BUCKLING ANALYSIS OF FUNCTIONALLY GRADED EPITROCHOIDAL SHELLS STRUCTURES

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Concepts of Stability and Instability: Instability is a universal phenomenon, which may occur in various material bodies. The- fundamental concepts of stability and instability are clarified through the followingdefini- tions: The state of a system is the collection of values of the system parameters at any instant oftime. For example,the positions of material points in a structure and the temperature field atvarious points constitute the state of that system. The state of a system depends on systemparameters and environmental conditions. For example, in a shell structure, the systemparameters are geometrical and material properties. And the environmental condi- tions are the applied forces and thermal conditions. Stability- The state of a system, at any instant of time, is called stable if the relatively small changes in system parameter and / or environmental conditions would bring about relatively small changes in the existing state of the system. Instability- The state of a system at any instant of time is called unstable if relatively small. Changes in system parameter and / or environmental conditions would cause major changes in the existing state. Stability and Instability of Equilibrium- The equilibrium state of a system is called stableif small perturbations in that state,caused by load changes or other effects would be confined to a vicinity of the existing equilibrium state. The equilibrium state of a system is called unstable if slight changes in conditions related to that state would force the systemawayfrom that equilibrium state; an unstable system would find other equilibrium state(s); the new equilibrium state(s) may be in the vicinity of the initial state or may be far away from the initial equilibrium configuration.The concepts of stable and unstable equilibrium are illustrated in Fig.1 [1]. This figureshows a small balllying on a smooth surface. According to theforgoing definitions,the equilibrium state 1 is stable while state 2 is unstable [2]. The relativity of the foregoing definitions is clearly demonstrated in this figure; the state 1 may be stable in a certain limited region, but may be unstable in a larger domain. Bucklingis a special mode of instability of equilibrium which may occur in deformable- bodies subjected mostly to compressive loadings. So far as the structural problems are- concerned, an existing state of equilibrium or trend of behavior of the structure subjected toapplied loadings and / or temperature variations could be altered and the structure coul- dacquire a new equilibrium state or a new trend of behavior. This phenomenon is termed thebuckling of that particular structure. A well-known example of elastic buckling insta- bility is the flexural buckling of an axially compressed slender elastic column subjected to a concentric compressive force.The type of applied loading affects the modes of elastic instability. Loading systems are classified as conservativeor nonconservative. Dead load- ings, such as the weight of structures, are conservative forces; time dependent loadings, and the forces which depend on the state of the system are generally nonconservative. Conservative loadings are derivable from a potential function whereas nonconservative forces have no generating potential.From this viewpoint, frictional forces are nonconser- vative.Elastic bodies subjected to conservative forces may loose their current equilibrium state and find other equilibrated configurations; this mode of elastic instability is normal- ly of thebucklingtype. The equilibrium of the same elastic bodies subjected to nonconser- vative forces may become dynamically unstable; the system could undergo oscillations withincreasing amplitude. This mode of elastic instability is called flutter. Thin panels or shells in contact with flowing fluids could develop a flutter mode of elastic instability. An Overview of Shell Buckling: The equilibrium of thin elastic shells subjected to certain force fields may become un- stableand the shell may undergo prebuckling, buckling, and postbuckling deformation. Theoccurrence of buckling in thin shells is quite probable due to the fact that the thick- ness tospan ratio of shells is usually much lower than other structural elements.The re- sponse of thin shells to compressive forces is essentially very different from thebehavior of other structural elements such as struts, columns, and plates; some types of thinshells are extremely sensitive to geometrical and loading imperfections.Geometrical imperfec- tions include all deviations in the shape of the structural member froman ideally assumed geometrical configuration. Thus, a slightly crooked column, in comparison with a per- fectly straight bar is considered imperfect. In the case of shells, thegeometrical imperfec- tion is marked by deviation of middle surface geometry from aconceived ideal shape. Loading imperfections are probable deviations of magnitudes and / or directions of ap- pliedForces from assumed values and / or directions. As an example, an