An analytical model of linear density foam protected structure under blast loading

Transcription

1 721543PRS / International Journal of Protective StructureXia et al. reearch-article2017 Article An analytical model of linear denity foam protected tructure under blat loading International Journal of Protective Structure 2017, Vol. 8(3) The Author() 2017 Reprint and permiion: agepub.co.uk/journalpermiion.nav DOI: journal.agepub.com/home/pr Ye Xia 1, Chengqing Wu 2 and Terry Bennett 1 Abtract Aluminium foam i widely known a an energy aborptive material which can be ued a a protective cladding on tructure to enhance blat reitance of the protected tructure. Previou tudie how that higher denity provide larger energy aborption capacity of aluminium foam, but reult in a larger tranmitted preure to the protected tructure. To lower the tranmitted preure without acrificing the maximum energy aborption, graded denity foam ha been examined in thi tudy. An analytical model i developed in thi article to invetigate the protective effect of linear denity foam on repone of a tructure under blat loading. The model i able to imulate tructural deformation with reaonable accuracy compared with experimental data. The enitivity of denity gradient of foam cladding on reinforced concrete tructure i teted in the article. Keyword Linear denity, graded foam, blat loading Introduction A a protective material, aluminium foam cladding are able to mitigate blat effect on the protected tructure by aborbing a ignificant amount of kinematic energy (Hanen et al., 2002; Schenker et al., 2005; Wu et al., 2010; Ye and Ma, 2007). The outtanding energy aborption capacity of aluminium foam i contributed by the cellular tructure with ga-filled pore which provide a plateau of tre after deformation. The volume proportion determine the denity and trength of aluminium foam which directly affect the protective performance. When blat load i applied, the foam cladding will aborb a ignificant amount of energy and tranfer the intenive load into a much lower preure with a longer duration onto the protected tructure. Theoretically, the plateau tre of the foam i equal to the preure tranmitted from the foam onto the target tructure (Li and Meng, 2002). Since the plateau tre exhibit a power law function with denity 1 School of Civil, Environmental & Mining Engineering, The Univerity of Adelaide, Adelaide, SA, Autralia 2 Centre for Built Infratructure Reearch, School of Civil and Environmental Engineering, Univerity of Technology Sydney, Sydney, NSW, Autralia Correponding author: Chengqing Wu, School of Civil and Environmental Engineering, Univerity of Technology Sydney, Sydney, NSW, Autralia. chengqing.wu@ut.edu.au

