Wei Wei , Yu-lei Zhng , Jin-jun Su , Yn Liu , Feng-lei Hung ,***

a State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing,100081, China

b Xi'an Modern Chemistry Research Institute, Xi'an, 710065, China

Keywords:SDOF model Close-in explosion Specific impulse equivalent method Flexural resistance

ABSTRACT In this paper, a modified single-degree-of-freedom (SDOF) model of reinforced concrete (RC) beams under close-in explosion is proposed by developing the specific impulse equivalent method and flexural resistance calculation method.The equivalent uniform specific impulse was obtained based on the local conservation of momentum and global conservation of kinetic energy.Additionally,the influence of load uniformity, boundary condition and complex material behaviors (e.g.strain rate effect, hardening/softening and hoop-confined effect) was considered in the resistance calculation process by establishing a novel relationship between external force, bending moment, curvature and deflection successively.The accuracy of the proposed model was verified by carrying out field explosion tests on four RC beams with the scaled distances of 0.5 m/kg1/3 and 0.75 m/kg1/3.The test data in other literatures were also used for validation.As a result,the equivalent load implies that the blast load near the mid-span of beams would contribute more to the maximum displacement,which was also observed in the tests.Moreover,both the resistance model and test results declare that when the blast load becomes more concentrated, the ultimate resistance would become lower, and the compressive concrete would be more prone to softening and crushing.Finally,based on the modified SDOF model,the calculated maximum displacements agreed well with the test data in this paper and other literatures.This work fully proves the rationality of the modified SDOF method, which will contribute to a more accurate damage assessment of RC structures under close-in explosion.

1.Introduction

In recent decades, frequent occurrences of terrorist bombing attacks on crowded buildings have caused major personnel and property losses, which arouse people's attention to building protection against accident explosion.Explosion can produce highpressure detonation products and shock waves, subsequently damaging building components and leading to the collapse of buildings.To better protect the buildings from damage, the dynamic responses of reinforced concrete(RC)structures under blast loading have been studied experimentally, numerically and theoretically.Due to high efficiency, the equivalent single-degree-offreedom (SDOF) method has been widely used in flexural response prediction of RC structures under blast load,especially for RC beams.

The traditional equivalent SDOF model was proposed by Biggs[1]in 1964.Based on the assumed deformed shape function and perfectly elastic-plastic resistance function, the uniformly loaded beam was transformed into the concentrated mass-spring-load system.This method was included by UFC 3-340-02 [2]as the protection design tool and widely used for flexural deformation calculation of RC structures under blast load.Stochino [3]carried out the equivalent SDOF method, considering the dynamic strain rate effects, to predict the flexural responses of RC beams under uniform shock waves driven by a shock tube, and the calculated results showed good agreements with the experimental results conducted by Magnusson [4].Alexander [5]investigated the bearing resistance of one-way reinforced ductile concrete plates under uniform blast loading by shock tube tests,and the structural responses were accurately predicted by the equivalent SDOF method.Liao [6]conducted three groups of high-strength RC beams and ordinary RC beams under uniform blast loads by a blast pressure simulator and calculated the P-I (Pressure - Impulse)curves based on the equivalent SDOF method.It indicated that the damage evaluation results of RC beams by the P-I curves agreed well with the test data.Evidences show that the traditional SDOF method is an effective method for calculating the flexural responses of RC structures under uniform blast load.Even so, the fatal disadvantage of this method is that it is only applicable to uniformly distributed load, that is, the medium or far field explosion scenarios.

Abbreviation

Recently, many scholars began to pay attention to the close-in blast effect of explosives on one-way RC structures and attempted to predict the structural flexural deformation through the traditional SDOF method.Oswald and Bazan[7]collected a large amount of data on blast loaded components,including 76 one-way RC walls with various dimensions, supports and load conditions.Then, the test data were compared to the SDOF analyses performed by the Excel based tool SBEDS (SDOF Blast Effects Design Spreadsheet),which was developed based on UFC 3-340-01[8]and UFC 3-340-02[2].The results showed that the ratios between the calculated maximum displacements to the measured ones ranged from 0.61 to 1.92.Jones [9]compared the displacement responses obtained by the traditional SDOF method and the close-in explosion tests conducted by Wu [10]on several one-way RC specimens with the scaled distance ranging from 0.93 to 3 m/kg1/3, where the scaled distance is generally defined by(His the distance between explosive and target andWcis charge weight).The calculation error was between 38%and 64%.Wang[11]carried out a series of close-in explosion tests on one-way RC plates with the scaled distance ranging from 0.518 to 0.684 m/kg1/3, and the traditional SDOF method showed a large error (1000-1387%) to predict the midspan displacements.Nagata [12]conducted several close-in explosion tests on simply supported RC beams using 110 g,160 g and 250 g C-4 explosives at a scaled distance of 0.20 m/kg1/3and predicted the bending behaviors using the traditional SDOF method.The results indicated that the maximum displacements were 10 times higher compared to the experimental results.It can be concluded that the traditional SDOF method failed to predict the flexural displacement of one-way RC components, and the calculation error seems to increase with the decrease of scaled distance.

It is speculated that the reasons for the predicted errors of traditional SDOF method under close-in explosion are as follows.In the case of far field explosion, the assumption of uniformly distributed blast loading is reasonable.However, under close-in explosion, the blast loading is highly non-uniform distributed temporally and spatially according to the experimental [10,12-16]numerical [14-17]and analytical results [18].Therefore, the assumption of uniform load failed to represent the real blast scenarios.Reasonable methods need to be proposed to convert the non-uniform load into uniform load.In addition,according to Biggs[1], the transformation factors (KL, KMandKLM=KM/KL) and ultimate flexural resistance are related to the load shape.The correlations between these parameters and the non-uniform distributed blast load need to be established.Aiming at the shortcomings,modifications of traditional SDOF method are necessary to adapt the non-uniform blast loads.Wang [11]proposed a method to convert the non-uniform peak pressure into the equivalent uniform one based on the principle of virtual work, and the mid-span specific impulse is selected as the equivalent specific impulse.Then, the traditional SDOF method was adopted by using the equivalent load.This method was adopted by Refs.[19-21]and the calculation results of flexural deformation showed agreements with the experimental and simulation results.In spite of this, the deficiency of this method is obvious because the equivalent specific impulse was not considered.In Ref.[12],the modified SDOF model considering the effect of load uniformity on ultimate resistance and transformation factors was derived.The accuracy of the modified method was verified with experimental results of simply supported RC beams.However,this model used the average peak pressure and specific impulse as the load input and ignored the weighting effect which was confirmed by Refs.[11,22].Moreover,in the research of Rigby et al.[23], the spatial load transformation factor was proposed to transform any spatially varying load into an energy equivalent uniform load.The accuracy of this model was verified with numerical [24]and experimental results [25].

