Buckling Analysis of Tapered Laminated Composite Plates Using Ritz Method

Thermo-mechanical post-buckling of FGM cylindricalpanels with temperature-dependent propertiesJ.Yang a ,K.M.Liew b ,Y.F.Wu a ,S.Kitipornchaia,*aDepartment of Building and Construction,City University of Hong Kong,Tat Chee Avenue,Kowloon,Hong Kong bNanyang Centre for Supercomputing and Visualization,School of Mechanical and Production Engineering,Nanyang Technological University,Nanyang Avenue,Singapore 639798,SingaporeReceived 27August 2004;received in revised form 1April 2005Available online 23May 2005AbstractThis paper presents thermo-mechanical post-buckling analysis of cylindrical panels that are made of functionally graded materials (FGMs)with temperature-dependent thermo-elastic properties that are graded in the direction of thickness according to a simple power law distribution in terms of the volume fractions of the constituents.The panel is initially stressed by an axial load,and is then subjected to a uniform temperature change.The theoretical formula-tions are based on the classical shell theory with von-Karman–Donnell-type nonlinearity.The effect of initial geometric imperfection is also included.A differential quadrature (DQ)based semi-analytical method combined with an iteration process is employed to predict the critical buckling load (where it is applicable)and to trace the post-buckling equili-brium path of FGM cylindrical panels under thermo-mechanical loading.Numerical results are presented for panels with silicon nitride and nickel as the ceramic and metal constituents.The effects of temperature-dependent properties,volume fraction index,axial load,initial imperfection,panel geometry and boundary conditions on the thermo-mechan-ical post-buckling behavior are evaluated in detail through parametric studies.Ó2005Elsevier Ltd.All rights reserved.Keywords:Thermo-mechanical post-buckling;Functionally graded materials;Temperature-dependent properties;Cylindrical panel1.IntroductionDue to their unique advantages of being able to withstand severe high-temperature environment while maintaining structural integrity,microscopically inhomogeneous functionally graded materials (FGM),0020-7683/$-see front matter Ó2005Elsevier Ltd.All rights reserved.doi:10.1016/j.ijsolstr.2005.04.001*Corresponding author.Tel.:+852********;fax:+852********.E-mail address:bcskit@.hk (S.Kitipornchai).International Journal of Solids and Structures 43(2006)307–324308J.Yang et al./International Journal of Solids and Structures43(2006)307–324whose material properties vary smoothly and continuously from one surface to the other by gradually changing the volume fraction of their constituent materials,have received considerable attention in many industries,especially in high-temperature applications such as space shuttle,aircraft,nuclear reactors and etc.,as is reflected in a great number of research papers published on this subject(e.g.,Koizumi,1993; Tanigawa,1995;Aboudi et al.,1997;Reddy and Chin,1998;Noda,1999;Reddy,2000;Reddy and Cheng, 2001;Liew et al.,2001b,2002,2003a,b,2004;Ng et al.,2002;Vel and Batra,2002;Kitipornchai et al.,2004; Yang and Shen,2003a,b;Yang et al.,2003,2004a,b).When FGMs are used as heat-shielding components with restrains against in-plane thermal expansions or contractions,significant thermally induced strains and stresses develop at elevated temperatures,which introduce a certain membrane pre-stress state that may ini-tiate buckling and post-buckling(Javaheri and Eslami,2002;Morimoto et al.,2003;Ma and Wang,2003; Na and Kim,2004).It is further envisaged that considerable variation in the material properties with tem-peraturefluctuations makes the mechanical response even more complicated,which calls for a thorough understanding of the buckling and post-buckling behavior of FGMs with temperature-dependent material properties being taken into account.Numerous investigations of the buckling and post-buckling responses of composite structures in thermal environments can be found in the literature,among which those incorporating temperature-dependent material properties are due to Chen and Chen(1989),Noor and Burton(1992a,b),Argyris and Tenek (1995),Feldman(1996),Deng et al.(2000),Lee(2001),Shen(2001)and Singha et al.(2003).Significant influence of temperature-dependent properties on the critical buckling temperature and post-buckling tem-perature–deflection curves has been reported.For FGM shell structures,studies on the buckling and post-buckling behavior under different thermal and compressive loading are limited in number.Shahsiah and Eslami(2003)discussed the instability of FGM cylindrical shells under two types of thermal loads based on thefirst order shell theory and the complete Sanders kinematic equations.Sofiyev(2004)dealt with the stability analysis of FGM truncated conical shells that are subjected to a uniform external pressure which is a power function of time.A Lagrange–Hamilton type variational principle was employed to solve the governing differential equations.In his work on the stability of functionally graded shape memory alloy sandwich panels that are subjected to the simultaneous action of a uniform temperature and a uniaxial compression,Birman(1997)found that,at elevated temperatures,the buckling load can be increased by using shape memory alloy(SMA)fibers in resin sleeves embedded within the core,at the midplane of the sandwich panel.When dealing with the post-buckling behavior of FGM shells,geometrically nonlinear kinematics will be involved and the analysis becomes much more complex.Shen(2002a,b,2003)and Shen and Leung(2003)conducted a series of studies on the post-buckling of FGM cylindrical shells and cylin-drical panels under pressure or axial compression.The effect of initial geometric imperfection was included in the analyses.Woo et al.(2003)presented an analytical solution for the post-buckling of FGM thin plates and shallow cylindrical shells under edge compression combined with a uniform temperaturefield by using mixed Fourier series.The above-mentioned work can be divided into two types:Thefirst type(those by Shahsiah and Eslami,2003and Woo et al.,2003)gave the thermal buckling or post-buckling analysis but the temperature-dependence of the material properties was not considered whereas the second type (those by Shen,2002a,b,2003;Shen and Leung,2003;Sofiyev,2004)used temperature-dependent material properties in compressive buckling or post-buckling analyses only.Within the framework of classical shell theory with von Karman–Donnell-type of kinematic nonlinearity,Shen(2004)recently investigated the thermal post-buckling behavior of imperfect FGM cylindrical thin shells offinite length.The material pro-perties were assumed to be nonlinearly dependent on the temperature,and a singular perturbation technique,together with an iteration process,was used to determine the buckling temperature and post-buckling temperature–deflection curves.Numerical results were presented for cylindrical shells that are made from two sets of material mixtures under a uniform temperature change.In the aforementioned stud-ies,the loading condition is either thermal or mechanical.The effect of a combined thermo-mechanical loading was not considered.To the best of our knowledge,no studies have been reported in the literaturethat concern the post-buckling behavior of functionally graded cylindrical panels with temperature-depen-dent material properties and subjected to a combined action of thermal and mechanical loadings.In this paper,the thermo-mechanical buckling and post-buckling of functionally graded cylindrical thin panels are investigated by using the classical shell theory.The material properties,which are modeled as nonlinear functions of the temperature,are assumed to vary smoothly along the thickness direction accord-ing to a power law distribution of the constituent materials.The theoretical formulations include the effects of thermal loads,initial geometric imperfection,and von-Karman–Donnell-type nonlinearity which consid-ers moderate deflections and small strains.The panel is subjected to a combined initial axial force and a uniform temperature change.Nonlinear governing differential equations are derived in terms of transverse deflection and stress function and are then transformed into a nonlinear algebraic system through a semi-analytical differential quadrature-Galerkin method followed by an iteration process that determines the buckling temperature and the post-buckling equilibrium path of FGM cylindrical panels under simply sup-ported,clamped,or mixed boundary conditions.Numerical results are provided for cylindrical panels made from silicon nitride and nickel to show the influence of temperature-dependent material properties,initial axial load,material composition,geometric imperfection,panel geometry,and boundary conditions on the thermo-mechanical buckling and post-buckling response of FGM cylindrical panels.2.Theoretical formulationsConsider an FGM cylindrical thin panel with radius of curvature R,thickness h,axial length L and arc length S that is subjected to an axial load p x combined with a uniform temperature change D T.The panel is made from a mixture of ceramics and metals,and is defined in a coordinate system(X,h,Z),as is shown in Fig.1,where X and h are in the axial and circumferential directions of the panel and Z is perpendicular to the middle surface and points inwards.Suppose that the material composition of the panel varies smoothly along the thickness in such a way that the inner surface is metal-rich and the outer surface is ceramic-rich by following a simple power law in terms of the volume fractions of the constituents asV c¼2Zþhn;V m¼1À2Zþhnð1Þwhere V c and V m are the ceramic and metal volume fractions and volume fraction index n is a nonnegative integer that defines the material distribution and can be chosen to optimize the structural response.The effective properties P eff,such as YoungÕs modulus E,PoissonÕs ratio m,and the coefficient of thermal expan-sion a,can be determined byP eff¼P mþðP cÀP mÞV cð2Þin which the subscripts‘‘c’’and‘‘m’’stand for ceramic and metal,respectively.As FGMs are most commonly used in high temperature environments in which significant changes in material properties are to be expected,the material properties of an FGM cylindrical panel are both posi-tion and temperature dependent.For example,YoungÕs modulus usually decreases,and the thermal expan-sion coefficient usually increases at elevated temperatures(Noor and Burton,1992b).It is essential to account for this temperature-dependence for reliable and accurate prediction of the structural response. Without loss of generality,the material properties are expressed as the nonlinear functions of environment temperature T(K)(Touloukian,1967)P¼P0ðPÀ1TÀ1þ1þP1TþP2T2þP3T3Þð3Þin which T=T0+D T and T0=300K(room temperature),and P0,PÀ1,P1,P2,and P3are temperature-dependent coefficients that are unique to the constituent materials.Thus,YoungÕs modulus E,PoissonÕs ratio m,and the coefficient of thermal expansion a can be written from Eqs.