2 Xia et al. 455 (Ahby et al., 2000; Hanen et al., 2002), the denity of the aluminium foam i one of the key propertie when deigning protective cladding. A demontrated by the natural denity-graded bone and wood, grading the denity of foam cladding may be an effective method to increae the protective effectivene and efficiency. According to everal numerical tudie in regard to double, triple and quadruple layer of aluminium foam (Chang et al., 2013; Li et al., 2011; Ma and Ye, 2007), it appeared that under dynamic loading, denity-graded foam with decreaing denity from loaded end to protected end outperform homogeneou foam with the ame average denity. When dynamic loading i applied, the low-denity layer at the bottom, which i in direct contact with the protected tructure, will tranmit a relatively lower preure to the tructure compared with homogeneou foam with the ame average denity. A a reult, the deformation of the protected tructure againt blat loading can be maller. Technique have been developed to manufacture continuouly graded denity aluminium foam (Brother and Dunand, 2006; Haani et al., 2012; He et al., 2014; Higuchi et al., 2012), which allow experiment to be undertaken. Similar to the finding from the aforementioned numerical tudie, experimental reult howed that the denity gradient increae the lope of the plateau tre range of the foam (Beal and Thompon, 1997; Brother and Dunand, 2008; Xia et al., 2016). Thi mean that the tranmitted preure onto the protected tructure will depend on how the graded denity foam deform, rather than a contant plateau tre of homogeneou foam. For example, if the blat load i relatively mall and only the lower denity part of the foam i compacted, the tranmitted preure will be the trength of the lower denity foam, which i maller than the plateau tre of the homogeneou foam with the ame average denity. However, when the blat load i high enough to fully compact the entire graded denity foam, the tranmitted preure will depend on the foam denity which i denified finally. Reearcher have found that graded denity foam will alway deform from the loaded end gradually through the thickne regardle the denity ditribution if the loading velocity i large enough (Li et al., 2011, 2012; Shen et al., 2013). Hence, theoretically, decreaing the foam denity from loaded end to protected end will improve the protective effectivene of foam cladding. Furthermore, a erie of blat tet conducted by Xia et al. (2016) alo indicate that the foam with decreaing denity from loaded end protect the reinforced concrete (RC) lab better than uniform denity foam. Analytical model i a convenient and economical method in invetigating the protective effect of aluminium foam againt blat loading. The one-dimenional hock wave theory ha been verified numerically and experimentally in the analyi of aluminium foam under blat loading (Hanen et al., 2002; Li and Meng, 2002; Reid and Peng, 1997; Xia et al., 2014a). Baed on the hock wave theory and ingle degree of freedom theory, loading-cladding-tructure model have been developed to predict the repone of tructural member with homogeneou foam cladding (Xia et al., 2014a; Ye and Ma, 2007). Recently, a theoretical model of continuou graded denity foam under blat loading wa etablihed by Zhou et al. (2015). In thi model, the foam denity wa deigned a a decreaing exponential function and the tudy conclude that the denity gradient make foam denify lower (Zhou et al., 2015). However, no foam tructure interaction wa conidered in thi theoretical model. In the current reearch, an analytical model, which contain a cover plate, a linear denity foam cladding and a protected tructure, i developed to invetigate the protective effect of linear denity foam on the tructure under blat loading. The propoed analytical model i verified by blat experimental reult conducted by Xia et al. (2016). Moreover, parametric tudie are carried out to tet influence of different parameter of linear denity foam on reducing the peak deflection of protected RC lab. In addition, two different lab are ued in the parametric tudy to tet the protective effect of linear denity foam on different tructure.

3 456 International Journal of Protective Structure 8(3) Figure 1. Graded denity foam protected RC model under blat loading. Derivation of analytical model The blat preure conidered in the model i implified a a triangular pule and uniformly act on the cover plate of the foam cladding, which i expreed by equation (1) P t ()= P t t d t t t > t d d (1) where P 0 i the initial peak preure of the blat and t d i the duration of the blat load. For linear denity foam, the denity can be calculated with the ditance x along the thickne ρ0 ρl ρ( x)= ρ0 x L (2) where ρ 0 i the initial denity contacted with cover plate, ρ L i the final denity attached on the tructure and L i the thickne of the foam cladding. The analytical model in thi article i developed baed on hock wave theory (Reid and Peng, 1997) which aume that the foam deform gradually from the loaded end through the croectional direction under blat load a illutrated in Figure 1. The foam i idealied a a rigid, perfectly platic locking (RPPL) material with a plateau tre σ pl and a denification tre σ D a hown in Figure 1. The plateau tre and denification tre of graded denity foam are written a function of foam denity which i calculated by the location of hock front

