JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 2, pp. 299-322, Warsaw 2006 SOLUTION TO THE STATIC STABILITY PROBLEM OF THREE-LAYERED ANNULAR PLATES WITH A SOFT CORE Dorota Pawlus Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biała e-mail: doro@ath.bielsko.pl The solutions to the static stability problem of a three-layered annular platewith a soft core and a symmetric cross-section structure are presented in this paper. The basic element of the solution is the formulation of a system of differential equations describing plate deflections and the use of the finite differencemethod in calculation of critical loads of buckling forms solving the eigen-value problem. The solution indicates theminimal values of static critical loads as well as the buckling forms of plates compressed on a selected edge. The obtained results have been compared with those obtained for plate models built by means of the finite element method. The final remarks concerning the forms of the loss of static stability of analysed plates with the sandwich structure have been formulated. This paper is a complement of the work by Pawlus (2005), which concerned calculations of the dynamic stability of plates, and it is an extension to cases of wave forms of the plate buckling problem earlier presented only for regular, axially-symmetrical forms of deformation in, eg., Pawlus (2002). Key words: sandwich, annular plate; static stability; buckling form; finite differences method; FEM 1. Introduction The evaluation of critical static loads with the indication of the minimal valueand the corresponding formofplatebuckling is thebasicprobleminplate stability analysis. The analysis of static critical loads precedes the evaluation of dynamic critical loads of plates and the observation of their supercritical behaviour. Buckling loads and geometrically nonlinear axisymmetric postbuckling be- haviour of cylindrically orthotropic annular plates under inplane radial com- pressive load applied to the outer edgewere undertaken byDumir andShingal 300 D. Pawlus (1985). Geometrically nonlinear, axisymmetric,moderately large deflections of laminated annular plates were presented by Dumir et al. (2001). Also in the work byKrizhevsky andStavsky (1996) laminated annular plates were exami- ned.Buckling loads of suchplates uniformly compressed in the radial direction were analysed, too. The axisymmetric dynamic stability of sandwich circular plates with viscoelastic damping layer under periodic radial loading along the outer edge was the subject of considerations by Wang and Chen (2003). Al- so the axisymmetric dynamic instability of a rotating sandwich annular plate with a viscoelastic core under periodic radial stress was examined by Chen et al. (2006). Solutions to the static analysis of plate stability presented in this paper refer to the solutions of the three-layer annular plate problem presented by Pawlus (2005). They exactly constitute the introduction to the dynamic stabi- lity plate problemundertaken in thementionedwork. The presented solutions do not limit the range of the examined plates only to such forms of deforma- tions which are regular and axially-symmetrical (see Pawlus, 2002), but they are global solutions for different circumferential wave forms of the loss of plate static stability. The presented solutions eliminate possible questions connected with forms of the plate buckling forminimal values of critical loads. They also show that the variability in number of waves of deformation plates strongly depends on geometric and material properties of layers in plate structures. Two solutions presented in this paper use approximation methods: finite dif- ferences and finite elements. The proposed solution to the static stability of analysed three-layer annular plate, which uses the finite difference method, refers to solutions of homogeneous elastic plates presented, eg. by Wojciech (1979) aswell as byTrombski andWojciech (1981). Additionally, somemodifi- cation of calculation algorithms to formulas necessary for sandwich structures is introduced. 