(1)–(3)asE¼ðE cÀE mÞ2Zþh2hnþE mm¼ðm cÀm mÞ2Zþh2hnþm ma¼ða cÀa mÞ2Zþh2hnþa mð4ÞIt is evident that E=E c,m=m c,a=a c at Z=h/2and E=E m,m=m m,a=a m at Z=Àh/2.LetðU;V;WÞbe the displacement components in the(X,h,Z)coordinates.The panel is assumed to be initially imperfect with geometric imperfection WÃ,and W is the additional deflection that is caused by ther-mo-mechanical loading.Denote the stress function by FðX;hÞwhich is related to the stress resultants byN x¼o2FR2o h,N h¼o2Fo X,N x h¼Ào2FR o X o h,and introduce the following dimensionless quantities:x¼XL;b¼LR;ðW;WÃÞ¼ðW;WÃÞðDÃ11DÃ22AÃ11AÃ22Þ1=4;F¼FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDÃ11DÃ22Þp;k1¼p x R2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDÃ11DÃ22Þpc12¼DÃ12þ2DÃ66DÃ11;c14¼DÃ22DÃ111=2;c22¼AÃ12þ0.5AÃ66AÃ22;c24¼AÃ11AÃ221=2;c5¼AÃ12AÃ22ðc30;c32;c34Þ¼1ðDÃ11DÃ22AÃ11AÃ22ÞBÃ21;BÃ11þBÃ22À2BÃ66;BÃ12ÀÁ;n¼LðDÃ11DÃ22AÃ11AÃ22Þðc M11;c M12;c M21Þ¼1ðAÃ11AÃ22DÃ11DÃ22Þ1=4ðBÃ21;BÃ11;BÃ22Þ;ðc M13;c M22Þ¼1DÃ11ðDÃ21;DÃ12Þð5ÞBased on the classical shell theory and von-Karman–Donnell-type kinematic relations,the dimensionless governing equations for an FGM cylindrical panel under mid-plane loading and a uniform temperature change can be derived as follows:310J.Yang et al./International Journal of Solids and Structures43(2006)307–324L11ðWÞþL12ðFÞÀc14bn o2Fo x2¼c14b2LðWþWÃ;FÞð6ÞL21ðWÞþL22ðFÞÀc24bn o2Wo x2¼À12LðWþ2WÃ;WÞð7Þwhere the partial differential operators areL11ðÞ¼o4o x4þ2c12b2o4o x2o hþc214b4o4o hL12ðÞ¼c14c30o4o x4þc32b2o4o x2o hþc34b4o4o hL21ðÞ¼Àc24c30o4o x4þc32b2o4o x2o hþc34b4o4o hL22ðÞ¼o4o x4þ2c22b2o4o x2o h2þc224b4o4o h4Àbno2o x2LðÞ¼o2o x2o2o h2À2o2o x o ho2o x o hþo2o h2o2o x2ð8ÞThe reduced stiffness elements AÃij ,BÃij,and DÃijare calculated fromAüAÀ1;BüÀAÀ1B;DüDÀBAÀ1Bð9ÞandðA ij;B ij;D ijÞ¼Z h=2Àh=2ðQ ijÞð1;Z;Z2Þd Zði;j¼1;2;6Þð10ÞFor FGMs with temperature-dependent material properties,the elastic stiffness Q ij are dependent on both temperature and position.The dimensionless moment resultants(M x,M h,M x h)areM x¼Àc14c M11o2Fo x2þc M12b2o2Fo hÀo2Wo x2þc M13b2o2Wo hþM Txð11aÞM h¼Àc14c M21o2F2þc34b2o2Fo hÀc M22o2W2þc214b2o2Wo hþM Thð11bÞM x h¼Àc14c M31o2Fo x2þc M32b2o2Fo h2Àc M33o2Wo x2þc M34b2o2Wo h2þM Tx hð11cÞwhereðM Tx ;M Th;M Tx hÞ¼ðM Tx;M Th;M Tx hÞL2=DÃ11ðAÃ11AÃ22DÃ11DÃ22Þ1=4ð12ÞThe thermal force resultantsðN Tx ;N Th;N Tx hÞand moment resultantsðM Tx;M Th;M Tx hÞare given byN Tx M TxN Th M ThN Tx h M Tx h2 66 43775¼ÀZ h=2Àh=2Q11Q12Q16Q12Q22Q26Q16Q26Q66264375100100264375aa&'ð1;ZÞD T d Zð13ÞJ.Yang et al./International Journal of Solids and Structures43(2006)307–324311The panel may be either simply supported or clamped on each of its edges.The associated out-of-plane boundary conditions readSimply supportedðSÞ:W¼0;M x¼0or M h¼0ð14aÞClampedðCÞ:W¼0;o Wo x¼0oro Wo h¼0ð14bÞDepending on the nature of constraints against mid-plane displacements,the following cases of in-plane boundary conditions,referred to as Case1and Case2respectively,are considered in the analysis:Case1:The uniform axial load is applied on the curved edges x=0,1.The panel is freely movable in the x-direction but is immovable at the unloaded straight edges h=0,h0.Case2:The panel is fully immovable at all edges.The in-plane boundary conditions require thato2F o x o h ¼0;Z h0o2Fo h2d hþk1¼0or V¼0ðCase1Þð15aÞoro2Fo x o h¼0;U¼0or V¼0ðCase2Þð15bÞin whichU¼1bZ h0Z1b2c224o2Fo h2þc5o2Fo x2Àc24c M12o2Wo x2þc34b2o2Wo h2Àc240.5o2Wo x2þo Wo xo WÃo xÀðc224N Txþc5N ThÞ!ð16aÞV¼1bZ h0Z1o2Fo x2þc5b2o2Fo hÀc24c30o2Wo x2þc M21b2o2Wo hÀc24b20.5o2Wo hþo Wo ho WÃo hþc24nb WÀðc5N Tx þN ThÞ!ð16bÞwhereðN Tx ;N Th;N Tx hÞ¼ðN Tx;N Th;N Tx hÞL2=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDÃ11DÃ22Þp.It is worth noting that the thermal effect terms,which vanish in the governing equations(6)and(7)due to the uniform distribution of D T,are present in the boundary conditions(14)and(15).3.Solution proceduresThe governing equations(6)and(7)together with the boundary conditions(14)and(15)form a nonlin-ear partial differential system that is also dependent on temperature.A differential quadrature(DQ)-based semi-analytical iteration approach is used to solve this nonlinear system.The essence of this method isfirst to convert the partial differential equations into a set of ordinary differential equations by making use of DQ approximation,and then to apply the Galerkin technique to obtain a nonlinear algebraic system from which the post-buckling equilibrium path is determined through an iterative process.312J.Yang et al./International Journal of Solids and Structures43(2006)307–324The solutions of W and F are constructed in the form ofW¼X Mm¼1a m W mðx;hÞð17aÞF¼Àh22k xÀx22k hþX Mm¼1b m F mðx;hÞð17bÞwhere M is the number of terms in the solution series,a m and b m are the unknown constants to be deter-mined,and k x and k h are the edge forces acting on the curved and straight edges which can be determined from in-plane boundary conditions(15).According to DQ rule,the k th partial derivative of an unknown function with respect to a coordinate, say,x,at a discrete point is approximated as the linear weighted sums of its values at all of the pre-selected sampling points along this coordinate.In this way,we haveo kðW m;F mÞo x kx¼x i¼X Nj¼1CðkÞijðW mj;F mjÞð18Þwhere N is the total number of sampling points,and CðkÞij are the weighting coefficients that are dependent on the sampling grid only and can be calculated from recursive formulae given by Bert and Malik(1996) and Liew et al.(2001a).In this study,the sampling points are unevenly distributed in x-axis asx1¼0.0;x2¼0.0001;x j¼121ÀcospðjÀ2ÞNÀ3!;x NÀ1¼0.9999;x N¼1.0ð19ÞIn Eq.(18),W mj=W m(x j,h)and F mj=F m(x j,h).They are to be further modeled in terms of orthogonal functions that satisfy the boundary conditions at straight edges h=0,h0and take the form ofF mj¼sinðmþ0.5ÞphÀsinhðmþ0.5ÞphÀn mðcosðmþ0.5ÞphÀcoshðmþ0.5ÞphÞn m¼sinðmþ0.25ÞpÀsinhðmþ0.25Þpcosðmþ0.25ÞpÀcoshðmþ0.25Þpð20ÞS–S:W mj¼sin m phð21aÞC–C:W mj¼F mjð21bÞS–C:W mj¼sinðmþ0.25ÞphÀn m sinhðmþ0.25Þph;n m¼sinðmþ0.25Þpsinhðmþ0.25Þpð21cÞSubstitution of Eqs.(17),(18),(20),and(21)into the governing equations(6),(7)and the boundary con-ditions(14)and(15),and then application of the Galerkin technique leads to a set of nonlinear algebraic equations in terms of the unknown coefficients a mj and b mj(m=1,...,M,j=1,...,N) GðT;DÞD¼UðTÞð22Þwhere D is an unknown vector which is composed of a mj and b mj,G is the nonlinear matrix that is dependent on both the temperature and D.The right-hand side term U comes from the thermally induced stress resul-tants and bending moments in simply supported boundary conditions at straight edges h=0,h0,and will automatically vanish when the panel is isotropic where the stretching–bending coupling is absent,or when the panel is clamped and only displacement boundary conditions are involved.It is obvious that the bifur-cational thermal buckling will take place only when U=0,otherwise transverse deflection will be induced, irrespective of the magnitude of the temperature change.J.Yang et al./International Journal of Solids and Structures43(2006)307–324313314J.Yang et al./International Journal of Solids and Structures43(2006)307–324 The thermal buckling temperature,when it exists,is determined from the nonlinear homogeneous equationGðT;DÞD¼0ð23Þby an iterative numerical procedure with the following steps.(1.1)Solve the buckling temperature D T cr from Eq.(23)by using temperature-independent material pro-perties,that is,the thermo-elastic properties at reference temperature T0.(1.2)Update G by using the property values at T=T0+D T cr to obtain a new buckling temperature. (1.3)Repeat step(1.2)until the thermal buckling temperature converges to a prescribed error tolerance.The nonlinear temperature–deflection curve,which is also known as the post-buckling equilibrium path, is traced by two different iterative schemes depending on the presence of U.When U=0,the following iter-ation process is applicable:(2.1)Begin with the dimensionless deflection W/h=0at a specific point;(2.2)Use the iterative procedures(1.1)–(1.3);(2.3)Specify a new value of W/h;(2.4)Calculate the thermo-elastic properties at T=T0+D T cr,and scale up the buckling mode that isobtained in step(2.2)to form a new G to determine the post-buckling temperature;(2.5)Repeat step(2.4)until the post-buckling temperature converges to a prescribed error tolerance; (2.6)Repeat steps(2.3)–(2.5)to obtain the post-buckling equilibrium path.The modified Newton–Raphson technique is used in case of U50,but the process is omitted here for brevity.4.Numerical resultsAs there are no suitable results on the thermo-mechanical buckling and post-buckling of FGM cylindri-cal panels for direct comparison,thermal buckling of a clamped anti-symmetric angle-ply laminated cylin-drical panel under a uniform temperature increment is solved and critical temperature results are compared in Table1with the existing ones obtained by Chang and Chiu(1991)as the validation of the present anal-ysis.The material properties are temperature independent in this instance,and are given as E1=E0¼21;E2=E0¼E3=E0¼G12=E0¼1.7;G13=E0¼0.65;G23=E0¼0.639;E0¼106psi m12¼m13¼0.21;m23¼0.33;a1¼À0.21Â10À6=F;a2¼a3¼10À6=FThe panel is fully clamped and immovable at all edges with lamination[±42.5°]3,and the geometry param-eters are L/S=1.0,S/R=0.25and S/h=200.Good agreement is achieved when the number of grid points Table1Comparisons of buckling temperatures D T cr(F)for a clamped antisymmetric angle-ply cylindrical panel subjected to uniform temperature changePresent Chang and Chiu(1991) (M,N)=(3,9)(M,N)=(5,13)(M,N)=(5,15)(M,N)=(7,19)784.76851.30846.24847.55831.88J.Yang et al./International Journal of Solids and Structures43(2006)307–324315 Table2Temperature-dependent thermo-elastic coefficients for silicon nitride and nickelThermo-elastic properties Material PÀ1P0P1P2P3E(Pa)Silicon nitride0348.43e9À3.070eÀ4 2.160eÀ7À8.946eÀ11Nickel0223.95e9À2.794eÀ4 3.998eÀ90m Silicon nitride00.2400000Nickel00.3100000a(1/K)Silicon nitride0 5.8723eÀ69.095eÀ400Nickel09.9209eÀ68.705eÀ400N and the number of series terms M are increased to(N,M)P(15,5).Our solutions are slightly higher than Chang and ChiuÕs results because the higher order shear deformation shell theory,instead of classical shell theory,was used in their work.Parametric studies are then conducted to supply information on both the buckling temperature and the post-buckling equilibrium paths of various FGM cylindrical shells with temperature-dependent material properties when they are subjected to axial pre-stress and uniform temperature change.Silicon nitride and nickel are chosen to be the constituent materials of the FGM panel,referred to as Si3N4/Ni.The tem-perature-dependent material constants for YoungÕs modulus E,PoissonÕs ratio m and the coefficient of ther-mal expansion a are listed in Table2.To highlight the effect of temperature-dependence of thermo-elastic properties,comparisons are made between the solutions using temperature-dependent and temperature-independent properties in all of the numerical examples(Tables3,4and Figs.2–9).In computation,the temperature-independent properties are(Gauthier,1995):E=310GPa,m=0.24,a=3.4·10À61/K for silicon nitride and E=204GPa,m=0.31,a=13.2·10À61/K for nickel.In what follows,a clockwise notation that starts from X=0is employed.‘‘SCSC’’,for example,stands for a panel simply supported at edges X=0,L and clamped at edges h=0,h0.Three axial loading cases are considered in the analysis,i.e.,(1)the panel is subjected to a tensile force(negative k1),(2)the panel is sub-jected to a compressive force(positive k1),and(3)the panel is free from axial force(k1=0).The error tolerance in the iteration process is defined as the relative difference between the two consec-utive solutionsTable3Buckling temperature parameter k cr=a0D T cr·103for clamped FGM cylindrical panels subjected to uniform axial load(S=0.3m, L/S=2.0,S/h=100)Material composition Temperature-dependent solutions Temperature-independent solutionsk1=À200p2k1=200p2k1=À200p2k1=200p2S/R=0.3Si3N4 3.1742 2.9445 3.7492 3.4393n=0.5 2.5941 2.4012 2.9739 2.7262n=2.0 2.3209 2.1373 2.6174 2.3888n=10.0 2.2018 2.0218 2.4628 2.2424Nickel 2.0284 1.8866 2.2554 2.0829S/R=0.5Si3N4 5.9231 5.67397.92597.5118n=0.5 4.9754 4.7585 6.6366 6.0295n=2.0 4.4242 4.2225 5.5009 5.2032n=10.0 4.1417 3.9467 5.0711 4.7919Nickel 3.9086 3.728 4.7516 4.4948。

