Accepted Manuscript The Similarity Law and Its Verification of Cylindrical Lattice Shell Model under Internal Explosion Shiqi Fu , Xuanneng Gao , Xin Chen PII: DOI: Reference: S0734-743X(18)30232-X https://doi.org/10.1016/j.ijimpeng.2018.08.010 IE 3155 To appear in: International Journal of Impact Engineering Received date: Revised date: Accepted date: 14 March 2018 18 August 2018 19 August 2018 Please cite this article as: Shiqi Fu , Xuanneng Gao , Xin Chen , The Similarity Law and Its Verification of Cylindrical Lattice Shell Model under Internal Explosion, International Journal of Impact Engineering (2018), doi: https://doi.org/10.1016/j.ijimpeng.2018.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights 1 2 4 ? The similarity law of the cylindrical lattice shell structure under internal explosion is deduced based on the dimensional analysis principle. CR IP T 3 5 ? It is verified that the scale model test of the cylindrical lattice shell structure 7 under internal explosion could accurately reflect the characteristics of the 8 original model. AN US 6 9 ? The reasonable and reliable scaling coefficients are proposed for the scale 11 model tests of the cylindrical lattice shell structure under internal explosion. M 10 AC CE PT ED 12 1 ACCEPTED MANUSCRIPT 2 The Similarity Law and Its Verification of Cylindrical Lattice Shell Model under Internal Explosion 3 Shiqi Fu a, Xuanneng Gao a,*, Xin Chen a,b 4 a. College of Civil Engineering, Huaqiao University, Xiamen, Fujian 361021, PR China 5 6 7 b. College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, PR China 8 Abstract: It is very important to ensure that the results of a scale model experiment of the large-span space steel 9 structure under internal explosion could accurately show the characteristics of its original model. For this purpose, 10 a one-way inclined bar type of single layer cylindrical lattice shell was taken as an example in this paper, and the 11 similarity law between the scale models and the original model under the internal explosion was deduced by the 12 dimension analysis principle. Then, the original model and the scale models under internal explosions were 13 simulated by ANSYS/LS-DYNA, and the correctness and the feasibility of the similarity law were compared and 14 confirmed. The research results showed that: if the scale models conform to the similarity law, the propagation 15 law and the overpressure distribution of shock waves to the scale models subjected to internal explosions were 16 consistent with the original model; the phenomena of critical destruction of the scale models were similar to the 17 experimental results. But the accuracy of the loading effects of blast (such as the overpressure peak of shock 18 waves, the positive pressure time and the impulse etc.) and the structural dynamic responses obtained by the scale 19 models was affected by the geometric incomplete similarity of the scale models and the strain rate effect of steel 20 respectively, which would reduce with the decrease of the scaling coefficient. Especially, the research results also 21 indicated that if the scaling coefficient was not less than 0.1, the maximum deviations of the overpressure peak of 22 shock waves, the positive pressure time after conversion and the specific impulse after conversion would be 10%, 23 4.78% and 29.8%, respectively, and if the scaling coefficient was less than 0.25, the stress responses and 24 displacement responses of the scale model under internal explosion would not be reliable. Based on the research 25 results, the reasonable scaling coefficients to the scale model experiment of cylindrical lattice shell under internal 26 explosion are proposed. 27 Key words: Internal Explosion, Similarity Law, Cylindrical Lattice Shell, Dimensional Analysis, Scale Model AC CE PT ED M AN US CR IP T 1 * Corresponding author. Xuanneng Gao E-mail: [email protected] Tel: (+86) 13599933263. ORCID ID:0000-0002-7905-7418. 2 ACCEPTED MANUSCRIPT 1 1 Introduction It is well known that irreversible casualties, property damage and serious social panic would 3 occur if a terrorist attack happens in a large span space structure that is always built as a city 4 landmark. With the increase of terrorist attacks in recent years, the research of explosion prevention 5 and resistance (especially the experimental research) for protecting large-span space structure has 6 been paid increasingly attention [1-2]. Ishikawa and Beppu [3] summarized three explosion tests of 7 building envelope that were presided over by the late Professor A. Johoji, they considered that the 8 impact of blast should be taken into account in the design of building envelope. Remennikov and 9 Uy [4] carried out the explosion test of hollow square steel tubes to study the dynamic responses of 10 the ones in contact explosion and close explosion. Their study provided the experimental data for 11 theoretical analysis and verification of the numerical simulation results. Dey and Nimje [5] adopted 12 the method of combining experiment with numerical simulation to study the dynamic responses of 13 metal sandwich plate subjected to explosive load. The results showed that the perfect coincidence 14 between the experimental and numerical simulation. If the detonation distance was 0.5m and the 15 mass of TNT was 8kg, the metal sandwich panel could consume the most explosive energy. 16 Spranghers et al. [6] conducted the scale tests and verified the correctness of aluminum plate model 17 as well parameter selection under explosive loading. Li et al. [7] carried out sixteen original models 18 to study explosion venting performance of non-reinforced brick masonry wall under internal 19 explosion. The results showed that the dynamic responses of masonry wall depended on the 20 overpressure peak of shock waves and the failure mode depended on typical yield curve. In 21 summary, researchers had focused on components or small-size concrete structures on the 22 experimental research of the explosion prevention and resistance, but very little research AC CE PT ED M AN US CR IP T 2 3 ACCEPTED MANUSCRIPT 1 concentrated on the large-span space structures. Thus, there is a great significance to carry out the 2 model test of the large-span space structure under internal explosion. Due to the actual constraints, the scale models are justifiably chosen by the model test of large 4 span space structure under internal explosion. The similar relationship between the scale models 5 and the original model should be conventionally defined before the the scale model test, namely the 6 similarity law of the model under internal explosion should