Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 1, pp. 85-107, Warsaw 2008 APPROACH TO EVALUATION OF CRITICAL STATIC LOADS OF ANNULAR THREE-LAYERED PLATES WITH VARIOUS CORE THICKNESS Dorota Pawlus University of Bielsko-Biała, Faculty of Mechanical Engineering and Computer Science, Poland e-mail: doro@ath.bielsko.pl The evaluation of computational results of annular, three-layered plates with a soft foam core of different thickness under lateral compressive loads acting in facings planes has been conducted in this work.The values of critical static loads of plates and the forms of critical buckling corresponding with them have been the examined results of calculations. The calculations have been carried out using two approximation methods: finite difference method and finite elementmethod.The solution to the problemof static stability of plates presented in both methods concerns the general problem of the loss of plate stability corresponding to possible circumferential wave forms of plate critical deformations. The plates with a thick core have been analysed in detail. Eva- luating the results, it has been noted that the observed significant decrease in the critical static loads of plates with thick cores. This observation has been made by carrying out calculations with the help of the finite elementmethod for plate models differing in the condition of the plate layers connection. The observed regions of good consistency and of essential differences of the results obtained using the assumed numerical methods for plates with middle and thick cores have been the main effect of the examinations. They seem to be practical importance in the modelling of plate structures. Key words: annular layered plate, critical static loads, buckling form, finite differences method, FEM 1. Introduction Calculations of critical, static loads of plates are the fundamental stage of examinations of their stability. The minimal loads and forms of critical plate deformations allow for determination of the range of practicable loads, cha- racter of plate operation and their supercritical behaviour. Static critical pa- rameters have the essential importance in the evaluation of dynamic stability 86 D. Pawlus of structures subjected to time-dependent loads. The plate stability problems are multi-parameter tasks strongly depending on the material and geometri- cal parameters of the structure. The problem become more complex, when the examined object is a multi-element structure. In this case, not only the mentioned parameters, but also the method of connection of components in the structure and their participation in the operation of the whole object af- fect the behaviour of such a complex structure.Multi-layer plates and, among them, widely applied three-layer plates with soft cores are typical objects in the field of plate structures. Three-layer annular plates analysed in the range of the loss of its static stability are the subject of the present considerations. The presented solution has a general form enabling analysis both the particu- lar axisymmetric form of plate buckling and circumferential wave forms, too. This solution refers to the solution presented by Pawlus (2006). Thecomputational resultspresented indetail eg. byPawlus (2003a,b, 2004, 2007) concern exactly a special example of a plate, which loses its stability in a regular, axially-symmetrical form. The results presented in these works in- dicate significant differences in values of critical loads of plates with thick cores modelled with differing assumptions on layer deformation. This sensiti- vity has not been observed inmodels of plates with thin or medium-thickness cores. The approach to the evaluation of critical loads found by the presented methods is the task of this examination. Platemodels, whose critical deforma- tions could be an the unlimited, radial and circumferential wave forms have been considered. It can be noticed that the field of undertaken problems for annular, three- layer plates with thick cores is still really limited. There are works considering thick annular plates, butwith a homogeneous structure (see, Dumir and Shin- gal, 1985a,b). Numerous works deal with the dynamic stability of sandwich annular or circular plates. Some of them were presented by Wang and Chen (2003, 2004), Chen et al. (2006). The dynamic stability problemwas also con- sidered by Pawlus (2005). The structure of the analysed sandwich plate with thin andmedium cores was similar to that presented in this work. 2. Problem formulation An annular, three-layer plate under a load acting in the surface of its facings is the object of the analysis. The examined structure of the plate with double slidable clamped edges is presented inFig.1.Examples of plates compressedon outer or inner edges are considered. The schemes are presented in Fig.2. Thus Approach to evaluation of critical static loads... 