some angle steels were severely deformed, and the tested beams obviously rebounded (detached from the supports). Apparently, the post-peak displacement is a combined behavior of the deflection and the rigid rebound of the beam, which is quite different from the elastic recovery response owing to structural resilience. It is a technology fault in the blast test that results in notable discrepancies between predicted values and experimental results, which should be avoided in future investigations.

The maximum deflections acquired by theoretical and experimental approaches are both summarized in Table 11.It is apparent that the relative error between the presented model and blastresistant test for simply supported beam is only 0.15%, but it tends to increase with restraint grade and goes up to 19.20% for beam A-36 with restraint grade III. Basically, the maximum deflections achieved using improved SDOF model are larger than that of the blast-resistant test,illustrating that the proposed model will underestimate the blast resistance of HFR-LWC beams to some extent.The reasons might be that:(ⅰ)for improved SDOF model,the deflection of the clamped beam is derived based on a simplysupported member with a longitudinal restraint spring and a rotational restraint spring at both beam-ends. It will result in somewhat overestimation of peak displacement, although the resistance function is achieved based on a fixed-end member with three in-span plastic hinges, one at the mid-span and two at the supports once its elastic limit is reached; (ⅱ) the membrane forces are represented by the peak tensile force of the steel rods in the theoretical model; actually the membrane effect is an deflectiondependent factor and will greatly lag to deflection when the beam-like member is subjected to severe dynamic load; (ⅲ) the enhancement effect of hybrid fibers is not considered in the theoretical approach for HFR-LWC beams under blast loadings,and thus the displacement of the beam will be overestimated in the improved SDOF model;(ⅳ)the stirrups have been neglected in the predicted deflections.

Generally, the membrane action is beneficial for enhancing the response limit of HFR-LWC beams under blast loadings. The dynamic strain of longitudinal reinforcement is significantly diminished by the membrane action until rupture occurs. On the other hand, the membrane action is helpful for the development of the interlocking effect of coarse aggregate and the bridging effect of hybrid fibers, which will also enhance the reverse capacity of the beam. While cracks penetrate the whole thickness, the load is carried mainly by reinforcing bars and steel rods acting as a tensile net. The propagation velocity of cracks of HFR-LWC beams under blast loadings can be effectively prevented by end-constrain clamps. The presence of hybrid fibers facilitates widening of cracks through the bridging effect of fiber in the macro-cracks,and allows ductile failure through pulling-out of the fibers. A higher restraint grade will give rise to larger crack width and fiber pull-out failure, and therefore result in the ductile behavior of HFR-LWC beams, which can be observed in Fig.15(d) and (h).

5. Conclusions

Fig.18. Comparisons of mid-span displacement-time histories.

Table 10 Characteristic times of dynamic response.

Table 11 Comparisons of predicted deflections and experimental results.

An improved SDOF model is developed to describe the dynamic response of clamped beam-like members under blast loadings,where the mass-load coefficient is determined according to the nonuniformly distributed load induced by close-range explosion,and the membrane action is represented by an in-plane force and a resisting moment. A specially built end-constrain clamp is presented to provide membrane action for structural members when they are subjected to blast load simultaneously. It is demonstrated that the analytical curves are in good agreement with the experimental results at the response branch before the maximum deflection,suggesting that the improved SDOF model is an acceptable tool to predict the peak responses of HFR-LWC beams under the combined action of blast load and membrane force.

It is confirmed that the ductility of LWC beams is significantly enhanced by hybrid incorporation of HPP fibers,and that there are no noticeable shear cracks close to the supports, unlike plain LWC members which failed abruptly with penetrating cracks and crushed concrete. Membrane action is helpful for development of the interlocking effect of coarse aggregate and the bridging effect of hybrid fibers, which will also enhance the response limit of the beam. The propagation velocity of the cracks of HFR-LWC beams under blast loadings can be effectively prevented by end-constrain clamps.The presence of hybrid fibers facilitates widening of cracks through the action of fiber in stitching the macro-cracks,and allows ductile failure through pulling-out of the fibers.

The development of membrane action for beams under blast loading shows significant difference from that under static loading.Membrane action grows abruptly with blast loads, and reaches its peak value in a short time,indicating that the maximum membrane force is synchronous with the peak deflection of the HFR-LWC beam. Membrane action can be characterized by the peak strain of steel rods.The contribution of membrane action to the deflection response of beam-like members cannot always be ignored under blast loadings. The resistance will be seriously underestimated if the membrane effect is taken as a “hidden” safety factor in the design manual, and attention must be given to a more precise definition of structural resisting mechanisms.

Declaration of competing interest

There are no conflicts of interest with other reports or authors.

Acknowledgements

We gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant: 51578541,51378498),and the Natural Science Foundation of Jiangsu Province(Grant:BK20141066).