Many works on the theory and analysis of plates are available. Most of them deal with the classical (Kirchhoff) theory, which is not adequate in providing accurate buckling. This is due to the effect of transverse shear strains. Shear deformation theories provide accurate solutions compared to the classical theory. During recent years many authors have developed this problem. At present, most classical heads of cylindrical tanks and vessels have a torispherical form and ellipsoidal heads. Blachut [1,2] presented, among others, optimization problems of internally pressurised cylindrical vessels [1] and elastic-plastic stability and optimization of domed pressure vessels subject to external or internal pressure [2]. Spence and Tooth [3] discussed the problems of strength and stability of torisperical and ellipsoidal heads. They investigated the reason of the stress concentration of pressure vessels end closures. Magnucki [4] discussed stability, strength as well as optimization problems of pressure vessels. He drew attention to stress concentration in heads and the cylindrical shells of pressure vessels and presented more comprehesive analysis of the disturbance of the stress membrane state. Magnucki and Lewinski [5] presented a problem of an optimal shape of the head. They minimized the stress concentration in the pressure vessel with a dished head that have a low relative convexity.

Instead, porous plates and beams with varying properties were described by Malinowski, Magnucki [6] and Magnucki, Stasiewicz [7,8] where also a non-linear hypothesis was assumed. Structural and functional applications for the different industrial sectors are discussed. Banhart [9] provided a comprehensive description of various manufacturing processes of metal foams and porous metallic structures.

In this paper the head of circular cylindrical vessel made of porous-cellular material is presented. The simplest heads are a flat homogeneous plate and a sandwich plate. Mechanical properties of the head vary across its thickness. The material is of continuous mechanical properties. The top and the bottom plate surfaces are non-porous material, while maximal porosity of the material occurs in the middle surfaces of the plate. The degree of porosity varies in the normal direction. The moduli of elasticity occurring here are variable and depend on the -coordinate. This head and cylindrical shell are described in a polar (cylindrical) coordinate system with the -axis in the plate-depth direction.

For such a case, the Euler-Bernoulli or Timoshenko plate theories do not correctly determine the displacements of the plate's cross-section.

The problem was calculated by means of analytical and numerical methods. The theory of the perturbation of membrane stress state of the thin cylindrical shell is used. The porous flat head is rigid in its plane, as compared with bending stiffness of the porous cylindrical shell. The middle plane of the plate is its symmetry plane. First of all, a displacement field of any cross section of the head was defined. Afterwards, the components of strain and stress states were found. The vessel is under internal uniform pressure.

The material properties varying through thickness of the cross-section of numerical model were discretized using 81 layers of constant properties. The problem was analyzed using the finite element method by means of the ANSYS computer system. Results given in this paper show the effect of coefficient of porosity on the strength of the plate and shell through the wall thickness. The results obtained for porous circular flat heads are compared to a homogeneous head.

1

Blachut J., "Elastic-plastic stability and optimization of pressure vessel end closures", Fundamental Technical Sciences, V.20, 83, Cracow University of Technology, Cracow, Poland (in Polish), 1996.

2

Blachut J., "Minimum weight of internally pressurised domes subject to plastic load failure", Thin-Walled Structures, 27, 127-146, 1997. doi:10.1016/S0263-8231(96)00036-5

3

Spence J., Tooth A.S. (eds), "Pressure Vessel Design-Concepts and principles", Chapman & Hall, London,1994.

4

Magnucki K., "Strength and Optimization of Thin-Walled Vessels", Scientific Publishers PWN, Warszawa-Poznan, (in Polish), 1998.

5

Magnucki K., Lewinski J., "Fully stressed head of a pressure vessel", Thin-Walled Structures, 38, 167-178, 2000. doi:10.1016/S0263-8231(00)00031-8

6

Malinowski M., Magnucki K., "Buckling of an isotropic porous cylindrical shell", Proc. Tenth Int. Conference on Civil, Structural and Environmental Engineering Computing, B.H.V, Topping (Ed.) Civil-Computer Press Stirling, Scotland, Paper 53, 1-10, 2005. doi:10.4203/ccp.81.53

7

Magnucki K., Stasiewicz P., "Elastic bending of an isotropic porous beam", Int. Journal of Applied Mechanics and Engineering, 9(2), 351-360, 2004.

8

Magnucki K., Stasiewicz P., "Elastic buckling of a porous beam", Journal of Theoretical and Applied Mechanics 42(4), 859-868, 2004.

9

Banhart J., "Manufacture, characterisation and application of cellular metals and metal foams", Progress in Materials Science, 46, 559-632, 2001. doi:10.1016/S0079-6425(00)00002-5

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