87 loaded plate loses its static stability. Forms of the loss of the plate stability, which correspondto theminimalvalueof critical loads arevarious,particularly for plates compressed on the outer perimeter. The deformation forms strongly depend on thematerial and geometrical parameters of the plate. Fig. 1. Scheme of the analysed plate Fig. 2. Scheme of the plate loaded: (a) on the outer perimeter, (b) on the inner perimeter Particularly, thecore thicknesshashere the significance.Thepossible forms of lossofplate stability areglobal or local andbasic,whichoccur forplateswith thinor thick core, respectively (Romanów, 1995; StammandWitte, 1983).The global formof stability loss is for the global displacement state, which includes the condition of equal deflections of three plate layers. This condition is not in force for other forms of the loss of stability, where the general displacement state exists. Then, the results for critical loads of plates with thick cores are 88 D. Pawlus lower thantheresults obtained for theassumedconditionof equaldeflectionsof three plate layers. To some extent, this condition extorts global plate buckling. The introduction of this condition, particularly for plates with thick cores, may essentially disturb the correct final result. Above all, it can influence the increase in values of the critical loads. The evaluation of the effect of taking into account the coupling between layers or not on computational results of critical loads is the main problem considered in this work. The problem has been solved using two approximation methods: finite difference and finite element. The solution obtained using the finite difference method is exactly based on the condition of equal deflection of layers, whereas, in the finite element method, the layers are connected by a surface contact interaction,which assures continuity of displacement and, additionally,mutual coupling as well. Results of plates with coupled layers and without coupling have been compared with the results obtained by making use of the finite difference method. Calculations have been carried out for the transverse symmetrical structu- re of plateswith the following geometrical parameters: inner radius ri =0.2m, outer radius ro =0.5m, facing thickness h ′ =0.0005m and 0.001m, various core thickness: h2 =0.005m, 0.01m, 0.02m and 0.04m, 0.06m. In the analy- sis, the core thickness keptwithin the range from h2 =0.005mup to 0.02m is generally treated asmedium.As a thick core, the thickness above h2 =0.02m is considered. 3. Numerical calculations Fundamental numerical calculations have been carried out for plates loaded on theouter edges of their facings.Theplates lose their stability in circumferential wave forms. Observation of the forms of plate buckling with the evaluation of minimal critical loads corresponding to them,particularly for plateswith thick cores, are the main goal of the undertaken numerical calculations. The examined plates are built of steel facings and treated as an isotropic polyurethane foam core. The material parameters are the following: Young’s modulus of facings E=2.1 ·105MPa andPoisson’s ratio ν =0.3, Kirchhoff’s moduli of core materials G2 = 5MPa (presented in work by Majewski and Maćkowski, 1975) and G2 = 15.82MPa (presented by Romanów, 1995). Po- isson’s ratio of the foam material equal to ν =0.3 is chosen according to the standard specification PN-84/B-03230. Approach to evaluation of critical static loads... 