JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 4, pp. 827-839, Warsaw 2004 DETERMINATION OF THE POST-CRITICAL STATE OF COMPRESSED PLATES WITHIN THE BI-NONLINEARY THEORY Franciszek Romanów Przemysław Najdychor Institute of Mechanics and Machinery Construction Fundamentals, University of Zielona Gora e-mail: F.Romanow@ibmp.uz.zgora.pl; P.Najdychor@ibmp.uz.zgora.pl In a majority of strength problems related to sandwich structures the concept of the linear displacement state is used. This state of displa- cements is defined with the help of the broken line hypothesis, which is used for the determination of critical states within the geometrical- ly linear theory and for analysis of post-critical states of geometrically non-linear theory. In thepaper, taking intoaccount thenon-lineardisplacement state of the core and the geometrically linear theory for faces and core (binonlinear theory), we carry out an analysis of post-critical loads. The problem is solved by means of the energy method. The post-critical stresses are described by a general formula applied also for a uni-axial compressed plate simply supported at both edges. Key words: sandwich plates, the bi-nonlinear theory, post-critical 1. Introduction The strength and stability of sandwich structures with a soft foam core is usually analysed by means of the linear displacement state of the core and faces (linear hypothesis) (Grądzki andKowal-Michalska, 2000; Hop, 1980;Ma- gnucki andOswald, 2001; Planterma, 1966; StamandWitte, 1973; Szyc, 1971; Twardosz and Hong-Thai, 1975). The above assumption confines calculation methods to thin wall structures. A survey ofPolish scientific results related to the calculation and investiga- tion of sandwich structures on the turn of the 20th century has beenpresented in Romanów (2001), Biliński and Kmita (2000). 828 F.Romanów, P.Najdychor Non-linearanalysis of stability of sandwichstructuresbaseduponthe linear hypothesis of the displacement state was already published in the seventies of the last century, (e.g. Szyc, 1971; Twardosz andHong-Thai, 1975). With the help of the idea of the linear displacement state, for the core and faces, papers taking into account the post-elastic state have also been presented (e.g. Grądzki and Kowal-Michalska, 2000; Zielnica, 1981). In a paper by Grądzki and Kowal-Michalska (2000) post-critical states in the elastoplasticity range were analysed. An example of a compressed three layer plate (steel-aluminium-steel) was given. In all the authors’ papers, in contrast to others, the non-linear core displacement state hypothesiswas used. By analogy with the linear hypothesis, this state can be shortly denoted as the hyperbolic hypothesis. Acomprehensive set of knowledge andvariedapplications of thehyperbolic hypothesis can be found in Romanów (1995). In the present paper, the post-critical loads for the three-layer plate are presented. The faces are made of steel, the core is made of foam. The plate is compressed inonedirectionwithaunit force Nxwhich isuniformlydistributed along simply supported edges (for x= 0; x= a). The other two sides of the plate are free. An example of a plate is given in Fig.1. Fig. 1. A compressed sandwich plate and compressive load Theproblem is characterized by the core non-linear displacement state and takes into account the geometrically non-linear theory for the faces and core. So, such an approach can be named a bi-nonlinear theory. A similar analysiswaspresented inRomanów et al. (2001).There, thepost- critical states for a compressed homogeneous plate fastenedwith two external Determination of the post-critical state of compressed plates... 