The previous studies provided some good ideas to improve the traditional SDOF method and tried to make it suitable for close-in explosion.However, some problems remain to be clarified and solved.Firstly, the previous studies differed greatly in equivalent uniform load, and the equivalence of specific impulse ignored the effect of weighting effect.Secondly,the existing flexural resistance function for non-uniform load ignored the influence of material hardening/softening,dynamic strain rate effect and hoop-confined effect, thus a more complete process is required to be proposed.Additionally, it is difficult to verify the accuracy of the modified SDOF models at present due to the lack of integral validation test data,especially the reflected pressure data under close-in explosion which is difficult to obtain.Generally, empirical formulas [11,12]and numerical simulation results [21]were mostly used as explosion load input to validate the SDOF model,which might incorrectly estimate the real explosion load.Therefore,more detailed test data should be obtained to ensure the reliability of validation data,including the blast load data and displacement data.

In this paper,a more comprehensive SDOF method considering the load uniformity, complicated material behaviors (e.g.material hardening and softening,strain rate effect and stirrup confinement effect) and boundary conditions (e.g.simply support and fixed support) for RC beams under close-in blast loading is proposed.In order to validate this model, near field explosion tests on four identical RC beams with different boundary conditions, scaled distances (0.5 m/kg1/3and 0.75 m/kg1/3) and charge weights (1 kg and 3 kg) were conducted.The blast load and displacement time histories were measured in the tests.Five pressure sensors were arranged to establish the near field spatial distribution model of peak reflective pressure and specific impulse.At last,a comparative analysis was carried out between the test results and calculation results obtained by this method and other modified SDOF models.The accuracy of the suggested SDOF model in this paper was also validated by the test data in other literatures.This study can provide support for damage assessment of reinforced concrete structures under close-in explosion.

2.Modified SDOF model

Based on Biggs's work [1], the continuous dynamic system can be equivalent to a concentrated mass-spring-load system,as shown in Fig.1P（x，t） denotes the pressure function related to timetand span coordinatex.The coordinate origin is located at the mid-span.

The kinetic equation of the equivalent SDOF model can be described by

where,Wis the total mass,Ris the resistance function,F（t） is the external load,ω（t）is the equivalent displacement which is equal to the original structural mid-span displacement,（t）= d2ω（t）/ dt2is the acceleration,Re=KLRis the equivalent resistance,Fe=KLFis the equivalent load,KLandKMare defined as the load transformation factor and the mass transformation factor,respectively.

Fig.1. Equivalent SDOF system.

The calculation methods of resistance function and transformation factors for uniform blast loading were given in Ref.[1].For non-uniform blast loading, the calculation methods of equivalent uniform blast load, resistance function and transformation factors are given below.It should be noted that the non-uniform blast load is considered symmetrical, which means that the explosives are located directly above the mid-span of RC beam.

2.1.Equivalent uniform blast load

Transforming the non-uniform blast load of continuous structures to the uniform one is the key step to establishing the SDOF system.For this purpose, the shock wave is assumed to reach the loaded surface at the same time,which is accepted by Refs.[11,12].Moreover, the negative pressure is not considered in this paper.

By assuming that the work done by the external force would remain unchanged before and after equivalence, the equivalent method of uniform peak reflected pressure proposed by Wang[11]is adopted first, referring to Eq.(2)

Subsequently, the fictitious definition of energy equivalent impulse proposed by Rigby[26]is adopted and developed to calculate the equivalent uniform specific impulse based on the local conservation of momentum and conservation of global kinetic energy.The applications of this method in Ref.[27]have strongly proved its rationality in predicting the response of plates under non-uniform impulse.Actually,the energy equivalent impulse was proposed for impulsive problem,which means that the load duration should be much less than the vibration period of the loaded structure.For the impulsive problem, the local conservation of impulse should be satisfied because the structures would remain almost stationary during the loading process, which means that the local impulse would be completely converted into the initial velocity.However,the close-in blast load in this research might not be the impulsive load for sure.For the non-impulsive problem, the actual local velocity depends not only on the local external impulse but also on the impulse caused by the shear resistance.Unfortunately, the effects of the two impulses on kinetic energy are coupled by ΔEk=（I1+I2）2/2ΔW,where ΔEkis the local kinetic energy,I1andI2are the external impulse and shear resistance impulse respectively,ΔWis the local mass.Based on the existing knowledge, the equivalent impulse could not be calculated under this condition.As a simplification and approximation, we assume that the virtual kinetic energy caused by the purely external impulse remains unchanged before and after the equivalence.The calculation steps are as follows.

For the non-uniform distributed specific impulse, the spatial distribution of velocity can be described by

whereI（x） is the non-uniform specific impulse distribution along the beam span,Bis the width of structures,ρ is the mass per length.The total kinetic energy before equivalence can be calculated by

For the equivalent uniform specific impulse, the spatial distribution of velocity can be described by

The total kinetic energy before equivalence can be calculated by

According to the assumption above,the total specific impulse is completely transformed into the initial kinetic energy.Therefore,the initial kinetic energy should remain unchanged before and after equivalence.Thus,letEk1equals toEk2,then the equivalent uniform specific impulse can be calculated by

Assuming that the external force decays linearly, then we can obtain the durationtdand load historyF（t） by Eq.(8) and Eq.(9)respectively.

Eq.(2) and Eq.(7) imply that the load near the mid-span contributes more to the equivalent uniform load,which means that the weighting effect is considered.Further, the definition of specific impulse enhancement factor proposed in Ref.[26]is used herein to characterize the physically-based load uniformity and the resulting weighting effect.The peak pressure enhancement factorKpand specific impulse enhancement factorKiare defined by

Then,we assume that the peak pressure and specific impulse are exponential decayed spatially, that is,P（x）=Pme-αp|x|andI（x） =Ime-αi|x|.Herein, αpand αiare the distribution coefficients that characterize the load uniformity.The larger αpand αiare,the more non-uniform the peak pressure and specific impulse will be.The relationship betweenKp,Kiand αp, αifor beams with a length of 1.4 m are given in Fig.2 and Fig.3 respectively.BothKpandKiincrease with the increase of αpand αi.Then, the enhancement factorsKpandKican be used to characterize the load uniformity.Kp,Ki=1 indicates a perfectly uniform load andKp,Ki>1 indicates a non-uniform load.The greater the values, the more non-uniform the blast load.

2.2.Dynamic resistance function

The non-linear dynamic resistance function is established considering the non-uniform blast load, complicated material behaviors(e.g.material hardening and softening,strain rate effect and stirrup confinement effect) and boundary conditions (e.g.simply support and fixed support).

2.2.1.Material model

The uniaxial compressive relations of concrete cover are given in Eq.(14) [28].

where εc, σcare the compressive strain and corresponding compressive stress,E0is the initial tangent modulus,ε0,σ0are the strain and stress at the peak compressive stress,Es= σ0/ε0.The tensile behavior of concrete is ignored because the tensile strength is much less than the compressive strength.

