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Journal Articles

Article Type: Technical Briefs

*. February 2021, 143(1): 014501.*

*J. Pressure Vessel Technol*Paper No: PVT-19-1069

Published Online: August 17, 2020

Abstract

Analytical approaches for cylindrical shell are mostly based on expansion of all variables in Fourier series in circumferential direction. This leads to eighth-order differential equation with respect to axial coordinate. Here it is approximately treated as a sum of two fourth-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 based on opposite assumption. The accuracy and applicability of this approach were demonstrated (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032) on example of action of one or two concentrated radial forces and compared with finite element method results. This paper is an improvement of our previous work (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032). 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 to 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 the case of distributed concentrated force acting along short segment on axial line.

Proceedings Papers

*Proc. ASME*. PVP2018, Volume 3A: Design and Analysis, V03AT03A033, July 15–20, 2018

Paper No: PVP2018-85032

Abstract

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 4 th order (biquadratic ones) instead of one equation of 8 th 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.

Proceedings Papers

*Proc. ASME*. PVP2018, Volume 3B: Design and Analysis, V03BT03A030, July 15–20, 2018

Paper No: PVP2018-85033

Abstract

The determination of stress intensity factors (SIF) and crack opening area or displacements (COA or COD) is important constituent when performing the “leak before break” analysis of piping systems in NPPs. The tabulated parametrical results of their calculation are widely presented in modern scientific and normative literature. Nevertheless, there is one aspect of crack behavior, at least in thin walled pipes, which still had not obtained its due attention. We mean here the geometrically nonlinear effect, which can be the big enough to be accounted for in practical applications. It is considered in geometrically linear analysis that only the inner pressure opens the crack, and COA and SIF are directly proportional to it. SIF is presented usually as solution for infinite plate multiplied by so-called bulging factor, BF, which depends on dimensionless crack length, i.e. ratio of crack length divided on square root of product of radius, R , and wall thickness, t . Two loading factors in thin walled pipes can contribute to geometrically nonlinear behavior. The first one is axial stresses induced by value of axial force or bending moment. The second one – is the inner pressure itself. The most attention in present paper is given to influence of the axial force. With this goal the numerical models were created for pipes with different ratios of R/t (20, 30, 40, 50) and different dimensionless crack length (2, 4, 6, 8). To exclude the nonlinearity due to circumferential stress the inner pressure is kept as a very small value and dimensionless SIF and COD values are calculated with respect to axial force. To prove the correctness of choosing the finite element types, meshing, number of elements along the thickness, loading steps the auxiliary problem of nonlinear modeling of transverse beam loaded additionally by very big axial force is considered. The very good correspondence was attained. For the pipe with axial crack the careful verification of numerical model was performed by comparison with linear results existing in literature. The results obtained are presented as a percentage of difference between the linear and nonlinear results. They show that influence of geometrical nonlinearity is fairly essential to be accounted in practice and can reach for practically real cases almost 3–10%. The change of SIF in percentages due to geometrical nonlinearity for different axial stress levels and for different crack lengths can be fairly well presented as unique dependence from product of stresses, radius to thickness ratio, and square root of dimensionless crack length. The change COD in central point of crack is slightly bigger than for SIF and the same unique dependence can be formulated for COD with only exception for small cracks λ < 3.

Proceedings Papers

*Proc. ASME*. PVP2017, Volume 7: Operations, Applications and Components, V007T07A019, July 16–20, 2017

Paper No: PVP2017-65769

Abstract

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.