829 layers,were analysed.Here, kinematic functionsaregiven at twoparallel edges. By using a simplified function φ(z) transversal non-linear displacements for core are taken into account. 2. Displacement and strain state The displacements U and W of the faces in the x and y directions must satisfy boundary conditions for the thus supported plate Ui =Bm cosβx Wi =Am sinβx β = mΠ a (2.1) where m is the number of half-waves of the displacement state of faces of a unilaterally compressed plate after stability loss (m = 1,2,3, ...). The- se expressions describe mid-face plane displacements. Hence, the mid po- int displacement O is equal to U+ = (U1 +U2)/2 and related to rotation U− = (U1 −U2)/2. The complete face displacement state is described by a linear function of the variable z U =Ui− ( z± c± t 2 )∂Wi ∂x (2.2) The supperscript ”+” refers to faces in the interval −c− t¬ z¬ c, the lower one ”-” refers to c¬ z¬ c+ t. The core displacement state of considered plate (biaxial displacement sta- te) is described by formulas (Appendix 1) Ur =U +− z c ( U−− t 2 ∂W ∂x ) F(z) Wr =WT(z) (2.3) which Result from (A8)2, where Vr =0. The functions F and T depend on the variable z only, and are expressed by hyperbolic functions (formulas (A6), (A7) in Appendix 1). The constants Hi appearing in these formulas are discussed in Appendix 2. When the func- tions F(z) and T(z) are equal to one, then formulas (2.3) describe the linear displacement state (broken line hypothesis). The face and core strains are non-linear relations, and are defined by for- mulas εx = ∂U ∂x + 1 2 ∂W ∂x 2 εxr = ∂Ur ∂x + 1 2 ∂Wr ∂x 2 εzr = ∂Wr ∂z + 1 2 (∂Wr ∂z )2 γxzr = ∂Ur ∂z + ∂Wr ∂x + ∂Wr ∂z ∂Wr ∂x (2.4) 830 F.Romanów, P.Najdychor 3. Elastic energy of the plate The total energy of the plate is composed of two terms: the first term corresponds to the core energy Er, the second one to the energy of faces Eo Ec =Er +Eo (3.1) Er =Gr a ∫ 0 c ∫ −c [( 1+ νr 1−2νr ) (ε2xr+ε 2 zr)+ 2νr 1−2νr εxrεzr + 1 2 γxzr ] dxdz where G and ν are Kirchoff’s modulus and Poisson’s ratio, respectively. Taking into account (2.3) and (2.4), one can get a formula for the core energy expressed by the displacements U and W Er =Grd1 a ∫ 0 ( 1 c2 ∂U ∂x 2 F1− t c2 ∂U ∂x ∂2W ∂x2 F1+ t2 4c2 ∂2W ∂x2 F1+W 2F5+ + 1 4 ∂W ∂x 4 F16+ W4 4 F19 ) dx+ +Grd2 a ∫ 0 ( − 2W c ∂U ∂x F9+ tW c ∂2W ∂x2 F9+ W2 2 ∂W ∂x 2 F22 ) dx+ (3.2) +Gr a ∫ 0 (U2 2c2 F10− tU 2c2 ∂W ∂x F10+ t2 8c2 ∂W ∂x 2 F10+ 1 2 ∂W ∂x 2 F11− U c ∂W ∂x F12 ) dx+ +Gr a ∫ 0 ( t 2c ∂W ∂x 2 F12+ 1 2 W2 ∂W ∂x 2 F22 ) dx where F1-F22 are certain integrals of hyperbolic functions, see Appendix 2, but the expressions described in italics contain non-linear elements. The elastic energy of the faces is described bymeans of the formula Eo =B ∗ a ∫ 0 ∂U ∂x dx+D a ∫ 0 ∂2W ∂x2 2 dx+ 1 4 B∗ a ∫ 0 ∂W ∂x 4 dx (3.3) where E∗ – Young’s modulus, t – face thickness, a – plate length, b – its width and B∗ = E∗t 1−ν2 D= E∗t3 12(1−ν2) Determination of the post-critical state of compressed plates... 