1 Introduction

Stiffened plates/shells consisting of ribs and a single panel(facesheet)are commonly found in large-scale structural constructions,such as transportation vehicles,ship hulls and architectural structures,due to convenience in terms of fabrication and high specific strength[1].However,compared with stiffened plates/shells,sandwich structures with honeycomb cores exhibit better bending strength because the second facesheet in a honeycomb sandwich significantly enhances its flexural rigidity[2].Consequently,honeycombcored sandwiches are widely used in architecture,transportation,aerospace,heat transfer,chemical Engineering,nanofabrication and,more recently,bio-medicine for their excellent mechanical properties such as high specific stiffness/strength and superior energy absorption[3].In particular,as regular honeycombs have long been identified as one of the most competitive candidates for lightweight constructions[2,4],extensive analytical,numerical and experimental researches have been carried out to characterize the mechanical properties of honeycomb-cored sandwich structures having different topologies and material makes[5–12].

More recently,a range of novel honeycomb structures have been proposed to not only further improve the mechanical properties of these structures but also explore their multifunctional application potentials.For instance,Yu et al.[13]introduced gradient distribution of mass to regular honeycomb sandwich structures and demonstrated that,for enhanced loading capacity,the materials could be more efficiently utilized by introducing the gradient.In another study,Han et al.[14,15]proposed sandwich constructions with honey comb-corrugation hybrid cores and demonstrated,both theoretically and experimentally,that the hybrid core exhibits superiority in both strength and energy absorption particularly in the low density regime.Subsequently,Zhang et al.[16]carried out free vibration analysis of sandwich beams with honeycomb-corrugation hybrid cores.Then too,Oftadeh et al.[17]studied the mechanical properties of anisotropic hierarchical honeycombs and found that increasing the hierarchy order could improve the mechanical properties.Finally,it is worth mentioning that,upon introducing small perforated holes to the facesheet and/or core webs,Han et al.[10]revealed that honeycomb-corrugation cored sandwich panels have superiority not only in specific strength and energy absorption but also in sound absorption and sound proofing.In other words,the proposed hybrid constructions may be classified as acoutomechanical metamaterials for multifunctional applications.

Meanwhile, the bending mechanism of sandwich beams having a variety of lightweight cellular cores has been widely studied. For a typical instance, Han et al. [18] performed design optimization of foam-reinforced corrugated sandwich beams under transverse three-point bending while Rubino et al. [19] carried out three-point bending tests of sandwich beams comprising Y-frame and corrugated cores. In addition, Yan et al. [20] carried out three-point bending tests of sandwich beams with aluminum foam-filled corrugated cores and demonstrated that foam insertions altered not only the failure mode but also increased dramatically its bending resistance.In another instance, Zhang et al. [21] studied experimentally the bending strength, stiffness and energy absorption of composite corrugated core sandwich structures with different fiber types, corrugation angles and core-sheet thicknesses. In addition, Xu et al. [22] proposed graded lattice core sandwich structures and estimated their mechanical strengths and failure mechanisms under bending load, both theoretically and experimentally.More research about the bending mechanism of lightweight sandwich structures can be found in Refs [23–27].

Fig.1 Unit cell of a sandwich beam without perforation,b sandwich beam with perforated bottom facesheet,and c stiffened beam

Nonetheless,in terms of fabrication cost and maintenance convenience,honeycomb sandwich structures are in general inferior to stiffened plates/shells.For instance,arc or laser welding extensively used in the construction of ship hulls/decks[28]can not be used to fabricate honeycomb sandwiches having relatively thick facesheets.Meanwhile, metal corrosion is always an important issue in ocean exploitation,but it is difficult to carry out maintenance checking for all-metallic honeycomb sandwich structures with totally enclosed cells.It is,therefore,natural to ask the question whether there exists a lightweight structure that combines the performance superiority of honeycomb sandwiches with the low fabrication cost and easy maintenance of stiffened plates/shells.The present study aims to answer this question by introducing perforations to one of the facesheets of a square honeycomb sandwich and comparing its resistance to three-point bending with that of the non-perforated one.

其中,t∈Τ，Τ为概周期时标，x(t)表示t时刻食饵总数，y(t)表示t时刻的捕食者总数，u(t)、v(t)为反馈控制项，ri(t)、ai(t)、b(t)、ci(t)、di(t)、ki(t)、αi(t)、βi(t)(i=1,2)为时标Τ上的非负有界概周期函数.从生态学的角度来看，假设系统(1.1)满足初始条件：x(0)>0，y(0)>0，u(0)>0，v(0)>0.