89 3.1. Calculations using the finite difference method (FDM) A detailed description of the solution to the formulated problem using the finite differencemethodwas presented inwork byPawlus (2006). The solution is based on the assumption of equal values of layer transverse deflections, the classical theory of sandwich plates with the broken line hypothesis and the normal distribution of plate stresses on the facings and shearing stresses carried by the core (Volmir, 1967). This solution comes down to a solution to the eigen-value problem with determination of the minimal plate load, being the critical static load pcr. The basic elements of the solution are as follows: • formulation of the equilibrium equations for each plate layer • determination of geometrical relations with equations for angles α, β of the circumferential and radial core deformation, respectively • formulation of physical relations of the layermaterials using the relations of Hook’s law, • on the strength of the equations of the sectional forces and moments and suitable equilibrium equations determination of the formulas for the resultant radial Qr andcircumferential Qθ forces andthe resultantmem- brane radial Nr, circumferential Nθ and shear Trθ forces determined by means of the introduced stress function Φ • usage of the equilibrium equations of projections in the z-direction of forces loading the plate layers (Fig.1), formulation of the basic differen- tial equation describing deflections of the analysed plate, which is in the following form 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+ (3.1) +H′rw′rr ) = 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 ) where: k1 =2D, k2 =4Drθ+νk1 D=Eh′ 3 /[12(1−ν2)], Drθ =Gh ′3/12 – flexural rigidities of the outer layers 90 D. Pawlus w – plate deflection δ=u3−u1, γ= v3−v1 u1(3), v1(3) – displacements of points of the middle plane of facings in the radial and circumferential directions, respectively H′ =h′+h2 Φ – stress function • determination of additional equilibrium equations of projections in the radial u and circumferential v directions of forces loading the undefor- med outer plate layers • determination of the boundary conditions described as follows w ∣ ∣ r=ro(ri) =0 w′r ∣ ∣ r=ro(ri) =0 δ ∣ ∣ r=ro(ri) =0 δ′r ∣ ∣ r=ro(ri) =0 γ ∣ ∣ r=ro(ri) =0 γ′r ∣ ∣ r=ro(ri) =0 (3.2) • determination of the following dimensionless quantities F = Φ Eh2 ζ = w h ρ= r ro δ= δ h γ= γ h (3.3) • assumption that the stress function Φ is a solution to the disk state, • application of the finite differencemethod for the approximation of deri- vatives with respect to the dimensionless radius ρ by central differences in discrete points and, after transformation, determination of the follo- wing forms of basic equations in the analysed problem MAPu+MADd+MAGg= p ∗ MACu MACPu=MACDd+MACGg (3.4) MPu=MDd+MGg where: p∗ = p/E – dimensionless load u, d, g – vectors of plate deflections and differences of the radial ui and circumferential vi displacements of facings Approach to evaluation of critical static loads... 91 MAP ,MAC,MACD,MACG,MD,MG –matrices of elements composed of geometric and material parameters of the plate and the quanti- ty b of the length of the interval in the finite differences method and the number m of buckling waves MAD – matrix of geometric parameters and the quantity b MAG – matrix of geometric parameters and the number m MACP – matrix with elements described by the quantity b/2 MP –matrix with elements described by the number m • solution to the eigen-value problemwith calculation of theminimal value of p∗ as the critical static load p∗ cr using the equation in the following form det[(MAP +MADMATD+MAGMATG)−p ∗ MAC] = 0 (3.5) and MATD, MATG – matrices expressed after some transformations by the matrices MP ,MD,MG,MACD,MACG,MACP . The calculations by means of the finite difference method have been pre- ceded by selection of the number N of the discrete points, which fulfils the accuracy of calculation up to 5% of the technical error. The computational results of critical loads and numbers m of circumferential waves showing the buckling forms of plates with thick cores (h2 = 0.04m and 0.06m) for dif- ferent numbers of discrete points: N = 11, 14, 17, 21, 26 are presented in Tables 1 and 2. Exemplary results of plates withmedium core thickness were presented by Pawlus (2006). The bold print results indicate the form of plate deformation (the number m of circumferential waves) corresponding to the minimal critical load pcr, which has been calculated for different numbers of discrete points N. The number m does not change with the increase in the number N of discrete points. The number, equal to N =14 has been accep- ted in the numerical calculations. The results show an essential increase in the number m of circumferential waves with the increase in the plate stiffness. This is exactly presented in Figs. 3 and 