In order to consider the influence of stirrups on the core area concrete,a hoop-confined uniaxial stress-strain model proposed by Mander et al.[29]is utilized in this paper.The expression of this model is as follows.

Fig.2. Relation between Kp and αp.

Fig.3. Relation between Ki and αi.

Bilinear stress-strain model is adopted for the longitudinal reinforcements.The strain hardening effect is considered.The stressstrain relationship for both tensile and compressive behaviors is given below

where εsand σsare the strain and stress of steel bar respectively,Esandare the Young's modulus and the hardening modulus,respectively,fyis the yield stress, εyis the yield strain, εuis the fracture strain.

The stress-strain curves of both concrete cover and core area include the ascending and descending segments, where the latter segment generally characterizes the softening behaviors.The hardening behaviors are characterized by the post-yield state of reinforcement bars.

Due to the enhancement of material strength under blast loading, the strain rate effect is considered by using the Dynamic Increase Factor(DIF).DIF of concrete under compressive strain can be calculated by[30].

wherefcis the dynamic compressive strength at,fcsis the static compressive strength at,is the strain rate in the range of 3E-5~3E+2 s-1,=30×10-6s-1, logγs= 6.145α- 2, α = 1/（5 +9fcs/fco）;fco= 10 MPa.

DIF of reinforcement bars can be calculated by[31].

2.2.2.Resistance function of simply supported RC beams

In this section, the resistance model of simply supported RC beams under non-uniform blast load is presented by improving the method proposed by Yi[32].It mainly includes the following steps.At first, the relation between mid-span curvature and mid-span deflection is established.Secondly, the sectional momentcurvature relation is obtained.Finally, the resistance can be calculated by the mid-span moment.The detailed calculation processes are given below.The slopes during the unloading and reloading processes are assumed to be the same as the slope of elastic segment.

(1) The curvature-deflection relation

The simply supported RC beams under non-uniform distributed load will undergo two stages,the elastic stage and the plastic stage.The formation of plastic hinge at the mid-span is the criterion of plasticity.The curvature-deflection relations are different for different stages.

Stage I: elastic

During the elastic stage, the deformed shape function can be obtained by calculating the static elastic deformation under the non-uniform peak pressure as follows.

The corresponding support reactionFLandFRon each support can be calculated by

The shear force distribution can be calculated by

Then we can obtain the bending moment and curvature distribution by

whereEIis the elastic bending stiffness, which is the slope of the elastic segment of the bending moment-curvature relation.

The global deformation can be obtained by integrating the curvature function

By normalization, the shape function of simply supported RC beam can be obtained by

wherey（0）is the mid-span displacement,φse（x）is the elastic shape function of simply supported beam.

Eq.(23) indicates that the relationship between the mid-span displacement and mid-span curvature during the elastic stage is linear, then:

wherekseandyseare the mid-span curvature and displacement during the elastic stage.kscis the curvature corresponding to the formation of plastic hinge, which can be obtained by the moment curvature relationship.yscis the mid-span displacement that corresponds withksc, which can be obtained based on the elastic deformed function.

Similar conclusion can be drawn for mid-span velocity and curvature rate:

Stage II: plastic

For the plastic stage,the shape function is generally assumed to be linear [1], as follows:

The mid-span displacement and mid-span curvature during the plastic stage should satisfy the following equation:

whereypandkpare the mid-span displacement and curvature during the plastic stage; Δypand Δkpare the additional plastic displacement and curvature caused by plastic development,such as the rotation of plastic hinge.In order to clarify the relationship between Δypand Δkp, the plastic hinge lengthlpis introduced by[33].

wheredis the effective height of beam section.

According to Fig.4, the rotation angle of plastic hinge is

Thus, the additional plastic displacement is

Similar to the elastic stage, the relationship between the midspan deflection and curvature, as well as the mid-span velocity and curvature rate during the plastic stage can be calculated by

(2) The moment-curvature relation

The layered section method is used to acquire the sectional bending moment based on specific curvature and curvature rate.The bond-slip between concrete and reinforcement bars is not considered.It mainly includes the following steps to calculate the bending moment during the loading process.

Step 1:The beam section is divided intonequal layers along the height direction, as shown in Fig.5,then define the height coordinated（i） = （i- 1）Δd,i=1，2，…，n，Δd=D/（n- 1）.

Step 2: Input a curvature valuekand a curvature rate value,then Step 3-Step 6 will calculate the corresponding bending momentM.

Step 3:Assume the depth of the neutral axis of sectiondc;then calculate the strain and strain rate distributions on all layers and the longitudinal reinforcement bars.

Step 4: According to the stress-strain relationship of concrete and steel,obtain the average stress of concrete on all layers σc（i），i=1，2，…，n, the stress of upper reinforcement σsuand the stress of lower reinforcement σsd.

Step 5: Calculate the bending momentMand axial forceN

wherebis the thickness of concrete cover.If|N|≤ε,outputM;Else,return Step 3.ε is the tolerant error.

Through the above steps, the bending moment of the section can be calculated by the given curvature and curvature rate,described as the function below:

Fig.4. Plastic deformation.

Fig.5. Layered section method.

Fig.6 displays the calculation process to obtain the momentcurvature relationship.The bisection method is used to search the depth of the neutral axis.

(3) The resistance-moment relation

The resistance-moment relation can be calculated by relating the external force with the mid-span bending moment through Eq.(19)and Eq.(21).Because the calculation formula is too complex, it is not shown here.Assuming that the non-uniform peak pressure is expressed byP（x） =Pme-αp|x|,then the relation between mid-span bending momentMand resistance forceRcan be simplified by

Assuming a perfectly elastic-plastic resistance function, Fig.7 displays the relation betweenRuL/Muand αp/L, whereRuis the ultimate resistance andMuis the ultimate moment.The calculation results are in good agreement with Ref.[1].With the increase of αpfrom 0.001 to 1000, the value ofRuL/Muranges from 8 to 4, corresponding to the uniform load and concentrated load respectively.

2.2.3.Resistance function offixed supported RC beams

The resistance of fixed supported RC beams is different from that of simply supported beams, which is mainly reflected in the calculation of curvature-deflection relation and resistance-moment relation.The moment-curvature relation is exactly the same so it will not be repeated in this section.

(1) The curvature-deflection relation

The fixed supported RC beams under non-uniform distributed load will undergo three stages, the elastic stage,the elastic-plastic stage and the plastic stage.The formation of plastic hinge at the supports is the criterion of elastic-plastic stage and the formation of plastic hinge at the mid-span is the criterion of plastic stage.

Stage I: Elastic stage

During the elastic stage, the deformed shape function can be obtained by calculating the static elastic deformation under the non-uniform load as follows.