4 Xia et al. 457 σ pl 2 ρ 3 = C1 σ ρ y (3) σ D ρ = σ pl + ( u y ) 2 (4) ε where ρ i the denity of the bae material of the foam, C 1 i a material contant, σ y i the compreive yield trength of the bae material and u i the velocity of cover plate which move together with the loaded face of the foam cladding. Cover plate i commonly ued in the application of protective foam cladding to ditribute the blat preure uniformly on the foam and avoid diintegrating; y i the velocity of the protected tructure; the denification train of the foam ε D i expreed a D ε D = 1 λ ρ (5) ρ where λ i a material contant. The ma of the denified part of the foam cladding can be calculated by the product of denified foam denity and denified foam volume ( ) m= ρ * A u+ y (6) where A i the loading area of the foam cladding which i equal to the loading area of the RC tructure, i the progreive location of the hock front in the foam from the loaded end, u i the diplacement of the cover plate which i the ame a the diplacement of the denified foam, y i the diplacement of protected tructure and the denity of the fully denified foam ρ * i written a ρ ρ* = ε 1 D ρ = (7) λ When conidering the denity of the originally undeformed foam ρ( x ) and the location of hock front, the ma of the denified part of the foam cladding alo can be calculated by the following equation due to the ma conervation m= A ρ( x) dx = ρ A 0 0 ρ ρl A 2L 0 2 (8) Equating equation (6) and (8), the diplacement of the cover plate can be acertained in a function of the hock front location λρ λ ρ ρl u = + y + L ρ ρ (9) With initial condition u 0 = 0 and u 0 = 0, the velocity and acceleration of the cover plate can be obtained by differentiating equation (9)

5 458 International Journal of Protective Structure 8(3) λρ λ ρ ρ 0 0 L u = + y + ρ ρ L ( ) (10) ( ) λ ρ0 ρl u = + y + (11) ρ L The ytem of the denified foam with cover plate in Figure 1 i conidered a a ingle degree of freedom ytem where damping i ignored. The dynamic equation of motion i written a ( ) + () ml + m u σ D P t A= 0 (12) where m l i the ma of cover plate. Subtituting equation (2) to (4) and (8) to (11), into equation (12), the dynamic equation of equilibrium can be written a ( ) ρ ρl λ ρ ρ L ml A A y L + ρ ρl + ρ ( ) C1 ρ 2 3 σ y ( ) ( ) ρ + ρ 1 λ ρ ( ) λ ρ ρl + ρl 2 λρ ρ () P t A = (13) The motion equation of the ytem of the undeformed foam with protected tructure (a hown in Figure 1) can be expreed a ( mf m+ me) y+ R( y) σ pl A = 0 (14) where m f i the ma of foam cladding, m e i the equivalent ma of the protected tructure and Ry ( ) i the reitance function of the tructure. Taking the equation (3) and (8) into account, equation (14) can be rewritten a m f ρ ρ + 2L 0 L 2 ρ 3 A ρ A + me 0 y+ R( y) C1 σ y ρ A = 0 (15) 2 From the imultaneou nonlinear differential equation (13) and (15), the deflection of the protected tructure y can be olved by uing the fourth-order Runge Kutta method with initial condition 0 = 0, y0 = 0, 0 = 0, y 0 = 0. The current analytical model i only applicable when atifying the Shock Wave Theory. Reearcher have proved that ufficient impact velocity will make the graded denity foam denify progreively from the loaded end (dener end) while keeping the other part of foam undeformed (Li et al., 2011, 2012; Shen et al., 2013). Therefore, an important aumption of the analytical model in the preent article i that the blat load i large enough to deform the foam from the loaded end progreively toward the tructure.

6 Xia et al. 459 Figure 2. Setup of the blat tet. When the foam cladding i totally denified along the thickne, the denified foam will move in union with the protected tructure. Therefore, the covered foam and the protected tructure can now be een a a ingle degree of freedom ytem ( ml + mf + me ) y+ R( y) P() t A = 0 (16) Equation (13), (15) and (16) form an analytical model which can theoretically calculate the deformation of linear denity foam cladding and protected tructure. The running cot of thi model i low and uitable for huge amount of parametric tudie on protective effect of linear denity foam. Thi propoed analytical model i applicable for different type of foam and different target tructure with different reitance function Ry ( ). In the following ection of thi article, RC will be conidered a the protected tructure to invetigate the protective effect of linear denity aluminium foam on RC lab under blat loading. The material contant of aluminium foam are taken a C 1 = 03. and λ = 15. (Ahby et al., 2000). Verification of analytical model The analytical model i verified by the full-cale blat tet conducted by Xia et al. (2016). In the blat tet, the blat load and the protected RC lab in all event are the ame. The 8-kg TNT (trinitrotoluene) charge ued in the tet are cylindrical with a diameter of 225 mm and a height of 120 mm. All charge were placed vertically on the top of the RC lab a hown in Figure 2. The tandoff ditance from the charge to the centre of the RC lab i 1.5 m. All the protected RC lab are 2000 mm long, 800 mm wide and 120 mm thick and more detail can be referred in Xia et al. (2016). Every aluminium foam teted in the experiment wa covered by a 1.13-mm-thick teel cover plate and wa glued on the RC lab by epoxy. Three major element are required by the analytical model to verify it prediction: (1) preure time hitory of the blat load, (2) reitance deflection function of the protected RC lab and (3) denity ditribution of the aluminium foam. Simulation of the blat load Due to the fact that the preure time hitory during the blat event (Xia et al., 2016) wa not uccefully recorded, the preure time hitory applied on the analytical model i therefore calculated by uing LS-DYNA.