4. Figures 3 and 4 present the distribution of the critical static loads of plates with different forms of their critical deformation expressed by the number m of waves in the circumferential direction for plates with the thickness of facing equal to: h′ =0.0005m and 0.001m, respectively. The points marked by • in the diagrams correspond to theminimal critical load for the determined form of plate buckling. The presented diagrams show an essential increase in the critical plate loading with an increase in the plate stiffness – it is along with 92 D. Pawlus Table 1. Critical loads pcr [MPa] for different wave numbers m of the plate with parameters: G2 =5MPa, h2 =0.06m, h ′ =0.001m H H H H HH m N 11 14 17 21 26 5 123.49 123.90 124.18 124.43 124.62 6 117.94 118.30 118.54 118.76 118.93 7 114.87 115.18 115.39 115.58 115.74 8 113.44 113.71 113.90 114.06 114.20 9 113.15 113.39 113.55 113.70 113.83 10 113.72 113.92 114.07 114.20 114.31 11 114.95 115.13 115.25 115.37 115.47 12 116.72 116.88 116.99 117.09 117.17 Table 2. Critical loads pcr [MPa] for different wave numbers m of the plate with parameters: G2 =15.82MPa, h2 =0.04m, h ′ =0.0005m H H H H HH m N 11 14 17 21 26 19 375.49 376.15 376.59 377.00 377.36 20 374.58 375.23 375.68 376.10 376.47 21 373.96 374.62 375.08 375.50 375.87 22 373.62 374.28 374.74 375.17 375.54 23 373.54 374.20 374.67 375.09 375.47 24 373.71 374.36 374.83 375.26 375.63 25 374.10 374.75 375.21 375.64 376.01 26 374.70 375.34 375.81 376.24 376.61 the increase in the core thickness andKirchhoff’smodulusof the corematerial. Themajornumberof circumferentialwaves correspondsto thecritical buckling of stiffer plates, too. It is particularly high for plates with thick cores. For these plates, the critical loads change insignificantly in a wide range of forms of plate deformations, eg. for plates with parameters: h2 = 0.06m, h′ =0.0005m,G2 =15.82MPa, the fluctuation of critical loads for the forms of buckling presented by the number m equal from 24 up to 29 are in the range in 543.54MPa to 543.37MPa with a decrease down to the minimal value: pcr = 542.70MPa for m = 27. It could be found that these forms of plate buckling are equivalent to each other. The similar plate behaviour is observed for other cases presented inFigs. 3 and 4, particularly for plates with thin (h′ =0.0005m) facings. Approach to evaluation of critical static loads... 93 Fig. 3. Critical static load distributions depending on the number of buckling waves for the plate compressed on the outer perimeter with facing thickness h′ =0.0005m Fig. 4. Critical static load distributions depending on the number of buckling waves for the plate compressed on the outer perimeter with facing thickness h′ =0.001m The distribution of the critical static loads pcr depending on the plate core thickness is shown in Fig.5. The presented results are for plates loaded both on the inner and outer perimeter of facings. It is obvious that the values 94 D. Pawlus of loads pcr increase with an increase in the plate stiffness, and the curve for plates with medium core thickness is near to a linear one. The observed, rather insignificant, decrease in the distribution of the critical loads for plates with thick cores (above h2 = 0.02m) could arise some questions about their excessive values. Therefore, the analysis has been supported by calculations carried out by the finite element method also for plate models without the simplification, which is connected with the assumption of the equal deflection of plate layers. Fig. 5. Critical static load distributions depending on core thickness for plates compressed on the inner and outer perimeter 3.2. Calculations using the finite element method (FEM) Simulating numerically the plate models built by means of the finite element method, it was possible to observe the essential differences in the critical loads of plates with the thick core. In this observation, the way of the plate layers connection, i.e. having or not having equal deflection was significant. Calculations using thefinite elementmethodwere carried out for platemo- dels built of shell and solid elements. The application of the shell elements for themesh of facings and the solid elements for the coremesh took into account different participation of the layers in carrying the plate loading: normal by the facings and shearing by the core. Approach to evaluation of critical static loads... 