2. Problem formulation A three-layer annular plate with a symmetric cross-section structure com- posed of thin steel facings and a soft foam isotropic core is considered. Plate edges are clamped. Compressive loads uniformly distributed on the plate pe- rimeter act on the outer or/and inner edge of the plate facings. A scheme of the plate is presented in Fig.1. In the solution based on the finite differencemethod, the classical theory of sandwichplateswith thebroken linehypothesis (Volmir, 1967) is adopted.The Solution to the static stability problem... 301 Fig. 1. Scheme of analysed plate classical participation of plate layers in carrying the plate load is assumed: the facings are loaded with normal but the core with shear stresses. Equal values of transverse deflections of plate layers are accepted. The minimal critical static load of the plate and the corresponding form of buckling are calculated analysing the minimal critical loadings found from the eigen-value problem for different numbers m of plate circumferential waves describing the form of plate deformation. 3. System of elementary equations In the group of presented basic equations – the obtained equation, (3.14), enabling calculation of transverse plate deflections is fundamental. The qu- antities describing the relative radial δ and circumferential displacements γ of plate facings coming from the sandwich structure of the analysed plate build additional expressions in equation (3.14) in relation to the formulas of homogeneous plates. 3.1. Equilibrium equations The system of forces acting on each of three layers of a single annular sector of the plate is presented in Fig.2. The system of equilibrium equations of each layer is presented by the formulas: 302 D. Pawlus — layer 1 Mr1 −Mθ1 r +Mr1′r + 1 r Mθr1′θ −Qr1 + h1 2 τr1 =0 1 r Mθ1′θ + 2 r Mrθ1 +Mrθ1′r −Qθ1 + h1 2 τθ1 =0 (3.1) (Tθr1w′r)′θ+(Tθr1w′θ)′r+(rNr1w′r)′r+ 1 r (Nθ1w′θ)′θ+ +Qθ1′θ +(rQr1)′r+rτr1w′r + τθ1w′θ =0 — layer 2 −Qr2 + h2 2 τr1 + h2 2 τr3 =0 −Qθ2 + h2 2 τθ1 + h2 2 τθ3 =0 (3.2) Qθ2′θ +(rQr2)′r−rτr1w′r+rτr3w′r− τθ1w′θ+ τθ3w′θ =0 — layer 3 Mr3−Mθ3 r +Mr3′r + 1 r Mθr3′θ −Qr3+ h3 2 τr3 =0 1 r Mθ3′θ + 2 r Mrθ3 +Mrθ3′r −Qθ3+ h3 2 τθ3 =0 (3.3) (Tθr3w′r)′θ+(Tθr3w′θ)′r +(rNr3w′r)′r+ 1 r (Nθ3w′θ)′θ+ +Qθ3′θ +(rQr3)′r −rτr3w′r− τθ3w′θ =0 where Nr1(3),Nθ1(3) – normal radial and circumferential forces acting on facings per unit length, respectively Qr1(2,3),Qθ1(2,3) – transverse forces acting on facings and core layer per unit length, respectively Mr1(3),Mθ1(3) – elementary radial and circumferential bendingmo- ments of facings, respectively Mrθ1(3) – elementary torsional moments of outer layers Tθr1(3) – shear forces per unit length acting on outer plate layers τr1(3),τθ1(3) – shearing radial and circumferential stresses, espec- tively w – plate deflection. Solution to the static stability problem... 303 Fig. 2. Loading of plate layers 3.2. Geometric relations Radial and circumferential cross-section deformations of the plate struc- ture are shown in Fig.3. The angles β and α determine the radial and cir- cumferential deformation of the plate core, respectively. They are expressed by equations β = u1−u3−w′rh ′ h2 α= v1−v3− 1 r w′θh ′ h2 (3.4) where 304 D. Pawlus u1(3),v1(3) – displacements of the points of the middle plane of fa- cings in the radial and circumferential directions, re- spectively h′ =h1 =h3 – thicknesses of the plate layers. Fig. 3. Cross-sectional geometry of sandwich plate: (a) in radial direction, (b) in circumferential direction 3.3. Physical relations Linear physical relations of Hooke’s law for the plane stress state in plate outer layers are given by the following formulas σri = Ei 1−ν2i (εri +νεθi) σθi = Ei 1−ν2i (εθi +νεri) (3.5) where i denotes the outer layer, i=1 or 3. Young’s moduli andPoisson’s ratios of the facingmaterial fulfil the condi- tions: E =E1 =E3 and ν = ν1 = ν3. The physical relations of the core material under shearing stress are as follows τrz2 =G2γrz2 τθz2 =G2γθz2 (3.6) where γrz2, γθz2 – shearing strain of the core in the radial and circumferential directions, respectively γrz2 =u (z) 2′z +w′r γθz2 = v (z) 2′z + 1 r w′θ Solution to the static stability problem... 