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Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov’s) Solutions on Example of Concentrated Radial Force

There is the general feeling among the scientists that everything what could be performed by theoretical analysis for cylindrical shell was already done in last century, or at least, would require so tremendous efforts, that it will have a little practical significance in our era of domination of powerful and simple to use commercial software. Present authors partly support this point of view.
 Nevertheless there is one significant mission of theory which is not exhausted yet, but conversely is increasingly required for engineering community. We mean the educational one, which would provide by rather simple means the general understanding of the patterns of deformational behavior, the load transmission mechanisms, and the dimensionless combinations of physical and geometrical parameters which governs these patterns. From practical consideration it is important for avoiding of unnecessary duplicate calculations, for reasonable restriction of the geometrical computer model for long structures, for choosing the correct boundary conditions, for quick evaluation of the correctness of results obtained.
 The main idea of work is expansion of solution in Fourier series in circumferential direction and subsequent consideration of two simplified differential equations of 4th order (biquadratic ones) instead of one equation of 8th order. The first equation is derived in assumption that all variables change more quickly in axial direction than in circumferential one (short solution), and the second solution is based on the opposite assumption (long solution).
 One of the most novelties of the work consists in modification of long solution which in fact is well known Vlasov’s semi-membrane theory. Two principal distinctions are suggested: a) hypothesis of inextensibility in circumferential direction is applied only after the elimination of axial force; b) instead of hypothesis zero shear deformation the differential dependence between circumferential displacement and axial one is obtained from equilibrium equation of circumferential forces by neglecting the forth order derivative.
 The axial force is transmitted to shell by means of short solution which gives rise (as main variables in it) to a radial displacement, its angle of rotation, bending radial moment and radial force. The shear force is also generated by it. The latter one is equilibrated by long solution, which operates by circumferential displacement, axial one, axial force and shear force.
 The comparison of simplified approach consisted from short solution and enhanced Vlasov’s (long) solution with FEA results for a variety of radius to wall thickness ratio from big values and up to 20 shows a good accuracy of this approach.
 So, this rather simple approach can be used for solution of different problems for cylindrical shells.