be determined, so that the test results 7 could really reflect the performances of the original model. Dimensional analysis is an appropriate 8 method for solving this problem. It is widely used in engineering tests, which is theoretically 9 applicable to all-type structures under any load [8-9]. For example, Coutinho et al. [10] reviewed 10 the similarity theory from rise to perfection and its application in major research areas, and pointed 11 out the further development direction; Bakhtar et al. [11] established a series of tunnel scale models 12 based on the data of KLOTZ tunnel explosion test, and provided the basis for the explosion test of 13 underground structure model with comparing original model and scale model test results; Li and 14 Zhang [12] employed the dimensional analysis to sum up the similarity law of the model under the 15 impact load of bird. The results showed that the bird collision process followed the model similarity 16 law, and the strain rate effect would affect the similarity law of model; Yao and other scholars [13] 17 built three groups of steel box models according to the geometric similarity law and implemented 18 corresponding internal explosion experiments. Then, the reason of the results deviation to 19 traditional scale test was discussed and the size effect and the strain rate effect were considered on 20 the basis of test results account to amend the similarity law of steel box model under internal 21 explosion; Oshiro and Mazzariol et al. [14-15] addressed distortion of the scaling laws. For 22 instance, they suggested the density scaling factor to handle the generated incomplete similarity AC CE PT ED M AN US CR IP T 3 4 ACCEPTED MANUSCRIPT between the original model and the scale model in the structural model test subjected to dynamic 2 loads. And in order to solve material strain rate sensitivity, they set a scaling numbers that allowed 3 perfect similarity between the original model and sensitive scale model. Yang and et al. [16-17] 4 utilized experimental method to study the similarity law between the original model and the scale 5 model of the small-size confined concrete structure under internal explosion, and it was 6 demonstrated that internal explosion damage effect of the original model could be predicted through 7 the scale model with scaling coefficient ? ? 0.1 in accordance with load parameters and critical 8 failure characteristics of concrete under internal explosion. Jiang et al. [18-19] used dimensional 9 analysis principle to deduce the similarity law of single layer spherical lattice shell model under 10 impact loading, and took advantage of numerical simulation to validate related presumption while 11 considering the factors of the strain rate effect, the geometric deviation and the gravity. Gao et al. 12 [20] carried out internal explosion test of K6 single layer spherical lattice shell and proved scale 13 model with scaling coefficient ? = 0.1 was similar to the original model, and the results of the 14 internal explosion model test was able to characterize the original model. PT ED M AN US CR IP T 1 However, plenty of factors would seriously affect the accuracy of experimental results of the 16 scale model under internal explosion. Firstly, the propagation law of shock waves under internal 17 explosion is pretty complicated because of numerous reflection phenomena and convergence effects 18 [21-26]. Secondly, the strain rate effect and the strain hardening effect of steel under the high-speed 19 impact loading are remarkable [27]. Finally, the scale model could not be completely equivalent to 20 the original model actually. In a word, the reliability of the results of internal explosion experiment 21 would be reduced (sometimes even cause the failure of the test) unless the similarity law between 22 the original model and the scale model is verified. AC CE 15 5 ACCEPTED MANUSCRIPT The one-way inclined bar type of single layer cylindrical lattice shell is a kind of typical large 2 span space structure [28]. In this paper, the similarity law of cylindrical lattice shell model under 3 internal explosion was deduced based on the dimensional analysis method. The dynamic analysis 4 software ANSYS / LS-DYNA was applied to simulate the original model and the scale models of 5 cylindrical lattice shell, then correctness and feasibility of the similarity law were verified by 6 comparing the calculation results. The purpose of all above is for the purpose of providing a 7 reasonable and effective scaling coefficient to the internal explosion model test of the large span 8 space structure. 9 2 Theoretical derivations AN US CR IP T 1 The loading effects and the structural dynamic responses are the main subjects to investigate 11 into the scale model tests of the large-span space steel structure under internal explosion. In order to 12 ensure that the research results of the scale model truly reflect the characteristics of its original 13 model, it is required that the physical process of the characteristics (loading effects and responses of 14 the internal explosion, etc.) of the scale models is the same as its original model. In other words, 15 there is a confirmed scaling coefficient of each parameter involved in physical process between the 16 original model and the scale model. The similarity criterion between the original model and the 17 scale model can reflect their physical relationships and can further determine the scaling coefficient 18 of each parameter. Therefore, the correct derivation of the similarity criterion is the key to extend 19 the research results of the scale models to the original model. 20 2.1 Loading effects of internal explosion AC CE PT ED M 10 21 Explosive impact load belongs to a kind of typical accidental load, and it has the characteristics 22 of loads that are different from collision, fire and earthquake. At explosion, the shock waves rapidly 6 ACCEPTED MANUSCRIPT change with time, instantly reach the pressure peak, gradually attenuate and eventually return to the 2 standard atmospheric pressure. In short, that procedure could be summarized as a high strength and 3 a short pulse impact. Usually, the shock waves generated by both internal and external explosions 4 have the similar characteristics. According to this performance, the overpressure peak of shock 5 waves, the positive pressure time and the specific impulse were generally applied to quantitatively 6 describe the loading effects of blast waves in engineering design. The parameters that would ideally 7 influence the overpressure peak of shock waves ?P (FL-2), the positive pressure time t+ (T) and the 8 specific impulse i (FL-2T) were listed in Table 1, respectively. AN US CR IP T 1 Table 1 The parameters involved in loading effects of internal explosion and their dimensions Influence Influence Parameters Dimensions Parameters b: Span of cylindrical lattice -1 2 [FL T ] [L] PT ED W: TNT mass ? e : Explosive density CE Explosive Dimensions factors M factors shell l: Length of cylindrical -4 2 [FL T ] [L] lattice shell AC Ee : Releasing energy per f: Vector height of 2 -2 [L T ] Structure unit mass [L] cylindrical lattice shell h: Height of structure for P0 : Initial pressure [FL-2] supporting cylindrical lattice [L] Air shell ? 