305 and u (z) 2 =u2−zβ, v (z) 2 = v2−zα – radial and circumferential displacements of a point with the z – coordinate, respectively (z is the distance between the point and the middle surface of the core). 3.4. Differential equations for plate deflections Based on the relations between sectional forces, moments and stresses for plate facings, equations of sectional forces andmoments have been established Nri = Ehi 1−ν2 ( ui′r+ ν r ui+ ν r vi′θ + 1 2 (w′r) 2+ ν 2r2 (w′θ) 2 ) Nθi = Ehi 1−ν2 (1 r ui+ 1 r vi′θ +νui′r+ ν 2 (w′r) 2+ 1 2r2 (w′θ) 2 ) Trθi =Ghi (1 r ui′θ +vi′r − 1 r vi+ 1 r w′rw′θ ) (3.7) Mri =−Di ( w′rr+ ν r w′r+ ν r2 w′θθ ) Mθi =−Di ( 1 r2 w′θθ+ 1 r w′r+νw′rr ) Mrθi =−2Drθi (w r ) ′rθ where Di, Drθi denote the flexural rigidities of the outer layers, and Di = Eh 3 i 12(1−ν2) Drθi = Gh3i 12 The transverse forces Qr2 and Qθ2 respectively expressed by formulas Qr2 = τrz2h2, Qθ2 = τθz2h2, have been obtained using equations (3.4) and (3.6) Qr2 =G2(δ+H ′w′r) Qθ2 =G2 ( γ+H′ 1 r w′θ ) (3.8) where δ=u3−u1 γ = v3−v1 H ′ =h′+h2 (3.9) Finding from equations (3.1)1,2, (3.2)1,2 and (3.3)1,2 formulas determining the radial Qr1(2,3) and circumferential Qθ1(2,3) forces, enables one to obtain the resultant forces Qr and Qθ as the sums of the individual layer forces 306 D. Pawlus Qr = 1 r (Mr1 +Mr3)− 1 r (Mθ1 +Mθ3)+(Mr1 +Mr3)′r+ + 1 r (Mθr1 +Mθr3)′θ+ H′ h2 Qr2 (3.10) Qθ = 1 r (Mθ1 +Mθ3)′θ+(Mrθ1 +Mrθ3)′r+ 2 r (Mθr1 +Mθr3)+ H′ h2 Qθ2 Inserting equations (3.7)4−6 and (3.8) into equations (3.10) yields the following formulas Qr =−k1w′rrr− k1 r w′rr+ k1 r2 w′r− k2 r2 w′rθθ+ k1+k2 r3 w′θθ+ +G2(δ+H ′w′r) H′ h2 (3.11) Qθ =− k1 r3 w′θθθ− k1 r2 w′θr− k2 r w′θrr+G2 ( γ+H′ 1 r w′θ )H′ h2 where k1 =2D, k2 =4Drθ +νk1. Adding the summands of equations (3.1)3, (3.2)3, (3.3)3 all together gives the following equation (Trθw′r)′θ+(Tθrw′θ)′r+(rNrw′r)′r+ 1 r (Nθw′θ)′θ+Qθ′θ +(rQr)′r =0 (3.12) In the above equation, (3.12) the resultant membrane forces Nr, Nθ, Trθ are expressed respectively: Nr =Nr1+Nr3,Nθ =Nθ1+Nθ3 and Trθ =Trθ1+Trθ3. They have been determined bymeans of the introduced stress function Φ Nr =2h ′ (1 r Φ′r+ 1 r2 Φ′θθ ) Nθ =2h ′Φ′rr (3.13) Trθ =2h ′ ( 1 r2 Φ′θ − 1 r Φ′rθ ) Inserting (3.11) and (3.13) into equation (3.12) yields a differential equation for deflections of the analysed plate Solution to the static stability problem... 307 k1w′rrrr+ 2k1 r w′rrr− k1 r2 w′rr+ k1 r3 w′r+ k1 r4 w′θθθθ+ 2(k1+k2) r4 w′θθ+ 2k2 r2 w′rrθθ+ − 2k2 r3 w′rθθ−G2 H′ h2 1 r ( γ′θ+ δ+rδ′r+H ′ 1 r w′θθ+H ′w′r +H ′rw′rr ) = (3.14) = 2h′ r ( 2 r2 Φ′θw′rθ − 2 r Φ′θrw′θr+ 2 r2 w′θΦ′θr− 2 r3 Φ′θw′θ+w′rΦ′rr+Φ′rw′rr+ + 1 r Φ′θθw′rr+ 1 r Φ′rrw′θθ ) 3.5. Boundary conditions The boundary conditions for the loading are expressed by equations σr ∣ ∣ ∣ r=ri =−pd1 σr ∣ ∣ ∣ r=r0 =−pd2 (3.15) where d1, d2 are some quantities being 0 or 1, which determine the loading of the inner or/and the outer plate edge (Wojciech, 1978). The boundary conditions for the clamped edges of the plate are as follows w ∣ ∣ ∣ r=r0(ri) =0 w′r ∣ ∣ ∣ r=r0(ri) =0 δ ∣ ∣ ∣ r=r0(ri) =0 δ′r ∣ ∣ ∣ r=r0(ri) =0 γ ∣ ∣ ∣ r=r0(ri) =0 γ′r ∣ ∣ ∣ r=r0(ri) =0 (3.16) 4. Problem solution The quantities δ and γ, unknown in equations (3.14), have been obtained by finding the differences in the radial and circumferential displacements u1, u3 and v1, v3 of points from themiddle surface of the