农村类型的划分是生态网络建立的基础，是根据原始的、连续的自然环境指定的，以保护现有的生态土地形态，结合地形地貌变化，制定出有利于生态系统恢复的生态网络体系。本文着重从平地式生态网络、丘陵式生态网络和山地式生态网络等三方面介绍农村土地规划。

The contribution of the back face to the bending response of sandwich beams with corrugated and Y-frame cores was investigated by St-Pierre et al.[29].They demonstrated that sandwich beams absent a back face are stronger than those with front-and-back faces in a short span beam while the opposite holds in the case of a long span beam.Inspired by this work,a square honeycomb-cored sandwich beam with square perforations on its bottom face is proposed,as shown in Figs.1 and 2.Relative to the small perforations(sub-millimeter scale)introduced by Han et al.[10]for the purpose of absorbing sound energy,the perforations of Figs.1 and 2 are sufficiently large to facilitate not only the fabrication but also maintenance during service.For engineering applications like the decks of cargo vehicles and transportation ships,the design of large-scale sandwich constructions with facesheet perforations requires a relatively thick top facesheet to avoid local indentation failure(due to,e.g.,large concentrated load transported via the wheel of a heavy vehicle)so that considerable bending strength can be achieved.Meanwhile,introducing perforations to the sandwich design facilitates not only the fabrication process(e.g.,laser welding)but also, for certain applications,manual service maintenance during severe marine corrosive environments.As a result,w ill these perforations positively affect the mechanical properties of the sandwich and,if so,whether there exists optimal perforations for maximum strength are the two major issues that concern the present study.

田朵接过她为小宁点的那份套餐时，才想起来她忘了备注免辣。小宁受伤住在医院，医生说得忌辣，伤口才能好得快一些，可是她给忘了。这要是放在过去，她和小宁肯定免不了为此吵上一架，说不定还会把外卖给摔了。好在现在俩人都收敛了，田朵默默地拿着筷子往外夹辣椒，小宁连说没事，没那么矫情。这样的一派和谐，让田朵心生感慨，唉，如果以前她能懂得夫妻之间要互相包容和忍耐，小宁也就不会受伤住院了，两个人也不会差点闹到离婚的地步。

The paper is organized as follows.Firstly,the concept of a perforated all-metallic sandwich beam with square honeycomb core is introduced and the key geometrical parameters describing its topology are presented.For comparison,hon-eycomb sandwich beams without perforation and absent a back facesheet are also considered,subject to the constraint of equal mass.An other constraint is that the sandwich beams with or without perforations have relatively thick top facesheets so that local indentation failure does not occur.Secondly, finite element(FE)simulations are carried out to quantify the influence of perforations on the strength of the sandwich beam subjected to three-point bending.Subsequently,analytical formulae are developed to predict the failure loads corresponding to different failure modes of the perforated sandwich.The analytical predictions are compared with the results obtained from full FE simulations,and optimal perforations for maximum strength enhancement are identified.Finally,the sensitivity of failure loads to the ratio of core height to beam span is discussed for varying perforation ratios.

Fig.2 a Perforated honeycomb sandwich beam subjected to three point bending.b Unit cell of perforated sandwich beam.c Bottom view of unit cell

2 Perforated sandwich beam in three-point bending

The width of the perforated bottom facesheet can be written as

Table 1 Initial geometrical parameters of a perforated sandwich beam under three-point bending

t c0(mm) t f(mm) L c(mm) H c(mm) N holes N cores 0.25 1.22 12.7 12.7 8 13

With reference to the unit cell shown in Fig.2c,the perforation ratio is defined by

To maintain the same mass as the non-perforated sandwich,the thickness of core webs in the perforated sandwich is increased as

where

白耙齿菌固体发酵物对庆大霉素所致大鼠急性肾衰竭的改善作用研究 …………………………………… 李 昂等（13）：1772

Figure 2 presents the geometric parameters of a square honeycomb-cored sandwich beam with square perforations on its bottom facesheet and the corresponding unit cell.As for a conventional sandwich beam without perforations,the beam span L,the thickness of facesheet t f,the thickness of core webs t c,the height of core webs H c,and the width of a unit cell L c(Fig.2b)are used to characterize the topology of the perforated sandwich beam.Furthermore,the perforation ratio ω(area ratio of perforations to the intact faceshteet)and the width of the bottom facesheet after perforation L b(Fig.2c)are introduced to describe the square perforation.Let V 2/(EM)denote the non-dimensional loading intensity,where E is the Young modulus of the material,M is the maximum moment,and V is the maximum shear force,both per unit width[30].

