Stress Concentration Factors of Various Adjacent Holes Configurations in a Spherical Pressure Vessel

Transcription

1 5 th Australasian Congress on Applie Mechanics, ACAM December 2007, Brisbane, Australia Stress Concentration Factors of Various Ajacent Holes Configurations in a Spherical Pressure Vessel Kh. Fua 1, R.A.Siregar 2, Ch. Rangkuti 1, B. Ariwahjoei 1, an M. Firaus 1 1 Universiti Teknologi PETRONAS, Malaysia 2 Universiti Malaysia Perlis, Malaysia aress: khairulfua@petronas.com.my Abstract: The analysis of stress concentration factor (SCF) of ajacent holes in a spherical pressure vessel can be approache by consiering a thin plate unergoing hyrostatic stresses. This work aopts the approach to investigate the SCF of various ajacent holes configurations in a spherical pressure vessel using finite element analysis. The von Mises stress is consiere to etermine the SCF. Various arrangements of ajacent holes are investigate i.e., two, three, four, an five ajacent holes are taken into account. The SCF curves with respect to the ratio of the istance between ajacent holes to the iameter of hole, /, an for a certain ratio of the iameter of hole to the plate thickness, /t, are then plotte. The results show that the ecreasing of / will affect the increasing of SCF, while for the case of five ajacent holes configuration, the increasing of /t oesnt make any significant effect to the increasing of SCF. Keywors: ajacent holes, circular thin plate, finite element analysis, spherical pressure vessel, stress concentration factor (SCF). 1 Introuction The higher local stress aroun the nozzle-vessel connection ue to internal pressure, temperature, an other external loas has postulate the occurrence of the crack initiation an attracte much attention of researchers an pressure vessel esigners. Though the tremenous avances in numerical simulation, all reporte papers were in the case of a single nozzle-vessel connection. Previous work on the use of von Mises equivalent stresses to compute the stress concentration factors (SCF) base on finite element calculation have been reporte [1-5]. The case of an opening in a spherical shell uner internal pressure without an with pressure acting at the bore was reporte by Schinler [1]. The results obtaine by the usage of thin shell theory was reporte elsewhere [2], threeimensional volume FEM elements was iscusse [], an axisymmetric ring-elements [4]. In the latter parametric investigation, nozzle to shell wel seams were inclue in the moel. The information on the SCF of various ajacent nozzles-vessel connections, either spherical or cylinrical vessel, has not been reporte yet. To overcome the complexities of investigating various ajacent nozzles-vessel connections, it is important to introuce another metho instea. This paper reports the stuy on SCF of various ajacent holes configurations in a spherical pressure vessel. The stuy uses a thin plate subjecte to a constant bening moment which creates hyrostatic stresses within the plate element. The finite element metho is use for solving the stress analysis to fin the values of SCF. In the earlier report [6], the stuy has been one by using an infinite thin element subjecte to hyrostatic stresses in the periphery of the element. However, this metho is effective for analyzing a single hole but it will prouce a greater eviation of stress istribution in the region between holes ue to the isturbances of stress inuce flow. In this stuy, the von Mises stress is also consiere to etermine the SCF of the ajacent holes. Various arrangements of ajacent holes are investigate i.e., two, three, four, an five ajacent holes are taken into account. The SCF results for each case are plotte with respect to the ratio of the istance between ajacent holes to the iameter of hole, an the ratio of the iameter of hole to the plate thickness are also taken into account. A formula for employing the results to calculate the allowable stress, σ all, in esigning a spherical pressure vessel is also given.

2 Uniform circumferential istribute oa x y Circumferential balls supports (a) Circumferential thin plate M Area of constant bening moment x y (b) Figure 1. (a) A circular thin having a circumferentially an symmetrically overhanging support an loae with a constant circumferential istribute force at its periphery, an (b) Bening moment iagram prouce by the circular thin plate. 2 Stress an Finite Element Analysis in a Circular Thin Plate. The analysis of stress istribution in a spherical pressure vessel can be approximate by consiering a thin plate unergoing hyrostatic stresses. This work aopts the approximation to investigate the SCF of various ajacent holes configurations in a spherical pressure vessel using finite element analysis. A circular thin plate is circumferentially an symmetrically overhanging supporte, an in the periphery of the plate is subjecte to a uniform circumferential istribute loa as shown in Figure 1(a). The segment of the plate which is constraine by the circumferential balls supports will unergo a constant bening moment, as shown in Figure 1(b), an therefore, the hyrostatic stresses will be prouce within the segment. Various ajacent holes configurations are then positione in the inner segment of the thin circular plate, an the stress istribution over the segment is analyze using ANSYS finite element software. The SCF,, is efine as the ratio of the maximum von Mises stress that occurs aroun the holes to the von Mises stress without hole, or it is expresse in the following formula: = Maximum von Mises stress aroun holes von Mises stress without hole (1)