plate facings (3.9) using the equilibrium equations for forces acting on the undeformed outer plate layers in the u and v direction, respectively: — layer 1 Nr1 +rNr1′r −Nθ1 +Tθr1′θ +rτr1 =0 (4.1) Nθ1′θ +2Trθ1 +rTrθ1′r +rτθ1 =0 — layer 3 Nr3 +rNr3′r −Nθ3 +Tθr3′θ −rτr3 =0 (4.2) Nθ3′θ +2Trθ3 +rTrθ3′r −rτθ3 =0 308 D. Pawlus Having calculated the above expressions, the summands in equations (4.1) and (4.2) have been subtracted and then expressions (3.7)1−3, which determine the sectional forces Nri,Nθi, Trθi, have been inserted into the obtained equations. The shearing stresses τr, τθ have been expressed by sums of stresses τr1, τr3 and τθ1, τθ3 using equations (3.2)1,2 τr1 + τr3 = 2 h2 Qr2 τθ1 + τθ3 = 2 h2 Qθ2 After some transformations, the following differential equations have been fo- und 2r h2 G2H ′w′r = Eh′ 1−ν2 ( rδ′rr+ δ′r− 1 r δ+νγ′rθ− 1 r γ′θ ) + +Gh′ 1 r (δ′θθ+rγ′rθ−γ′θ)− 2r h2 G2δ (4.3) 2 h2 G2H ′w′θ = Eh′ 1−ν2 (1 r δ′θ +νδ′rθ+ 1 r γ′θθ ) − 2r h2 G2γ+ +Gh′ 1 r (δ′θ+rδ′rθ+r 2γ′rr+rγ′r−γ) Using the followingdimensionlessquantities andthe expressions in the solution F = Φ Eh2 ζ = w h ρ= r r0 δ= δ h ζ(ρ,θ)=X(ρ)cos(mθ) γ= γ h δ(ρ,θ)= δ(ρ)cos(mθ) γ(ρ,θ)= γ(ρ)sin(mθ) (4.4) where m is the number of circumferential waves corresponding to the form of plate buckling, h = h1 +h2 +h3 – total thickness of plate, equations (3.14) and (4.3) can be presented in the following form W1X′ρρρρ+ 2W1 ρ X′ρρρ− W3 ρ2 X′ρρ+ W3 ρ3 X′ρ+ W4 ρ4 X− 2W1 ρ4 m2X+ − W5 ρ H′ ( mγ+ δ+ρρ′ρ− m2 ρ H′ r0 X+ H′ r0 X′ρ+ H r0 ρX′ρρ ) = = 2W25W2 ρ ( X′ρY0′ρ+Y0X′ρρ− m2 ρ XY0′ρ ) X′ρ = δ ( A 1 ρ2 +B+ m2 ρ2 C ) −A 1 ρ δ′ρ−Aδ′ρρ−m K2 ρ γ′ρ+m K1 ρ2 γ (4.5) mX =−m K1 ρ δ−mK2δ′ρ+ρCγ′ρρ+Cγ′ρ−γ ( m2 A ρ + C ρ +Bρ ) Solution to the static stability problem... 309 where Y0 =F′ρ A=− Eh′ 1−ν2 h2 G2 1 2H′r0 B=− r0 H′ C =− Gh′ r0 h2 2G2H′ D=Aν K1 =A+C K2 =D+C W1 = k1 h′ h h2 G2 1 r30 W2 =E h2 G2 h3 r30 W12 = k2 h′ h h2 G2 1 r30 W3 =W1+2m 2W12 W5 = h′ h W4 =m 4W1−2m 2W12 Assuming that the stress function F is a solution to the disk state and using the boundary conditions for the clamped edges, based on the work by Wojciech (1978), the following expression has been obtained Y0 =K10p ∗ ( e1ρ+ e2 ρ ) (4.6) where K10 = r2z h2 p∗ = p E e1 = d2 ρi −d1ρi ρi− 1 ρi e2 = d1ρi−ρid2 ρi− 1 ρi and ρi is the dimensionless inner plate radius. In the solution, the finite difference method has been used for the appro- ximation of the derivatives with respect to ρ by central differences in discrete points. Transformed equations (4.5) have the forms MAPU+MADD+MAGG= p ∗ MACU (4.7) MACPU =MACDD+MACGG MPU =MDD+MGG where: U, D, G – vectors of plate deflections and differences of the radial ui and circumferential vi displacements of facings (3.9), respectively MAP , MAC, MACD, MACG, MD, MG – matrices of elements composed of geometric andmaterial parameters of the plate and the quantity b of the length of the interval in the finite difference method and the number m of buckling waves MAD – matrix of geometric parameters and the quantity b 310 D. Pawlus MAG – matrix of geometric parameters and the number m MACP – matrix with elements described by the quantity 1/(2b) MP –matrix with elements described by the number m. Solving the eigen-value problem, the minimal value of p∗ as the critical static load p∗cr has been calculated det[(MAP +MADMATD+MAGMATG)−p ∗ MAC] = 0 (4.8) and MATG,MATD arematrices obtained from transformed equations (4.7)2,3 in the forms MATG =M −1 TG(MP −MDM −1 ACDMACP) (4.9) MATD =M −1 ACDMACP −M −1 ACDMACGMATG where MTG =MG−MDM −1 ACD MACG 5. Numerical calculations Exemplary numerical calculations of a plate loaded on the inner or/and outer edges have been carried out by analysing the influence of geometric and material parameters on the critical static loads and corresponding forms of buckling. The calculations have been carried out for plates with the following geo- metrical dimensions: inner radius ri =0.2m, outer radius r0 =0.5m, various core and steel facing thicknesses in the range of: h2 =0.005m, 0.01m, 0.02m and h′ =0.0005m, 0.001m, respectively; accepting a polyurethane foamas an isotropic core material with Kirchhoff’s moduli G2 = 5MPa (Majewski and Maćkowski, 1975) and G2 = 15.82MPa (Romanów, 1995) and the Poisson’s ratio ν =0.3 (PN-84/B-03230). 