For the current study,the initial geometrical parameters of the sandwich beam are listed in Table 1.In consideration of specific applications(e.g.,the deck of transport ships and cargo carriers)that require a relatively thick facesheet to avoid local failure such as yielding and buckling,once the initial thicknesses of the facesheet and core webs are specified,the core would most likely fail ahead of the facesheet.As the main purpose of this study is placed upon the concept of the face perforation and its effects on sandwich strength under three-point loading,systematic variation of geometrical and loading parameters is not performed.Further,with reference to Fig.2,it is assumed that the rigid puncher has a sufficiently large width 2a relative to cell width Lc so that local indentation failure does not occur,and that the puncher is placed directly above the node connecting the honeycomb core to the top face.Under such conditions,the sandwich under three-point bending would fail in two main modes:(1)bending failure,i.e.,yielding of beam cross-section and buckling of top face caused by the bending moment;and(2)shear failure,i.e.,yielding and buckling of core webs due to shear force.Meanwhile,as above mentioned,local indentation is not considered.

Here,t c0 is the thickness of honeycomb core webs when ω=0,and N holes and N cores are the number of perforations in the facesheet and the number of unit cells in the sandwich beam shown in Fig.2,respectively.

Fig.3 Thickness of core webs and width of perforated bottom face plotted as functions of ω

Due to the shear force applied in the beam under bending,the honeycomb core can fail in either shear yielding or shear buckling[31].For a perforated sandwich beam under three point bending,its failure load of shear yielding can be written as

3 FE simulations

3.1 FE model

FE simulations of perforated sandwich beams in three-point bending as illustrated in Fig.2 are carried out with the commercial FE code ABAQUS/Explicit.The sandwich beam consisting of two unit cells along its width is employed for the simulation.Four-node shell elements(S4R)with five integration points through the thickness are used to model the core webs,and there are 30 layers of elements along the height direction.Eight-node solid elements(C3D8R),1mm×1mm×0.2mm in size,are used to model the facesheets.Four-node rigid elements(R3D4),2 mm×2 mm in size,are used to model both the puncher and roller in three point bending.A mesh sensitivity study shows that further mesh refining does not improve the numerical accuracy of the results.

Fig.4 Sandwich model for FE simulation

Both the facesheets and core webs of the sandwich beam are made of 304 stainless steel(density ρs =8000 kg/m3,Young modulus E s= 210GPa,and yield strength σs=210 MPa),which is assumed to obey the J2-fl ow theory once yielding is initiated,with strain hardening ignored for simplicity.

根据图2和图3研究区边坡的有效应力、最大剪应力分布情况可以得出研究区边坡初始应力场具有以下特点：边坡在自重应力场作用下，应力分布符合预期的应力分布规律。边坡浅表处应力分布较小，最大主应力方向与坡面接近平行，最小主应力方向与坡面基本垂直。地应力分布主要为垂直向，从地表到岩层深部随着深度增大，主应力相应增大。在边坡稳定的深度范围内，应力场连续分布无突变，从图2可以看出，边坡初始最大剪应力为1.56MPa，分布在边坡底部，其分布规律与自然边坡初始有效应力分布基本相似。

Quasi-static three-point bending is applied to the sandwich beam with a rigid puncher.The whole FE model is shown in Fig.4.Note that the bottom facesheet directly above the rollers is not perforated.

3.2 Results of FE simulations

与自然和谐相处，才是保持人长久生存的唯一途径，千万不要试图逃脱自然的掌控，只有与自然在遥相呼应中保持最佳距离，积极发挥人类的能动性与创造性，才是人之为人存在的生活状态。人类既是自然的产物，具有动物般的特质，但同时又区别于动物，属于有意识的存在者。人类所具备的主观能动性，让人成为主体的人，并与自然界的其他生物区别开来。“体育是人动物性行为的投射”[14]，体育项目在动物性生存活动中不断创造与丰富，在文明演进过程中，体育是让人摆脱功利价值取向并脱离绝对精神束缚的方式，保持人类与自然的统一性。

Fig.5 Sandwich beam with perforated bottom face(rear panel)under three-point bending.a Load versus deflection curves for selected values of perforation ratio ω.b Normalized sandwich beam strength plotted as a function of ω (Load:reaction force in puncher;δ:displacement of puncher;M:moment per unit width;V:transverse shear force per unit width.)