Fig.6. Flow chart of moment-curvature relation.

Fig.7. RuL/Mu -αp/L relation of simply supported beam.

Fig.8 indicates that the fixed beam is a statically indeterminate structure.The support moment should be determined first, referring to Appendix A.

Then we can obtain the bending moment and curvature distribution as follows:

Fig.8. Load characteristics of fixed supported beams.

The global deformation can be obtained by integrating the curvature function

By normalization, the shape function of simply supported RC beam can be obtained

Similar to the simply supported beams, the relationship between the mid-span displacement and mid-span curvature,as well as the mid-span velocity and curvature rate,for the fixed supported beams during the elastic stage is also linear

wherekfeandyfeare the mid-span curvature and displacement during the elastic stage.kfc1andyfc1are the mid-span curvature and displacement corresponding to the yield moment of the support.

Stage II: Elastic-plastic stage

During the elastic-plastic stage, the plastic hinges at both supports will develop with the increase of deformation.The structural deformed mode will be similar to the elastic stage of simply supported beam[1].Thus,the shape function φf(shuō)ep（x）during the elasticplastic stage of fixed supported beams is generally assumed to be the same as that of the elastic stage of simply supported structures.

The mid-span displacement and mid-span curvature should satisfy the following equation:

whereyfepandkfepare the mid-span displacement and curvature during the elastic-plastic stage;Δyfepand Δkfepare the additional plastic displacement and curvature caused by support rotation,thus the following condition should be met:

Thus, the relation between mid-span displacement and curvature is given below:

When the mid-span curvaturekfepreaches the yield curvature of the beam section,the elastic-plastic stage will come to an end.The corresponding mid-span critical curvature and displacement arekfc2andyfc2, respectively.

Stage III: Plastic stage

During the plastic stage,the plastic hinges will form at the midspan.The shape function φf(shuō)p（x）during the plastic stage is generally assumed to be the same as that of the plastic stage of simply supported structures.

The mid-span displacement and mid-span curvature should satisfy the following formula:

whereyfpandkfpare the mid-span displacement and curvature,Δyfpand Δkfpare the additional plastic deformation and curvature caused by the rotation of plastic hinge,thus the following condition should be met:

Thus, the relation between mid-span displacement and curvature is given below:

(b) The resistance-moment relation

If the non-uniform peak pressure satisfiesP（x） =Pme-αp|x|,then the mid-span bending momentMand resistance forceRwill satisfy the following linear relationship:

whereRcyandMcyare the resistance and mid-span moment respectively when the support plastic hinges appear.Fig.9 displays the relation betweenRuL/Muand αp/Lof the elastic-perfectly plastic beam.The calculation results are in good agreement with Ref.[1].With the increase of αp/Lfrom 0.001 to 1000,the value ofRuL/Muranges from 16 to 8,corresponding to the uniform load and concentrated load respectively.

Fig.9. relation of fixed supported beam.

2.3.Transformation factors

The transformation factors can be calculated by Eq.(51)and Eq.(52) based on the shape function and non-uniform load function.

For the non-uniform pressure functionP（x） =Pme-αp|x|, Fig.10 draws the trends of load transformation factor and the mass transformation factor with the increase of αpduring the elastic stage for both simply supported and fixed supported beams.The calculation results show good agreement with Ref.[1].With the increase of αpfrom 0.01 to 100, the value ofKLM（KLM=KM/KL）ranges from 0.78 to 0.49 for simply supported beam and from 0.77 to 0.37 for fixed supported beam, corresponding to the uniform load and concentrated load respectively.Similar conclusions can be drawn for elastic-plastic stage and plastic stage, which are not given here.

2.4.Calculating process

A complete theoretical model has been established above,then the last step is to solve Eq.(1).Considering the complexity of the equations, the analytical solution does not exist definitely.Therefore, the finite difference method is adopted to solve Eq.(1).The main steps are as follows.

Step 1: Define the calculation time seriesti=iΔt， wherei=0，1，2，3… and Δtis the time interval;

Step 3:Consider the strain rate effect,R（ω（t））in Eq.(1)should be replaced byR（ω（t），（t））,where（t）denotes velocity,calculated by

Step 5:According to the initial velocity and acceleration of the lumped mass,the virtual displacement at timet-1can be obtained by

Step 6: Iteratively calculate Eq.(53) to obtain ω（ti），i=1，2，3…untiltireach an end time.

It should be noticed that the values ofRandFwill change at each time step,meanwhile the value ofKLMis related to structural states(e.g.elastic,plastic or elastic-plastic).The overall flowchart to calculate the time-displacement histories is given in Fig.11.

3.Experiment

3.1.Experiment setup

A total of four reinforced concrete beams were built in this research program.All the specimens have the same specifications and sizes.The width and depth of the RC beams were both 130 mm,and the total length was 1600 mm.Details of reinforcement arrangements are given in Fig.12.The spacing of stirrups is 100 mm.The thickness of concrete cover is 20 mm.The compressive strength of concrete is obtained by compression test of three cubic concrete specimens with the dimension of 150 × 150 × 150 mm3.The average static compressive strength of concrete is 41.7 MPa.The average yield strength and ultimate strength of reinforcement bars are 420 MPa and 535 MPa,respectively.

In the tests,the RC beams were fixed on two reinforced concrete bases through screws.Two boundary conditions including simply and fixed supported were conducted and shown in Fig.13.For the fixed supported condition, the upper and lower surfaces of both ends of RC beams were constrained in all degrees of freedom by four 40 mm thick steel plates.The design of simply-simply supported condition is based on Ref.[34].Steel roller bars and 10 mm thick steel plates were placed between the 40 mm thick steel plates and RC beams to ensure that the boundary could rotate freely.Thus,the effective length of RC beams was 1200 mm and 1400 mm for fixed and simply supported structures,respectively.

Cylindrical TNT explosives with the mass of 1 kg and 3 kg were used to produce explosion load.The diameter and height of the explosives are listed in Table 1.The explosives were detonated at the central point of the upper surface through electric detonators and booster charges.The explosives were situated directly above the center of RC beams.

A total of four blast tests were carried out on four identical specimens.The details of the test cases are listed in Table 2.

Fig.10. Transformation factors during the elastic stage: (a) Simply supported; (b) Fixed supported.

Fig.11. Overall calculation flow chart.

Fig.12. Geometric details of specimens (Length Unit: mm).

Fig.13. Boundary conditions: (a) Fixed supported boundary; (b) Simply supported boundary.

Table 1Dimensions of explosives.