0 : Initial density [FL-4T2] R: Distance between 7 [L] ACCEPTED MANUSCRIPT ? 0 : Adiabatic coefficient explosive center and [1] structural inner surface According to the Table 1, the functional relationships of the overpressure peak of shock waves 2 ?P, the positive pressure time t+ and the specific impulse i with each parameter could be expressed 3 as follows. ?P ? f (W , ?e , Ee , P0 , ?0 , b, l , f , h, R) 5 t ? ? g (W , ?e , Ee , P0 , ?0 , b, l , f , h, R) 6 i ? ? (W , ?e , Ee , P0 , ?0 , b, l , f , h, R) AN US 4 CR IP T 1 ?1? ?2? ?3? Now, taking the overpressure peak of shock waves ?P as an example, the similarity criteria 8 would be deduced based on the principle of dimensional harmony. From Eq.(1), the dimensional 9 relationship between the overpressure peak of shock waves ?P and each parameter could be expressed as follows. ED 10 [?P] ? [W ]x1 [ ?e ]x2 [ Ee ]x3 [ P0 ]x4 [ ?0 ]x5 [b]x6 [l ]x7 [ f ]x8 [h]x9 [ R]x10 11 14 15 PT Substituting the dimensions in Table 1 into Eq.(4), the following equation was obtained. [FL-2 ] ? [FL-1T 2 ]x1 [FL-4 T 2 ]x2 [L2T -2 ]x3 [FL-2 ]x4 [FL-4 T 2 ]x5 [L]x6 [L]x7 [L]x8 [L]x9 [L]x10 ?5? CE 13 ?4? where, x j ( j ? 1,2,3......10) was the dimensional index of each parameter. According to the dimensional uniformity of the physical equation, the dimension index of the AC 12 M 7 16 same parameters on both sides of the Eq.(5) should be equal. So, there were the following 17 equations, corresponding to the dimensions F, L and T, respectively. 18 x1 ? x 2 ? x 4 ? x5 ? 1 ? ? ?? x1 ? 4 x 2 ? 2 x3 ? 2 x 4 ? 4 x5 ? x6 ? x7 ? x8 ? x9 ? x10 ? ?2 ? 2 x1 ? 2 x 2 ? 2 x3 ? 2 x5 ? 0 ? 8 ?6? ACCEPTED MANUSCRIPT follows. 1 0 1 1 ?1 ? ?1 ? 4 2 ? 2 ? 4 ? ?? 2 2 ?2 0 2 3 4 5 6 7 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1? ? 2?? 0 ?? After the finite elementary row transformations, it could be transformed into a row simplest CR IP T 2 The Eq.(6) was a system of linear equations, its coefficients could be written into a matrix, as matrix, as below. ? 1 ? 0 ? ?? 0 0 1 0 0 0 1 0 1 1 0 1 0 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 ?1/ 3 ?1/ 3 ?1/ 3 ?1/ 3 ?1/ 3 0 0 0 0 0 AN US 1 0 1 1 ? ? ? ?? Therefore, the Eq. (6) could be written as the Eq. (7), as follows. ? x1 ? ?1/ 3x6 ? 1/ 3x7 ? 1/ 3x8 ? 1/ 3x9 ? 1/ 3x10 ? ? x2 ? ? x4 ? x5 ? 1/ 3x6 ? 1/ 3x7 ? 1/ 3x8 ? 1/ 3x9 ? 1/ 3x10 ? 1 ?x ? ? x ? 1 4 ? 3 ?7? M 8 where, the x j ( j ? 4,5...10) was independent and arbitrary, could take any value but not all zero, and 10 the x1 , x2 and x3 could be determined by x j ( j ? 4,5...10) . Substituting the Eq.(7) into Eq.(4), the 11 following equation could be obtained. PT ED 9 x4 x5 x6 x7 x8 x9 ? ?P ? ? P0 ? ? ?0 ? ? ? ? ? ? ? ? ? ? ? b l f h R ? ??? ? ? ? ? 1/ 3 ? ? 1/ 3 ? ? 1/ 3 ? ? 1/ 3 ? ? 1/ 3 ? ? ?e Ee ? ? ?e Ee ? ? ?e ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ? ? (W / ?e ) ? x10 ?8? CE 12 According to the ? theorem of explosion mechanics, the W, ? e and Ee would be taken as 14 elementary variables. Taken the x j ( j ? 4,5...10) as equal 1, the mathematical relationships of the 15 dimensionless term 16 17 18 AC 13 ?P with other seven dimensionless groups could be obtained, as follows. ? e Ee P ? ?P b l f h R ? f( 0 , 0, , , , , ) 1/ 3 1/ 3 1/ 3 1/ 3 ?e Ee ?e Ee ?e (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e )1/ 3 ?9? Similarly, the positive pressure time and the specific impulse could be expressed by the seven dimensionless groups, as follows. 9 ACCEPTED MANUSCRIPT t? 1 W 1/ 3 ?e -1 / 3 Ee -1 / 2 ? g( P0 ?0 b l f h R , , , , , , ) ?10? 1/ 3 1/ 3 1/ 3 1/ 3 ?e Ee ?e (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e ) (W / ?e )1 / 3 2 i P ? b l f h R ? ?( 0 , 0 , , , , , ) ?11? 2/3 1/ 2 ?e Ee ?e (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3 (W / ?e )1 / 3 W 1 / 3 ?e Ee 3 The similarity criterion ? j ( j ? 1,2,3......10) could be obtained from Eqs.(9), (10) and (11), as ?P ?1 ? ? e Ee 5 ?4 ? 6 7 ?7 ? l (W / ? e ) 1/ 3 ?2 ? t? W P0 ? e Ee ?8 ? 1/ 3 ? ?1 / 3 e ?0 ?e ?5 ? f (W / ?e ) E ?1 / 2 e CR IP T below. ?3 ? ?6 ? i W 1/ 3 ?9 ? 1/ 3 ? e2 / 3 Ee1/ 2 b (W / ?e )1 / 3 AN US 4 h (W / ?e ) 1/ 3 ? 10 ? R (W / ? e )1/ 3 Thus it can be seen, that the similar conditions should be satisfied in the scale model design 9 were ? jm ? ? jp (j = 1, 2......10). Here, the scale model and the original model were represented by subscript m and subscript p, respectively. ED 10 M 8 Assuming the scaling coefficient was a constant and no greater than 1, which could be 12 represented by the ?, and the geometric relationship between the original model and the scale 13 models would be easily obtained as follows. 15 16 CE ?b ? ?l ? ? f ? ?h ? ? ?12? AC 14 PT 11 If the explosive type was the same and the propagating medium did not change, the scaling relation for the overpressure of shock waves and the explosive mass would be expressed as below. 17 ??P ? ?? ? 1 ?13? 18 ?W ? ?3 ?14? e 10 ACCEPTED MANUSCRIPT 1 According to the similar conditions above, the magnitude of overpressure in the scale model 2 should be equal to that of the original model and the amount of explosive should be cubic of scaling 3 coefficient. 5 Similarly, the scaling relation of the positive pressure time and the specific impulse should be showed as follows. 6 ?t ? ? 7 ?i ? ? 2.2 Internal explosive responses ?15? ?16? AN US 8 ? CR IP T 4 In fact, researchers tend to pay more attention to the motion and the deformation of lattice shell 10 structure subjected to the shock waves of internal explosion. However, the material parameters and 11 the structural dynamic characteristics are different from ones of the static, because the structural 12 inertia and the propagation of the stress waves cannot be neglected under the short and strong loads. 13 Usually, the stress ? (t ) and displacement ? (t ) are the main parameters that represent the 14 structural dynamic responses. For the isotropic elastic-plastic materials, if the strain rate effect and 15 the strain hardening effect were neglected during material deformation process, the parameters 16 involved in the stress ? (t ) ([FL-2]) and the displacement ? (t ) ([L]) and their dimensions would 17 be shown in Table 2. 