95 Themodel is composed of 9-node 3D shell elements and 27-node 3D solid elements, which create the facing and core mesh, respectively. The outer sur- faces of facing elements are tied with surfaces of the core elements by surface contact interaction. The platemodel of the full annulus form supported in sli- dable clamped edges has been used in the analysis. Some results for the plate loaded on the inner perimeter, have been obtained using a quite simplemodel built of axisymmetric elements. The structure of such a plate model could be based on axisymmetric elements, because the plate loaded on the inner edge and supported in double slidable clamped edges loses its stability in regular, axially-symmetric form,which corresponds to theminimal value of the critical load. This observation has been confirmedby the general solution to the static stability problem of plates with the wave forms of buckling exactly presented in the work by Pawlus (2006). The structure of this model is the same as the annulus platemodel. The facingmesh is built of shell 3-node elements but the core mesh is built of solid 8-node elements. A scheme of both plate meshes is presented in Fig.6. Fig. 6. The scheme of plate mesh: (a) annulus model, (b) model built of axisymmetric elements The calculations were carried out at the Academic Computer Center CYFRONET-Cracow (MNiSW/SGI3700/Płódzka/016/2007) using the ABAQUS system (ABAQUS, 2000). Figures 7-10 present results of calculations of critical loads for plates with various core thickness obtained using the Finite Difference Method (FDM) and the Finite Element Method (FEM). The calculations in FEM have been carried out for the full annulus plate model without the condition of equal deflection of the layers. Figures 7 and 8 present results for plates loaded on the outer edge, while Figures 9 and 10 concern plates loaded on the inner edge. Diagrams presented in Figs. 9 and 10 show the results obtained by FEM for two kinds of plate models: full annulus – denoted by ”line 1” (see the le- gend) and the model built of axisymmetric elements – denoted by ”line 2”. 96 D. Pawlus Fig. 7. Critical static load distribution depending on the core for the plate loaded on the outer perimeter with facing thickness h′ =0.0005m Fig. 8. Critical static load distribution depending on the core for the plate loaded on the outer perimeter with facing thickness h′ =0.001m The consistency of theses curves (lines 1 and 2) informs about the correctness of the mesh structure, and confirms the observations presented in this work. A decrease in the critical static loads of plates with the thick core (above h2 =0.02m) is observed. Their forms of bucklingmaintain the global form, or particularly for plates with thin facings (h′ =0.0005m) the form are connec- ted with strong layer deformations. Marked by times points in Figs.7 and 9 Approach to evaluation of critical static loads... 97 Fig. 9. Critical static load distribution depending on the core for the plate loaded on the inner perimeter with facing thickness h′ =0.0005m Fig. 10. Critical static load distribution depending on the core for the plate loaded on the inner perimeter with facing thickness h′ =0.001m exactly concern such cases of deformations of plates loaded on the outer and inner edges, respectively. These forms of deformations are shown in Figs.11 and 12, respectively. Theobserved critical buckling of plates loaded on the outer perimeter has a formof circumferential waves concentrated near the inner plate perimeter (see Fig.11). At the same time, the critical deformation of the plate compressed 98 D. Pawlus Fig. 11. Critical buckling form of the plate with parameters: h′ =0.0005m, h2 =0.06m and G2 =15.82MPa, loaded on outer perimeter Fig. 12. Critical buckling form of the plate with parameters: h′ =0.0005m, h2 =0.06m and G2 =15.82MPa, loaded on inner perimeter and built: (a) as full annulus, (b) of axisymmetric elements on the inner perimeter has a form of a strongly, radially deformed structure in the region of the loaded edge (Fig.12) with the observed local form of the loss of stability (see Fig.12b). Table 3 presents detailed results of calculations by FDMand FEM for the plate with facing thickness equal to: h′ = 0.001m compressed on the outer perimeter. Exemplary forms of plate buckling are presented in Fig.13. The results presented in Table 3 show consistency in the critical loads pcr and forms of plate deformations found by numerical methods: FDM and FEM for pla- tes with medium cores. For plates with the core thickness above 0.02m, one can observe a decrease in the critical loads obtained through the finite ele- mentmethod. For some plates, e.g. characterised by parameters: h2 =0.04m, G2 = 15.82MPa, it is also observed that the number of circumferential buc- klingwaves is lower for platemodels calculated byFEM(the number m is 10) than by FDM (m=11). Approach to evaluation of critical static loads... 