Fig.14 shows the details of test setup.Dynamic pressure sensors were installed to measure the distribution of blast loading close to the upper surface of RC beams.Five pressure gauges(P1,P2,P3,P4,P5), with the same interval of 100 mm, were located on the centerline of a steel beam close to the RC beams.Honestly, the placement of the steel beam might lead to the inconsistent decaying process of the shock wave along both sides of the beam due to the influence of clearing effect [23-25], which is related to the beam width [35].In spite of this, Refs.[12,21]adopted similar test setup and had successfully predicted the structural response.The reason might be that the clearing effect is weak under close-in explosion due to the extremely short duration of shock wave.Therefore, the clearing effect in this paper is not considered.The width of the steel beam was 100 mm.The upper surfaces of the RC beam and the steel beam were at the same horizontal level.The left gauge“P1”was situated at the mid-span.For tests No.2 and 4 with identical blast scenarios compared with No.1 and 3 respectively,the blast load was not measured repeatedly.For test No.1 and 3, five displacement sensors(Y1,Y2,Y3,Y4,Y5),with the same interval of 110 mm, were installed on the lower surface of RC beams to measure the deflection of multiple positions along the span direction of the structure.The left gauge“Y1”was situated at the mid-span.For test No.2 and 4, only one displacement gauge“Y1” was placed.

Table 2Test details.

Fig.14. Test setup.

3.2.Test results

3.2.1.Time-pressure histories

Fig.15 and Fig.16 show the pressure and specific impulse time histories of Load-1 and 2 respectively.The time lines take the initiation time as zero.The specific impulse curves are obtained by integrating the time-pressure histories.The pressure gauge“P1”of Load-2 failed to collect valid data.

Fig.17 illustrates the values of peak pressure and positive phase specific impulse at all gauges.It is observed that, the spatial distribution of peak pressure and specific impulse is highly nonuniform.For Load-1, the peak pressure and specific impulse of gauge“P5”are 87.9%and 87.1%lower than gauge“P1”,respectively.For Load-2, the peak pressure and specific impulse of gauge “P5”decrease by 70.4% and 53.8% compared with gauge “P2”, respectively.Fig.18 shows the arrival time of shock wave front.The shock wave arrived at the mid-span firstly and traveled from the midspan to the supports.It takes about 0.1 ms for the shock waves to propagate from“P1”to“P5”.The natural vibration period of beams can be calculated byT=[1], whereKis the spring stiffness and approximately equals tofor simply supported beams andfor fixed supported beams.After calculation,the periods of simply and fixed supported specimens are 18.6 ms and 7.4 ms respectively,which means that the travelling time of shock wave is extremely small compared with the natural vibration periods.

It is confirmed that the near field blast wave parameters of spherical explosives and cylindrical explosives are quite different under near field explosion.Fig.17 compares the differences of peak pressure and specific impulse between the experimental results and the empirical algorithm of spherical explosives.The keyword*LOAD_BLAST_ENHANCED(LBE)in Ls-dyna is used to calculate the blast parameters of spherical explosive.Fig.17 implies that the peak pressure and specific impulse of cylindrical explosive are higher than spherical explosives right below the explosives.In spite of this,as the horizontal offset distance increases, the peak pressure and specific impulse of spherical explosives decrease more slowly and gradually exceed that of cylindrical explosives.This is because the energy distribution of spherical explosive is uniform in different azimuths while that of cylindrical explosive is more concentrated near the axial direction[36].Therefore,under near field explosion,the effect of explosive shape on blast loading should not be neglected.

3.2.2.Displacement responses and damage effect

Fig.19 shows the global displacement time histories of beams B-1 and B-3 at five displacement gauges.Fig.20 shows the mid-span displacement time histories of beams B-2 and B-4.Table 3 summarizes the damage effects and structural response parameters,including the maximum displacement (ym) and residual displacement (yr).The crack distribution was redrawn by black marker.

3.3.Discussion

3.3.1.Determination of blast load

The peak pressure and positive phase specific impulse obtained in the tests are used to establish the corresponding spatial distribution models on the RC beams through mathematical fitting,and then taken as the load inputs of SDOF models.Generally,the closein blast load profile can be approximated by the exponentially decaying function [12,37].Therefore, according to the spatial distribution characteristics of blast load in the tests,the second-order exponential decay function is adopted to fit the peak pressure and specific impulse functions of Load-2, while for Load-1 the firstorder one is adopted due to the smaller scaled distance.

Fig.15. Pressure and specific impulse histories of Load-1 (1 kg-0.5 m/kg1/3): (a) Gauge “P1”; (b) Gauge “P2”; (c) Gauge “P3”; (d) Gauge “P4”; (e) Gauge “P5".

Fig.21 and Fig.22 show the fitting results of Load-1 and Load-2,respectively, whereRddenotes the horizontal distance from the projection point of explosive center on the upper surface of the RC beam.Due to the lack of data in Load-2, supplementary tests are carried out to obtain the load data in Gauge“P6”and“P7”,as shown in Fig.23.

Fig.23 also exhibits the mapping relation between the locations of pressure gauges and corresponding areas on the loaded RC beams by drawing circles taking the center of the upper surface of the structure as the center of the circle.As can be seen from Fig.23,the blast loading in the area within the radius of‘P1’is unknown.In addition,the blast load along the width direction of beam section is completely different near the mid-span due to the rapid attenuation of shock wave.For the first issue, because it is difficult to accurately calculate the near-field blast load by numerical simulation, both the peak pressure and specific impulse are extrapolated based on the fitting curves above.For the second one, the average peak pressure and specific impulse on each beam section are calculated by integrating the fitting curves along width direction,as given in Eq.(54)and Eq.(55),where（x）and（x）are the sectional average peak pressure and specific impulse.The calculation results of blast load are shown in Fig.24.

By refitting the blast loading model shown in Fig.24, Table 4 summarizes the key parameters including the maximum peak pressure （Pm）, maximum specific impulse （Im） and the corresponding distribution coefficients αpand αi.In addition, the peak pressure enhancement factors and specific impulse enhancement factors referred in Eq.(12) and Eq.(13) are calculated to characterize the load uniformity.

3.3.2.Effect of load uniformity

Fig.16. Pressure and specific impulse histories of Load-2 (3kg-0.75 m/kg1/3): a) Gauge “P2”; (b) Gauge “P3”; (c) Gauge “P4”; (d) Gauge “P5".

Fig.17. Spatial distribution of blast load parameters: (a) 1 kg-0.5 m/kg1/3; (b) 3 kg-0.75 m/kg1/3.

Fig.18. Arrival time of shock wave front: (a) 1 kg-0.5 m/kg1/3; (b) 3 kg-0.75 m/kg1/3.

Fig.19. Global displacements histories of beam (a) B-1 and (b) B-3.

Fig.20. Mid-span displacements histories of Beam B-2 and B-4.