7 460 International Journal of Protective Structure 8(3) Figure 3. ALE model employed for determination of blat preure. In LS-DYNA, the Arbitrary Lagrangian Eulerian (ALE) Method i ued to accurately model the wave tructure interaction. It i previouly mentioned that a cylindrical exploive wa ued during the experiment; however, the current built-in ConWep function in LS-DYNA are only able to account for pherical or hemipherical exploive. Therefore, the TNT exploive and the urrounding air domain are both explicitly modelled in the ubequent numerical imulation. A hown in Figure 3, hexahedral element are ued for all numerical component to achieve the bet computational tability and accuracy. The air meh extend approximately 10 cm beyond the lab edge and bottom. To maximie the efficiency, an eighth-ymmetry model i ued by impoing appropriate boundary condition to the node lying on the x y, y z and x z plane which interect at the exploive centre while nonreflecting boundary condition are forced on the remaining exterior urface of the air domain. To output the preure time hitory, a preure tranducer i placed at the centre of the lab. The contitutive model *MAT_HIGH_EXPLOSIVE_BURN i ued to model TNT exploive and the equation of tate i *EOS_JWL. Standard input parameter (Kingery and Bulmah, 1984) for the contitutive model and equation of tate are ued and the detonation point i aumed at the centroid of the exploive. The ambient air domain i modelled by *MAT_NULL with the denity of air being the ole material input. The equation of tate i *EOS_LINEAR_POLYNOMIAL. In the current tudy, the only purpoe of the numerical imulation i to obtain the reflected preure time hitory for the propoed analytical model, the repone/behaviour of the tructure itelf in LS-DYNA i not of interet. Therefore, the lab i modelled a a rigid body with *MAT_ RIGID to minimie the computational effort. The imulated preure time hitory on the central point of the lab i preented in Figure 4. To apply to the analytical model, the preure time hitory i idealied into a triangular load a hown by the dahed line in Figure 4. The idealied preure ha a peak reflected preure of 18 MPa to match the imulated peak preure. A duration of 0.6 m i adopted to match the impule neglecting

8 Xia et al. 461 Figure 4. Preure time hitory in the blat tet. the low-preure tail. In the idealiation, the extenive low-preure range after i ignored o that the idealied impule i lightly maller than the imulated preure. The idealied preure will be uniformly applied on the lab in the analytical model wherea the actual preure will experience clearing, therefore, the influence of the neglecting the low-preure tail will compenate the overetimated preure on the ide of the lab. Reitance function of the RC lab According to Xia et al. (2016), the bilinear reitance deflection diagram (Figure 5) of the RC lab ued in the blat tet i calculated by R yield M yield = 8 (17) L lab R y ult yield M ult = 8 (18) L lab Ryield = (19) K y ult L = θ 2 lab (20) where M yield and M ult are the yield and ultimate moment of the RC lab, and dynamic increae factor are conidered in the calculation baed on ASCE Guideline; L lab i the length of the RC lab; K i the tiffne of the RC lab calculated by K = 384EI / 5L 3 lab ; θ i the ultimate rotation of the RC lab taken a 2 in thi tudy. Aluminium foam in the blat tet Three aluminium foam cladding in the blat tet (Xia et al., 2016) are referred in the current tudy: one uniform denity foam UD300 and two linear denity foam LD300 and LD400. Due to the