19 ED PT CE AC 18 M 9 Neglected the dimensionless parameters in Table 2, the functional relationships of the stress and the displacement with each parameter were as follows. 20 ? (t ) ? ? ( E, ? , t , ? s , P(t ), L(t ), E(t )) ?17? 21 ?(t ) ? ? ( E, ? , t , ? s , P(t ), L(t ), E (t )) ?18? 11 ACCEPTED MANUSCRIPT 1 2 Taking the stress ? (t ) as an example, the dimension relation between it and each parameter was as follows. [? (t )] ? [ E] y1 [ ? ] y2 [t ] y3 [? s ] y4 [ P(t )] y5 [ L(t )] y6 [ E(t )] y7 3 (19) Influence factors parameters dimensions Time t : Time [T] CR IP T Table2 The parameters involved in internal explosive responses and their dimensions Influence factors E : Elastic [FL-2] [FL T ] density [FL-2] Time variable Material M L(t ) : Structural [1] ED ? : Poisson ratio [L] dimensions E (t ) : Tangential -2 [FL-2] [FL ] modulus PT ? s : Yield stress CE Substituting the dimensions in Table 2 into Eq.(19), the following equation was obtained. [FL?2 ] ? [FL?2 ] y1 [FL?4T 2 ] y2 [T] y3 [FL?2 ] y4 [FL?2 ] y5 [L] y6 [FL?2 ] y7 (20) where, y j ( j ? 1,2,3......7) was the dimensional index of each parameter. AC 7 [1] the unit area -4 2 6 ? (t ) : Strains AN US ? : Material 5 dimensions P(t ) : Pressure on modulus 4 parameters According to the dimensional uniformity of the physical equation, the dimension index of the 8 same parameters on both sides of the Eq. (20) should be equal. So, there were the following 9 equations, corresponding to the dimensions F, L and T, respectively. 12 ACCEPTED MANUSCRIPT y1 ? y2 ? y4 ? y5 ? y7 ? 1 ? ? ?? 2 y1 ? 4 y2 ? 2 y4 ? 2 y5 ? y6 ? 2 y7 ? ?2 ? 2 y 2 ? y3 ? 0 ? 1 follows. 1 ?1 ?? 2 ? 4 ? ?? 0 2 4 5 6 1 1 ?2 ?2 0 0 0 1 0 1 1? ? 2 ? 2?? 0 0 ?? After finite elementary row transformations, it would be transformed into a row simplest matrix. ? 1 ? 0 ? ?? 0 7 0 1 0 0 0 1 1 0 0 1 0 0 1/ 2 ?1/ 2 1 1 0 0 1 0 0 ? ? ? ?? Therefore, the Eq. (21) could be written as the Eq. (22), as follows. M 8 0 0 1 CR IP T 3 The Eq.(21) was a system of linear equations, its coefficients could be written into a matrix, as AN US 2 (21) ? y1 ? ? y4 ? y5 ? 1/ 2 y6 ? y7 ? 1 ? ? y2 ? 1/ 2 y6 ?y ? ?y 6 ? 3 ?22? ED 9 where, the y j ( j ? 4,5...7) was independent and arbitrary, could take any value but not all zero, and 11 the y1 , y 2 and y3 could be determined by y j ( j ? 4,5...7) . Substituting the Eq.(22) into Eq.(19), the 12 following equation could be obtained. 14 CE AC 13 PT 10 y6 4 5 ?? (t ) ? ?? s ? ? P(t ) ? ? L(t ) ? ? E (t ) ? ? ? E ? ? E ? ? E ? ? ( E / ? )1 / 2 t ? ? E ? ? ? ? ? ? ? ? ? ? ? y y y7 ?23? According to the ? theorem of explosion mechanics, the E , ? and t would be taken as 15 elementary variables. Taken the y j ( j ? 4,5...7) as equal 1, the mathematical relationships of the 16 dimensionless term 17 ? (t) E with other four dimensionless groups could be obtained, as follows. ? (t ) E ? ?( ? s P(t ) E , E 13 , L(t ) E (t ) , ) 1/ 2 (E / ? ) t E ?24? ACCEPTED MANUSCRIPT 1 Similarly, the displacement could be expressed by four dimensionless groups, as follows. 2 ? P(t ) ? (t ) L(t ) E (t ) ?? ( s , , , ) 1/ 2 1/ 2 (E / ? ) t E E (E / ? ) t E 3 The similar criterion ? j ( j ? 11,12,13......16) could be obtained from Eq. (24) and Eq. (25), as below. ? 11 ? 5 ? 14 ? E P(t ) E ? 12 ? ? (t ) ( E / ? )1/ 2 t ? 15 ? L(t ) ( E / ? )1/ 2 t ? 13 ? ?s ? 16 ? E E (t ) E AN US 6 ? (t) CR IP T 4 ?25? 7 If the material types were consistent, namely ?E ? ?? ? 1 , and ? jm ? ? jp (j=11, 12......16) 8 could be satisfied, the scaling relation to the stress and the displacement would be expressed as 9 follows. M ?? (t ) ? ?? ? ?P (t ) ? ?E (t ) ? ?E ? 1 10 s ?? (t ) ? ?L(t ) ? ?t 13 ?L(0) ? ?? (t ) ? ?t ? ? 3 Verification of the similarity law 15 3.1 Numerical Model ?28? AC CE 14 16 ?27? At the t equal to zero, the scaling relation of the displacement would be as below. PT 12 ED 11 ?26? A one-way inclined single-layer cylindrical lattice shell was selected as the original model. Its 17 geometric size was as follows: the span b =30 m, the rise to span ratio f / b = 1/5, the length to span 18 ratio l / b = 1.6 and the supporting structure height h = 8 m. The original model and the scale 19 models with ? = 0.5, 0.25, 0.1 and 0.05 were established by the ANSYS/LS-DYNA. 14 ACCEPTED MANUSCRIPT The numerical model was shown in Figure 1 (b). The ground was considered rigid and the 2 bottom of supporting column was fixed to the ground. Besides, the major structure and building 3 envelope were connected by the connecting components. Among them, the major structure and the 4 connecting component were simulated by Beam161, and the building envelope and the ground were 5 stimulated by Shell163. The air of surrounding the entire structure model was set up and meshed, 6 which was simulated by Solid164 with 8 nodes and 6 faces, and the boundary was defined as the 7 transmission boundary to simulate the explosion in an infinite area. The TNT explosive was defined 8 by the volume fraction method in the cylindrical lattice shell structure, which was located at the 9 center above the ground 1.5m. In the whole process of explosion, the ALE (Arbitrarily 10 Lagrange-Euler) algorithm was adopted to consider the interaction between the air shock waves and 11 both the building envelope and the ground [29]. CE PT ED M AN US CR IP T 1 ?b?Numerical model 12 AC (a) Experimental model Figure 1 Cylindrical lattice shell structure The structural size, mesh size and TNT mass of the scale models were scaled down according 13 to Eqs.(12) and (14), and the element attributes and process of numerical calculation of the scale 14 models were the same as the original model. It was necessary to point out that material size of the 15 scale models theoretically should also satisfy the similarity law. However, when ? was small 15 ACCEPTED MANUSCRIPT enough, the scaled-down material size would not be according with the true condition. Therefore, 2 according to the actual material size, when the scale models were established, the materials of 3 similar size should be selected as shown in Table 3. This performance that the geometric 4 characteristics of model or the form of construction could not be completely satisfied the same 5 proportion was defined as the geometric incomplete similarity of model [30-31]. CR IP T 1 Table3 Material sizes of original model and scale model?unit: mm? ? 