The load uniformity will influence the displacement responses of RC beams from two aspects, that is, the equivalent load and flexural resistance.Fig.20 shows that the maximum mid-span displacement of beam B-2 under Load-1(1 kg,0.5 m/kg1/3)is 33.3%higher than that of beam B-4 under Load-2 (3 kg, 0.75 m/kg1/3),respectively.However, according to Fig.17, except for gauge “P1”(Not record),all the peak pressures and specific impulses of gauge“P2~P5” of Load-2 are larger than that of Load-1.Thus, it is confirmed that the peak pressure and specific impulse of Load-1 near the mid-span are far more than that of Load-2,which is proved by the fitting curves displayed in Fig.24.Summarily, Table 4 lists the spatial distribution parameters of peak pressure and specific impulse, including the average values, enhancement factors and equivalent uniform values.Taking beams B-2 and B-4 as examples,the average peak pressure and specific impulse of B-4 are 19.3%and 47.0% higher than B-2.However, because the blast load of B-2 is more non-uniform than B-4, the load enhancement factors are greater and both the equivalent uniform peak pressure and specific impulse are close to B-4 with the small derivation of 0.57 MPa and 0.02 MPa · ms.Undoubtedly, it is certain that the structural response depends more on the equivalent uniform load rather than the average values because the maximum mid-span displacement of B-2 is higher than B-4, which means that the weighting effect works and the blast load near the mid-span of the beams would contribute more to flexural deformation.

Evidences also show that the flexural resistance of RC beam is greater under more uniform blast load,which was also declared in Refs.[1,12].Fig.25 compares the spatial distribution of peak displacements and the normalized ones of beams B-1 and B-3.It can be observed that although the peak displacement of B-1 at gauge“Y1”is greater than B-3 by 12.6%,the corresponding value at gauge“Y5”shows a 34%lower.The normalized curves clearly describe the deformation uniformity differences,that is,the spatial distribution of deformation of B-1 was more concentrated near the mid-span,while that of B-3 was more global.Subsequently, although the maximum displacements of beams B-1 and B-3 were very close with a deviation of 2.2 mm,beam B-1 was obviously crushed whileB-3 was almost intact which means that the local curvature of B-1 near the mid-span was higher.Meanwhile, Table 3 shows that beams B-3 and B-4 under more uniform load had more tensile cracks than B-1 and B-2.All the observed phenomena confirm that the beams under more concentrated blast load will bend more concentrated, thus the concrete in the compression zone will be more prone to softening and crushing.Additionally, the structures with more global deformation can effectively take advantage of the strain energy and obtain larger resistance.

Table 3Post-blast observation and displacement response.

Fig.21. Fitting curves of Load-1 (1 kg, 0.5 m/kg1/3): (a) Peak pressure; (b) Specific impulse.

Fig.22. Fitting curves of Load-2 (3 kg, 0.75 m/kg1/3): (a) Peak pressure; (b) Specific impulse.

Fig.23. Location mapping relation between pressure gauges and beam surface.

3.3.3.Effect of scaled distance

Table 3 shows that spalling occurred at the scaled distance of 0.5 m/kg1/3.By comparison, at the scaled distance of 0.75 m/kg1/3,although the explosive mass was increased to 3 kg, spallation was not observed.Generally, when the blast wave in the air interacts with the structure,compressive stress wave forms at the interface,and propagates to the bottom of structure.When the stress wave reaches the bottom free surface,it will reflect and then turn into a tensile wave.If the dynamic tensile strength of concrete is not sufficient to resist the tensile waves,a mount of concrete fragments will peel off and spallation appears.Fig.24 compares the blast load of 1 kg-0.5 m/kg1/3and 3 kg-0.75 m/kg1/3, and the result indicates that the peak pressure and specific impulse of the former are almost twice of the latter right below the explosive.Therefore,spalling occurred in the case of 1 kg-0.5 m/kg1/3.This fully proves the importance of scaled distance in determining whether spallation failure will occur because spall damage criterion is more related to peak pressure, which depends more on scaled distance than charge weight.

Fig.24. Blast loading model.

3.3.4.Effect of boundary condition

Fig.26 displays the influence of boundary condition on midspan deflection-span ratio.It shows that, compared with the fixed supported beams, the peak and residual deflection-span ratios of simply supported beams increase by 39.7%and 91.6%in the case of 3 kg-0.75 m/kg1/3,67.1%and 45.8%in the case of 1 kg-0.5 m/kg1/3.This phenomenon is well understood because fixed beams have stronger flexural capacity.

Further, because crushing and spalling occurred in beams B-1 and B-2,the effect of boundary condition on crushing and spalling is compared and discussed.Fig.27(c) and Fig.27(d) indicate that the spalling area size and shape of B-1 and B-2 are almost identical.This implies that the boundary condition might not influence the formation of spalling.The reason is obvious.The formation of spalling is caused by the travelling of shock wave in concrete at the speed of sound (over 3 km/s).Therefore, it takes the shock wave less than 43 μs to reach the lower face from the upper face,which is extremely small compared with the flexural response time.It is also observed that the left and right boundaries of the spallation region are“V"shaped approximately,which means that the spalling width close to the side surface is larger while that close to the center is smaller.The possible reason is that the reinforcement might dissipate the shock wave,resulting in the reduction of the reflected wave.This may also be caused by the angular crack formed by the reflection and convergence of the shock wave from the side and lower surfaces.

Fig.25. Deformed shape of beams B-1 and B-3.

Fig.27(a) and Fig.27(b) show obvious difference of crushing areas of B-1 and B-2.Generally,there are two reasons for crushing damage.When the reflected blast load on the upper surface is larger than the dynamic compressive strength,crushing will occur.In addition, with the development of bending deformation, the structural sectional curvature will exceed the elastic bearing capacity, both concrete crushing and reinforcement yielding will happen.Fig.27(a)and Fig.27(b)imply that the crushing zone sizes of simply supported beams are clearly larger than that of fixedfixed supported beams under identical blast loading.Concrete crushing resulted in the formation of inverted triangle hollow area on the upper surface of beam B-2,while beam B-1 showed a lower crushing damage degree with only a few concrete scabs falling off from the upper surface.Therefore,it is confirmed that the crushing damage of all beams were caused by local bending deformation.Compared with fixed supported beams, the simply supported beams are more likely to be crushed due to greater local bending deformation.

3.3.5.Synchronization

Fig.28 shows the rise time of peak displacements at five gauges of beams B-1 and B-3.The results showed good synchronization along the beam span direction.Therefore,it is reasonable to use the equivalent SDOF method for structural response calculation under near field explosion.

4.Validation and comparison

4.1.Validation with the test results in this work

Based on the non-uniform blast load models given in Table 4,the mid-span displacement histories are calculated by the modified SDOF model.Fig.29 compares the displacement histories of the experimental and calculated results.Table 5 summaries the maximum displacement (ym), residual displacement (yr) and rise time of maximum displacement (tm).(Note:are the deflection-span ratio ofymandyrrespectively.are the corresponding growth rate of ηm,ηrof simply supported beams relative to fixed supported beams.The subscript ‘s’ and ‘f’ identify the simply and fixed boundary conditions respectively.).