The physical mechanism underlying the variation trend in Fig.5b can be explained as follows.For a given perforation ratio,the beam fails with the smaller load of either shear or bending.For instance,the beam with intact bottom facesheet fails in shear,since the bending failure load is larger than that of shear failure.When the perforation ratio ω is increased,the area of perforations in the bottom facesheet becomes larger and the thickness of the core webs increases,resulting in a weaker facesheet but a stronger core.Consequently,the bending failure load decreases while the shear failure load increases.At a specific point,simultaneous bending failure and shear failure of the perforated beam occur,thus allowing the beam to reach the highest strength.

The influence of perforation ratio on the strength of a perforated sandwich beam is firstly discussed,with the strength interpreted as the reaction force of the puncher when the beam fails.Figure5 a plots the normalized reaction for ceversus the displacement of the puncher.The deformation of the sandwich is elastic at the beginning.Once the elastic limit is reached,the sandwich enters the plastic regime,which can be regarded as total failure as strain hardening is not ignored.In Fig.5a,the value of the perforation ratio(ω)is marked on each load-displacement curve.The beam with ω=0.25 has the highest strength,whereas that having ω=1 exhibits the lowest strength.More details are shown in Fig.5b,which displays the normalized strength as a function of ω.When ω≤0.25,the strength increases with increasingω and peaks when ω =0.25.Afterwards,the strength decreases as ω is further increased.Ultimately,the strength reaches the minimum as ω approaches its limiting value 1 that corresponds to the case when the whole bottom facesheet disappears(i.e.,the sandwich beam degrades to a stiffened beam).

To explore the failure modes,the distribution of Mises stress in sandwich beams with different perforation ratios at the occurrence of failure is presented in Fig.6.When ω=0,the beam fails in shear.As ω is increased,the area of the perforated holes and the thickness of the core webs both increase,resulting in a stronger core and a weaker bottom facesheet,which in turn leads to a higher shear strength and a lower bending strength.When ω is increased to 0.25,the shear failure load is equal to the bending failure load:correspondingly,the beam acquires its maximum strength.This maximum strength and its dependence on beam configuration will be investigated further in the next section using analytical predictions.

4 Analytical predictions

In this section,analytical formulas for the failure loads of perforated sandwich beams under three-point bending are presented.

Fig.6 Distribution of Mises stress in honeycomb-cored sandwich beams with different perforation ratios at the occurrence of failure

4.1 Bending failure

When subjected to three-point bending,the sandwich beam can fail in bending failure,i.e.,yielding of its cross-section or buckling of its top face.Conventionally,in the absence of perforation,only face yielding is induced by the bending moment.In this study,however,yielding of honeycomb core due to bending moment is taken into consideration because the presence of perforations causes the moment acting on core webs to be non-negligible.As shown in Fig.7,when ω is zero or small,the neutral axis is located in the core;when ω is large,however,the neutral axis is shifted to the top facesheet.

Fig.7 Schematic of sandwich cross-section under three-point bending.a When ω is zero or small,the neutral axis is located in the core.b When ω is large,the neutral axis is shifted to the top face

In Fig.7,d is the location of the neutral ax is which divides the beam cross-section into two regimes:the blank regime is in compression while the shaded regime is undergoing tension when the beam is subjected to three-point bending.When the whole cross-section enters yielding,the beam loses the capacity of loading.Stress equilibrium in the cross section dictates that

Due to stress equilibrium, the neutral axis divides the sandwich cross-section into two regions of identical area so that the location of the neutral axis can be determined using simple geometrical calculations,as

The failure loads of sandwich beams with varying perforation ratios are obtained using both analytical predictions and FE simulations,and the results are shown in Fig.8.For the sandwich beams investigated in the current study,panel buckling and core shear buckling do not happen and thus are neglected in Fig.8.

Correspondingly,the moment applied in the cross-section when the beam fails is given by

The failure load for the yielding of perforated beam subjected to three-point bending can then be obtained as

In addition to face or core yielding,panel buckling(PB)occurs in the top face,which is also a possible failure mode[31].The corresponding failure load can be expressed as

4.2 Shear failure

From Eq.(2),the thickness of core webs varies linearly with ω.Meanwhile the width of the perforated face,L b,varies linearly withω as shown in Fig.3.When ω =0,the sandwich is intact without perforation,as shown in Fig.1a.When ω=1,the sandwich has no bottom facesheet,i.e.,it degrades to the stiffened plate of Fig.1c.With ω taking values in the range of 0−1,the perforated area of the bottom face increases as ω is increased,resulting in the increasing thickness of core webs t c and decreasing width of bottom face L b.