0.5 Latticed shell ?140� ?70� (Seamless steel tube) Supporting beam or column General beam or column ED (square steel tube) 400�0� Wall beam PT (rectangular steel tube) 200�0� M 400�0� 200�0� 200� CE 400� 0.25 AN US 1 Material type AC 3 0.05 ?14�8 ?7�4 (?14� (?14� ?35� 100�0� 20��40��(20�� 40�� 20�� (40��2) (20�� 40�4 20�2 (40� (20� 0.75 0.3 0.15 (1.2) (1.2) (1.2) 100�0� 100� (angle iron) Enclosure structure 0.1 1.5 (hot rolled steel plate) Notes: The parameters in parentheses were the material type and the material size of the physical models. 6 In the original model and the scale models, the Q235 steel was selected and the modified 7 Johnson-Cook constitutive relation was used, which considered the strain rate effect and strain 16 ACCEPTED MANUSCRIPT 1 strengthening effect of steel subjected to high-speed impact loading [29]. Specifically, it was shown 2 in Eq.(29). ? ? ( A1 ? A2? n )(1 ? A3 ln ??* )(1 ? T *m ) ?29? t 3 where, ? , ? , ?? * and T* represented the equivalent flow stress, the equivalent plastic strain, the 5 relative strain rate and the relative temperature, respectively. A1, A2, A3, n and mt were undetermined 6 parameters that could be calibrated by experiment. Because the experiment and simulation were 7 carried out at room temperature (the T* is zero), the value of mt was arbitrary and A1, A2, n and A3 8 were shown in Table 4. AN US CR IP T 4 Table 4 J-C constitutive model parameters of the Q235 [27,32] 10 320.7556�6 582.102�6 n A3 0.3823 0.0255 M A2 The explosive adopted the high energy explosive model (MAT_HIGH_EXPLOSIVE_BURN) ED 9 A1 and the JWL (Jones-Wilkins-Lee) state control equation as shown in Eq.(30). p ? A(1 ? PT 11 ?E0,exp ? ? RV RV )e 1 exp ? B(1 ? )e 2 exp ? R1Vexp R2Vexp Vexp ?30? where, p was the pressure of shock wave, A, B, R1, R2 and ? were input parameters, Vexp was the 13 relative volume of explosive. Besides, V0,exp and E0,exp represented the initial relative volume and 14 the initial internal energy of explosive, respectively. The parameters of the explosive were shown in 15 Table 5. AC CE 12 17 ACCEPTED MANUSCRIPT Table 5 Material parameters of explosive ? exp /kg? m ?3 D /m? s ?1 PCJ /GPa A/GPa B/GPa R1 R2 ? E0,exp /J? m ?3 V0,exp 1630 6713 18.5 540.9 9.4 4.5 1.1 0.35 8 ? 109 1.0 respectively. The air was simulated by the CR IP T Notes: ? exp , D and PCJ were the density of the explosive, the detonation velocity and the detonation pressure, MAT_NULL EOS_LINEAR_POLYNOMIAL was adopted, as follows. and the state P0,air ? C0 ? C1? ? C2? 2 ? C3? 3 ? (C4 ? C5? ? C6? 2 ) Eair AN US ? ? 1 / Vair ? 1 equation ?31? ?32? where, P0,air was the initial pressure of air, C0, C1, C2, C3, C4, C5 and C6 were the material parameters of air, Eair was the internal energy per unit volume, Vair was the relative volume of air, M ? air was the density of air. Besides, V0,air and E0,air were the initial relative volume and the initial ED internal energy of air, respectively. The material parameters of air were shown in Table 6. Table 6 Material parameters of air C0 1.29 0 C1 C2 C3 C4 C5 C6 E0,air /J? m ?3 V0,air 0 0 0 0.4 0.4 0 2.5 ? 105 1.0 CE PT ? air /kg? m ?3 AC In order to verify the correctness and reliability of the numerical modeling method, the numerical model of the air explosion was established by using the same method and the simulate results were compared with ones of the air explosion tests. For the air explosion tests, the No. 2 rock emulsion explosive (its mass was 40g) were employed and the tests were carried out in an open spaces. The overpressure of the shock waves were obtained by the dynamic pressure sensor KD2009A that shown in Figure 2(a). The layout of the measuring points was shown in Figure 2(b). 18 CR IP T ACCEPTED MANUSCRIPT (a) Dynamic pressure sensor KD2009A (b) Layout of the measuring points Figure 2 The Layout of the measuring points and dynamic pressure sensor AN US The comparison between experimental and simulated phenomenon at an explosion moment was showed in Figure 3. The overpressures of shock waves obtained by the air explosion tests and the simulation as well as their comparison results were showed in Table7. The compared results M from the Table7 showed that the simulated results agreed well with ones of the air explosion tests in ED general. The errors between the simulation and the experiments were within 30%, the biggest error was 27.32%. On the other hand, the existed literature also suggested that the simulated results PT agreed well with the empirical formula, were closest to the J. Henrych?s empirical formula [33-36]. CE This indicated that the parameter selections of the numerical model were correct and the calculation AC results were reasonable and credible. (a) The experimental phenomenon (b) The simulated result Figure 3 The comparisons between experimental and simulated results 19 ACCEPTED MANUSCRIPT Table 7 The comparisons of overpressure peak values of shock waves between experiment and simulation Measuring Measuring ?Pexp /MPa ?Psim /MPa ?air / % ?Psim /MPa ?air / % 0.0391 0.0325 16.88 point No. 1 0.0703 0.0604 14.08 5 2 0.0388 0.0313 19.33 6 3 0.0831 0.0604 27.32 7 4 0.0240 0.0223 7.08 8 0.0232 0.0184 20.69 invalid 0.0325 / invalid 0.0142 / AN US Notes? ?Pexp were the overpressure peak values by air explosion tests. CR IP T point No. ?Pexp /MPa ?Psim were the overpressure peak values by numerical simulation. ?air were the errors between ?Pexp and ?Psim , ? air ? (?Pexp ? ?Psim ) / ?Pexp . 3.2 Critical TNT mass and Critical failure phenomena M For the analysis of convenience, the critical TNT mass was here defined as the mass of the initial venting of a cylindrical lattice shell structure subjected to internal shock waves. In the ED numerical simulation of the original model and scale models under internal explosion, gradually PT increasing the TNT mass until the cylindrical lattice shell structure occurred the first explosion CE venting. The corresponding TNT mass was the critical TNT mass. Then the explosion characteristics of the original model and scale models could be compared and analyzed according to AC the phenomena of the initial venting. The major structure and the building envelope were connected by the connecting components. With the increase of the TNT mass, the tensile failure of the connecting components would gradually occur. Therefore, the different stages of the structure subjected to internal shock waves would be directly reflected by the failure ratio of the connecting components. The failure ratio of the connecting components could be defined as the percentage of that the failing numbers of 20 ACCEPTED MANUSCRIPT connecting components took up the total numbers of connecting components. Statistical results of the failure ratio of the connecting components were shown in Table 8. Table 8 The proportion of the connection component failure ? dam /% 1 ? 