基于经典层合板理论的简支梁力学分析陈玲俐;赵晓昱【摘要】与传统的金属材料相比,复合材料具有比强度高、比模量大,耐疲劳性、耐腐蚀性好,且热性能和电性能良好等优点.本文选择复合材料对简支梁进行铺层设计,首先从简支梁的受力分析入手,根据复合材料力学的经典层合板理论和弹性力学的基本方程,建立简支梁的数学模型;然后对简支梁进行结构设计,确定简支梁的铺层角度与铺层数目,构建复合材料结构,对复合材料简支梁进行静力学分析;最后利用蔡吴张量准则进行强度校核.【期刊名称】《力学与实践》【年(卷),期】2019(041)002【总页数】5页(P152-156)【关键词】复合材料;简支梁;数学模型;静力学分析;强度比【作者】陈玲俐;赵晓昱【作者单位】上海工程技术大学机械与汽车工程学院,上海 201600;上海工程技术大学,上海 201600【正文语种】中文【中图分类】O342复合材料具有高模量、高强度、轻量化等优点，已经广泛应用于航空航天、军事、汽车等工业领域。

为了提高复合材料的结构性能，许多研究学者也在复合材料结构优化方面给予了较多的关注。

Kazemi等[1]提出了弯曲刚度和弹性模量约束的优化，设计了最大弯曲刚度和层压板有效弹性模量，通过层合板的最优取向，使层合板具有最大的屈曲强度。

Shimoda等[2]提出自由形状的优化方法，从理论上解决了形状梯度函数中柔度最小化问题，并在无形状参数的情况下，确定了可以显著降低由不同材料组成的复合结构的柔度的最佳界面和表面形状。

Ruoshui等[3]将非均质的蜂窝材料做成均匀正交各向异性体进行建模，并通过研究细胞壁厚度和弹性模量对结构整体力学响应的影响，实现了复合材料性能的优化。

针对复合材料组合梁的设计优化，Nguyen等[4]基于高阶剪切变形梁理论，提出了一种新的复合材料层压梁在热载荷作用下的屈曲和振动混合形状函数，分析了梁的跨高比、边界条件、材料各向异性和温度变化对组合梁屈曲载荷和固有频率的影响。

铰接中心支撑钢框架阶形柱计算长度系数于海丰;张文元【摘要】With the research background of the concentrically braced steel frame with pinned connections for a typical main factory building of a large thermal power plant, the effective length factors for stepped columns were studied. Based on the small⁃deformation elastic stability theory, the effective length factor for the two⁃stepped column with one side pinned and another placed elastic support was derived and verified in comparison with the value calculated by the eigenvalue buckling analysis. The derived and calculated results are in good a⁃greement. Taking the 3⁃axis transverse frame selected from the large thermal power plant as an example, the effective length factor and stability bearing capacity for columns were verified by the eigenvalue buckling analy⁃sis andelasto⁃plastic buckling analysis. The results showed that the stability bearing capacity calculated by the numerical analysis is in good agreement with the theoretical value. For the columns in the concentrically braced steel frame with pinned connections, it is suggested that the effective length factor should be calculated by the derived formulas in this paper. The advantages are that the lateral displacement type of frame does not need to be judged firstly, and that the restriction effect of the upper and lower columns on the research column can be considered for the sideway columns.%以某火力发电厂主厂房铰接中心支撑钢框架体系为研究背景，对该体系中阶形柱计算长度系数进行了研究。

大型储罐基础沉降及罐体变形检验方法及应用刘民【摘要】大型石油储罐基础沉降及罐体变形是影响储罐安全服役的重要因素,介绍了储罐基础沉降及罐体变形检验方法,包括基准点设计、观测点布置和观测方法.为了有效监测石油储罐的基础沉降及罐体的椭圆化变形,以某双盘式浮顶油罐为例,采用全站仪巡视监测储罐外表面及环墙相对坐标的方法检验储罐沉降及变形情况.根据观测数据计算储罐基础沉降量,采用最小二乘拟合法计算储罐椭圆化变形情况,可指导储罐的运行管理.【期刊名称】《石油化工设备》【年(卷),期】2018(047)004【总页数】4页(P97-100)【关键词】储罐;基础沉降;变形;椭圆度;检验【作者】刘民【作者单位】中国石油大连石化分公司,辽宁大连 116011【正文语种】中文【中图分类】TQ053.2;TE972石油和天然气存储及装卸场区是连接产、运、销各环节的纽带，其安全、高效运营尤为重要。