while its failure load of shear buckling can be given as

where

研究数据来源于调研组于2016年7-8月在江阴市、滁州市、上海市以及南京市等地制造业工厂所进行的问卷调查，该调查主要聚焦于农民工的新媒体使用现状，内容涵盖人口统计学基本特征、工作现状以及农民工新媒体使用的方式和具体内容等。本次调查共发放问卷800份，其中有效问卷686份，问卷有效率为86%。对于新老两代农民工的划分则采用国家统计局发布的农民工调查监测报告中关于新生代农民工的定义，即将30岁作为新老两代农民工的划分标准，共筛选出365份新生代农民工样本。

and tc is defined in Eq.(2).

4.3 Comparison between analytical predictions and FE simulations

Fig.8 Comparison between analytical predictions and FE simulations for failure loads of perforated sandwich beam plotted as functions of perforation ratio

When ω is zero or small,the neutral axis is located in the central portion of the core;when ω is large,the neutral axis is shifted to the top face.The critical perforation ratio ω0 corresponding to the case when the neutral axis is just shifted to the top face can be obtained as

随着我国改革开放以后国民生活水平的不断提高，劳动人民的工作环境以及工作方式都得到了不同以往的巨大良性变化。加之国民人均寿命的不断延长，我国劳动人民想工作、能工作的欲望和能力都在增长，因此，在延迟退休的基础上，要进行与之相关的养老金和养老保险的配套改革。在人口老龄化加剧的今天，社会保险基金面临的压力不言而喻，延迟退休是国际趋势，想要完成政策目标，同步推进养老金与养老保险的改革势在必行。要从费率与费基以及劳动人民的工作年限、劳动人民的缴费年限以及目前所希望的延迟退休年龄进行综合考虑，完成配套性的改革。

As can be seen from Fig.8,for both bending and shear typesof failure,the failure loads obtained from analytical predictions and FE simulations compare well with each other.For a given perforation ratio,the beam fails with the smaller load of shear or bending.For instance,the beam with intact bottom face(ω=0)fails in shear,since the bending failure load is larger than that of shear failure.Whenω increases,the area of perforations in the bottom face becomes larger and the thickness of the core webs increases,resulting in a weaker face and a stronger core.Consequently,the bending failure load decreases while the shear failure load increases.At a certain stage,the perforated beam initiates bending failure and shear failure simultaneously,thus reaching its maximum(optimal,though somewhat naive)strength.Let ωopt represent the corresponding perforation ratio.As the perforation ratio is further increased,bending failure dominates and the beam failure load decreases.

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4.4 Effects of core height to beam span ratio on maxim um strength and optimal perforation ratio

For the perforated sandwich beams considered in the present study,it has been demonstrated that their failure loads can be adequately evaluated using analytical predictions of both the bending and shear failures.Next,the dependence of bending strength and shear strength upon beam aspect ratio H c/L is quantified.Figure 9a plots the bending and shear failure loads as functions of perforation ratio ω for selected values of H c/L in the range of 0.07–0.16.Corresponding results obtained from FE simulations and theoretical analysis are presented in Table 2.These results show good agreement between simulations and theoretical analysis.Initially,the failure load is seen to increase nearly linearly with increasing ω and the beam fails in shear at its infancy.Then,the failure load peaks when the perforation ratio is optimized at ωopt.Beyond this optimal perforation ratio,the failure load decreases with increasing ω and the beam fails in bending for all selected values of H c/L.

Figure 9b shows that ωopt increases almost linearly with H c/L.For a long beam with small H c/L,ωopt is small and tends to zero,which means the facesheet perforations do not significantly strengthen the beam.In contrast,for a short beam with large H c/L,ωopt is large and tends to 1,which means the perforated holes significantly enhance the beam strength.To be more specific, for a perforated sandwich beam with ωopt,the ratio of its failure load to that of the nonperforated one is plotted as a function of H c/L in Fig.9b.Again,as far as bending strength is of concern,it is seen that facesheet perforations are more beneficial to short beams rather than the long ones.