9 462 International Journal of Protective Structure 8(3) Figure 5. Reitance deflection diagram of RC lab in the blat tet. Table 1. Denity ditribution of aluminium foam. Foam type Initial denity, ρ 0 (kg/m 3 ) Final denity, ρ L (kg/m 3 ) Average denity, (kg/m 3 ) Denity gradient, (kg/m 3 /mm) UD LD LD manufacturing proce, the denity cannot be perfectly controlled a per the deigned value. For a clear comparion, the approximate denity ditribution introduced in Xia et al. (2016) will be ued in the analytical model a lited in Table 1. Figure 6 how the configuration of aluminium foam before and after exploive in the blat tet. Reult comparion Four event of the blat tet (Xia et al., 2016) are imulated by the analytical model. Table 2 ummarie the reult from the experiment and the analytical model. The firt event in the blat tet i an unprotected RC lab under the exploive loading. The deflection time hitory i calculated by a baic ingle degree of freedom model with the reitance function hown in Figure 5. The comparion with experimental reult in Figure 7 how a good agreement until the peak deflection i reached. Thi indicate that the idealied preure time hitory i reaonable. After the peak deflection, the predicted curve keep at the high level becaue damping effect i not conidered in the model. The analytical model introduced in thi article i verified by comparing the deflection time diagram of RC lab with experimental reult where foam UD300, LD300 and LD400 were employed, repectively, in the tet (Xia et al., 2016). The uniform denity foam UD300 i modelled by etting the initial denity equal to the final denity of the foam. The model yield an error of 8.3% on peak deflection (Figure 8) which i acceptable in blat analyi. For linear denity foam LD300 and LD400, the predicted reult are hown in Figure 9 and 10. The imulated peak deflection and period of the protected lab are matched; however, the imulated deflection after

10 Xia et al. 463 Figure 6. Configuration of foam in blat tet. Table 2. Experimental and imulated reult of peak deflection. Foam type Peak deflection of RC lab (cm) Difference % Experiment Model Unprotected lab UD LD LD peak are not reliable due to ignoring the damping effect. Therefore, the current model i only applicable in the calculation of peak deflection. It wa oberved that the linear denity foam did not how any advantage in the experiment (Xia et al., 2016). Thi i becaue the blat load wa too intenive and caued all of the RC lab to fail with ignificant deformation, o that the actual foam effectivene i hard to diplay. Therefore, in the next ection, a lower blat load i applied to thi verified model to invetigate the protective improvement of linear denity foam. Parametric tudie Parametric tudie are carried out to invetigate the influence of different parameter on the protective effect of linear denity foam. A blat load of a peak preure of 12 MPa and duration of 0.3 m

11 464 International Journal of Protective Structure 8(3) Figure 7. Deflection time diagram of the unprotected RC lab. Figure 8. Deflection time diagram of the UD300-protected RC lab. i applied to all cenario hereafter. RC lab ued in thi ection are the ame a the one ued during the blat experiment (Xia et al., 2016), and all foam are 80 mm thick unle pecified. Influence of denity gradient with the ame average denity Three foam with ame average denity of 300 kg/m 3, but different denity gradient (0, 2.5 and 5.0 kg/m 3 /mm, repectively) are teted by uing the propoed analytical model. The detail of the foam are lited in Table 3. A hown in Figure 11 and 12, the homogeneou foam reduce the peak deflection of RC lab by 59% while the linear denity foam LD300G2.5 mitigate a further 11.7% on the peak deflection. With an increae of denity gradient, linear denity foam LD300G5.0 provide an even better protection with a 75% reduction on the peak deflection. The reult indicate that the improvement on the protective effect of uing denity-graded foam i evident, epecially when the blat load i large enough to compact the foam from loading end.