6 15.35 0.5 ? 6 13.39 0.25 ? 6 12.99 0.1 ? 6 12.99 0.05 ? 6 Crucial TNT mass/kg CR IP T ? bef /% AN US ? 12.60 300 37.5 4.6875 0.3 0.0375 Notes: ? bef was the failure ratio of the connecting components before the initial venting. M ? dam was the failure ratio of the connecting components at the initial venting. ? bef or ? dam were determined by the percentage of that the failing numbers of connecting components took up the total numbers of connecting components. ED From the Table 8, it was clearly found that the failure ratios of connecting components of the PT original model (?=1) and the scale models (?=0.5, 0.25, 0.1 & 0.05) were all less than 6% before explosion venting and rapidly increased up to more than 2 times once the explosion venting CE occurred. But in general, the multiplier value of the failure ratios decreased with the decrease of ?, AC and accordingly, the critical TNT mass of the scaled models decreased cubically with the decrease of ?. The phenomena of the initial venting of the original model and scale models were showed in Figure 4. As shown in Figure 4, the initial venting was occurred at the wall-shell junction of the cylindrical lattice shell because of the gathering and enhanced effect of shock waves at the structural corners. Remarkably, the positions appeared the initial venting of the original model and the scale models were identical and then their venting phenomena were similar. 21 ACCEPTED MANUSCRIPT AN US CR IP T ?=1 ?=0.25 ?=0.05 PT ?=0.1 ED M ?=0.5 CE Figure 4 Initial venting phenomena of scale models under internal explosion Following similarity law deduced by the Section 2, a physical model of the ?=0.1 cylindrical AC latticed shell was established. Its material type and material size were consistent with those in Table 3 parentheses. Remarkably, the roof panel steel and the major structure were welded with a small steel bar. The wall panel and the major structure were connected with rivets. To ensure that the experimental environment was consistent with the actual structure, the internal explosion experiment of the cylindrical latticed shell was carried out in an open quarry. To ensure that the explosion venting of the cylindrical latticed shell model could occur during the experiment, the 22 ACCEPTED MANUSCRIPT internal explosion experiment used 1kg explosive. The failure process and the final failure phenomena were shown in Figure 5. Figure 5 (a) and Figure 5 (b) demonstrated the failure process of the physical model of the cylindrical latticed shell. Figure 5 (c) and Figure 5 (d) showed the final failure phenomenon of the roof panel steel and wall panel, respectively. It was obviously found that CR IP T the roof panel steel was torn locally due to the strong connection stiffness and the connection component of wall was broken totally and the whole steel plate was separated from the major structure due to the weak connection. Although the 1kg explosive was larger than the critical mass, AN US it could be clearly seen from the final destruction that the initial venting area was at the joints of the PT ED M wall-shell. Which all above was good agreement with the simulation results shown in Figure 4. (b) Blasting moment AC CE (a) Scaled model before explosion (c) Failure phenomenon of the roof panel steel (d) Tear failure of the roof panel steel Figure 5 The experimental phenomena under internal explosion 23 ACCEPTED MANUSCRIPT 3.3 Propagation law of shock wave The propagation law of shock waves under internal explosion was different from that under external explosion. Firstly, the shock waves reflected on the rigid ground and the reflected waves and the incident waves stacked up to form the Maher waves. Subsequently, the incident waves and CR IP T the Maher waves spread to the building envelope of roof and wall successively, and the shock waves were gradually converged on the top of the cylindrical lattice shell and the joints of the wall-shell owing to the obstruction of the building envelope. At the end, the overpressure of shock AN US waves reached the venting threshold and the cylindrical lattice shell began explosion venting. At this moment the overpressure distribution of shock waves became disordered, because a part of the shock waves were reflected back to inside and converged in the rigid ground center for starting over M the reflections. ED Figure 6 showed the pressure contours of internal explosive shock waves in original model and scale models under the critical TNT mass. It could be clearly observed that the impact of the Maher PT reflection, the convergence phenomena at the top of the cylindrical lattice shell and the junction of CE the wall-shell, and the second reflection of the shock waves. Remarkably, the internal explosive phenomena in original model and scale models were identical at the same time point, which AC demonstrated that the propagation law of the shock waves in the scale models resembled that of the original model under internal explosion. t=8ms t=19ms t=26ms ?=1 24 t=58ms ACCEPTED MANUSCRIPT t=4ms t=9.5ms t=13ms t=29ms t=2ms t=4.75ms t=6.5ms t=14.5ms t=2.6ms t=5.8ms t=1.3ms t=2.9ms M t=1.9ms AN US ?=0.25 t=0.8ms CR IP T ?=0.5 t=0.95ms AC CE t=0.4ms PT ED ?=0.1 ?=0.05 Figure 6 Propagation law of shock waves under internal explosion 3.4 overpressure distribution of shock waves To study the similarity of the specific distribution of the internal explosive shock waves in the model, according to the propagation law of the shock waves, the overpressure time history curves of A, B, C, D and E points of the model (as shown in Figure 1) were chosen for analyzing the characteristics of the blast waves in the internal surface. As shown in Figure 7, the point A was 25 ACCEPTED MANUSCRIPT closest to the explosion position, and the shock waves reached first. Then the shock waves arrived PT ED M AN US ?=1 CR IP T at the two side walls. ?=0.25 AC CE ?=0.5 ?=0.1 Figure 7 ?