99 Table 3.Critical loads of plates with facing thickness h′ =0.001m calculated by FDM and FEM h2 [m] /G2 [MPa] FDM FEM pcr [MPa] m pcr [MPa] m 0.005 / 5.0 20.52 5 16.48 5 0.005 / 15.82 46.53 6 35.04 6 0.02 / 5.0 46.95 7 43.71 7 0.02 / 15.82 125.11 9 115.10 9 0.04 / 5.0 80.68 8 76.29 8 0.04 / 15.82 224.50 11 207.65 10 0.06 / 5.0 113.39 9 102.86 9 0.06 / 15.82 320.63 12 274.23 12 Fig. 13. Exemplary critical buckling forms of plates loaded on the outer perimeters Table 4 presents the critical loads of plates with thin facings (h′ = = 0.0005m) and medium cores of the thickness h2 = 0.02m. The values are consistent, whereas the forms of buckling found by FEM mostly have a lower number of circumferential waves than in plates solved by FDM. The results of calculations byFEMof platemodelswith coupled layers and the results obtained by FDM are presented in Figs.14-17. Figures 14 and 15 show the results for plates loaded on the outer edge, but Figures 16 and 17 present the results of plates compressedon the inner edgewith the facing thick- ness equal to h′ =0.0005m and h′ =0.001m, respectively. The introduction of the condition of equal deflections of three layers essentially changed the critical results of plates with thick cores. Strong critical deformations did not occur and the correspondence of results obtained by the two computational methods really increased. 100 D. Pawlus Table 4. Critical loads of plates with the facing thickness h′ = 0.0005m calculated by FDM and FEM h2 [m] /G2 [MPa] FDM FEM pcr [MPa] m pcr [MPa] m 0.005 / 5.0 22.37 8 19.16 7 0.005 / 15.82 61.51 9 52.03 9 0.01 / 5.0 38.56 9 35.53 9 0.01 / 15.82 109.75 13 101.72 10 0.02 / 5.0 69.49 12 66.56 10 0.02 / 15.82 200.62 18 193.15 12 Fig. 14. Critical load distribution depending on the core for the plate with equal layer deflections, loaded on the outer perimeter and with facing thickness h′ =0.0005m Tables 5 and 6 present the critical loads and critical forms of buckling of plates with coupled layers compressed on the outer perimeter andwith facing thickness equal to h′ =0.001m and h′ =0.0005m, respectively. The presented results indicate a tendency of the number m of circum- ferential waves for critical buckling of plates calculated with coupled layers to decrease. It is particularly observed for plate models calculated by FEM. Exemplary forms of deformation are presented in Fig.18 for plates with para- meters: h′ =0.0005m, h2 =0.06m,G2 =5MPa and G2 =15.82MPa. Detailed computational results for plates compressed on the inner perime- ter were presented by Pawlus (2007). This study includes analysis of other Approach to evaluation of critical static loads... 101 Fig. 15. Critical load distribution depending on the core for the plate with equal layer deflections, loaded on the outer perimeter and with facing thickness h′ =0.001m Fig. 16. Critical load distribution depending on the core for the plate with equal layer deflections, loaded on the inner perimeter and with facing thickness h′ =0.0005m than annulus plate models. Themodels are in the form: • of an annular sector with a single or double layer of core elements • of a model built of axisymmetric elements with a single, double or qu- aternary layer of core elements. 102 D. Pawlus Fig. 17. Critical load distribution depending on the core for the plate with equal layer deflections, loaded on the inner perimeter and with facing thickness h′ =0.001m Table 5. Critical loads of plates with equal layer deflections calculated by FDM and FEM, with facing thickness h′ =0.001m h2 [m] /G2 [MPa] FDM FEM pcr [MPa] m pcr [MPa] m 0.005 / 5.0 20.52 5 16.49 5 0.005 / 15.82 46.53 6 35.06 6 0.02 / 5.0 46.95 7 43.97 7 0.02 / 15.82 125.11 9 116.31 8 0.06 / 5.0 113.39 9 111.60 8 0.06 / 15.82 320.63 12 318.34 10 Theobservations of these plates are similar to those of theplates loaded on the outer edge. The global formof critical buckling for theminimal critical load of plates compressedonthe inner edge is regular axially-symmetric (m=0).This form for the full annulus platemodel and for themodel built of axisymmetric elements, which was used in FEM calculations (presented in Figs.16, 17 and by line 2 in Figs.9, 10), is shown in Fig.19. Generally, it could be noticed that the examinations indicate quantitative and qualitative compatibility of the results for critical loads of plates with medium core thickness (in the range of h2 around 0.02m). For these plates, the introduction of the condition of equal layer deflections or its absence do Approach to evaluation of critical static loads... 