12 Xia et al. 465 Figure 9. Deflection time diagram of the LD300-protected RC lab. Figure 10. Deflection time diagram of the LD400-protected RC lab. Table 3. Detail of foam with the ame average denity but different denity gradient. Foam type Initial denity (kg/m 3 ) Final denity (kg/m 3 ) Denity gradient (kg/m 3 /mm) UD LD300G LD300G Influence of denity gradient with the ame final denity Once the final-layer denity of the foam (the end that i in direct contact with the protected tructure) i kept contant, raiing the denity gradient increae the average denity of the foam and therefore, the energy aborption capacity of the foam can be enhanced.

13 466 International Journal of Protective Structure 8(3) Figure 11. Deflection time diagram of RC lab protected by foam with the ame average denity but different denity gradient. Figure 12. Peak deflection of RC lab protected by foam with different denity gradient and reponding percentage of reduction. Table 4. Detail of foam with the ame final denity but different denity gradient. Foam type Initial denity (kg/m 3 ) Final denity (kg/m 3 ) Denity gradient (kg/m 3 /mm) Average denity (kg/m 3) UD FD200G FD200G FD200G Four foam with the ame final-layer denity but different denity gradient are tudied a hown in Table 4. The equal final-layer denity mean the tranmitted preure on the tructure

14 Xia et al. 467 Figure 13. Deflection time diagram of RC lab protected by foam with the ame final denity but different denity gradient. Table 5. Detail of foam with different thicknee. Foam type Thickne (mm) Initial denity (kg/m 3 ) Final denity (kg/m 3 ) Denity gradient (kg/m 3 /mm) LD300T LD300T LD300T will be the ame when the foam are fully denified. Thi i becaue theoretically, the tranmitted preure i equal to the plateau tre of the foam that i in direct contact with the protected tructure. However, for a given blat load, the denification tatu of different foam can be different o that the actual tranmitted preure on the protected tructure can be different. Figure 13 how the deflection time diagram of the RC lab protected by the foam with the ame finallayer denity but different denity gradient a introduced in Table 4. Thoe reult indicate that increaing the denity gradient from 2.5 to 3.75 kg/m 3 /mm improve the mitigation effect. However, the further growth of denity gradient from 2.5 to 5.0 kg/m 3 /mm amplifie the peak deflection of the protected RC lab. Thi i becaue the FD200G5.0 foam ha a larger average denity o that it deform le compared to other type of foam under the ame given blat load. Baed on the reult, it can be concluded that when the final-layer denity i contant, the effect of increaing denity gradient of the linear denity foam i dependent on the intenity of the blat load which determine the extent of the foam deformation. Influence of thickne with the ame denity ditribution The effect of foam thickne i examined by uing three foam of the ame denity arrangement a ummaried in Table 5. Since the three foam have the ame initial- and final-layer denitie but different thicknee, the thicker foam therefore ha a lower denity gradient. From Figure 14, it