Fig.9 a Normalized failure load plotted as a function of perforation ratio (ω) for selected values of beam aspect ratio(H c/L).b Optimal perforation ratio and failure load ratio of perforated sandwich beam to an intact one plotted as functions of H c/L

Table 2 Error analysis of results obtained from simulations and theoretical analysis

H c/Lω FE simulations(V2/(EM)(×10−6))Analytical predictions(V2/(EM)(×10−6))Error(%)0.16 0 4.2 3.8 −9.8 0.3 5.4 5.0 −7.9 0.6 5.8 6.0 3.0 0.7 5.2 5.3 0.8 0.8 4.5 4.6 1.7 1 3.3 3.4 1.6 0.12 0 3.1 2.8 −9.5 0.3 4.2 3.9 −6.4 0.4 4.0 4.2 3.2 0.5 3.7 3.7 0.7 0.7 2.9 2.9 0.0 1 1.9 2.0 3.9 0.1 0 2.5 2.8 11.4 0.15 3.0 3.1 1.4 0.25 3.2 3.1 −3.6 0.35 2.8 2.8 −2.2 0.5 2.3 2.3 −0.7 0.75 1.7 1.8 3.4 1 1.2 1.3 9.7 0.08 0 2.1 1.9 −9.4 0.1 2.4 2.2 −7.4 0.2 2.3 2.3 −2.5 0.3 2.0 2.0 −0.6 0.5 1.6 1.6 1.7 0.7 1.3 1.3 0.0 1 0.9 0.9 6.5 0.07 0 1.8 1.6 −9.2 0.1 2.0 1.9 −3.9 0.2 1.6 1.6 −0.1 0.3 1.4 1.5 2.1 0.5 1.2 1.2 0.7 0.7 0.9 0.9 1.3 1 0.7 0.7 6.4

Finally,it should be pointed out that because the main focus of the present study is placed upon practical applications like the decks of cargo vehicles and transportation ships,the failure of the upper facesheet is not allowed by specifying sufficiently large initial thickness of the upper facesheet.Under such conditions,the core would most likely fail ahead of the facesheet.In a future study, systematic variation of geometrical and loading parameters will be carried out to reveal if the proposed design still has advantages if the failure of the upper facesheet occurs.

5 Concluding remarks

The concept of facesheet perforations has been proposed to enhance the resistance of square honeycomb-cored sandwich beams to three-point bending,in addition to facilitating fabrication procedures and access for maintenance during service.Both FE simulations and analytical modeling are used to evaluate the concept,with good agreement achieved between predictions obtained using the two approaches.Targeting applications like the decks of cargo vehicles and transportation ships,failureof the top facesheet such as local indentation is not allowed by specifying sufficiently large initial thickness of the upper facesheet.Two main failure modes,bending failure and shear failure,then occur in the perforated sandwich beams.The corresponding failure loads are characterized as functions of the perforation ratio.

随着公众社会意识的加强，大众对企业的要求不再仅是其提供的产品和服务，同时，还要求企业对社会有更大程度的贡献。如中化国际对当地动植物等环保领域的关注，以及他们对民生等方面的可持续支持。这些行为属于伦理层面和慈善层面的企业社会责任行为，对于国际化企业获得当地社区、政府、非政府组织以及大众等利益相关者的信任和支持，对当地社会环境的保护以及社会和谐都有着重要意义。因此，好的伦理责任和慈善责任履行能够进一步加强国际化企业在当地公众心中的品牌形象，对于企业品牌影响力的提升具有深远影响。

The shear failure load is found to increase with increasing perforation ratio because of a stronger core,for the mass of the perforated sandwich beams is fixed as the perforation ratio is varied.In contrast,the bending failure load decreases with increasing perforation ratio due to a weaker bottom facesheet.Consequently,when the two types of failure load are equal,the perforated sandwich beam reaches its maximum strength,which can be up to60%higher than that of the non-perforated one with equal mass:the corresponding perforation ratio is then optimal.The optimal perforation ratio increases as the beam aspect ratio is increased,and so is the strength enhancement relative to the non-perforated beam.In other words,with the core height fixed,facesheet perforations are more beneficial to short sandwich beams rather than long ones.Results obtained in the current study can help the design of large-scale sandwich constructions for applications like the decks of cargo vehicles and transportation ships where sandwich plates/shells with relatively thick facesheets are required.

Acknowledgements This work was supported by the National Natural Science Foundation of China(Grants 11472209,11472208),the China Postdoctoral Science Foundation(Grant 2016M 600782),the Postdoctoral Scientific Research Project of Shaanxi Province(Grant 2016BSHYDZZ18),the Fundamental Research Funds for Xi’an Jiaotong University(Grant xjj2015102),and the Jiangsu Province Key Laboratory of High-end Structural Materials(Grant hsm1305).

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