103 Table 6. Critical loads of plates with equal layer deflections calculated by FDM and FEM, with facing thickness h′ =0.0005m h2 [m] /G2 [MPa] FDM FEM pcr [MPa] m pcr [MPa] m 0.005 / 5.0 22.37 8 19.18 7 0.005 / 15.82 61.51 9 52.10 8 0.02 / 5.0 69.49 12 67.39 9 0.02 / 15.82 200.62 18 198.65 12 0.06 / 5.0 185.68 17 189.45 12 0.06 / 15.82 542.70 27 571.69 14 Fig. 18. Critical buckling form of the plate loaded on the outer perimeter with equal layer deflections and with parameters h′ =0.0005m, h2 =0.06m: (a) G2 =5MPa, (b) G2 =15.82MPa Fig. 19. Axially-symmetric buckling form of the plate compressed on the inner edge: (a) full annulus model, (b) model built of axisymmetric elements not significantly influence the obtained results. At the same time, this con- dition really influences the final results for plates with core thickness above h2 =0.02m. 104 D. Pawlus Table7.Comparisonof values of critical loads pcr [MPa] obtained fordifferent plate models h2 [m] /G2 [MPa] / h ′ [m] I II III 0.005 / 5.0 / 0.0005 57.52 57.48 57.84 0.005 / 15.82 / 0.001 119.94 119.92 120.30 0.02 / 5.0 / 0.001 144.16 143.20 143.77 0.02 / 15.82 / 0.001 328.30 324.01 326.41 0.06 / 5.0 / 0.001 317.34 292.68 293.90 I – Plate model built of axisymmetric elements with coupled layers II – Plate model built of axisymmetric elements without coupled layers III – Annulus plate model without coupled layers Additionally, while evaluating the correctness of the plate structure, analy- sis of critical loads convergent for different numbers ofmesh elements has been undertaken. The plate model built of axisymmetric elements has been exami- ned. The results are given for the plate loaded on the inner perimeter. The plate parameters are as follows: h2 = 0.02m, G2 = 5MPa, h ′ = 0.001m. A Fig. 20. Convergence of critical loads diagrampresented inFig.20 shows results for two analysed platemodels: with coupled layers and without this condition. The convergence has been achie- ved for 15 elements in the radial direction. In the calculations, 30 elements have been accepted. The comparison and good correspondence of the results of plates modelled in the form of full annulus and composed of axisymmetric elements are presented in Table 7. The results are given for the plate loaded on the inner perimeter. Approach to evaluation of critical static loads... 105 4. Conclusions Numerical results of the critical static loads of three-layer annular plates with a foam core of medium thickness and a thick core are presented in this paper. The solution to the static stability problem is general. It includes circumfe- rential wave forms of the critical buckling, observed in particular in plates compressed on the outer perimeter. The critical loads of such plates have been specially considered. The calculations have been carried out by means of the finite difference and finite elements method. The model of plate calculated by the finite difference method uses the classical theory of sandwich plates with the broken line hypothesis and the assumption on the equal deflection of three plate layers. The plate models built for the finite element method have essentially an annular plate structure and differ by condition of equal layer deflections. The influence of this condition on the critical loads in thick- core plates is very significant. The results for plates with core thickness above h2 =0.02m, treated as thick, indicate a great decrease in the critical loads and the possibility of occurrence of different buckling forms than those obtained in models of plates calculated by the finite difference method or models built for the finite element method with coupled layers. It could be stated that the critical loads of plates with thick cores obtained with the use of the finite dif- ferencemethod are too high. Particularly, it is observed