15 468 International Journal of Protective Structure 8(3) Figure 14. Deflection time diagram of RC lab protected by foam with the ame denity gradient, but different thicknee. can be een that the thinnet foam LD300T60 mitigate the peak deflection of the RC lab mot effectively. The other two thicker foam LD300T80 and LD300T100 reult imilar peak deflection, which indicate that they were not fully denified under the given blat load. Therefore, the thinner foam LD300T60 i more effective and more efficient in thi cae ince it denifie more and tranmit a relatively lower preure to the protected RC lab. Influence of average denity with the ame denity gradient The influence of average denity of linear denity foam i invetigated on two different RC lab. Slab I i the RC lab ued during the blat experiment (Xia et al., 2016) and Slab II ha a thickne of 200 mm which i 80 mm thicker than Slab I. All other configuration are exactly the ame. The reitance deflection curve of both lab are plotted in Figure 15, which are ued in the following parametric tudie. Table 6 lit the detail of foam with different average denitie, and the protective effect of thee foam on two type of lab are hown in Figure 16. For a contant denity gradient of 2.5 kg/m 3 /mm, increaing the foam denity from 300 kg/m 3 to 400 kg/m 3 and 500 kg/m 3 increae the peak deflection of the Slab I under the given blat load. Thi i becaue the foam LD400G2.5 and LD500G2.5 are too trong and the tranmitted preure are too large for Slab I. On the contrary, for the tronger tructure Slab II, increaing foam denity up to 500 kg/m 3 can actually improve the mitigation effect a Slab II i capable of withtanding higher tranmitted load. Further increaing the foam denity to LG600G2.5 lead to an oppoite effect ince the tranmitted load of LG600G2.5 exceed the blat reitance of Slab II. Thee outcome agreed with the finding from the previou tudie on homogeneou foam (Xia et al., 2014a, 2014b), which uggeted that foam acted differently on different RC lab and the foam denity hould be choen baed on the reitance of the lab to achieve the optimal protection. Within the trial in thi tudy, LG300G2.5 i cloer to the optimal protection for Slab I while LG500G2.5 i the cloet foam to the optimal deign for Slab II. Therefore, the average denity of linear denity foam i an important parameter to find the optimal deign for RC lab.

16 Xia et al. 469 Figure 15. Reitance deflection diagram of Slab I and Slab II. Table 6. Detail of foam with the ame denity gradient, but different average denitie. Foam type Initial denity (kg/m 3 ) Final denity (kg/m 3 ) Average denity (kg/m 3 ) Denity gradient (kg/m 3 /mm) LD300G LD400G LD500G LD600G Figure 16. Deflection time diagram of Slab I and Slab II protected by foam with the ame denity gradient, but different average denitie. Comparion of uing foam cladding and thickening the lab Figure 17 how the comparion of the protective effect of uing foam cladding and thickening the RC lab. The olid line i the deflection time curve of the unprotected Slab II, and the dahed line preent the deflection time curve of a Slab I protected by a 80-mm foam of LD300G2.5, which ha

17 470 International Journal of Protective Structure 8(3) Figure 17. Deflection time diagram of foam-protected Slab I and unprotected Slab II. the ame total thickne a Slab II. It can be een that Slab I with foam cladding outperform the thicker Slab II without foam cladding under the ame blat load. Thu, in thi cae, uing foam cladding provide a better blat load bearing protection than imply thickening the lab. Thi finding i conitent with the tudy on homogeneou foam in Xia et al. (2014b). It hould be noticed that uing a foam cladding i not alway more effective than thickening the lab. Baed on the reult from Xia et al. (2014b), when the peak blat preure i lower than a certain value, the extra lab thickne can perform better. Therefore, the proper foam hould be deigned baed on the expected blat load. Concluion An analytical model i developed to predict the protective effect of linear denity foam on RC lab under blat loading. The model i verified againt ome experimental reult and yield acceptable accuracy. A range of parametric tudie that decribe the enitivitie of different parameter of linear denity foam are alo carried out. The reult ugget an improved protective ability of linear denity foam over uniform denity foam which mean linear denity foam i capable of mitigating the repone of the RC tructural member more effectively under tronger exploion: 1. For a given average denity, the increae of denity gradient further improve the performance on mitigating the peak deflection of the protected RC lab. 2. When the final-layer denity (protecting end) of the linear denity foam i given, larger denity gradient provide better energy aborption ability. However, for a given blat load and RC lab, enlarging the denity gradient can increae the average denity, which can reduce the protective effect on the lab. 3. The effect of average denity of linear denity foam on RC lab i imilar to that of homogeneou foam: dener foam can protect tronger RC lab better and vice vera. 4. Uing a linear foam cladding on a thinner lab may outperform a thicker lab of the ame total thickne in ome circumtance. The finding from thi article are valuable for future tudie on optimiation of linear denity foam deign for RC lab under blat loading.

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