场区内的储罐是原料储备、油品调合及成品油输转的重要设备，其内部储存着易燃、易爆、会对环境产生污染的介质，一旦发生安全事故后果严重[1]。

因此，储罐的安全服役对于石油储运系统至关重要。

大型储罐基础沉降及罐体变形是影响储罐安全服役的重要因素，也是评价储罐罐体安全性的重要指标。

大型储罐在服役过程中会发生一些沉降及变形，一般分为均匀沉降、整体平面倾斜沉降、不均匀沉降和底板沉降[2]。

其中不均匀沉降对储罐的影响最大，主要会引起储罐的径向变形和罐壁的超应力。

储罐的径向变形通常称为椭圆化变形，过大的椭圆化变形将引起无顶盖储罐罐顶产生很大的径向位移，对浮顶罐则会影响到浮顶的自由升降，使顶盖活动失灵[3]。

同时，罐体周围的不均匀沉降会引起很大的罐底局部轴向压应力和檐梁周向应力，导致罐壁局部屈曲和檐梁屈服，使罐壁撕裂，储液溢出，甚至造成安全事故。

国内外大量储罐工程事故分析表明，地基不均匀沉降是导致储罐破坏的主要原因，因此应在储罐服役过程中监测储罐的不均匀沉降及罐体的椭圆化变形。

Topology Optimization of Continuum Structures Under Buckling andDisplacement ConstraintsBIAN Bing-chuanDepartment of Applied Science and TechnologyTaishan UniversityTaian , ChinaE-mail:bianbingchuan@Abstract—In this paper, the topology optimization model for the continuum structure was constructed. The model had the minimized weight as the objective function subjected to the buckling constraints and displacement constraints. Based on the Taylor expansion and the filtering function, the objective function and the constraints were approximately expressed as an explicit function. The optimization model was translated into a dual programming and solved by the sequence second-order programming. All the corresponding numerical procedures are developed by the PCL toolkit in the MSC.Patran/Nastran software platform. Numerical examples show that this method can solve the topology optimization problem of continuum structures with the buckling and displacement constraints efficiently and give more reasonable structural topologies.Key words- buckling constraints; displacement constraints; ICM method; topology optimization; filtering functionI.I NTRODUCTIONThe intensity, the stiffness and the buckling are very important factors determining the safety of a structure. Structural failure due to buckling of one part is catastrophic. Buckling has attracted more attention in recent years.As the structural static optimization and the dynamical optimization, there are three development phases of the buckling structural optimization. They are cross-sectional, shape, and topology optimizations. The cross-sectional and shape optimization considering buckling constraints were developed by many scholars.The Optimum design with size and shape variables and sensitivity analysis for buckling stability of complex built-up structures and composite material were made byGu yuanxian et al [1]. The plate’s shape and frame’scross-sectional optimization were studied by D. ManickarSUI Yun-kangThe Centre of Numerical Simulation for EngineeringCollege of Mechanical Engineering and AppliedElectronics TechnologyBeijing University of TechnologyBeijing, China-ajah et al [2] using the Evolutional Structural Optimization (ESO) method. The studies by Rong Jian-hua et al [3] reported that the cross-sectional optimum designs of the frame under the maximum bucking critical load using ESO method.The topological optimization studies considering buckling is seldom due to difficulty. The topological optimization of truss studies by Guo Xu[4] reported that the model for the truss structure is constructed, which the minimized weight as the objective function has subjected to the local buckling constraints and global buckling constraints. The optimum designs reported by Zhou Ming [5] are the shell’s topological optimization under linear buckling response using variable-density method.From above we can see that several mechanical properties were difficultly considered and the weight’s upper limit was difficultly confirmed when the object was minimum buckling eigenvalue. In order to overcome these difficulty, based on the independent continuous and mapping idea [6, 7], the continuous independent topological variables are used in this problem. The topology optimization model for the continuum structure is constructed, which the minimized weight as the objective function has subjected to the buckling and displacement constraints.2009 International Conference on Information Technology and Computer ScienceII.F ORMULATION AND SOLUTION OF THE TOPOLOGICALOPTIMIZATION MODELA.The topological optimization model of the continuum structureThe buckling critical load is one structure’s inherent character. It is not alter with the outer load alter. Buckling analytic eigenvalue equation can be represented as)1(,J ,j j j0u G K ˅˄O (1)The symbols K ,G ,j O and u j denote stiffness matrix, thegeometric stiffness matrix, the j-th eigenvalue, and the corresponding eigenvector, respectively.The structure’s displacement which is the key point’s displacement must be limited. So the displacement constraints can be represented asrir u u d (2)Where ir u is the i-th element’s r -th displacement andr u is upper limit of r -th displacement, respectively.In the present study, the filter functions of weight, stiffness matrix, geometric stiffness matrix for one element, are denoted as followsa i i w t t f )(b i i k t t f )( bi i g t t f )((3)Where a and b are constants, respectively. Numerical example demonstrate that the optimal results with most of elements having value 0 or 1can be obtained using the filter functions in which a =1,b =3.3.As a consequence, the variables such as weight, stiffness matrix, geometric stiffness matrix, are identified by the filter functions mentioned previously0)(i i w i w t f w ˈ0)(i i k i t f k k ˈ0)(i i g i t f g g (4)Where 0i w is the weight,0i k is the stiffness matrix,0i g is the geometric stiffness matrix, of the original structure element before topology optimization, respectively. Finally, the topology optimization model for the continuum structure is constructed, which the minimized weight as the objective function has subjected to the buckling and displacement constraints.°°°°¯°°°°®­ d d d d ¦¦ )1;1;1(10))(())()((s.t.)(min find 1J ,...,j R ,...,r N ,...,i t t f u t f ,t f w t f W E i r N i i k ir j i g i k j N1i ii w N O O t (5)Where W is the total weight of the structure, w i and N are the weight of i -th element and the total number of element,j O is the upper limit of the critical load ,J i s the total number of the buckling mode ,ir u isthe i -th element’s r -th displacement and r u is upper limit of r -th displacement, respectively.B.Explicit approximate function of the constraint 1) Explicit approximate function of the buckling constraint: Since eigenvalue O is implicitly related with topology variable t , eigenvalue can be explicit expression by the first-order Taylor expansion if the first-order derivative of eigenvalue O has been obtained. So to obtain the first-order derivative of eigenvalue O is the most important approach to optimum design.The expression for the eigenvalues is obtained by Eq. (1); i.e.jj j j j Guu Ku u TT / O (6)The stiffness matrix and geometric stiffness matrix can be obtained)(10t K k K ¦ Ni bi i t ˈ)(10t G gG¦ Ni bi it (7)The inverse variable is importedbi i t x /1 (8)From all of above, we can obtain)2//()2/2/(/TT T i j j j i j j j i j i j x x Gu u u g u u k u O O w w ij ij j ij x V V U 6O /)( (9)Where ji j ij U u k u T5.0 i s the mode strain energy for the i -th element under j -th mode .ji j ij V u g u T5.0 is the mode geometric strain energy for the i -th element under j-th mode .i j j j x V Gu u T 5.0 6is the total mode geometric strain energy for the structure under j-th mode. For bucking constraint j j O O d , using first-order Taylor expansion, theapproximate expression of eigenvalue can be obtained ˖¦¦ 66 d Ni jk ij j k ij k j j k i j k ij j k ijNi i V V U x V V Ux 1)()()()()()(1/)()(/)(O O O O x (10)Where superscript k is the number at the k -th iteration. Ifwe define j k ij j k ij k ij V V U A 6 /)()()()(O ,the left-hand of inequation re-written as ˖¦ Ni k i j k ij j k ij i x V V Ux 1)()()(/)(6O iNi ijNi i k ij b k i Ni k i i k ij x dx A t x x A ¦¦¦11)()(1)()()(/(11)The right-hand of inequation re-written as ˖jNi k ij k j j Ni j k ij j k ij k j j e A V V U ¦¦ 1)()(1)()()()(/)()(x x O O O O O 6(12)Then all kinds of buckling constraints can be simplified by the formj i Ni ije x dd ¦ 1(13)2) Explicit approximate function of the displacement constraint: The generalized displacement of one node can be expressed as¦³ Ni iir dvu 1R T V )(˅˄İı(14)Where Vi ıis the stress component of i-th element under the unit virtual load .R i İ is the strain component of i-th element under the unit real load.So the Eq. (14) can be written as follow¦¦³ Ni i i Ni i i r dv u 1VT R 1R T V )()(u F İı˅˄(15)Based on the Mohr’s theorem we can obtain ˖¦¦ N i i i k i i Ni i i r x x u 1VT R )(1VT R )()(u F u F (16)Let ir k iii c x)(VTR /)(u F ˈthe displacement constraintwas expressed as explicit approximate function ˖r i Ni ir Ni i i k ii r x c x x u d ¦¦ 11VT R )()(u F (17)C. Explicit approximate function of the objective function The object function can be expressed as¦N1i i i w w t f W 0)((18)Where the filter function is a i i w t t f )( the filter function can be re-written asb a i i w x t f //1)((19)If we define E b a /ˈthe Eq.(19) is translated into E i i w x t f /1)( ˈ then, the object function is re-written as¦Ni i i x wW 10/E(20)The object function is translated into explicit expression by second-order Taylor expansion}]/)2([]2/)1({[1)(0212)(0i k i i i Ni k i i x x w x x w W ¦E E E E E E (21)Where )(k ix is the initial design variable of i-th element at the k -thiteration.We define 2)(02/)1( E E E k i i i x w A ,1)(0/)2( E E E k i i i x w B The topology optimization model can be re-written as°°°°°¯°°°°°®­ d d d d ¦¦¦)1(1)1()1(s.t .)(min find112N ,...,i x R ,...,r ux c J ,...,j e x dx B x A WE ii ri Ni ir ji Ni ijN1i i i i i Nx(22)D.Solution of the topological optimization modelIn order to reduce the number of the variable and the model’s scale. The optimization model is translated into a dual programming°¯°®­t 0s.t.)(max find ȤȤȤĭ(23)In Eq. (23) ˈ»¼º«¬ª ȘμȤ,),,(min ),(1Șμx ȘμL ĭii x x d d We can get the second-order programming model°°¯°°®­t 0s.t.)(21)(min find T 0ȤȤD ȤH D ȤȤȤȤT T ĭ(24)Where ĭ2D ˈĭH The second-order programming model was solved ˈwecan get Ȥ. The optimum values of topological variables *tare accordingly obtained. Then the structural analysis and topological optimization are renewably begun untilsatisfying the following convergence condition which isH'd |/)(|)1()()1(k k k W W W W (25)Where )(k W and )1( k W is the structural weight ofprevious iteration and current iteration, respectively, H theconvergence precision. In the present study, 001.0 H is employed.III.N UMERICAL EXAMPLESA.Example 1Illustrating in Figure.1 (a), the base structure is a plane elastic body with size 80.0u 50.0u 1.0mm. The elastic modulus of structure material is E=1.0u 106MPa, and thePoisson's ratio is J =0.3, and the material density is U =1.0h 10-3kg /mm3. The right-hand points of the structure are clasped. A concentrated force P=9000N along with the up-to-down (-y) direction is located in right-bottom of structure. Quadrilateral element with 4-node is used and number of total elements is 48u 30.The base structure’sweight is 4kg.(a)Base structure(b)Optimal topology configuration subjected tobuckling constraints(c)Optimal topology configurationsubjected to displacement constraints of reference [7](d)Optimal topology configuration subjected to buckling and displacementconstraintsFigure 1.Definition and optimal results of example 1In this example the maximal value of the critical load is 1O ˙1100N and the maximal value of the displacement is 0.5mm.After 47 times iterations the optimal topology configuration depicted in Figure 1 can be get. From figure 1, we can see that the topologyconfiguration subjected to displacement constraints in reference [7] is thin and uniformity. The topologyconfiguration with two constraints is not same as that with single constraint. The middle bar is shortened and the whole structure’s stability is strengthened. B.Example 2Illustrating in Figure.2 (a), the base structure is a solid elastic body with size 120u 45u 8mm. The elastic modulus of structure material is E =7.0u 104MPa, and the Poisson's ratio is J =0.3, and the material density is U =2.7×10-6kg /mm 3. The right-hand and left-hand bottom corner points of the structure are clasped. A distributed force P =12500N along with the up-to-down (-y ) direction is located five nodes in top of surface. Hexahedron element with 8-node is used and number of total elements is 40u 15u 4.The base structure’s weight is 0.117kg.In this example the maximal value of the critical load is1O ˙21000N and the maximal value of the displacement is 0.4mm. After 34 times iterations the optimal topology configuration depicted in Figure 2 can be get.(a) Finite element model(b)Optimal topology configurationsubjected to buckling constraints(c)Optimal topology configuration subjected to displacement constraints (d)Optimal topology configuration subjected to buckling and displacement constraintsFigure 2. Definition and optimal results of example 2IV.C ONCLUSIONS(1) The topological optimization of the continuum structures with two kinds of constraints was solved. The study constraints scope to topological optimization of the continuum structures was expanded. So the structure’s constraints are similar to the technical structure.(2)The buckling and displacement of the continuum structures is one of important aspect of structure design. The studies of topology optimization problem of continuum structures with the buckling and displacement constraints can provide design consult to engineer in conception phase.R EFERENCES[1] Y.X. Gu, G.Z. Zhao, H.W. Zhang, Z. Kang, and R.V. Grandhi.Buckling design optimization of complex built-up structures with shape and size variables. Struct Multidisc Optim 19Springer-Verlag 2000, 183-191.[2] D. Manickarajah, Y.M. Xie, G.P. Steven. Optimization of columns andframes against buckling. Computers and Structures, 75 (2000), 45-54.[3] J.H. Rong, Y.M. Xie, X.Y. Yang .An improved method forevolutionary structural optimization against buckling. Computers and Structures, 79 (2001), 253-263.[4] X. Guo · G.D. Cheng · N. Olhoff. Optimum design of truss topologyunder buckling constraints. Struct Multidisc Optim (2005), DOI 10.1007/s00158-004-0511-z.[5] M.Zhou. Topology Optimization for Shell Structures with LinearBuckling Responses. Computational Mechanics WCCMVI Sept.5-10, 2004.Beijing.China.[6] Sui Yunkang, Yang Deqing, et al. Uniform ICM theory and methodon optimization of structural topology for skeleton and continuum structures. Chinese Journal of Computational Mechanics, 2000, 17(1): 28-33.[7] PENG Xi-rong, SUI Yun-kang. Topological Optimization ofContinuum Structure with Static Displacement and Frequency Constraints by ICM method. Chinese Journal of Computational Mechanics, 2006, 23(4):391-396.。