Table 4Blast load distribution function.

Fig.26. The deflection-span ratio: (a) Peak deflection-span ratio; (b) Residual deflection-span ratio.

Fig.27. Concrete crushing and spalling near the mid-span:(a)Crushing of B-1(Upper surface);(b)Crushing of B-2(Upper surface);(c)Spalling of B-1(Lower surface);(d)Spalling of B-2 (Lower surface).

It is evident that the calculated results of maximum displacement by the modified SDOF model agree well with the test data,with an average error of 8.25% and a maximum error of 26.3%.However,although the calculation error of maximum displacement by this work is acceptable, the modified SDOF model seriously overestimates the residual displacement by 111.9%-168.1%.The reason might be that the bending stiffness of structures will decrease with the developments of plastic deformation and failure[38], while the bending stiffness is assumed to be constant during the unloading and reloading process in this paper because no suitable stiffness degradation model was available.Additionally,the rise time of maximum displacement of fixed supported beams is underestimated by 39.6%and 51.7%in the cases of beams B-1 and B-3 respectively, yet that of the simply supported beams is only 7.4% and 13.2% for beams B-2 and B-4 respectively.It is supposed that the stiffness of fixed supported beams is overestimated,resulting in the increase of natural frequency and decrease of vibration period.

Fig.28. Rise time of maximum displacement.

4.2.Validation with the test results in other literatures

4.2.1.Uniform blast load scenarios

Although the research in this paper is carried out for nonuniform distributed blast load, in fact, uniform distribution is only a typical case of non-uniform distribution.In order to validate the adaptability of the modified SDOF model for uniform blast load scenarios, the experimental research conducted by Magnusson et al.[4]is introduced here.The main reasons for choosing this literature are as follows.Firstly, the uniform distributed blast load was generated through the shock tube to ensure the load uniformity.Secondly, the experimental time-displacement results had been compared with the theoretical results obtained by Ref.[3]through a modified version of traditional SDOF model considering the strain rate effect.

Fig.30 displays the sketches of explosion test.The test RC beam was simply supported with an effective span of 1.5 m.The cross section of the RC beam was rectangular.Longitudinal steel bars and stirrups were arranged in the specimen.The detailed structural and material parameters are listed in Table 6.Explosive was suspended in the shock tube 10 m away from the loaded surface of the test RC beams.The blast load and mid-span displacement histories were measured during the test.More details can be found in the literature.

Fig.29. Displacement time histories of test beams: (a) Beam B-1; (b) Beam B-2; (c) Beam B-3; (d) Beam B-4.

Table 5Comparison of test results and modified SDOF model.

Fig.30. Experimental set-up of uniform loaded RC beam [4].

Table 6Structural and material parameters [4].

Fig.31. Displacement histories comparison.

Fig.31 compares the mid-span displacement-time curves obtained by the test, Ref.[3]and the modified model in this paper.Compared with the test results, the calculation error of mid-span ultimate deflection by this work is about 1.7%.Additionally, it can be clearly seen that the two calculated curves almost coincide within 3.7 ms.This shows that in the application scenario of uniform distributed load, the modified SDOF model in this paper is almost consistent with Ref.[3], which represents the traditional SDOF method.It's a shame that the last part of time-displacement curve was not calculated by Ref.[3]because the material softening behavior was not considered.

Actually, it can be seen from the derivation of the modified model in this paper that it is fully applicable to the uniformly distributed load case.The reasons are as follows.Firstly,Eq.(2)and Eq.(7) imply that the weighted equivalent uniform peak pressure and specific impulse equal to the corresponding average values for uniform blast load,which are consistent with the traditional SDOF method.Secondly, Fig.7, Fig.9 and Fig.10 show that as the load becomes more uniform,the ultimate resistance and transformation factors(KM，KL)will gradually tend to the corresponding values for uniformly distributed load given in Ref.[1].Therefore, under uniform distributed load, the modified model in this paper would completely degenerate into the traditional SDOF model.

4.2.2.Non-uniform blast load scenarios

In order to further verify the rationality of this work in close-in explosion scenarios, the experimental research conducted in Ref.[39]is adopted here.The reasons for choosing this literature are as follows.Firstly, in the experiment, three pressure gauges were arranged to capture the non-uniform distributed blast load.The non-uniform blast load model was established under the test condition afterwards [40], and the application of the blast load model in predicting the structural failure [41]had fully proved its rationality.Secondly, the mid-span time-displacement histories were measured.

Fig.32. Scenario of blast tests in Ref.[39].

Fig.33. Non-uniform blast load in Ref.[40].

Table 7Structural and material parameters [39].

Fig.34. Displacement histories comparison.

Fig.32 shows the setup of the explosion test.The 7 kg rock emulsion explosive was suspended directly above the center of the simply supported RC beam with the height of 1.5 m.Three pressure gauges (P1, P2, P3) were located close to the RC beam to establish the non-uniform blast load model given in Fig.33.The detailed structural and material parameters are listed in Table 7.Fig.34 compares the mid-span displacement histories between test results and calculated results obtained by the modified SDOF model.It shows that the ultimate deflection and the corresponding rise time obtained by this work agree well with the test results,with the predicted error of 4% and 8.7% respectively.

4.3.Comparison with other modified SDOF models

Refs.[11,12]also proposed similar modified SDOF models for close-in explosion scenarios.Table 8 summarizes the differences of the modified versions of SDOF model, including the equivalent load, resistance and transformation factors.Fig.29 illustrates the comparison of the time-displacement curves calculated using different modified SDOF models and Table 9 summarizes the equivalent load and calculated errors.The results show that the modified SDOF model in Ref.[11]overestimates the maximum displacement except test No.3, and the average and maximum errors are 156.1% and 360.3% respectively.Otherwise, the modified model in Ref.[12]underestimates the maximum displacement by an average value of 38%.It also shows that all the predicted maximum displacements by this work are less than the values of Ref.[11]and greater than that of Ref.[12].

The difference of calculated results is mainly due to the difference of equivalent load, which is strongly related to the load enhancement factors (KpandKi).As the blast load becomes more non-uniform, bothKpandKiwould become larger.As a result of neglecting the load enhancement factors in Ref.[12],the equivalent load is less than that of this work by 19.8%-47.9%for peak pressure and 8.9%-41.1%for specific impulse respectively.Subsequently,the displacement response is underestimated because of the underestimated equivalent load.However, although the equivalent pressure in Ref.[11]was calculated considering the pressure enhancement factor, the maximum specific impulse was used directly as the equivalent uniform specific impulse which seriously overestimated the real equivalent specific impulse.Compared with this work,the equivalent specific impulses obtained by Ref.[11]are 212.5%,236.9%,46%and 57.6%higher for test No.1-4,respectively.These additional specific impulses would inevitably lead to more severe deformation.The above comparison demonstrates the need for reasonably calculating the equivalent load.

