Analytical approaches for cylindrical shell are mostly based on expansion of all variables in Fourier series in circumferential direction. This gives 8th order differential equation with respect to axial coordinate. Here it is approximately treated as a sum of two 4th order biquadratic equations. First one assumes, that all variables change more quickly in circumferential direction than in axial one (long solution), while the second (short) one is an opposite one. The accuracy and applicability of this approach was demonstrated [8] on example of action of one or two concentrated radial forces, and compared with FEM results. This paper is improvement of our previous work [8]. Two amendments are made. The first is insignificant one and use slightly modified expressions for bending strains, while the second one relates to the short solution. Here we do not consider any more that circumferential displacement is negligible as compared with radial one. Eventually this improves the accuracy of results, as compared with previous work. For example, for cylinder with radius, R, to wall thickness, h, ratio equal to 20, the maximal inaccuracy for radial displacement in point of force application, decreases from 5% to 3%. For thinner cylinder with R/h=100 this inaccuracy decreases from 2.5% to 1.25%. These inaccuracies are related with larger terms in Fourier expansion, the significance of which decrease, when length or area of outer loading becomes greater. The last conclusion is demonstrated for case of distributed concentrated force acting along short segment on axial line.
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.
Traditionally, the brittle strength evaluation of reactor pressure vessel was the central issue in lifetime assessment of Ukrainian nuclear power plants (NPPs). The problem of swelling of the reactor core baffle only recently got due attention from the side of operator. Here the most efforts were given on numerical modeling of austenitic steel 08Kh18N10T swelling and its effect on induced stresses in core baffle and distortion of its geometry. The calculation shows that essential changing of core baffle dimensions is expected after 35–40 years of operation. Eventually this can lead to the contact with the core barrel. Yet, these predictions contain the big number of uncertainties related to the input data used in analysis: fluence distribution; temperature variation due to heat release induced by neutron and gamma radiation; thermal-hydraulic boundary condition between the baffle and coolant; and, especially, the adopted law of swelling in dependence with above factors as well as mechanical stresses. So, the second task was to measure the real geometry of baffle after 27 years of operation, to determine its change and compare these results with the numerically calculated data with accounting for the design tolerances. Thus, the spatial measurement system (SMS) equipped with ultrasonic gages was designed. It contains the central vertical beam which can move in vertical direction and rotate. To the lower end of the beam four horizontal levels are attached, which are equipped with device resistant to the hot water and radiation. The gages are used to measure the shortest distances to the edges of baffle. Two types of results were obtained. The first one are the measurements in the different horizontal planes obtained by rotation the SMS around the vertical axis with angular steps equal to 1 degree. These results were difficult to handle with and required a special mathematical treatment due to the possible shift of the centre of measurement. The second set of measurements was performed by moving the SMS in vertical direction. These data demonstrate the change of distance with the height. The results clearly show that problem of swelling do exists, and, in general, the measured patterns of the distortions along the vertical and angular coordinates correspond to numerically obtained results. Further work on baffle integrity is however needed.
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