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1001-344X CN 11-4250/C索书号：C91/3 馆藏地：本部四楼江宁二楼本期目录内容理论研究中国社会发展范式的转换：普遍性与特殊性……………………………………………刘新刚(3) 欧洲社会模式的反思与展望……………………………………………(英)安东尼·吉登斯(10) 分支学科自我行动与自主经营——理解中国人何以将自主经营当作其参与市场实践的首选方式……………汪和建(21) 声望危机下的学术群体——当代知识分子身份地位研究……………………………刘亚秋(37) 中国城市教育分层研究(1949-2003) …………………………………………………郝大海(51) 法律执行的社会学模式——对法律援助过程的法社会学分析………………王晓蓓郭星华(63) 社会发展系统打造农村现代职业体系的创新探索——武汉农村家园建设行动计划和实践的社会学分析之一……………………郑杭生(69) “活着的过去”和“未来的过去”——民俗制度变迁与新农村建设的社会学视野…………………………………杨敏(76) 社会问题解决社会问题的关键：协调好社会各群体之间的关系…………………………………李强(87) 环球视窗美国式的贫困与反贫困...........................................................................张锐(89) 索引 (92)英文目录 (96)领导科学[半月刊]=Leadership Science—第21期，2007年.—河南：领导科学杂志社编辑出版，（450002）.3.80元ISSN1003-2606 CN41-1024/C索书号： C93/8 馆藏地：本部四楼江宁二楼本期目录内容领导科学界的首要政治任务…………………………………………………………本刊编辑部（1）学习贯彻十七大精神把思想和行动统一到党的十七大精神上来——在2007年度省领导与社科葬专家学者座谈会上的讲话…………………………………………………………………………………徐光春（4）新一届中央领导集体治国理政的新方略(上) 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(44)名家在线科学大家平民本色——记中国测绘科学研究院名誉院长、中国工程院院士刘先林眉 (56)群星闪烁 (59)公务员管理提高公务员考试科学性 (61)加强县乡公务员队伍建设必须从优化人员结构入手 (63)“在线学习”引领干部培训“网络化” (66)事业单位聘用制推行中常见问题与对策分析 (68)如何做好当前外国专家管理与服务工作 (68)西部地区海外引才的一道亮丽风景线 (70)巧用职称评聘“杠杆”支撑人才活力 (72)人才资源配置流动人员人事档案管理难点与对策 (74)搞好毕业生人事档案管理 (77)县域人才开发如何与国际化接轨 (79)集团文化建设落地的关键点 (80)国有企业“二线”人员的开发 (81)简明定位“薪”事不再重重 (83)公备员在受处分期间受到新的处分，处分期如何计算 (85)未满服务期辞职应如何承担违约责任 (86)27 经济、经济学（F 经济）中文期刊世界经济[月刊]=The Journal of World Economy /中国经济学会中国社会科学院世界经济与政治研究所．—第2期，2008年2月．—北京：《世界经济》编辑部，（100732）．15.00元ISSN1002-9621 CN11—1138/F索书号：F1/45本期目录内容国际贸易与国际投资研发全球化与本土知识交流：对北京跨国公司研发机构的经验分析…………………………………………………………………梁正，薛澜，朱琴，朱雪炜(3) 区际壁垒与贸易的边界效应…………………………………………赵永亮，徐勇，苏桂富(17) 国际金融国际分工体系视角的货币国际化：美元和日元的典型事实……………………徐奇渊，李婧(30) 存在金融体制改革的“中国模式”吗…………………………………………………应展宇等(40) 宏观经济学习惯形成与最优税收结构…………………………………………………………邹薇，刘勇(55) R&D溢出渠道、异质性反应与生产率：基于178个国家面板数据的经验研究……………………………………………………………………………高凌云，王永中(65) 中国经济三种自主创新能力与技术进步：基于DEA方法的经验分析……………李平，随洪光(74) 经济史明代海外贸易管制中的寻租、暴力冲突与国家权力流失：一个产权经济学的视角……………………………………………………………………………………郭艳茹(84) 会议综述第一届青年经济学家研讨会(YES)综述 (95)经济与管理研究[月刊]=Research on Economics and Management/首都经济贸易大学．—第2期，2008年2月．—北京：《经济与管理研究》编辑部，（100026）．10.00元ISSN1000-7636 CN11—1384/F索书号：F2/8本期目录内容会议纪要努力探索中国特色国有公司治理模式——中国特色国有公司治理高层论坛综述 (5)专题论坛改革开放与国有经济战略性调整………………………………………………………王忠明(13) 股权多元化的国有控股公司治理结构特点及其构建………………………………魏秀丽(21) 剩余权的分配与国企产权改革……………………………兰纪平，罗鹏，霍立新，张凤环(28) 企业创新需求与我国自主创新能力的形成：基于收入分配视角………………………张杰，刘志彪(33) 集成创新过程中的三方博弈分析……………………………………………宋伟，彭小宝(38) 创新与企业战略制定模式的演进………………………………………刘鹏，金占明，李庆(43) 企业管理大型国际化零售企业经营绩效的影响因素分析………………………………蔡荣生，王勇(49) 跨团队冲突与组织激励机制分析………………………………………………李欣午(54) 运用基尼系数增强企业薪酬制度的公平性……………………………王令舜，马彤(59) 三农研究乡村旅游发展的公共属性、政府责任与财政支持研究……………单新萍，魏小安(64) 论失地农民长效保障机制的构建………………………………………………魏秀丽(69) 资本市场外资银行进入与东道国银行体系的稳定性：以新兴市场国家为例………………张蓉(74) 挤兑风险与道德风险的权衡：显性存款保险制度下最优保险范围的制定…冯伟，曹元涛(80) 贸易经济在反倾销中对出口商利益的考量………………………………………………金晓晨(86) 我国加征出口关税政策思辨…………………………………………夏骋祥，李克娟(90) 名刊要览公司治理和并购收益 (94)规制——自由化的必由之路：以色列电信市场1984-2005 (94)金融与收入分配不平等：渠道与证据 (94)特别主题论坛：重新审视组织内部和组织自身的“污名”问题 (95)经济理论与经济管理[月刊]= Economic Theory and Business Management.—第11期，2007年.—北京：《经济理论与经济管理》编辑部，（100080）.8.00元ISSN 1000-596X CN 11-1517/F索书号：F2/12 馆藏地：本部四楼江宁二楼本期目录内容深入贯彻落实科学发展观的经济视阈……………………………………………………张雷声(5) 科学发展观与中国特色社会主义经济理论体系的创新与发展…………………………张宇(9) 关于转变经济发展方式的三个问题……………………………………………………方福前(12) 统筹城乡协调发展是落实科学发展观的重大历史任务………………………………秦华(16) 中国进出口贸易顺差的原因、现状及未来展望………………………………王晋斌李南(19) 劳动力市场收入冲击对消费行为的影响………………………………………杜凤莲孙婧芳(26) 中国经济增长中土地资源的“尾效”分析……………………………………………崔云(32) 货币需求弹性、有效货币供给与货币市场非均衡模型解析“中国之谜”与长期流动性过剩……………………………………………………………………………………李治国 (38) 全流通进程中的中国股市全收益率研究………………………………………陈璋李惊(45) 金融体系内风险转移及其对金融稳定性影响研究……………………………………许荣(50) 金融衍生品交易监管的国际合作……………………………………………………谭燕芝等(56) 税收饶让发展面临的挑战及我国的选择………………………………………………张文春(61) 区域产业结构对人民币升值“逆效应”的影响………………………………………孙伯良(66) 企业社会责任管理新理念：从社会责任到社会资本……………………………………易开刚(71) 关于建设创新型国家的讨论综述………………………………………………………杨万东(76)经济理论与经济管理[月刊]= Economic Theory and Business Management.—第1期，2008年.—北京：《经济理论与经济管理》编辑部，（100080）.8.00元ISSN 1000-596X CN 11-1517/F索书号：F2/12 馆藏地：本部四楼江宁二楼本期目录内容经济热点中国宏观经济形势与政策：2007—2008年………………………中国人民大学经济学研究所(5) 理论探索出口战略、代工行为与本土企业创新——来自江苏地区制造业企业的经验证据…张杰等(12) FDI在华独资化的动因——基于吸收能力的分析……………………………秦凤鸣张中楠(20) 学术前沿主权财富基金的发展及对21世纪初世界经济的影响………………………宋玉华李锋(27) 公共经济环境税“双重红利”假说述评……………………………………………………………司言武(34) 基于合谋下的税收征管激励机制设计………………………………………………岳朝龙，等(39) 金融研究内生货币体系下房价波动对货币供求的冲击…………………………………丁晨屠梅曾(43) 基于DEA的中小企业债务融资效率研究………………………………………曾江洪陈迪宇(50) 区域经济地区经济增长中的金融要素贡献的差异与金融资源配置优化——基于环北部湾(中国)经济区的实证分析…………………………………范祚军等(54) 工商管理基于价值链的预算信息协同机制研究………………………………………………张瑞君，等(59) 公司特征、行业特征和产业转型类型的实证研究……………………………王德鲁宋学锋(64) 国际经济基于市场体系变迁的中国与欧洲银行业发展比较……………………………胡波郭艳(70) 动态与综述我国发展现代农业问题讨论综述………………………………………………………王碧峰(75) 全国马克思列宁主义经济学说史学会第十一次学术研讨会纪要……………………张旭(80)国有资产管理[月刊]= State assets management /中国人民共和国财政部.-第1期，2008年.—北京：《国有资产管理》杂志社，(100036) .10.00元ISSN 1002-4247 CN 11-2798索书号：F2/51 馆藏地：西康校区四楼本期目录内容贯彻落实科学发展观开创中央企业又好又快发展新局面.................................李荣融（4）努力做好新形势下的监事会工作..................................................................黄丹华（8）资产评估行业发展的重要里程碑...............................................................朱志刚（13）加快评估立法步伐加强评估法律体系建设................................................石秀诗（15）资产评估行业将进入新的发展时期............................................................李伟（16）提升资产评估执业质量促进资本市场健康发展..........................................李小雪（17）评估准则对中国不动产及相关资产评估的作用..............................埃尔文.费南德斯（18）发挥评估准则对中国资产评估行业健康发展的作用.......................................林兰源（20）正确发展适合中国国情的评估准则..........................................格来格.麦克纳马拉（21）财政部国资委关于印发《中央企业国有资本收益收取管理暂行办法》的通知 (22)财政部关于印发《中央国有资本经营预算编报试行办法》的通知 (25)力口快建立国有资本经营预算推动国民经济又好又快发展..............................贾谌（28）关于国有企业改制和整体上市..................................................................季晓南（30）加快建立科学规范的财务监督体系............................................................孟建民（39）贯彻科学发展观’开创财务监督管理工作新局面..........................................赵杰（43）寓监管于服务之中——对做好四川国资监管工作的思考.................................李成云（47）努力实现广西区国资国企健康发展............................................................尹建国（49）新企业会计准则对国资监管可能带来的影响................................................安玉理（52）2007年宏观经济形势分析及2008年展望 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(55)国有公司治理结构存在的问题及其法律风险防范……………………………………王玉宝（61）加强对外投资及多种经营监管…………………………………………………………张凯（64）盈余管理对企业有益性的探讨…………………………………………………………葛晓红（66）国有企业引进战略投资者的策略……………………………………………屈艳芳郭敏（68）促进我国企业内部控制的建设………………………………………………张宜霞文远怀（71）企业年金信托管理的治理结构研究(一) ……………………………………李连仁周伯岩（74）美英国家政府绩效考评对我国的启示与借鉴…………………………………………聂常虹（76）复印报刊资料·外贸经济、国际贸易[月刊]=Economy of Foreign Trade And Internaional Trade.—第1期，2008年.—北京：中国人民大学书报资料中心，（100086）.11.00元ISSN 1001-3407 CN 11-4289/F索书号：F7/17 馆藏地：西康校区四楼本期目录内容本刊综述2007年国际贸易与我国对外贸易问题综述………………………………………………王亚星(3) 研究与探讨试论新贸易理论之新……………………………………………………………郭界秀朱廷捃(9) 比较优势理论的有效性：基于中国历史数据的检验……………………………………管汉晖(14) 制度分析视角中的贸易开放与经济增长——以投资效率为中心……………………盘为龙(23) 国际贸易、外国直接投资、经济增长对环境质量的影响——基于环境库兹涅茨曲线研究的回顾与展望…………………………胡亮潘厉(30) 贸易政策贸易模式与国家贸易政策差异…………………………………………………………曹吉云(37) 分工演进对贸易政策的影响分析——基于交易成本的考虑…………………张亚斌李峰(44) 中国贸易结构的变化特点、决定要素以及政策建议……………………………………章艳红(50) 专题：进出口贸易二元经济结构、实际汇率错位及其对进出口贸易影响的实证分析……………………吕剑(56) 人民币汇率波动性对中国进出口影响的分析……………………………………谷宇高铁梅(66) 中国对外贸易出口结构存在的问题……………………………………………………魏浩(75) 服务贸易国际知识型服务贸易发展的现状、前景及我国对策分析……………………潘菁刘辉煌(80) 国际服务外包趋势与我国服务外包的发展……………………………………李岳云席庆高(86) 文摘加快我国资本输出和经济国际化的建议......................................................裴长洪(90) 双赢的中美经贸关系缘何被扭曲...............................................................李若谷(91) 索引 (93)英文目录 (96)复印报刊资料·市场营销 [月刊]=Marketing.—第2期，2008年.—北京：中国人民大学书报资料中心，（100086）.6.00元ISSN 1009-1351 CN 11-4288/F索书号：F7/26 馆藏地：西康校区四楼视点营销资讯 (4)特别关注激情燃烧的岁月——行将远去的2007…………………………………………………刘超等(6) 营销创新数字营销上路……………………………………………………………………………岳占仁(12) 手机广告：精准营销的黄金地段…………………………………………………………王浩(15) 论坛营销分析中小企业网上营销安全问题分析...............................................................潘素琼(17) 如何克服电子邮件营销中的广种薄收.........................................................郝洁莹(19) 国产洗发水何以走出迷局? (21)营销人物校长茅理翔………………………………………………………………………………叶丽雅(23) 营销策略博客营销策略……………………………………………………………………………缪启军(26) 企业社会责任标准下的出口营销策略转变……………………………………于晓玲胡日新(29) 品牌管理品牌管理的价值法则……………………………………………………………………辰平(30) 品牌延伸：中国企业需要补课……………………………………………………………曾朝晖(33) 渠道管理渠道模式：一半是火焰一半是海水………………………………………………………钱言(36) 弱势品牌渠道拓展之路…………………………………………………………吴勇毅陈绍华(39) 销售管理销售经理管控销售队伍的四种工具……………………………………………………谢宗云(41) 遭遇难题，见招拆招……………………………………………………………虞坚老树(44) 销售冠军是怎样炼成的——专访苏宁朝阳路店店长刘玉君…………………………齐鹏(47) 成功策划“右手之戒”成就戴比尔斯 (49)拉芳舍一个休闲餐饮王国扩张之谜……………………………………………………王翼(51) 阿尔迪最赚钱的“穷人店”………………………………………………………………杨育谋(53) 个案解读LG巧克力手机得失之间…………………………………………………………………林景新(56) 南京菲亚特：四面楚歌……………………………………………………………………陈宇祥(58) 奥克斯：反思“三大败笔”…………………………………………………………………刘步尘(62)财经科学[月刊]=Finance And Economics—第4期，2007年.—成都：《财经科学》编辑部，（610074）.8.00元ISSN1000-8306 CN51-1104/F索书号： F8/19 馆藏地：本部四楼江宁二楼。