Flexural resistance is another important factor affecting the maximum displacement.Fig.35 displays the time-resistance histories calculated by different SDOF models.It is observed that the ultimate resistances obtained by Ref.[11]are 62.6%, 59.8%, 35.2%and 36.24% higher in test No.1-4 respectively because the influence of load uniformity on resistance was not considered.Nevertheless, because the overestimated degree of resistance is far less than that of equivalent specific impulse, the maximum displacements obtained by Ref.[11]are totally greater than that of thispaper.Additionally, Fig.35 shows that the ultimate resistance between Ref.[12]and this paper are very close, with a maximum deviation of 5.6%.The deviation is mainly caused by the difference of ultimate moment.The ultimate moment capacity in Ref.[12]was calculated by[1].

Table 8Comparison of the modified versions of SDOF model.

Table 9Comparisons of equivalent load and maximum displacements.

Fig.35. Comparison of resistance time histories: (a) Beam B-1; (b) Beam B-2; (c) Beam B-3; (d) Beam B-4.

where ρsis reinforcement ratio, σdyand σdcare the dynamic strength of reinforcement bars and concrete with a constant strain rate of 10/s respectively,banddare the width and effective height of beam section respectively.In contrast,the moment in this paper is obtained by sectional analysis method, which could consider complicated material behaviors (strain rate effect, hardening/softening and hoop-confined effect).In general, although the resistances between Ref.[12]and this work are close, deviations of the predicted maximum displacements still exist due to the difference of equivalent load.

Further, the load-mass transformation factorsKLMcould affect the calculation results by changing the equivalent mass,which was calculated byWe=KLMW.Table 10 compares the values ofKLMbetween this work and in Ref.[11]during elastic,elastic-plastic and plastic stages.In Ref.[11], the transformation factors for uniform load [1]were adopted, and the values remain unchanged for blast load with different uniformity.Instead, the transformation factors vary dynamically with load uniformity in this paper.As the blast load becomes more concentrated, the values ofKLMget lower.During the plastic stage, the values ofKLMunder Load-2 (3 kg,0.75 m/kg1/3)are 25.8%and 30.3%lower than Ref.[11]for beams B-3 and B-4 respectively, while that of the percentage under Load-1(1 kg, 0.5 m/kg1/3) increases to 39.4% and 40.9% for B-1 and B-2 respectively.This indicates that the influence of load uniformity onKLMshould not be ignored.

Table 10Comparisons of load-mass transformation factors.

Fig.36. Influence of strain rate effect on time-displacement histories.

4.4.Strain rate effect

Under short duration dynamic load, the strain rate has a significant effect on the material properties of structural members[42].To examine the effect of strain rate effect in the SDOF model,the mid-span time-displacement curves of beams B-2 and B-4 with and without strain rate effect are given in Fig.36.The calculation result illustrates that ignoring the strain rate effect would increase the bending deformation by 18.8% and 19.0% for B-2 and B-4,respectively.The reason is that stress of concrete and reinforcement will be amplified by the rate related dynamic increase factor(DIF),consequently resulting in the increase of resistance.

5.Conclusions

This paper proposed a modified SDOF model for the flexural response analysis of RC beams under close-in explosion.The accuracy of the modified model was validated by explosion tests.The conclusions are drawn as follows:

(1) The spatial distribution of peak pressure and specific impulse on the loaded surface of RC beam is highly non-uniform under close-in explosion, which can be well described by the exponential decay function.The shape of explosive is a non-negligible factor affecting the characteristics of load distribution.Empirical algorithms of spherical explosives could not represent the real blast scenarios of cylindrical explosives.

(2) The modified SDOF model adopted a new specific impulse equivalent method based on the local conservation of momentum and global conservation of kinetic energy.Both the equivalent load and test results imply that the blast load near the mid-span would contribute more to the structural deformation.

(3) The modified SDOF model established a novel process to obtain the flexural resistance function of simply and fixed supported RC beams considering the load uniformity, dynamic strain rate effect, material hardening/softening and hoop-confined effect.The resistance model indicates that when the blast load becomes more concentrated, the ultimate resistance would become lower, and compressive concrete would be more prone to softening and crushing.

(4) The accuracy of the modified SDOF model is validated by conducting close-in explosion test.The calculated results of maximum displacement by the modified SDOF model agree well with the test data, with a maximum predicted error of 26.3%and an average value of 8.25%.However,the rise time to the peak displacements of fixed supported RC beams is severely underestimated by 39.2%-52.7%, which might be due to the overestimation of bending stiffness.

(5) The adaptability of the proposed model is validated through the test data in other literatures, including the uniform and non-uniform blast load scenarios.

In general, the proposed SDOF model in this paper fully proves the rationality of the modified SDOF method in the close-in explosion scenarios, which will not only largely broaden the application scenes of SDOF method but contribute to a more accurate damage assessment of one-way RC structures (beam, column and plate) under close-in explosion.Subsequently, the Pressure-Impulse (P-I) curves of RC structures under close-in explosion can be obtained based on the modified model.

There are still deviations between the calculated results and the test results.The possible reasons for the deviations are as follows.At first, it is difficult to measure the close-in blast load accurately under the influence of detonation products,thus the measurement error and the subsequent fitting error of blast load might lead to inaccurate results.Ref.[15]has guiding significance for accurately obtaining close-in explosion load.Secondly, the calculation processes of resistance and transformation factors only consider the spatial distribution of peak pressure.Ignoring the specific impulse distribution is a compromise because resistance and transformation factors are calculated based on static analysis.In spite of this,due to the close trend of spatial distributions of peak pressure and specific impulse referring to Fig.24, acceptable results are obtained in this work.Third, spallation may decrease the local bending stiffness and sectional ultimate bending moment,which is not considered in this paper.With the further reduction of the scaled distance,the structural failure will be dominated by spalling and punching [37].At this time, the modified SDOF model will no longer be applicable.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to acknowledge National Natural Science Foundation of China (Grant No.12102337) to provide fund for conducting experiments.In addition,the first author would like to thank his best friend Dr.Gao Zhao for his advices on paper writing.

Appendix A.Calculation process of MR for fixed supported beams

The right boundary constraint is released firstly, as shown in Fig.A1.

Fig.A1. Boundary released beam.

Then, the summary of right support rotations caused by the external loadp（x）, the right support reactionFRand the bending momentMRshould equal to zero:

The calculation process of φMRinclude the following steps:

Step 1: calculate the support bending moment

Step 2: calculate the support reaction:

Step 3: calculate the shear force distribution:

Step 4: calculate the bending moment distribution

Step 5: calculate the curvature distribution

Step 6: calculate the rotation angle distribution by integrating the curvature

Substitute Eq.(A2), Eq.(A3), Eq.(A10) into Eq.(A1),MRwill be obtained.