5.1. Calculations by finite difference method Calculations of plates using the Finite Difference Method (FDM) have been preceded by analysis of the accuracy of values of the critical loads for different numbers N of discrete points: N = 11, 14, 17, 21, 26. Tables 1, 2, 3, 4 show the critical plate loads pcr for different buckling forms determined Solution to the static stability problem... 311 by the number m of circumferential waves. Theminimal critical loadwith the wave number mhave beenmarked. The analysis of critical loads pcr indicates that the number N = 14 of discrete points fulfils the accuracy up to 5% of technical error. The calculations were carried out for this number (N = 14) of discrete points in FDM. The results show that for a higher number N of discrete points, N = 21, 26, the form of plate buckling has an additional circumferential wave for the minimal critical plate load pcr. The influenceof coreKirchhoff’smodulusand layer thicknesses, particular- ly the core on the distribution of critical loads and the forms of plate buckling are presented in Fig.4-Fig.6. Table 1.Critical plate loads pcr for different wave numbers m d1 =0 d2 =1 E =2.1 ·10 5 MPa ri =0.2m r0 =0.5m h ′ =0.001m ν =0.3 G2 =5MPa h2 =0.005m pcr [MPa] m N 11 14 17 21 26 0 32.78 32.89 32.94 32.98 33.01 1 30.95 31.06 31.12 31.16 31.19 2 26.89 27.01 27.09 27.14 27.18 3 23.32 23.45 23.53 23.59 23.63 4 21.25 21.37 21.44 21.50 21.54 5 20.41 20.52 20.58 20.63 20.67 6 20.44 20.53 20.58 20.62 20.65 7 21.05 21.13 21.17 21.21 21.24 8 22.09 22.16 22.22 22.24 22.26 All analysed examples of plates loaded on the inner perimeter of facings confirmed the observation earlier noticed in homogeneous plates (Wojciech, 1978; Pawlus, 1996) that the buckling of plates with double clamped edges for the minimal critical static load has a regular, axi-symmetrical form. Figure 4 shows a suitable distribution of the critical loads. Detailed results for such lo- adedplates, including their behaviour,were presentedbyPawlus (2002, 2003). Diagrams 5, 6 present the distribution of critical loads for the plate compres- sed at outer perimeters depending on the number m of buckling waves. The pointsmarked by ∗ in the diagrams correspond to forms of buckling of plates loaded withminimal critical loads. The presented results indicate a change in 312 D. Pawlus Table 2.Critical plate loads pcr for different wave numbers m d1 =0 d2 =1 E =2.1 ·10 5 MPa ri =0.2m r0 =0.5m h ′ =0.0005m ν =0.3 G2 =5MPa h2 =0.02m pcr [MPa] m N 11 14 17 21 26 0 118.77 118.95 119.04 119.11 119.16 9 70.31 70.44 70.54 70.64 70.72 10 69.70 69.83 69.92 70.01 70.09 11 69.41 69.53 69.62 69.71 69.79 12 69.38 69.49 69.58 69.67 69.74 13 69.55 69.67 69.75 69.83 69.90 14 69.91 70.02 70.10 70.18 70.25 Table 3.Critical plate loads pcr for different wave numbers m d1 =0 d2 =1 E =2.1 ·10 5 MPa ri =0.2m r0 =0.5m h ′ =0.001m ν =0.3 G2 =15.82MPa h2 =0.005m pcr [MPa] m N 11 14 17 21 26 0 76.10 76.19 76.23 76.27 76.30 3 54.84 55.08 55.22 55.33 55.42 4 49.81 50.04 50.18 50.29 50.37 5 47.31 47.50 47.62 47.72 47.80 6 46.37 46.53 46.64 46.72 46.79 7 46.42 46.56 46.65 46.72 46.78 8 47.15 47.27 47.34 47.40 47.45 9 48.37 48.47 48.53 48.59 48.63 10 49.98 50.06 50.12 50.16 50.12 the deformations for plateswith stiffer structures.With an increase in the core thickness and Kirchhoff’s modulus or with a decrease in the facing thickness, the form of plate deformation has an additional buckling wave. Solution to the static stability problem... 