下部结构 substructure桥墩 pier 墩身 pier body墩帽 pier cap, pier coping台帽 abutment cap, abutment coping盖梁 bent cap又称“帽梁”。

重力式[桥]墩 gravity pier实体[桥]墩 solid pier空心[桥]墩 hollow pier柱式[桥]墩 column pier, shaft pier单柱式[桥]墩 single-columned pier, single shaft pier 双柱式[桥]墩 two-columned pier, two shaft pier排架桩墩 pile-bent pier丫形[桥]墩 Y-shaped pier柔性墩 flexible pier制动墩 braking pier, abutment pier单向推力墩 single direction thrusted pier抗撞墩 anti-collision pier锚墩 anchor pier辅助墩 auxiliary pier破冰体 ice apron防震挡块 anti-knock block, restrain block桥台 abutment台身 abutment body前墙 front wall又称“胸墙”。

翼墙 wing wall又称“耳墙”。

U形桥台 U-abutment八字形桥台 flare wing-walled abutment一字形桥台 head wall abutmentT形桥台 T-abutment箱形桥台 box type abutment拱形桥台 arched abutment重力式桥台 gravity abutment埋置式桥台 buried abutment扶壁式桥台 counterfort abutment, buttressed abutment 衡重式桥台 weight-balanced abutment锚碇板式桥台 anchored bulkhead abutment支撑式桥台 supported type abutment又称“轻型桥台”。

Buckling of Plate ElementsContents8.1.Introduction 3738.2.Differential Equation of Plate Buckling 3748.2.1.Plate Bending Theory 3748.2.2.Equilibrium Equations 3818.2.3.Stationary Potential Energy 3848.3.Linear Equations 3908.3.1.Adjacent-Equilibrium Criterion 3918.3.2.Minimum Potential Energy Criterion 3938.4.Application of Plate Stability Equation 4018.4.1.Plate Simply Supported on Four Edges 4018.4.2.Longitudinally Stiffened Plates 4058.4.3.Shear Loading414 Plate Simply Supported on Four Edges415 Plate Clamped on Four Edges415 Plate Clamped on Two Opposite Edges and Simply Supportedon the Other Two Edges 4168.5.Energy Methods 4188.5.1.Strain Energy of a Plate Element 4188.5.2.Critical Loads of Rectangular Plates by the Energy Method 4208.5.3.Shear Buckling of a Plate Element by the Galerkin Method 4228.5.4.Postbuckling of Plate Elements 4268.6.Design Provisions for Local Buckling of Compression Elements 4318.7.Inelastic Buckling of Plate Elements 4328.8.Failure of Plate Elements 433 References434 Problems 4378.1.INTRODUCTIONEquilibrium and stability equations of one-dimensional elements such asbeams, columns, and framed members have been treated in the precedingchapters. The analysis of these members is relatively simple as bending,the essential characteristic ofbuckling, can be assumed to take place in oneplane only. The buckling ofa plate, however, involves bending in two planesand is therefore much more complicated. From a mathematical point of view,the main difference between framed members and plates is that quantitiessuch asStability of Structures© 2011 Elsevier Inc. ,ISBN 978-0-12-385122-2, doi:10.1016/B978-0-12-385122-2.10008-9 M rights reserved. 373 deflections andbending moments, which are functions of a single independent variable inframed members, become functions of two independent variables in plates. Consequently, the behavior of plates is governed by partial differential374 Chai Yooequations, which increases the complexity of analysis.There is another significant difference in the buckling characteristics of framed members and plates. For a framed member, buckling terminates the member’s ability to resist any further load, and the critical load is thus the failure load or the ultimate load. The same is not necessarily true for plates.A plate element may carry additional loading beyond the critical load. This reserve strength is called the postbuckling strength. The relative magnitude of the postbuckling strength to the buckling load depends on various parameters such as dimensional properties, boundary conditions, types of loading, and the ratio of buckling stress to yield stress. Plate buckling is usually referred to as local buckling. Structural shapes composed of plate elements may not necessarily terminate their load-carrying capacity at the instance of local buckling of individual plate elements. Such an additional strength of structural members is attributable not only to the postbuckling strength ofthe plate elements but also to possible stress redistribution in the member after failure of individual plate elements.The earliest solution of a simply supported flat-plate stability problem apparently was given by Bryan (1891), almost 150years afterEulerpresented the first accurate stability analysis of a column. At the beginning of the twentieth century, plate buckling was again investigated by Timoshenko, who studied not only the simply supported case, but many other combinations of boundary conditions as well. Many of the solutions he obtained are given in Timoshenko and Gere (1961). Treatments of flat-plate stability analysis that are much more extensive than those given here may be found in Bleich (1952) and Timoshenko and Gere (1961). Other references such as Allen and Bulson (1980), Brush and Almroth (1975), Chajes (1974), and Szilard (1974) are reflected for their modern treatments of the subject.8.2. DIFFERENTIAL EQUATION OF PLATE BUCKLING8.2.1.Plate Bending TheoryThe classical theory offlat plates presented here follows the part ofmaterials leading to the von Karman equations entailed by Langhaar (1962) excluding the energy that results from heating. Consider an isolated free body of a plate element in the deformed configuration (necessary for stability problems examining equilibrium in the deformed configuration, neighboringBuckling of Plate Elements 3752 whereNx, N^xy^ 丄 transverse shearin equilibrium). The plate material is assumed to be isotropic and homoge-neous and to obey Hooke ’s law. The plate is assumed to be prismatic, and forces expressed per-unit width of the plate are assumed constant along the length direction. The plate is referred to rectangular Cartesian coordinates x, y, z, where x and y lie in the middle plane of the plate and z is measured from the middle plane. The objective of thin-plate theory is to reduce a three-dimensional (complex) problem to an approximate (practical) one based on the following simplifying assumptions:1. Normals to the undeformed middle plane are assumed to remain normal tothe deflected middle plane and inextensional during deformations, so that transverse normal and shearing strains may be ignored in deriving the plate kinematic relations.Transverse normal stresses are assumed to be small compared with theother normal stresses, so that they may be ignored.Novozhilov (1953) referred to these approximations as the Kirchhoff assumptions. The first approximation is tantamount to the typical plane strain assumption, and the second is part of plane stress assumption.Internal forces (generalized) acting on the edges of a plate element dxdy are related to the internal stresses by the equations(8.2.1)Nx V , N V x = in-plane normal and shearing forces, Qx, Qy =forces, Mx, My = bending moments, M xy ,M yx = twistingmoments.The barred quantities, Ox ，Txy ； etc., stand for stress components at any point through the thickness, as distinguished from s x ，Sxy ，etc., which denote-h /2 ay dzaxz dz Q 1axy dzayz dzxy-h/2axzdz czdzx-h /2：dz Q ：Oyzdzh /2〒xyzdz h /2-h/2376 Chai Yoocorresponding quantities on the middle plane (z = 0). The positive in-plane normal and shearing forces, transverse (also referred to as bending) shearing forces, bending moments, and twisting moments are given in Figs. 8-1, 8-2, and 8-3, respectively.Since s X y = Ty X, Eqs. (8.2.1) reveal that N X y = Ny X and = M yx. The directions ofthe positive moments given in Fig. 8-3 result in positive stresses at the positive end of the z-axis. As a result of the Kirchhoff’s first approximation, the displacement components at any point in the plate, u； v; w； can be expressed in terms of the corresponding middle-plane quantities, u, v, w, by the relationsdxdxQyFigure 8-2 Bending shearBuckling of Plate Elements 377d 2wdxdy_dw _ dw u = u —a = v —w = w (8.2.2)dxvywhere positive rotations are shown in Fig. 8-1.Ignoring negligibly small higher order terms, the components of the strain-displacement relations for a three-dimensional body as given by Novozhilov (1953) are_ Vu 1 /Vw\ ^ _ Vv 1 /Vw\ ^ _ Vu Vv dw dw3x =VX +3y = Vy+式石」T x y =Vy+Vy+vxVy(8.2.3)The strain components in Eqs. (8.2.3) are the Green-Lagrange strain (Sokolnikoff 1956; Bathe 1996), which is suited for the incremental analysis based on the total Lagrangian formulation. Substituting Eq. (8.2.2) into Eq. (8.2.3) yieldsV 2w V 2w~3x — £x — z d 2 = 3x + zK x Oy = 3y — z d 2 = £y + zky gxy — g xy — 2z。