3 2 holes holes (Right angle triangular configuration) holes (Triangular configuration) 4 holes (Square configuration) 5 holes (Square configuration) 5 holes (Pentagonal configuration) Figure 2. Various ajacent holes configurations for analysis. Note that the von Mises stress that occurs over the surface of the segment without hole is constant at any point an it is similar with the hyrostatic stress which is taken as a reference stress. A number of -imensional finite element computations were performe on various ajacent holes by consiering the ratio of the iameter of the holes to the thickness of the plate an the ratio of the istance between two holes centre to the iameter of the holes. The arrangement of various ajacent holes configurations taken into account for the current analysis is epicte in Figure 2. Numerical Results an Discussion The SCF for two, three, four, an five ajacent holes configurations have been analyze numerically using the finite element analysis. The results are then plotte to ientify the stress concentration factor,, an the ratio of the istance between the holes centre to the iameter of the holes, /, for various thickness of the circumferential plate. The guieline in employing the results to calculate the allowable stress, σ all, in esigning a spherical pressure vessel is given by the following expression, σ all = (pr/2t) (2) where p is the flui pressure within the spherical pressure vessel, r is the mean raius of sphere, an t is the wall thickness of the sphere. All analyses are conucte with the variation of / ranging from 1.5 to 2.4 with the value of /t equals to,, an 4. The ratio / less than 1.5 tens to give rise of the SCF value higher exponentially, an, therefore, it is beyon our consieration. While the ratio / greater than 2.4 tens to inicate constant SCF value. The SCF values are plotte with respect to / for respective three v alues of the aforementione /t. The interpolation metho can be applie to fin the value of SCF which is not provie in this stuy. Numerical results inicate that for the same value of /, the greater the value of /t the greater the value of SCF, however, it oes not show any significant increasing of SCF.

4 Table 1. The stress concentration factor for / = 1.5 an 2.4, an for /t =, an 4. Number of holes Holes Configuration Stress Concentration Factor, / = 1.5 / = 2.4 /t = /t = /t = 4 /t = /t = /t = Right angle triangle Triangle Square Square Pentagon 2.12 Our investigation has shown that the value of SCF for a single hole is 2.0. Figures to 8 show the results of various ajacent holes configurations taken for analysis in this stuy. The SCF for each case for / = 1.5 to 2.4 are tabulate in Table 1. For three holes configurations, the SCF for right angle triangular configuration is greater than the triangular one. For the five holes configuration, it is foun that the SCF for square configuration is slightly greater than the pentagonal one. Again, the increasing of SCF with respect to the increasing of /t oes not exhibit any significant effect. It can be observe that the more the number of holes, the more insignificant the effect of the increasing of /t to the value of SCF. In fact, for the case of five ajacent holes configurations, the increasing of SCF ue to the increasing of /t is very small an, therefore, it can be neglecte. The results for two an three ajacent holes configurations show a goo agreement with the results reporte by Nishia as referre in [6] using an infinite thin element subjecte to hyrostatic stresses in the periphery of the element. 4 Conclusion The SCF of various ajacent holes configurations have been stuie using the finite element analysis. It is foun that the ecreasing of / will affect the increasing of SCF significantly, while the more the number of holes, the more insignificant the effect of the increasing of /t to the value of SCF. In turn, for the case of five ajacent holes configurations, the effect of increasing /t to SCF can be neglecte. Eq.(2) gives a guieline in employing the SCF results to calculate the allowable stress, σ all, in esigning a spherical pressure vessel. References [1] S. Schinler an J.. Zeman, Stress concentration factors of nozzle-sphere connections, Int J Pressure Vessels an Piping, Vol.80, Issue 2, 200, pp [2] C.J. Dekker an H.J. Bring, Nozzles on spheres with outwar area uner internal pressure analyse by FEM an thin shell theory, Int J Pressure Vessels an Piping, Vol.77, 2000, pp [] I.J.S. Attwater, J. Anerson an G.E. Finlay, Three-imensional finite element analysis of sphere/cyliner intersections uner axisymmetric loaing, Int J Pressure Vessel an Piping, Vol.57, 1994, pp [4] E. Weiss an J. Ruolph, Finite element analyses concerning the fatigue strength of nozzle to spherical shell intersections, Int J Pressure Vessel an Piping, Vol.64, 1995, pp [5] Y.S. ee an Y.S. Sohn, Moifie solution of raial nozzle with thick reinforcement in spherical vessel hea subjecte to raial loa, Nuclear Engineering, Vol.178, 1997, pp [6] Walter D.Pilkey, Petersons stress concentration factors, 2 n E., John Wiley & Sons, Inc., 1997, pp

5 2.70 /t = / Figure. Stress concentration factor for two ajacent holes. /t = Figure 4. Stress concentration factor for three ajacent holes with right angle triangular configuration. / /t = / Figure 5. Stress concentration factor for three ajacent holes with triangular configuration.

6 /t = / Figure 6. Stress concentration factor for four ajacent holes with square configuration /t = Figure 7 Stress concentration factor for five ajacent holes with square configuration. / /t =, an / Figure 8. Stress concentration factor for five ajacent holes with pentagonal configuration.