313 Table 4.Critical plate loads pcr for different wave numbers m d1 =1 d2 =0 E =2.1 ·10 5 MPa ri =0.2m r0 =0.5m h ′ =0.001m ν =0.3 G2 =5MPa h2 =0.005m pcr [MPa] m N 11 14 17 21 26 0 74.70 75.61 76.05 76.39 76.57 1 86.78 87.72 88.16 88.48 88.69 2 121.70 123.47 124.27 124.87 125.27 3 159.50 165.46 167.18 168.42 169.26 4 195.78 214.02 217.68 219.94 221.45 5 236.00 263.92 275.70 279.59 282.12 Fig. 4. Critical static load distributions depending on number of buckling waves for plates compressed on inner perimeter 314 D. Pawlus Fig. 5. Critical static load distributions depending on number of buckling waves for plates compressed on outer perimeter with different core thicknesses andmaterial parameters Fig. 6. Critical static load distributions depending on number of buckling waves for plates compressed on outer perimeter with different core and facing thicknesses 5.2. Calculations by finite element method The presented results of examined plates have been comparedwith the re- sults obtained using the Finite ElementMethod (FEM). For this purpose, the computational plate models consistent with the analysed models in the finite Solution to the static stability problem... 315 difference method have been built. The fundamental computational model of the plate is a full annulus plate model presented in Fig.7. Additionally, the plate models in the form of an annular sector as the 1/8 or 1/6 part of the annulus have been built – see Fig,8a,b, respectively. Fig. 7. Full annulus plate model Fig. 8. Annular sector plate model, (a) 1/8 part of annulus (α=45◦), (b) 1/6 part of annulus (α=60◦) The facing mesh has been built using 3D, 9-node shell elements, but the core mesh was made of 3D, 27-node solid elements. The outer surfaces of facingmesh elements have been connected with the outer surfaces of core ele- ments using the surface contact interaction. The deformations of the inner and outer plate edges have been limited by the support conditions witho- ut the possibility of relative displacements of facings in their clamped edges. There is no limitation to the deformation, which was earlier formulated by the condition of equal deflections of each plate layer. The calculations we- re carried out at the Academic Computer Center CYFRONET-CRACOW (KBN/SGI ORIGIN 2000/PŁódzka/030/1999) using the ABAQUS system. The symmetry conditions enabling the observation of such forms of plate deformations for which the length of a single circumferential wave is included or is amultiple of the angle of an annular sector have been imposed on the side 316 D. Pawlus edges of the annular sector plate models. The computational results of plate models built in the form of annular sectors (Fig.8) compared with the results of the full annulus plate model (Fig.7) allow the evaluation of correctness of the FEM-based calculations. The computational capability of the program enabled creation of plate meshes for the annular sector model thicker than those for the full annulus model, hence the accuracy of the results could be greater. The results presented inFig.9 and inTables 5, 6, 7 for plates with the facing thickness h′ =0.001m show some quantitative discrepancy in values of the critical loads. Fig. 9. Distribution of critical static loads of plates modelled as annular sectors or full annulus; (1) annular sector model, (2) full annulus model The presented inTables 5, 6, 7 critical loads pcr and forms of buckling are given in the increasing order up from the minimal value to numbers obtained for the full annulus platemodel. All results concern the platemodels loaded at the outer edge of facings. The presented results are comparable, however some differences in the range of higher values of critical loads depending on the kind of computational platemodel are observed. One can notice some sensitivity of numerical results with depend on the computational model using FEM. Some detailed remarks concerning calculations of plates loaded at inner facing edges, which are differently modelled were presented by Pawlus (2002, 2004, 2005). The numerical calculation of the full annulus plate model loaded at the inner edges confirms the observation that the minimal critical loads corresponds Solution to the static stability problem... 