Journal Articles

Article Type: Research-Article

*. April 2017, 139(2): 021210.*

*J. Pressure Vessel Technol*Paper No: PVT-16-1012

Published Online: January 11, 2017

Abstract

Consideration of a geometrical nonlinearity is a common practice for thin-walled pressurized structures, especially when their cross section is not a perfectly circular one due to either initial imperfections or distortions caused by the nonsymmetrical loading. The application of inner pressure leads to so-called rerounding effect when decreasing of local flexibilities takes place. The crack can be also treated as the concentrated flexibility, so the goal of this work is the investigation of dependence of stress intensity factor (SIF) on applied pressure. Two cases of SIF calculation for 1D long axial surface crack in a pipe loaded by inner pressure are considered here: (a) cross section of pipe has an ideal circular form and (b) the form has a small distortion and crack is located at the place of maximal additional bending stresses. The theoretical analysis is based on: (a) well-known crack compliance method (CCM) (Cheng, W., and Finnie, I., 1986, “Measurement of Residual Hoop Stresses in Cylinders Using the Compliance Method,” ASME J. Eng. Mater. Technol., 108 (2), pp. 87–92) and (b) analytically linearized solution for deformation of the curved beam in the case of action of uniform longitudinal stresses. It is shown that for moderately deep crack (crack depth to the wall thickness ratio of 0.5 and bigger) in thin-walled pipe (radius to thickness ratio of 25–40) and inner pressure which induce hoop stress up to 300 MPa, the effect investigated can be quite noticeable and can lead to 5–15% reduction of calculated SIF as compared with the linear case. The analytical results are supported by the geometrically nonlinear finite element method (FEM) calculations.

Proceedings Papers

*Proc. ASME*. PVP2016, Volume 6A: Materials and Fabrication, V06AT06A003, July 17–21, 2016

Paper No: PVP2016-63304

Abstract

The general approach of numerical treatment of integro-differential equation of the flat crack problem is considered. It consists in presenting the crack surface loading as the set of the polynomial functions of two Cartesian coordinates while the corresponding crack surface displacements are chosen as the similar polynomials multiplied by the function of form (FoF) which reflects the required singularity of their behavior. To find the relations matrixes between these two sets a new effective numerical procedure for the integration over the area of arbitrary shape crack is developed. In based on the classical hyper-singular method, i.e. Laplace operator is initially analytically applied to the integral part of equation and the resulting hyper singular equation is subsequently considered. The presented approach can be implemented with any variant of FoF, but Oore-Burns FoF, which was earlier suggested in their famous 3D weight function method, is supposed to be the most accurate and universal. It takes into account all points of crack contour, which provides perfect physical conditionality of the solution, but such FoF is relatively heavy in implementation and of low computational speed. The special procedure is developed for the approximation of the crack contour of arbitrary shape by the circular and straight segments. It allows to easily obtain analytical expression for Oore-Burns FoF, which greatly increases the calculation speed and accuracy. The accuracy of the considered method is confirmed by the examples of the circular, elliptic, semicircular and square cracks at different polynomial laws of loading. The developed methods are used in the implemented procedure for crack growth simulation. It allows to model growth of crack of arbitrary shape at arbitrary polynomial loading, at that all contour points are taken into account and can expand with their own speeds each. Procedure has high accuracy and don’t need complex and high-cost re-meshing process between the iterations unlike FEM or other numerical methods. At that usage of Oore-Burns FoF provides high flexibility of the presented approach: unlike similar theoretical methods, where FoF calculation procedure is rigidly connected with the crack shape, which complicates the adequate crack growth modeling, the used FoF automatically takes into account all points of crack contour, even if its shape became complex during the growth. Presented crack growth procedure can be effectively used to test accuracy and correctness of correspondent numerical methods, including the newest XFEM approach.

Proceedings Papers

*Proc. ASME*. PVP2015, Volume 1A: Codes and Standards, V01AT01A008, July 19–23, 2015

Paper No: PVP2015-45838

Abstract

The development of powerful commercial computer programs made the concept of J-integral as computational parameter of fracture mechanics to be a very attractive one. It is equivalent to SIF in linear case, it converges in numerical calculation and the same results are obtained by different codes (programs). Besides, it is widely thought that elastic-plastic analysis gives bigger values than elastic SIF ones what is good from regulatory point of view. Such stand was reflected in the recommended by IAEA TECDOC 1627 (February 2010) devoted to pressurized thermal shock analysis of reactor pressure vessels, where the embedded crack in FEM mesh, elastic-plastic analysis with simultaneous determination of J-integral was stated as the best option of analysis. But at that time all the most widely used software were not able to treat the residual stresses, the thermal stresses in case of two different materials. Such a contradiction between requirements and the possibilities made a lot of problems for honest contractors especially in countries where the regulator had no own experience in calculation and completely relied on the authority of international documents. This means that at that time the said recommendations were harmful. The main reason of such a situation was the absence of the carefully elaborated examples. Now the capabilities and accuracy of such software are increasing. Nevertheless, some principal ambiguities and divergences of computations results in various J-integral contours around the crack tip still exist. They are exhibited when the large plastic zone emerges at the crack tip. Other problem is influence of the history of loading and the specification of the time of crack insertion in the mesh including the time of emergence of residual stress. This paper is invitation for discussion of the accuracy and restriction of computational J-integral. With this aim the detailed analysis of some simplified 2D examples of calculation of elastic -plastic J-integral for surface crack with accounting for residual stress, thermal stress and inner pressure are performed and commented. The attainment of consensus among the engineering society for treating the outcome results is the prerequisite for practical application of computational elastic plastic J-integral.