831 The work of external forces is described by Lz =− 1 2 a ∫ 0 Nx ∂W ∂x 2 dx (3.4) The potential energy of the whole plate is given by the sum of (3.2), (3.3) and (3.4) P =Eo+Er+Lz (3.5) From theminimum condition of the potential, two equations are obtained ∂P ∂Am =0 ∂P ∂Bm =0 (3.6) resulting in an expresion for the loading Nx Nx = A2mR6+(R1−R1R5) R4 (3.7) Formula (3.7) can be transformed into α= Nx Nxkr = A2mR7 Nxkr +1 (3.8) The last expression defines a relation between the post-critical loads Nx and deflection amplitude of the face Am. This is a quadratic equation for the independent variable Am. If Am =0, then eqyation (3.8) can be used to obtain the critical load of the plate Nxkr = R1−R1R5 R4 (3.9) where R1 =A 2 mGr [ β2 at c ( F1d1β 2 t 4c −F9d2+F10 t 8c + 1 2 F12 ) +β2F11 a 2 ] +A2mDβ 4a R2 =AmBm [ Grβ ( d2F9 a c − at 2c2 d1F1β 2− at 4c2 F10− a 2c F12 )] R3 =B 2 m [ Gr ( F1d1β 2 a 2c2 +F10 a 4c2 ) +B∗β2 a 2 ] R4 = ab 2 β2 R5 = R1 R3 R7 = R6 R4 R6 =A 4 m [ Gr ( d1F16β 43a 8 +d1F19 3a 8 +F22β 2a 4 +d2F22β 2a 4 ) +B∗β4 3a 8 ] 832 F.Romanów, P.Najdychor 3.1. Examples For given values of the plate parameters a=0.285m b=0.185m νr =0.17 ν =0.3 E∗r =58.9 ·10 6 Pa E∗ =68694.8 ·106 Pa t=0.001m the following results have been obtained, see (3.9). Table 1.Critical loads versus core thickness c [m] Nxkr [N/m] m 0.004 28961 1 0.005 36034 1 0.008 60040 1 0.010 77525 1 0.0176 99271 11 0.0185 99350 11 0.020 142754 13 0.050 143383 13 For these data, the dependence of the critical force, (see (3.9), on the core thickness is presented in Table 1. A similar relation is presented in Fig. 2. Fig. 2. The critical force Nxkr versus core thickness A dependence of the coefficient α on the amplitude Am, Eq. (3.8), where m=1 (thin plates) and m=13 (thick plates), is presented in Table 2. The criteria for the division into thin, medium and thick sandwich plates are described by Romanów (1995). Determination of the post-critical state of compressed plates... 833 Table 2.Dependence of the coefficient α on the amplitude Am, Eq. (3.8) c=0.004m; m=1 Am +0.0005 +0.0004 +0.0003 +0.0002 +0.0001 0 α 1.07 1.04 1.024 1.01 1.003 1 c=0.02m; m=13 Am +0.0005 +0.0004 +0.0003 +0.0002 +0.0001 0 α 1.009 1.0063 1.0035 1.0016 1.0004 1 Fig. 3. Dependence of the coefficient α on the amplitude Am In Fig.4 the dependence of the coefficient α on the core thickness c is presented. Fig. 4. Influence of the core thickness on the coefficient α It is seen in the diagram that the whole region of the core thickness can be divided into three sub-regions: • In the first sub-region 0 ¬ c ¬ cgr1 = 0.017, α increases with the core thickness, but does not significantly differ from the unit. The plates 834 F.Romanów, P.Najdychor belonging to that region are called thin plates and the number of half- waves m is equal to one. In that region, the broken line hypothesis and hyperbolic hypothesis give similar results. • In the second sub-region the number of half waves depends on the core thickness. It corresponds to cgr1 ¬ c ¬ cgr2 = 0.0235, 1 < m < 13. These are medium thickness plates. • The third sub-region is defined by the core thickness c ­ cgr2. In this zone, the half-wave number m has a constant value (m= 13) and the coefficient α is slowly changing with increase in the core thickness and tends asymptotically to a constant value. From that it can be seen that assumption of c> cgr2 is uneconomical, because the plate weight incre- ases, while α is practically constant. With the help of formulas (3.9) and (3.7), one can calculate the resultant force acting upon the plate Nkr =Nxkrb Nxn =Nxb (3.10) or related stresses σkr = Nxkr 2t σxn = Nx 2t (3.11) A. Appendix 1 Upto thepresent,methodsof calculation