=0.05 The overpressure distribution under internal explosion 26 ACCEPTED MANUSCRIPT It could be seen from the Figure 7 that the overpressure peaks of the points A, B and D, whether in the original model or in the scale models, were the following relations: peak B larger than peak D larger than peak A. That was reasonable because of the convergence effects of the shock waves at the structural top and corners. And it could also be explained that the initial venting CR IP T occurred at the joints of the wall-shell and some connection components of wall destroyed first as shown in Figure 4. Since then the shock waves gradually spread to both end walls and the overpressure constantly attenuated. AN US On the one hand, when the explosive waves arrived at the point E, the overpressure peak was only half of that of the point A, which was not powerful enough to destroy the connecting components of the end walls. On the other hand, the shock waves within the structure did not leak M and congregated as shown in Figure 8, which made the second overpressure peak at the point E ED would be larger than the first overpressure peak. The above comparative analysis presented that the overpressure time history curves of each PT characteristic point of the scale models were highly consistent with the original model. And it was CE implied that the overpressure distribution of each scale model was similar to that of the original AC model under internal explosion. Figure 8 End effect of the structure under internal explosion 27 ACCEPTED MANUSCRIPT 4 Discussions In summary, it could be concluded that the scale models of cylindrical lattice shell under internal explosion obeyed the similarity law from the perspectives of the critical TNT mass, the CR IP T phenomena of initial venting, the propagation law and the overpressure distribution of shock waves. However, the loading effects and the structural dynamic responses under internal explosion were the most concerned characteristic in practical engineering. In order to verify the similarity of the loading effects and the explosion responses between the original model and the scale models and to AN US explore the influence of scaling coefficient ? on similarity of the scale models, the data of the loading effects and structural dynamic responses were obtained, and it was necessary to further discuss. M 4.1 Loading effects of internal explosion ED Theoretically, the overpressure of shock waves, the positive pressure time and the specific impulse of the scale models should meet Eqs. (13), (15) and (16). For the sake of more intuitively PT explain the matter, the positive pressure time and the specific impulse divided by cube of the TNT CE mass was defined as the positive pressure time after conversion and the specific impulse after conversion. From the data outside brackets of Table 9, it could be seen that the overpressure of AC shock waves, the positive pressure time after conversion and the specific impulse after conversion of the scale models of the cylindrical lattice shell under internal explosion were similar to that of the original model, yet the degree of the similarities were different with each other. In general, the similarities reduced with the decrease of the scaling coefficient and the deviation was controlled within -1.18%, 1.26% and 2.0%, respectively. However, due to the limitation of material modulus on projects, the scale models with the scaling coefficient less than or equal to 0.25 existed the 28 ACCEPTED MANUSCRIPT geometric incomplete similarity, which would evidently affect the similarity of the loading effects of internal explosion. Table 9 Loading effects of internal explosion A B C D E ? ? value ? value ? value ? value ? 1 195528 0 247705 0 149632 0 218659 0 104996 0 0.5 195604 0.026 247135 -0.164 146675 -1.18 218032 -0.197 104749 -0.120 195661 0.045 246803 -0.259 146983 -1.06 217706 -0.299 104711 -0.139 (201267) (1.94) (258779) (3.18) (158783) (3.67) (224978) (1.98) (108295) (1.61) 195866 0.114 246685 -0.293 147108 -1.01 217627 -0.324 104502 -0.241 (208007) (4.22) (273037) (7.29) (174708) (10.0) (235106) (5.16) (112075) (3.45) 196058 0.179 246709 -0.286 147176 -0.984 217676 -0.308 105000 0.002 (210607) (5.10) (280174) (9.34) (181404) (12.7) (239796) (6.63) (113533) (4.16) 1 2.09 0 2.84 0 3.88 0 2.39 0 3.14 0 0.5 2.09 0 2.84 0 3.88 0 2.42 1.26 3.14 0 2.09 0 2.84 0 3.88 0 2.39 0 3.14 0 (2.09) (0) (2.85) (0.35) (3.88) (0) (2.39) (0) (3.29) (4.78) 2.09 0 2.84 0 3.88 0 2.39 0 3.14 0 (2.09) (0) (2.88) (1.41) (4.03) (3.87) (2.39) (0) (3.29) (4.78) 2.09 0 2.84 0 3.88 0 2.39 0 3.17 0.955 (2.09) (0) (2.96) (4.23) (4.03) (3.87) (2.39) (0) (3.29) (4.78) 0.25 ?P / Pa CE ED PT 0.05 M 0.1 CR IP T value AN US Item 0.25 + -1/3 / AC t 稺 ms穔g-1/3 0.1 0.05 29 ACCEPTED MANUSCRIPT -1/3 i稺 / 1 173302 0 237026 0 175154 0 214425 0 130326 0 0.5 173345 0.025 237477 0.190 174943 -0.120 213713 -0.332 129833 -0.378 173451 0.086 236020 -0.424 176055 0.514 213452 -0.454 129856 -0.361 (180296) (4.04) (250640) (5.74) (193596) (10.5) (224385) (4.64) (137175) (5.26) 173734 0.249 234914 -0.891 177885 1.56 213331 -0.510 129752 -0.440 (189323) (9.24) (275345) (16.2) (227283) (29.8) (238075) (11.0) (146328) (12.3) 173879 0.333 234826 -0.928 178659 2.00 213315 -0.518 130259 -0.051 (192933) (11.3) (287512) (21.3) (244777) (39.7) (243505) (13.6) (149951) (15.1) 0.25 kg -1/3 0.1 AN US 0.05 CR IP T Pa穖s� Notes: The values in parentheses were computed from the actual material size. ? represented the relative deviation between the scale model and the original model, expressed as percentile. The geometric incomplete similarity of the scale models were taken into account from the data M inside brackets of Table 9. Compared with the original model, it could be informed that the ED similarities of the overpressure of shock waves, the positive pressure time after conversion and the specific impulse after conversion obtained by the scale models still reduced with the decrease of the PT scaling coefficient but the deviations were further amplified. If the scaling coefficient equal to 0.1, CE the maximum deviation of the shock wave overpressure was 10.0% and the maximum deviation of the specific impulse after conversion was 29.8%. The two above errors were still within the AC acceptable range. But if the scaling coefficient equal to 0.05, the maximum deviation of the shock waves overpressure reached to 12.7% and the maximum deviation of the specific impulse after conversion was 39.7%, which were not reasonable and credible to accurately present the characteristics of the original model. It was indicated that the similarities of the loading effects of internal explosion were reduced if the geometric incomplete similarity was extended, especially presented to the scale models with the small scaling coefficients. 