Proceedings Papers

*Proc. ASME*. PVP2015, Volume 1B: Codes and Standards, V01BT01A005, July 19–23, 2015

Paper No: PVP2015-45276

Abstract

The consideration of a geometrical nonlinearity is a common practice for the thin-walled structures. The relevance here are two well-known cases treated in ASME codes. First one is accounting for reduction of the pipe bends flexibility due to the inner pressure. The second one is the retarded increasing (and subsequent saturation) of additional local bending stress with increasing of inner pressure in a pipe with initial cross section form distortion. In both cases the rerounding effect and decreasing of local flexibilities take place. The crack can be treated as the concentrated flexibility and it is quite natural to expect that the stress intensity factor should grow nonlinearly with applied load. Two cases of SIF calculation for 1-D long axial surface crack in a pipe loaded by inner pressure are considered here: a) cross section has an ideal circular form: b) the form has a small distortion and crack is located in the place of maximal additional bending stresses. The theoretical analysis is based on: a) the well known crack compliance method [1] and b) analytical linearized solution obtained for deformation of the curved beam in case of action of fixed circumferential stress due to pressure written in the form convenient for transfer matrix method application. It was shown that for moderately deep crack (crack depth to the wall thickness ratio is 0.5 and bigger) and typical dimensions of pipes used for oil and gas transportation (radius to thickness ratio is 25–40) and loading which can reach up to 200 to 300 MPa, the effect investigated can be quite noticeable and can lead to 5–15 percent reduction of calculated SIF as compared with linear calculation. The analytical results are supported by nonlinear FEM calculation.

Proceedings Papers

*Proc. ASME*. PVP2014, Volume 3: Design and Analysis, V003T03A044, July 20–24, 2014

Paper No: PVP2014-28383

Abstract

The exact analytical approach for stress intensity factor calculation for an arbitrary shape mode I crack loaded by the polynomial stresses is proposed. The approach is based on the calculation of the crack faces displacement at given loading. The displacement field is presented as a shape function multiplied by an adjustment polynomial. At that the key problem is the solution of well-known inverse task: obtaining the stresses field at the crack faces on the base of a given displacements field. Multiply solution of such task for a whole set of certain displacements base functions (e.g., for the single terms of the adjustment polynomial) allows to get analytical expression which connects stresses and displacements fields. The original semi-analytical technique for integration with subsequent differentiation of well-known singular integral equation of the flat crack problem is developed. The excellent accuracy of the method is confirmed for an elliptic crack as well as for a rectangular one in the infinite 3D body. New results are given for an inner semi-elliptic crack in the infinite body which surfaces are loaded by polynomial stresses up to the 6 th order. The importance of choosing the appropriate shape function is demonstrated.

Proceedings Papers

*Proc. ASME*. PVP2013, Volume 3: Design and Analysis, V003T03A062, July 14–18, 2013

Paper No: PVP2013-97561

Abstract

Brittle strength calculation of RPV nozzle is the central point of the integrity assessment of the reactor pressure vessel when extending its life. The important part of this calculation is a determination of the stress intensity factor, SIF, for the postulated crack of partly elliptical form in a nozzle under inner pressure, bending moments (from the main circulating pipe) and difference of temperatures. In this paper we use method of influence functions as the most convenient one for solution of similar tasks. Eight basic laws of the crack surface loading are introduced which account for real stress distribution in the depth and length direction of a crack including the jump of stress between cladding and main metal due to the difference in the thermal expansion factors. To determine the dimensionless SIF under chosen laws of loading were developed the FEM models of nozzle with crack of different ratios of axes. For all possible modes (regimes) of operation were carried the detailed calculations of the temperature field in the nozzle, which were used later for determining the stress state at each time point. The stress field defined in 120 discrete points of the crack surface was treated by the method of least squares for the presention as a linear combination of eight basic load laws with defined coefficients. The procedure for determination of the temperature brittle strength margin which employs the presentation of critical values of SIF (fracture toughness) in the exponential function form is described.