of sandwich structureshave taken into account linear displacement states of faces and core. This is the so-called broken line hypothesis. In Fig.5, the displacements in the x-axis direction are presented by means of segments AC−, CE− (broken line) and EG−. This means that only displacements in the x-direction and constant displacements of the three layers in the vertical direction are taken into account for the faces and core. In the x-direction, the core displacement is described by a linear function in z U∗r =U +− z c ( U−− t 2 ∂W ∂x ) (A.1) but in the direction vertical with respect to the plate surface, the core displa- cement (deflection) is constant along the whole thickness W∗r =W (A.2) Determination of the post-critical state of compressed plates... 835 Fig. 5. Longitudinal displacement of the core (illustration of the hyperbolic hypothesis) (a) before strain, (b) after strain It is assumed that the core in the vertical direction cannot be strained. This means that it is infinitely rigid (E∗r =∞). The geometrical meaning of this is as follows: a randompoint J (Fig.5), located upona cross-section of the core, is displaced parallel to the x-axis up to the point K. The geometrical set of points K forms the segment COKE. For real core parameters (E∗r 6=∞), J is displaced into point J ′, and its projection upon the x-axis is denoted by D. 836 F.Romanów, P.Najdychor From geometrical relations, the formula JK− = z c ( −U−+ t 2 ∂W ∂x ) (A.3) is obtained. A variable segment JD− can be defined by the function JD− = JK−F(z)=− z c ( U−− t 2 ∂W ∂x ) F(z) (A.4) The total displacement of the point D in the x-direction will be equal to the sum of the average displacement U+ of the point O and the segment JD−. In other words Ur =U +− z c ( U−− t 2 ∂W ∂x ) F(z) (A.5) The set of points D related to the core thickness forms the curve EDOC, the shape of which depends on the function F(z). In the general case, taking into account the tri-axial core displacement state, from the three equilibrium equations for the core element and from the boundary conditions z=0, z= c, ∂F(0)/∂x=0 and F(c) = 1, one can get the hyperbolic function F(z)=H1 coshλz+H2 sinhλz z (A.6) In the same way, in the plane yz, one can get S(z)=H3coshλz+H4 sinhz z (A.7) and in the direction perpendicular to the plate plane T(z)=H5coshλz+H6z sinhλz (A.8) λ= √ β2+ρ2 = √ (Πm a )2 + (Πn b )2 (A.9) Ultimately, displacements of a random point J in the x, y, z directions are defined by the functions Ur =U +− z c ( U−− t 2 ∂W ∂x ) F(z) Vr =V +− z c ( V−− t 2 ∂W ∂y ) S(z) (A.10) Wr =WT(z) Determination of the post-critical state of compressed plates... 837 B. Appendix 2 H1 =−C2 c r H2 = ( 1+C2 c r coshλc ) H4 =(1−H3coshλc) c sinhλc H3 =−C2 ρc βp1 H5 = ( 1−C2 cλ β sinhλc ) 1 coshλc H6 = λ β C2 C2 =− βC[λsinhλc+(βr+ρp1)coshλc] λ(2+C)sinhλccoshλc+λ2cC r= Bmn Amn − tβ 2 p1 = Cmn Amn − tρ 2 C = 1 (1−2νr) For the given task the value amout λ=β, Vr =0 F1 =H 2 1X1+2H1H2X2+H 2 2X3 F5 =(H 2 5β 2+H26 +2H5H6β)X3+(2H5H6β 2+2H26β)X2+H 2 6β 2X1 F9 =(H1H5β+H1H6+H2H6β)X2+(H2H5β+H2H6)X3+H1H6βX1 F10 =(H 2 1 +H 2 2β 2+2H1H2β)X5+(2H 2 1β+2H1H2β 2)X2+H 2 1β 2X4 F11 =H 2 5X5+2H5H6X2+H 2 6X4 F12 =(H1H5+H2H5β)X5+(H1H6+H1H5β+H2H6β)X2+H1H6βX4 F16 =H 4 5X6+4H 3 5H6X7+6H 2 5H 2 6X8+4H5H 3 6X9+H 4 6X10 F19 =(H 4 5β 4+H46 +4H 3 5H6β 3+6H25H 2 6β 2+4H5H 2 6β)X11+ +(4H35H6β 4+12H25H 2 6β 3+12H5H 3 6β 2+4H46β)X12+ +(6H25H 2 6β 4+12H5H 3 6β 3+6H46β 2)X8+ +(4H5H 3 6β 4+4H46β 3)X13+H 4 