for plateswith greater Kirchhoff’s modulus (G2 = 15.82MPa) and thinner facings (h ′ = 0.0005m). Generally, it could be determined that the presented numerical results show sensitivity of analysed plates to their geometric andmaterial parameters. He- re, the thickness of the foam core has themain importance. For thinner cores, the use of the simplifying condition of equal layer deflections in the solution is possible, whereas for thicker cores this simplification may cause incorrect results. It can be taken into consideration in themodeling of sandwich plates. This fact was pointed out by Romanów (1995) who indicated the necessity of implementing a hyperbolic form of deformation of plates with thicker cores. References 1. ABAQUS/Standard. User’s Manual, version 6.1, 2000, Hibbitt, Karlsson and Sorensen, Inc. 2. 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Pawlus D., 2003b, Obliczenia metodą elementów skończonych krytycznych obciążeń statycznych trójwarstwowychpłyt pierścieniowych,Czasopismo Tech- niczne, 6-M,Wydawnictwo Politechniki Krakowskiej, 137-150 9. PawlusD., 2004,Obliczeniametodą elementów skończonych trójwarstwowych płytpierścieniowychpoddanychobciążeniomwichpłaszczyźnie, III Sympozjum ”Kompozyty. Konstrukcje warstwowe”, Wrocław-Karpacz, 119-128 10. PawlusD., 2005,Dynamic stability problemof three-layeredannularplate un- der lateral time-dependent load,Journal of Theoretical andAppliedMechanics, 43, 2, 385-403 11. Pawlus D., 2006, Solution to the static stability problem of three-layered an- nular plates with a soft core, Journal of Theoretical and Applied Mechanics, 44, 2, 299-322 12. Pawlus D., 2007, Critical static loads calculations in finite element method of three-layered annular plates,Archives of Civil and Mechanical Engineering, VII, 1, 21-33 13. Romanów F., 1995,Wytrzymałość konstrukcji warstwowych, WSI w Zielonej Górze 14. Stamm K., Witte H., 1983, Sandwich Constructions, Moskwa [in Russian] 15. Volmir C., 1967, Stability of Deformed Systems, Nauka,Moskwa [in Russian] 16. Wang H.J., Chen L.W., 2003, Axisymmetric dynamic stability of sandwich circular plates,Composite Structures, 59, 99-107 17. Wang H.J., Chen L.W., 2004, Axisymmetric dynamic stability of rotating sandwich circular plates, Journal of Vibration and Acoustics, 126, 407-415 Approach to evaluation of critical static loads... 107 Ocena krytycznych obciążeń statycznych pierścieniowych płyt trójwarstwowych o różnych grubościach rdzenia Streszczenie W pracy poddano ocenie wyniki obliczeń pierścieniowych płyt trójwarstwowych z miękkim, piankowym rdzeniem o różnej grubości. Płyty obciążano równomiernie rozłożonym ciśnieniem ściskającymwewnętrzny lub zewnętrzny obwód okładzin pły- ty. Badanymi wynikami obliczeń są wartości krytycznych obciążeń statycznych płyt i odpowiadające im postaci deformacji krytycznych. Analizie poddano płyty o syme- trycznej strukturze poprzecznej i utwierdzonych przesuwnie krawędziach. Obliczenia prowadzono dwoma metodami przybliżonymi: metodą różnic skończonych i meto- dą elementów skończonych. Przedstawione w obumetodach rozwiązanie zagadnienia stateczności statycznej dotyczy ogólnego problemu utraty stateczności płyty, w któ- rym możliwe formy krytycznej deformacji określa liczba m-fal poprzecznych na jej obwodzie. Szczegółowej analizie poddano płyty z rdzeniem traktowanym jako gru- by. Oceniając wyniki, zwrócono uwagę na obserwowany znaczący spadek wartości obciążeń krytycznychpłytwłaśnie z rdzeniemgrubym. Spostrzeżenia te ujawniływy- niki obliczeń prowadzonemetodą elementów skończonychmodeli płyt różniących się wprowadzeniemdodatkowego, upraszczającegowarunkuwiążącegowarstwypłyty za- łożeniem ich jednakowych ugięć. Założenie to należy do formuł opisujących globalny stanuprzemieszczeńpłyty, której deformacja krytycznamapostać globalną. Inne spo- dziewane właśnie dla płyt z rdzeniem grubym formy utraty stateczności występują dla ogólnego stanu przemieszczeń pozbawionego upraszczającego warunku równości ugięć trzech warstw płyty. Wpływ tego warunku na wyniki krytyczne płyt zbadano prowadzącobliczenia numerycznemodeli płytw obumetodach.Obserwacja obszarów dobrej zgodności oraz istotnych różnicwynikówpłyt odpowiednio z rdzeniem średniej grubości i grubym jest zasadniczym efektem podjętej analizy, która w zagadnieniach modelowania struktur płyt możemieć istotne znaczenie praktyczne. Manuscript received May 23, 2007; accepted for print October 15, 2007