Effect of manufacturing defects on mechanical properties and failure features of 3D orthogonal woven C/CcompositesAi Shigang a ,Fang Daining b ,⇑,He Rujie b ,1,Pei Yongmao ba Institute of Engineering Mechanics,Beijing Jiaotong University,Beijing 100044,PR ChinabState Key Laboratory of Turbulence and Complex System,College of Engineering,Peking University,Beijing 100871,PR Chinaa r t i c l e i n f o Article history:Received 8September 2014Received in revised form 1November 2014Accepted 3November 2014Available online 10November 2014Keywords:A.Carbon-carbon composites (CCCs)B.DefectsC.Damage mechanics C.Numerical analysisD.Non-destructive testinga b s t r a c tFor high performance 3D orthogonal textile Carbon/Carbon (C/C)composites,a key issue is the manufac-turing defects,such as micro-cracks and voids.Defects can be substantial perturbations of the ideal archi-tecture of the materials which trigger the failure mechanisms and compromise strength.This study presents comprehensive investigations,including experimental mechanical tests,micron-resolution computed tomography (l CT)detection and finite element modeling of the defects in the C/C composite.Virtual C/C specimens with void defects were constructed based on l CT data and a new progressive dam-age model for the composite was proposed.According to the numerical approach,effects of voids on mechanical performance of the C/C composite were investigated.Failure predictions of the C/C virtual specimens under different void fraction and location were presented.Numerical simulation results showed that voids in fiber yarns had the greatest influences on performance of the C/C composite,espe-cially on tensile strength.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionCarbon fiber reinforced carbon composites (C/C)have high ther-mal stability,thermal shock resistance,strength and stiffness in non-oxidizing atmosphere.Due to its superior specific strength and toughness,C/C composites can be considered as favourite materials for highly demanding thermostructural lightweight applications e.g.in aerospace and nuclear industry [1–6].Nowa-days C/C components are leading candidates for applications under extreme conditions.C/C composites are produced by chemical vapor infiltration (CVI)of a textile fiber preform.After the CVI pro-cess and high temperature heart-treatments,generally,manufac-turing defects exist inner the materials.In particular,porosity/voids and micro-cracks are typical defects in C/C composites,and seriously affect the performance of the composites [7–9].So,it is mandatory to account for the effects of defects and their evolution,even in the early stages of the design process.With the increasing use of C/C composites as advanced structural materials,the deter-mination of damage criticality and structural reliability of compos-ites has become an important issue in recent years.Defects–mechanical property relationships of fiber reinforced composites have always been of interest to scientists addressing the composite performance.In Gowayed et al.’s work [10],defects in an as-manufactured oxide/oxide and two non-oxide (SiC/SiNC and MI SiC/SiC)ceramic matrix composites were categorized and their volume fraction quantified using optical imaging and image analysis.Aslan and Sahin [11]investigated the effects of delamin-ations size on the critical buckling load and compressive failure load of E-glass/epoxy composite laminates with multiple large del-aminations by experiments and numerical simulations.In Masoud et al.’s work [12]effects of manufacturing and installation defects on mechanical performance of polymer matrix composites appear-ing in civil infrastructure and aerospace applications were studied.Damage onset and propagation were studied used time-dependent nonlinear regression of the strain field.In Refs.[13–17],the finite element method (FEM)was followed by various authors to study the delamination problems.FEM is preferred than analytical solu-tions because it can handle various laminate configurations and boundary conditions.In recent decades,high-fidelity X-ray micro-computed tomog-raphy (l CT)technology has been used to characterize defects and reconstruct meso-structure of textile composites [18].In Cox et al.’s work [19–21],three-dimensional images of textile com-posites were captured by X-ray l CT on a synchrotron beamline.Based on a modified Markov Chain algorithm and the l CT data,/10.1016/positesb.2014.11.0031359-8368/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail addresses:sgai@ (F.Daining),rujh@ (H.Rujie).1Co-corresponding author.a computationally efficient method has been demonstrated for generating virtual textile specimens.In Fard et al.’s work [22],manufacturing defects in stitch-bonded biaxial carbon/epoxy composites were studied through nondestructive testing (NDT)and the mechanical performance of the composite structures was investigated using strain mapping technique.In Desplentere et al.’s work [23],X-ray l CT was used to characterize the micro-structural variation of four different 3D warp-interlaced fabrics.And the influence of the variability of the fabric internal geometry on the mechanical properties of the composites was estimated.In Guillaume et al.’s work [24]effects of porosity defects on the interlaminar tensile (ILT)fatigue behavior of car-bon/epoxy tape composites were studied.In that work,CT mea-surements of porosity defects present in specimens were integrated into finite element stress analysis to capture the effects of defects on the ILT fatigue behavior.In Thomas et al.’s work [25]X-ray microtomography technology was adopted to measure the dimensions and orientation of the critical defects in short-fiber reinforced composites.Generally,geometry reconstruction based on l CT data is a huge and complex work,sometimes,virtual specimens explored through this approach are difficult to use for numerical analysis.For 3D fabric composites,because of the 2.Material and experimentsMaterial studied in this article is C/C 3-D orthogonal woven ceramic composite (fabricated by National Key Laboratory of Ther-mostructure Composite Materials,Northwestern Polytechnical University,China)in which T300carbon fiber (Nippon Toray,Japan)tows rigidified by carbon matrix.The C/C composite was prepared using chemical vapor infiltration (CVI)method.T300car-bon fiber was used as reinforcement of the C/C composites with the fiber volume fraction was 56.5%.The fiber preforms,as shown in Fig.1a,were infiltrated with carbon matrix using multiple cycles of infiltration and heat treatment at 1373K,0.03MPa (the thick-ness of the fiber preforms is about 5mm).With increasing cycles,a matrix with near full density can be asymptotically approached,generally,it was about 10cycles (1200h).The C/C specimens are illustrated in Fig.1b (the thickness of the tensile specimen is 5.0mm).However,from the l CT images of the C/C materials,it was found that manufacturing defects such as voids and micro-cracks appeared inner the composites.It is because of the special material preparation process.The manufacturing defects are illus-trated in Fig.1c.Uniaxial tensile experiments were carried out under a Shima-Fig.1.C/C 3-D orthogonal woven composite.Fig.2.Stress–strain curve of the C/C composite under uniaxial tension.114 A.Shigang et al./Composites:Part B 71(2015)113–121In the tensile experiments,five specimens in total were tested and the tensile strengths were217.3,185.1,219.8,176.5and 187.3MPa correspondingly.The average value of the tensile strength was197.2MPa and the dispersion of the experimental results was less than11.5%.Other more,the fracture behaviors of thefive specimens were similar with the failure locations almost all located in the middle of the specimens.From the experiments, deformation of the C/C3-D orthogonal composite under uniaxial tension comprises with three stages:linear elastic stage,damage initiation/evolution stage and the material fracture stage.In the first stage the stress–strain curve increased linearly and in the sec-ond stage the stress–strain curve increased nonlinearly.In the frac-ture stage the stress–strain curve rapidly declined.3.Numerical programmer3.1.3Dfinite element modelFiber tows in the3-D orthogonal architecturesfit together snugly in the woven pattern by a system of periodic motions, and approximately in the same cross-sectional geometry.In this study,cross-sections of the warpfiber yarns and weftfiber yarns werefitted as rectangle.The cross-sections of the z-binder tows werefitted as circular.Geometric parameters of thefiber yarn cross-sections were recorded.For the warp yarns and weft yarns the side lengths of the cross-section rectangle were0.786mm and0.340mm.For the z-binderfiber yarns the diameter of the cross-section circular was0.790mm.The smallest repeatable rep-resentative volume element(RVE)of the textile architecture was constructed and shown in Fig.3.The lengths of the RVE model in X and Y direction both were1.96mm and the height of the RVE model in Z direction was0.76mm.To reveal the internal defects in thefinite element model,l CT technology was used to investigate the meso-structures of the fore,three local coordinates were constructed to identify the mate-rial directions.Then,an interface zone with a constant thickness 0.01mm was generated based on the geometrical model of the fiber yarns,as shown in Fig.3d.Finally,a solid block model with the same size of the composite specimen was constructed.Boolean operation were carried out among the solid block,interfaces and thefiber yarns to generate the geometrical model of the carbon matrix,which is shown in Fig.3b.A whole RVE model of the com-posite is illustrated in Fig.3a.A Monte Carlo algorithm was adopted to choose elements one-by-one randomly as‘‘void defects’’until the volume fraction of the voids satisfied the threshold values in the three zones respectively. For the C/C composite studied in this paper,the void fractions of thefiber yarns,matrix and the interfaces are0.51%,0.47%and 1.94%respectively.It must be noted out that those elements which identified as‘‘void defects’’were not moved away from the FE model,but the stiffness was degenerated by10eÀ6times in the simulation process.The void defects in the three zones are high-light as‘‘red’’,as shown in Fig.3.3.2.Progressive damage modelThe failure criterion proposed here is a strain-based continuum damage formulation with different failure criteria applied for matrix andfiber yarns.A gradual degradation of the material prop-erties is assumed.This gradual degradation is controlled by the individual fracture energies of matrix andfiber yarns,respectively. Thefiber yarn is in the X(1)–Y(2)–Z(3)Cartesian coordinate sys-tem,and the X direction corresponds to thefiber longitudinal direction.For thefiber yarns,two different modes of failure are considered:fiber failure in longitudinal direction and matrix fail-ure in transverse direction.The damage mechanism consists of two ingredients:the damage initiation criteria and the damage evolution law.orthogonal textile C/C composite,(a)RVE model,(b)carbon matrix,(c)fiber yarns and(d)fiber yarns-matrixinterpretation of the references to colour in thisfigure legend,the reader is referred to the web version of thisA.Shigang et al./Composites:Part B71(2015)113–121115failure strains infiber direction in tension and compression,F f;tX and F f;cXare the tensile and compressive strength of thefiberyarns in X direction,respectively.Once the above criterion is sat-isfied,thefiber damage variable,f Xf,evolves according to the fol-lowing equation law:d X f ¼1Àe f;t11f XfeÀC11e f;t11f X fÀe f;t11ðÞL c=G fðÞð2Þwhere L c is the characteristic length associated with the material point.For matrix failure the following failure criterion is used:f Y m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie f;t22e f;c22ðe22Þ2þe f;t22Àe f;t222e f;c22B@1C A e22þe f;t22e f;s12!2ðe12Þ2>e f;t22v uu uu tð3Þf Z m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie f;t3333ðe33Þ2þe f;t33Àe f;t33233B@1C A e33þe f;t3313!2ðe13Þ2>e f;t33v uu uu tð4Þwhere e f;t22;e f;t33;e f;c22and e f;c33are the failure strains perpendicular to the fiber direction in tension and compression,respectively.The failure strain for shear are e f;s13and e f;s12.Failure occurs when f Y m exceeds its threshold value e f;t22or f Z m exceeds its threshold value e f;t33.The evolu-tion law of the matrix damage variable,d m,is:d Ym¼1Àe f;t22f YmeÀC22e f;t22f Y mÀe f;t22ðÞL c=G mðÞð5Þd Zm¼1Àe f;t33fmeÀC33e f;t33f Z mÀe f;t33ðÞL c=G mðÞð6ÞAs damage progressing,the effective elasticity matrix isreduced as functions of the three damage variables f Xf,d Ymand d Zm, as follows:3.2.2.Failure criterion for matrixDamage in thefiber is initiated when the following criterion is reached:where e f;t and e f;c are the failure strains in tension and compression respectively and e f,t=r f,t/C11,e f,c=r f,c/C11.Once the above criterionis satisfied,thefiber damage variable,f XðY=ZÞm,evolves according to the equation:d XðY=ZÞm¼1Àef;tfmeÀC11e f;t f XðY=ZÞmÀe f;tL c=G mð9ÞThe modulus matrix of the matrix will be reduced according to:In user subroutine UMAT the stresses are updated according to the following equation:C f d ¼1Àd XfC111Àd Xf1Àd YmC121Àd Xf1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xf1Àd YmC4400Symmetric1Àd Xf1Àd ZmC5501Àd Ym1Àd ZmC66ð7Þf XðY=ZÞm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffief;tef;cðe11ð22=33ÞÞ2þe f;tÀðe f;tÞ2ef;c!e11ð22=33Þþef;tef;s2ðe12ð23=13ÞÞ2þef;tef;s2ðe13ð12=23ÞÞ2v uu t>e f;tð8ÞC m d ¼1Àd XmC111Àd Xm1Àd YmC121Àd Xm1Àd ZmC130001Àd YmC221Àd Ym1Àd ZmC230001Àd ZmC330001Àd Xm1Àd YmC4400Symmetric1Àd Xm1Àd ZmC5501Àd Ym1Àd ZmC66ð10Þ116 A.Shigang et al./Composites:Part B71(2015)113–121r ¼C d :eTo improve convergence,a technique based ization (Duvaut–Lions regularization [27])of is implemented in the user subroutine.In this age variables are ‘‘normalized’’via the _d t ;X ðY =Z Þf ðm Þ¼1gd X ðY =Z Þf ðm ÞÀd t ;X ðY =Z Þf ðm Þwhere d X f and d X ðY =Z Þm are the fiber and matrix culated according to the damage evolution d t ;X f and d t ;X ðY =Z Þm are the ‘‘normalized’’the real calculations of the damaged elasticity bian matrix,and g is the viscosity parameter.and d t ;X ðY =Z Þm can be calculated according to the d t ;X ðY =Z Þf ðm Þt 0þD t¼D t t 0þt d X ðY =Z Þf ðm Þ t 0þD t þg g þt dt ;X ðY =f ðm ÞTherefore,for the fiber yarns and matrix,can be further formulated as Eqs.