6β 4X14 F22 =(H 4 5β 2+2H35H6β+H 2 5H 2 6)X15+(2H 3 5H6β 2+2H25H 2 6β)X7+ +H25H 2 6β 2X16+(2H 3 5H6β 2+4H25H 2 6β+2H5H 3 6)X12+ +(4H25H 2 6β 2+4H5H 3 6β)X8+2H5H 3 6β 2X13+ +(H25H 2 6β 2+2H5H 3 6β+H 4 6)X17+(2H5H 3 6β 2+2H46β)X19+H 4 6β 2X18 X1 = c ∫ −c z2 cosh2βz dz X2 = c ∫ −c z sinhβz coshβz dz X3 = c ∫ −c sinh2βz dz X4 = c ∫ −c z2 sinh2βz dz 838 F.Romanów, P.Najdychor X5 = c ∫ −c cosh2βz dz X6 = c ∫ −c cosh4βz dz X7 = c ∫ −c z sinhβzcosh3βz dz X8 = c ∫ −c z2 sinh2βz cosh2βz dz X9 = c ∫ −c z3 sinh3βzcoshβz dz X10 = c ∫ −c z4 sinh4βz dz X11 = c ∫ −c sinh4βz dz X12 = c ∫ −c z sinh3βz coshβz dz X13 = c ∫ −c z3 sinhβz cosh3βz dz X14 = c ∫ −c z4 cosh4βz dz X15 = c ∫ −c sinh2βz cosh2βz dz X16 = c ∫ −c z2 cosh4βz dz X17 = c ∫ −c z2 sinh4βz dz X18 = c ∫ −c z4 sinh2βzcosh2βz dz References 1. Biliński T., Kmita J., 2000, Dorobek nauki polskiej w zakresie konstrukcji zespolonych, Wydawnictwo Zachodnie Centrum Organizacji, Politechnika Zie- lonogórska 2. GrądzkiR.,Kowal-MichalskaK., 2000,Nośność ściskanychwielowarstwo- wych płyt prostokątnych, IX Sympozjon Stateczności Konstrukcji, Zakopane, 53-60 3. Hop T., 1980,Konstrukcje warstwowe, ArkadyWarszawa 4. Magnucki K., Oswald M., 2001, Stateczność i optymalizacja konstrukcji warstwowych, Poznań-ZielonaGóra 5. Planterma F.J., 1966, Sandwich Construction, John Wiley-Sonc, Inc. New York 6. RomanówF., 1995,Wytrzymałość konstrukcji warstwowych,WSI ZielonaGó- ra 7. RomanówF., 2001,Wkład polskich uczonych w rozwój metod obliczeniowych i badań konstrukcji warstwowych, PolskaMechanika u progu XXI wieku, Oficyna Wydawnicza PolitechnikiWarszawskiej. Kazimierz Dolny-Warszawa, 123-134 Determination of the post-critical state of compressed plates... 839 8. Romanów F., Najdychor P., Bejenka S., 2001, Stany nadkrytyczne ści- skanych jednorodnych płyt usztywnionych zewnętrznymi warstwami, II Sym- pozjon Kompozyty-Konstrukcje Warstwowe, PTMTS oddział we Wrocławiu„ Wrocław-Karpacz, 199-204 9. Stam K., Witte H., 1973, Sandwichkonstruktionen, Springer-Verlag, Wien, NewYork 10. SzycW., 1971,Nieliniowe zagadnienia stateczności sprężystej, trójwarstwowej otwartej powłoki walcowej,Rozprawy inżynierskie, 19 11. Twardosz F., Hong-Thai D., 1975, Stateczność trójwarstwowej otwartej powłoki walcowej poddanej ścinaniu,Archiwum Budowy Maszyn, 3 12. Zielnica J., 1981,Wyboczenie trójwarstwowej powłoki stożkowej poza zakre- sem sprężystym przy obciążeniu złożonym,Rozprawy inżynierskie, 453-470 Określenie stanu naprężeń nadkrytycznych ściskanych płyt warstwowych z uwzględnieniem teorii binieliniowej Streszczenie Większośćproblemówwytrzymałościowychdotyczącychkonstrukcjiwarstwowych analizowana jest na podstawie liniowego stanu przemieszczeń. Ten stan przemiesz- czeń zdefiniowano za pomocą hipotezy linii łamanej. Hipoteza wykorzystywana jest zarówno do określenia stanów krytycznychw ujęciu geometrycznie liniowej teorii, jak i w ujęciu geometrycznie nieliniowej teorii – do analizy stanów nadkrytycznych. Wniniejszej pracy przedstawiono analizę nadkrytycznych obciążeń z uwzględnie- niem nieliniowego stanu przemieszczeń rdzenia oraz geometrycznie nieliniowej teorii dla okładzin i rdzenia (binieliniowa teoria). Problem rozwiązano przy pomocy metody energetycznej, a nadkrytyczne naprę- żenia opisano ogólnymwzorem, na podstawie którego przeanalizowano przykład jed- noosiowo ściskanej płyty przegubowo podpartej na dwóch krawędziach. Manuscript received January 12, 2004; accepted for print March 30, 2004