30 ACCEPTED MANUSCRIPT It was known from the above analysis that if the scaling coefficient was not less than 0.1, the loading effects of the scale models under internal explosion were similar to that of the original model, namely the Eqs. (13), (15) and (16) were met. With the decrease of the scaling coefficient, the influence of the geometric incomplete similarity of the scale models on the loading effects of CR IP T internal explosion became bigger. However, this conclusion was not universally applicable, only suitable for the cylindrical lattice shell under internal explosion which was discussed in this paper. On the one hand, the degree of the similarity between the scale model and the original model also AN US depends on different structure type, geometry and loading. On the other hand, in the practical engineering, especially for large-span space structures with variety of structural types, the scale model with a smaller scaled coefficient would be often adopted. Therefore, to get more accurate and M more general conclusions, it is necessary to carry out more theoretical, experimental and numerical ED research on different structural types and loadings, to analyze the essential reasons for the errors caused by the geometrically incomplete similarity model under internal explosion and to put PT forward the solutions. 4.2 Internal explosive responses CE The element stress time-history curves and node displacement time-history curves of the model AC center were showed in Figure 9 (a) and (b), respectively. It could be seen from Figure 9 (a) that the maximum stress of element was more than 400MPa, much larger than the yield stress of the Q235. This implied that the strain rate effect and the strain hardening effect of material of the cylindrical lattice shell under internal explosive impact loading were significant. Similarly, it was clear stated from Figure 9 (b) that the peak of displacement responses of the scale models was less than that of the original model, and the deviation increased with the decrease of the scaling coefficients. For 31 ACCEPTED MANUSCRIPT example, as ? ??0.25, the peak of displacement responses was only 5.56% less than that of the original model, and as ? ??0.05, the peak of displacement responses was 44.31% less than that of the original model. This implied that the scaling coefficient greatly affected the displacement (a) Stress time-history curves in the middle point AN US CR IP T responses of the cylindrical lattice shell under internal explosion. (b) Displacement time-history curves of the central member M Figure 9 Influence of scaling coefficients on internal explosion responses ED The Existed literature indicated that the scale model subjected to the high-speed impact PT loading would cause the similar distortion because of the strain rate effect and the strain hardening effect. And with the decrease of the scaling coefficient, the influence of the similar distortion of the CE scale models on the internal explosive responses became bigger [37-38]. In this paper, those AC opinions just had been verified by the above discussions because the Johnson-Cook constitutive model could consider the influence of the strain rate effect and strain hardening of steel. However, owing to spatial confined, the errors of internal explosive responses caused by the similar distortion had not been discussed in this paper. In order to ensure the accuracy of the internal explosive responses, the authors suggested that the scaling coefficient of the scale models was not less than 0.25, for dealing with the dynamic responses. For the purpose of getting better results, it is 32 ACCEPTED MANUSCRIPT necessary to conduct further research on the similar distortion. 5 Conclusions The similarity law of the cylindrical lattice shell model under internal explosion was deduced CR IP T based on the principle of dimensional analysis, and the original model and the scale models with different scaling coefficients were simulated by ANSYS/LS-DYNA. The correctness and the feasibility of the similarity law were compared and confirmed. By studying the results of theoretical AN US derivation and numerical simulation and by discussing the influence of scaling coefficient ? on loading effects and structural dynamic responses, the conclusions were obtained as followings. (1) The scale models and the original model of the cylindrical lattice shell under internal explosion can satisfy the similarity law. Specifically, the propagation law and the overpressure M distribution of explosive shock waves, the critical TNT mass and the initial venting phenomena of ED the scale models are similar to the ones of the original model. PT (2) In ideal status, the loading effects of the scale models of the cylindrical lattice shell under internal explosion are similar to that of the original model, namely can meet the similarity law. CE (3) The influence of the geometric incomplete similarity of the scale models on the loading AC effects of internal explosion cannot be ignored with the scaling coefficient less than or equal to 0.25. (4) The strain rate effect and the strain hardening effect of material have serious influence on the accuracy of the stress and displacement responses of the cylindrical lattice shell model under internal explosive impact loading. If the scaling coefficient is less than 0.25, the stress responses and displacement responses of the scale models will not accurately reflect the characteristics of the 33 ACCEPTED MANUSCRIPT original model. (5) The scaling coefficient greatly affects the stress and displacement responses of the cylindrical lattice shell under internal explosion. If the scaling coefficient is too small, the errors of the stress responses and the displacement responses between the scale model and the original model CR IP T will be too large to meet the similarity law. Therefore, for the scale models of the cylindrical lattice shell affected by geometrical incomplete similarity and strain rate effect of material, the authors suggest that the scaling coefficient should not be less than 0.1 for obtaining the loading effects of obtaining the reliable dynamic responses. Acknowledgments AN US the structure under internal explosion by scale model test, and it should not be less than 0.25 for M The authors are very grateful to the National Natural Science Foundation of China (Grant ED no.51278208), the Science and Technology Project of Fujian Province (Grant no.2018Y0063), and PT the Subsidized Project for Postgraduates? Innovative Fund in Scientific Research of Huaqiao University (Grant no.17011086003) for the financial support of this work. CE References AC [1] Corley WG, Sozen MA, Thornton CH. The Oklahoma City Bombing: Analysis of blast damage to the Murrah Building. Journal of Performance of Constructed Facilities 1998;12(3):113-119. [2] Taveau J. The Buncefield explosion: Were the resulting overpressures really unforeseeable. Process Safety Progress 2012;31(1):55-71. [3] Ishikawa N, Beppu M. Lessons from past explosive tests on protective structures in Japan. International Journal of Impact Engineering 2007;34(9):1535-1545. 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