(14)and (15)correspondingly@r e ðf Þ¼C f d þ@C f d@d f:e !@d X f @f f Á@f Xfe!þ@C f d @d Y m :e !@d Y m @f Y m Á@f Y m @e !þ@C f d @d Z m :e !@d Z m @f Zm Á@f Z m@e !ð14Þ@r @e ðm Þ¼C m d þ@C md @d m :e !@d X m @f m Á@f X m@e!þ@C md @d m :e !@d Y m @f m Á@f Y m @e !þ@C m d @d m :e !@d Z m @f m Á@f Z m @e!ð15Þ3.3.Material parameters3D orthogonal C/C composites are composed by T300fiber yarns and carbon matrix.The fiber yarns can be regarded as unidi-rectional fiber-reinforced C/C composites and are assumed to be one transversely isotropic entity in each local material coordinate system.The mechanical properties of the fiber yarns can be calcu-lated using the properties of the component materials (fibers and matrix):E 1¼e E f 11þð1Àe ÞE mE 2¼E 3¼E m1Àffiffie p 1ÀE m =E f 22ðÞG 12¼G 13¼G m 1Àffip 1ÀG m =G f 12ðÞG 23¼G m1Àffip 1ÀG m =G f23ðÞl 12¼l 13¼e l f 12þð1Àe Þl m l 23¼E 222G 23À19>>>>>>>>>>>>=>>>>>>>>>>>>;ð16Þwhere e is the yarn pack factor,for the C/C composite studied in this paper,e =0.81.E f 11,E f 22are the Young’s elastic modulus of the fiberin the principal axis 1and 2,respectively.Axis 1is the longitudinal direction of the fiber yarns.G f 12,G f 23are the shear modulus of the fiber in the 1–2and 2–3plane,respectively.l f 12is the primary Pois-son’s ratio of the fiber,E m ,l m and G m represent the Young’s elastic modulus,Poisson’s ratio and shear modulus of the matrix,respec-tively.Materials parameters are listed in Table 1.It should be noted that the mechanical parameters of the carbon matrix and the T300fibers changed after the CVI process.In particular,strength of the fiber will had a greater decline.The tensile and com-pressive strength of the T300fiber yarns were tested with the values listed in Table 1.The elasticity modular of the carbon matrix was tested by a nanoindentor system,which developed by Fang’s research team from Peking University [28].In the carbon matrix modular tests,the experiments repeated 20times for statistical averaging.The val-ues in the 20measurements were 7.18,9.77,8.58,10.01,11.92,5.30,9.98,8.14,8.63,7.31,6.19,10.69,11.10,13.45,9.15,9.13,11.27,9.20,10.14and 10.06GPa;average value was 9.36GPa.Mate-rial parameters of the fiber/matrix interface are not very clear so far,in this study the Young’s elastic modulus and Poisson’s ratio of the inter-faces were assumed as same as the carbon matrix.G f is one of the key parameters which control the failure pro-gress of the fiber yarns,however,different values were recom-mended in reported articles.In this study,influences of G f on the mechanical properties of the C/C composite were investigated firstly.Based on the values reported in Refs.[29,30],five virtual specimens with different G f values (0.5,2.0,6.0,10.0,14.0)were constructed and numerical tested.Simulation results were com-pared with the experimental result,as illustrated in Fig.4.It was found that G f has influences on tensile strength and fracture strain of the C/C composite.When G f were 0.5,2.0,6.0,10.0and 14.0,ten-sile strengths of the specimens were 200.5,205.1,205.3,214.2and 214.8MPa.When G f were 0.5,2.0,6.0,the failure strains were 0.36%,0.43%and 0.57%correspondingly.When G f is bigger than 10.0,failure strains of the C/C specimens were bigger than 1.0%.So,by the simulation results,in the present study the value of G f was set to 6.0.Table 1Materials parameters.E 11(GPa)E 22(GPa)ˆ12G 12(GPa)G 23(GPa)F t (MPa)F c (MPa)S (MPa)G f (m )(N/mm)gT300fiber 230400.262414.389075650 6.00.001C matrix 9.360.338210050 1.00.001Interface9.360.3382100501.00.001Fig.4.Stress–strain curves of the C/C virtual specimens under different G f .4.Simulation results and discussionThe anisotropic damage model of thefiber yarns and the isotro-damage model of the matrix and the interface were carriedmaterial constitutive equations by User subroutine UMAT ABAQUS nonlinearfinite element codes.Static uniaxial tensile sim-ulations were carried out.In order to keep forces continuity and displacements compatibility of the opposite faces of the unit cell, periodic boundary conditions were imposed in the simulation. Because the opposite faces of the unit cell have the same geomet-rical features,the nodes on the faces were controlled in the same position to form the corresponding nodes in the process of meshing.The periodic BCs were imposed on the corresponding nodes by FORTRAN pre-compiler code,detailed in Ref.[26].The RVE model subjected to a constant displacement load in Y direction and the loading strain is1%.4.1.Effects of the void defectsIn order to investigate the void defects on the mechanical prop-erties and failure behaviors of the C/C composite,two RVE models of the C/C composite were numerical simulated.In one RVE model (FE_D),thefiber yarns,interface and matrix all had void defects with the void fractions are0.51%,1.94%and0.47%respectively. For the other RVE model(FE_Intact),no defect inside.The stress–strain curves of the C/C composite in the simulations and experi-ment are illustrated in Fig.5.By the experimental results,the elas-ticity modular of this C/C composite was58.4GPa.By the numerical simulations,for the intact model,the elasticity modular was56.6GPa;for the‘defected’model the modular was56.3GPa. In the view of modular,the simulation error of the two models were3.1%and3.6%compared with the experimental results.The difference between the two FE models was only0.53%,so,void defects have relatively limited effects on the elastic modular of the C/C composite.The uniaxial tensile strength of the C/C compos-ite was197.2MPa by the experiments.In the simulations,the ten-sile strengths were231.4MPa and205.3MPa corresponding to the intact model and the‘defected’model.It was about17.3%and4.1% difference compared with experimental results.It is clear that,theFig.5.Stress–strain curves of the C/C composite under uniaxial tension.Fig.6.Damage evolution infiber yarns,(a)RVE model with voids defects,(b)intact model.Part B71(2015)113–121yarns are corresponding to the three pictures‘o’,‘p’and‘q’in Fig.6. For the RVE model with defects,it was found that damages were firstly generated besides the defects.During the loading process, damages were growing in several sections in thefiber yarns.How-ever,for the intact model,damages were almost generated in one section in thefiber yarns.Damage evolution in carbon matrix and the interface zone are illustrated in Figs.7and8.From the simulation results,in all of the three zones,damages werefirstly generated in the‘defected’RVE model.For the‘defected’model,when e=0.27%damages appeared in the interface zone,while for the intact model the strain was0.33%.In the matrix zone,the strains in the two models were0.31%and0.33%,respectively,when damages appeared.In fiber yarns,the strains when damages appear for the two models were0.37%and0.44%.So,because of the internal defects,in load-ing progress damages will generate early inner the material.Fail-ure strain of the materials which with defects is comparatively small when compared with the materials without defects.4.2.Influence of void locationBy the l CT images,it is clear that voids and micro cracks exist in fiber yarns,carbon matrix and the interface zones.By statistical analysis for those defects,fraction of the voids in those three zones was calculated.To study the influence of void location on the mechanical properties of the C/C materials,threefinite elementFig.7.Damage evolution in carbon matrix,(a)model with voids defects,(b)intact model.Fig.8.Damage evolution in interface,(a)model with void defects,(b)intact model.models were constructed and numerically analyzed.In the three RVE models,one model has defects only in thefiber yarns(FE_DF) and another model has the defects only in carbon matrix(FE_DM), while the other one has defects only in the interface zone(FE_DI). Simulation results were compared with the experimental results. Stress–strain curves in the numerical simulations and experiments are illustrated in Fig.9.Tensile strength calculated by the simulations were208.5MPa, 229.7MPa and230.1MPa,corresponding to the threefinite ele-ment models:FE_DF,FE_DI and FE_DM.By the simulation results of thefinite element models FE_D and FE_Intact,as mentioned in above section,the tensile strengths were205.3MPa and 231.4MPa.It can give the conclusion that,defects infiber yarns has the biggest effects on the mechanical properties of the C/C composite.If thefiber yarns are perfect and defects only exist in carbon matrix and interfaces,void defects have limited influences on the mechanical properties of the C/C composites under the cur-rent void volume fractions.4.3.Influence of void volume fractionBy the statistical analysis in Section3.1,volume fractions of voids in thefiber yarns,matrix and the interface zones are0.51%, 0.47%and1.94%.Under this defect fraction,as calculated in Section,tensile strength of the C/C composite declined12.7%compared with the material which contains no defects.So,it is important and meaningful that if we can make sure about the mechanical behav-iors of C/C composites when we exactly know the void defect frac-tion.If so,it will be helpful for the performance evaluation of C/C composites and structures.To investigate the influence of the void defect fraction on the mechanical performance of the C/C composite,five RVE models were constructed and the defect fractions of thefiber yarns were 0.25%,0.5%,1.0%,2.0%and4.0%.In this study,void defect was assumed only exist infiber yarns.Because,as calculated in Section 4.2,voids in carbon matrix and interfaces zone had very little effects on the mechanical properties of the C/C composite.Uniaxial tension simulations were carried out and the stress–strain curves of thefive C/C virtual specimens are illustrated in Fig.10.From the simulation results,it is clear that as the defect fraction increased tensile strength of the C/C composite decreased.For the intact FE model,the tensile strength was231.4MPa.For thefive FE models with voids defects,the tensile strengths were214.8MPa, 206.6MPa,197.1MPa,182.3MPa and152.8MPa.For the FE model under the defect density0.25%,tensile strength decreased7.2% compared with the intact model.So,if there exist defects inner the C/C materials,even if the volume fraction of the defects was small,it will has obvious effects on the mechanical performance of the composite,especially on the tensile strength.When the defect density was4.0%,tensile strength of the C/C virtual speci-men declined33.9%compared with the intact specimen.5.ConclusionUniaxial tensile properties and meso-structure of the3D orthogonal C/C composite were studied by experimental approaches.Manufacturing defects inner the C/C composite were investigated though a micron-resolution computed tomography (l CT)approach.From the l CT photos of the3-D orthogonal car-bon/carbon composite,it was found that voids and microcracks are two classic type of manufacture defects inner the C/C materials. Base on the statistical analysis of the l CT data,finite element mod-els of the C/C composite were constructed.According to a new pro-gressive damage model,failure behaviors and mechanical properties of the C/C composites were studied by ABAQUS code. Effects of the void defects on the mechanical performances of the C/C material were numerically investigated.From the numerical simulation results,manufacturing defects such as voids have great effects on the mechanical performance of the carbon/carbon com-posite,especially on the tensile strength.With0.51%void volume fraction,tensile strength of the carbon/carbon composite has 13.2%declines compared with the intact material.When void defects exist infiber yarns,even if the volume fraction of the defects is small it still will has great influence on tensile strength of the C/C composite.However,the defects which exist in carbon matrix and interface have limited effects on the mechanical prop-erties of the C/C materials.So,keep the continuity and improve the density of the carbonfiber yarns in C/C composite manufacture process is the key to improve the mechanical properties of the C/ C composites.AcknowledgementsFinancial support from the National Natural Science Founda-tions of China(Nos.11202007,11232001,11402132)and the Foundation of Beijing Jiaotong University(KCRC14002536)are gratefully acknowledged.Fig.9.Stress–strain curves of the C/C composite in experiment and simulations.10.Stress–strain curves of the virtual specimens with different void defectfraction.Part B71(2015)113–121。