eccentrically ap- plied axial force to a straight column can be considered an imperfect loading. Load- ingImperfections, may be quantified by the so-called "imperfection parameters"; in the column problem, the axial force eccentricity could be chosen as an imperfections parame- ter. Experiments performed on column and plates, under in-plane compressive conservative forces, have shown that such elements are relatively insensitiveto slight geometric and loading imperfections. This is not the case in shell problems. Buckling experiments carried out on shells have shown that some shells are very sensi- tiveto geometrical and loading imperfections. Thus the buckling load of laboratory shell samplesis normally smaller than the critical loads that a perfect system could sustain. This is, onone hand due the fact that the actual shells are, by production, never geometri- cally perfectand also that an ideally perfect conceived loading can never be produced and, on the otherhanddue to imperfection sensitivity of real shells. The imperfection sensitivity of shells has important analysis and design implications; toobtain a realistic estimate of buckling strength of shells, geometrical and loading imper- fections must be taken into account. Finite element modelling An epitrochoidal shell structure [3] withfixed supports is depicted in Fig 2. It has been analyzed under pressure and thermalloading. Thickness of the shell (h=1.0 cm) including two layers ( , shown in Fig.3.The mechanical and thermal material properties used in the present study have been listed in the Table 1. Table 1. The mechanical and thermal material properties: Material Ceramic( Metallic(Steel) Thermal expansion coefficient Poisons' ratio ( 0.25 0.25 Young's' modulus, 390 210 Density (kg/m3) 3890 7850 Conductivity( 25 40 In this study, finite element modelling of functionally graded cyclic shell (Epitrochoidal shells) structures with uniform thickness h is considered as shown in Figure 2. Here, FG epitrochoidal shell is modeled and analyzed in ANSYS through ANSYS parametric de- sign language (APDL) code. Ashell element (SHELL181), defined in the ANSYS library, isutilized to discretize the FG Epitrochoidal shell. This shell element has total six degrees of freedom per node i.e., translations and rotations in the x, y and zdirections. 4. Results and Discussions In this section, the stability behavior of FG epitrochoidal shell is performed under pres- sure and uniform temperature field (T =700?K). The FG epitrochoidal shell is discretized and solved using finite element steps in ANSYS APDL platform [4]. Block Lanczos me- thod is used to obtain the eigenvalue bucking responses. Fig.3.A discretized layers of the epitrochoidal shell model Fig.4. Displacement variation for FG epitrochoidal shell Fig.5. Rotation variation for FG epitrochoidal shell Fig.6.Von Mises Stress variation for FG epitrochoidal shell Fig.7.Von Mises of total mechanical and thermal strain variation for FG epitrochoidal shell Fig.8. X-component total mechanical and thermal strain variation for FG epitrochoidal shell Fig.9. Y-component total mechanical and thermal strain variation for FG epitrochoidal shell Fig.10. Z-component total mechanical and thermal strain variation for FG epitrochoidal shell Fig. 4 shows Displacement variation for FG epitrochoidal shell under pressure and ther- mal loading. The overall displacement varies from Fig. 5 shows Rotation variation for FG epitrochoidal shell under pressure and thermal loading. The overall Rotation varies from Fig. 6 shows the Von Mises Stress variation for FG epitrochoidal shell under pressure and thermal loading. The stress varies from Fig. 7 shows Von Mises of total mechanical and thermal Strain variation for FG epitro- choidal shell under pressure and thermal loading. The strain varies from Fig. 8 shows X- component of total mechanical and thermal variation for FG epitrochoi- dal shell under pressure and thermal loading. The strain varies from Fig. 9 shows Y- component of total mechanical and thermal Strain variation for FG epi- trochoidal shell under pressure and thermal loading. The strain varies from Fig. 10 shows Z- component of total mechanical and thermal variation for FG epitrochoi- dal shell under pressure and thermal loading. The strain varies from 5. Conclusions In this study, the thermal bucking behavior of FG epitrochoidal shell under pressure and uniform temperature field are investigated. In addition, temperature dependent material properties of FGM constituents are considered. Finite element solution for the buckling behavior of present FG model is proposed using Block Lanczos method. The influences of different material and geometrical parameters on the thermal buckling of FG epitro- choidal shell are illustrated. Finally, the change of the stresses, displacements, rotations and stains were investigated and presented.

About the authors

MATHIEU GIL-OULBE

Peoples' Friendship University of Russia

associate professor

SMAEL TAHA FARHAN

Peoples' Friendship University of Russia

student

DAU TYEKOLO

Peoples' Friendship University of Russia

lecturer

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