317 Table 5.Critical stresses calculated bymeans of FEM for plate models with parameters: h2 =0.005m, G2 =5MPa pcr [MPa] m Full annulus Annular sector of plate model plate model α=45◦ α=60◦ 5 16.48 – – 6 16.75 – 17.92 4 17.02 18.76 – 7 17.68 – – 3 18.65 – 20.22 8 19.25 19.74 – 9 21.49 – 22.48 2 21.52 – – 10 24.71 – – 1 24.87 – – 0 26.44 27.06 29.98 Table 6.Critical stresses calculated bymeans of FEM for plate models with parameters: h2 =0.005m, G2 =15.82MPa pcr [MPa] m Full annulus Annular sector of plate model plate model α=45◦ α=60◦ 6 35.04 – 36.74 5 35.46 – – 7 35.57 – – 8 36.94 38.35 – 4 37.19 38.37 – 9 38.98 – 39.86 3 40.87 – 44.24 10 42.48 – – 2 47.01 – – to the regular axi-symmetrical form (m = 0) of plate buckling. Exemplary critical loads in the increasing order for a plate with layer thicknesses equal: h′ = 0.001m, h2 = 0.005m and with core Kirchhoff’s modulus G2 = 5MPa are presented in Table 8. 318 D. Pawlus Table 7.Critical stresses calculated bymeans of FEM for plate models with parameters: h2 =0.02m,G2 =15.82MPa pcr [MPa] m Full annulus Annular sector of plate model plate model α=45◦ α=60◦ 9 115.10 – 123.23 8 115.26 124.43 – 7 116.52 – – 10 118.41 – – 6 119.53 – 130.13 11 122.55 – – 5 125.01 – – 12 129.98 – 124.03 4 134.29 147.01 – Table 8. Critical stresses and forms of buckling of plates loaded at inner perimeter of facings (d1 =1, d2 =0) pcr [MPa] 64.08 75.75 107.04 m=0 m=1 m=0, n=1 109.89 113.95 141.35 m=2 m=1, n=1 m=2, n=1 n – number of waves in radial direction Solution to the static stability problem... 319 Table 9. Critical stresses and forms of buckling of plates loaded at outer perimeter of facings (d1 =0, d2 =1) Parameters of plate pcr [MPa] h′/h2/G2 FEM Form of [m]/[m]/[MPa] FDM Full annulus Annular sector buckling plate model of plate model 0.001/0.005/5.0 20.52 16.48 – m=5 0.001/0.01/5.0 29.42 25.85 27.93 m=6 0.001/0.02/5.0 46.95 43.71 – m=7 0.001/0.005/15.82 46.53 35.04 36.74 m=6 0.001/0.02/15.82 125.11 115.1 123.23 m=9 0.0005/0.005/5.0 22.37 – – m=8 – 19.6 – m=7 320 D. Pawlus 6. Conclusions Comparing the results obtained using th two presented methods: Fini- te Difference Method (FDM) and Finite Element Method (FEM), quanti- tative correctness and qualitative consistency have been observed. Suitable results of critical static loads of plates calculated in the FDM and FEM with their forms of buckling are presented in Table 9. The critical loads of the annular sector plate model built in FEM show better consistency with the results of plates calculated in the FDM. For plates with thin facings (h′ =0.0005m), a difference in the buckling form calculated in the FDM and in FEM is observed. The number of waves is equal to m=8 and m=7, res- pectively. Analysing the results of critical static loadswith forms of buckling of annu- lar sandwichdouble-clampedplates determinedby the twopresentedmethods, it can be concluded: • in the case of loading of the inner plate perimeter, the minimal value of compressive static critical load is found for a regular axi-symmetrical form of loss of the plate static stability • in the case of loading of the outer plate perimeter, the minimal critical static loads and the numbers of buckling waves depend on the geome- trical and material parameters: with the increase in the plate stiffness, the critical loads and numbers of circumferential buckling waves incre- ase, too. References 1. ABAQUS/Standard. User’s Manual, version 6.1, 2000, Hibbitt, Karlsson and Sorensen, Inc. 2. Chen Y.R., Chen L.W.,Wang C.C., 2006, Axisymmetric dynamic instabi- lity of rotating polar orthotropic sandwich annular plates with a constrained damping layer,Composite Structures, 73, 2, 290-302 3. Dumir P.C., Shingal L., 1985, Axisymmetric postbuckling of orthotropic thick annular plates,Acta Mechanica, 56, 229-242 4. Dumir P.C., Joshi S., Dube G.P., 2001, Geometrically nonlinear axisym- metric analysis of thick laminated annular plate usingFSDT,Composites: Part B, 32, 1-10 Solution to the static stability problem... 321 5. Krizhevsky G., Stavsky Y., 1996, Refined theory for vibrations and buc- kling of laminated isotropic annularplates, International Journal ofMechanical Sciences, 38, 5, 539-555 6. Majewski S., Maćkowski R., 1975, Pełzanie spienionych tworzyw sztucz- nych stosowanych jako rdzeń płyt warstwowych, Inżynieria i Budownictwo, 3, 127-131 7. Pawlus D., 1996, Zachowanie pierścieniowych płyt lepkosprężystych przy ob- ciążeniach zmiennych,Ph.D.Thesis, PolitechnikaŁódzkaFiliawBielsku-Białej 8. Pawlus D., 2002, Obliczenia statycznych obciążeń krytycznych trójwarstwo- wych, osiowosymetrycznych płyt pierścieniowych, Czasopismo Techniczne, 6- M, Wydawnictwo Politechniki Krakowskiej, 71-86 9. Pawlus D., 2003, Calculations of three-layered annular plates under lateral loads, Studia Geotechnica et Mechanica, XXV, 3/4, 89-109 10. PawlusD., 2004,Obliczeniametodą elementów skończonych trójwarstwowych płytpierścieniowychpoddanychobciążeniomwichpłaszczyźnie, III Sympozjum Kompozyty. Konstrukcje warstwowe, Wrocław-Karpacz, 119-128 11. PawlusD., 2005,Dynamic stability problemof three-layeredannularplate un- der lateral time-dependent load, Journal of Theoretical and AppliedMechanics, 43, 2, 385-403 12. TrombskiM.,WojciechS., 1981,Płytapierścieniowaoortotropii cylindrycz- nej obciążonaw swej płaszczyźnie ciśnieniem zmiennymw czasie,The Archive of Mechanical Engineering,XXVIII, 2 13. Romanów F., 1995,Wytrzymałość konstrukcji warstwowych, WSI w Zielonej Górze 14. VolmirC., 1967,Stability of Deformed System, Science,Moskwa (in Russian) 15. Wang H.J., Chen L.W., 2003, Axisymmetric dynamic stability of sandwich circular plates,Composite Structures, 59, 99-107 16. Wojciech S., 1978, Stateczność dynamiczna ortotropowejpłytypierścieniowej obciążonej w swojej płaszczyźnie ciśnieniem zmiennymw czasie, Ph.D. Thesis, Politechnika Łódzka 17. Wojciech S., 1979, Numeryczne rozwiązanie zagadnienia stateczności dyna- micznej płyt pierścieniowych, Journal of Theoretical and Applied Mechanics, 17, 2 322 D. Pawlus Rozwiązanie zagadnienia stateczności statycznej pierścieniowych płyt trójwarstwowych z rdzeniem miękkim Streszczenie Wpracyprzedstawiono rozwiązania stateczności statycznej trójwarstwowychpłyt pierścieniowych o symetrycznej strukturze poprzecznej z piankowym rdzeniemmięk- kim. Zasadniczą częścią rozwiązania jest wyprowadzenie układu równań różniczko- wych opisujących ugięcia płyty oraz wykorzystaniemetody różnic skończonych i wy- znaczenie krytycznych obciążeń płyt poprzez rozwiązanie zagadnienia wartości wła- snych.Wyznaczonymwartościomciśnień krytycznychpłyt obciążonychnawybranym brzegu ich okładzin odpowiadają postacie deformacji płyt, które określa liczbam fal poprzecznych na obwodzie płyty. Otrzymane wyniki pod względem ilościowym i ja- kościowymporównano zwynikami obliczeńmetodą elementów skończonychprzedsta- wionychmodelipłyt. Sformułowanouwagikońcowedotyczące formutratystateczności statycznej analizowanychpłyt o strukturzewarstwowej.Artykuł stanowi uzupełnienie pracy Pawlus (2005) dotyczącej obliczeń stateczności dynamicznej płyt i rozwinięcie na przypadki sfalowanych form deformacji rozwiązania problemu stateczności sta- tycznej płyt rozpatrywanegowcześniej, min. w pracy Pawlus (2002) w zakresie tylko obrotowych, osiowo-symetrycznych form utraty ich stateczności. Manuscript received February 16, 2006; accepted for print March 20, 2006