UDK 669.14.018:539.5:620.11 Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 37(6)341(2003) P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES ASYMMETRICAL NOTCHES IN PLATES NESIMETRIČNE ZAREZE V PLOŠČAH Philippe Jodin1, Christophe Cera2 1Laboratoire de Fiabilité Mécanique, University of Metz, Ile du Saulcy, F-57045 Metz cedex, France 2Ecole Polytechnique, Palaiseau, France jodinŽsciences.univ-metz.fr Prejem rokopisa - received: 2003-09-24; sprejem za objavo - accepted for publication: 2003-12-11 Stress concentration due to notches in machine parts is nowadays well known, and design codes provide data to take into account this phenomenon. Several shapes of notches are referenced and, in the case of plates, one-side- or two-sides-notched plates are considered. In the latter case, only the symmetrical case (same shape) is presented. This work analyses the case of non-symmetrical plates loaded in tension, i.e., that have notches of different shape on opposite sides. Unexpected results relating to the stress-concentration factor have been obtainedandhave been analysedusing the global stress distribution. Key words: stress concentration factor, asymmetrical notches, FEM basedcalculations andsimulations V inženirski praksi se pogosto medkonstruiranjem in oblikovanjem srečamo z različnimi oblikami zarez. Dobro je poznano, da so zareze izvir koncentracije napetosti. Velikost koncentracije napetosti izrazimo s faktorjem koncentracije napetosti. Faktor koncentracije elastičnih napetosti je definiran kot razmerje med maksimalno in globalno (bruto) napetostjo (bruto faktor koncentracije napetosti). Lahko pa ga definiramo tudi kot neto faktor koncentracije napetosti. V referenčni literaturi so ti faktorji koncentracije napetosti zbrani tabelarično ali grafično za različne značilne geometrije oziroma oblike zarez. V glavnem najdemo v literaturi podatke le za simetrične zareze, kot so na primer polkrožne, U- in V-zareze. V literaturi skoraj nikjer ne najdemo podatkov za različne vrste nesimetričnih zarez. Zato moramo uporabiti računalniške simulacije, ki nam omogočajo izračun faktorjev koncentracije napetosti nesimetričnih problemov. V prispevku je predstavljenih nekaj primerov računalniških izračunov nesimetričnih zarez in primerjav s simetričnimi zarezami. V nekaj primerih smo dobili nepričakovane rezultate, ki smo jih analizirali z uporabo globalne porazdelitve napetosti. Ključne besede: elastične napetosti, asimetrične zareze, metode končnih elementov 1 INTRODUCTION Notches are frequently encounteredby engineers when they are designing a mechanical part. These notches are necessary for joining systems, for clamping shafts, etc. Notches are important in design problems because they induce stress concentration. There are two effects: the first is the reduction of the global section relatedwith the depth of the notch, the secondis the stress gradient related to the shape of the notch. In this work we are particularly interestedin the case of two opposite notches in plates subjectedto uniform tension andthe stress-concentration effect in relation to the shape of the notches. In the case of two notches they are generally identical, and the problem is symmetrical. Of course, in the case of a shaft, the problem is axisymmetrical. However, in the case of plates, non-symmetrical problems can be encountered. This means that both notches are not identical, e. g., a semi-circular and a U-shapednotch, or a U-shapedanda V-shapednotch, etc. Solutions for symmetrical problems are easily found in the literature. However, non-symmetrical problems are not referencedandit is necessary to make a computation for these cases. This is the object of the work here presented. 2 COMPUTATIONS The problems under investigation are the cases of non-symmetrical notches in plates subjectedto tension. Symmetrical cases will be computedfirst, as a basis for a comparison andfor the verification of the precision of the computation with respect to the data found in the literature1. Then, using similar numerical models, non-symmetrical cases will be computedandcompared to the results obtainedfor symmetrical cases. A typical geometry is shown in Figure 1. The cases computedand presentedhere are summarizedin Table 1. Table 1: Computedgeometrical combinations Tabela 1: Izračunavane geometrične kombinacije Semi-circular (SC) U-shaped (U) V-shaped (V) Semi-circular (SC) SC-SC SC-U SC-V U-shaped(U) U-U U-V V-shaped(V) V-V Throughout this paper, the total width of the plate is referredto as D, the ligament (the remaining section between the two notch tips) is d, andthe radius of the notch tip is r. The stress-concentration factors are usually referredto as ktg for the global stress-concentration factor, and ktn for the net stress-concentration factor. These factors are defined as follows: MATERIALI IN TEHNOLOGIJE 37 (2003) 6 341 P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES u U-shaped notch V-shaped notch y\ Figure 1: Typical geometry of a non-symmetrical notch problem Slika 1: Tipična geometrija obravnavanega problema plošče z nesime- tričnima zarezama ktg = k = maximum stress at crack tip global stress maximum stress at crack tip net stress (1) (2) Where the global stress is the stress far from the notchedsection andthe net stress is the stress computed in the notchedsection. Moreover, all the computations are made assuming the material behaves elastically, although elasto-plastic stress-concentration factors are of major importance for design. Finally, in the case of non-symmetrical problems, the geometry is built so that the depth of each notch is the same and the radius at the crack tip is also identical. The only varying factor is the shape of the notch. The total width of the plate is fixed at 10 mm. 3 PREPARATION OF THE COMPUTATIONS All the computations were made with the CASTEM2000® programme, available from the French Atomic Energy Agency CEA. This programme is very suitable for research problems andis easily adaptable to our specific problem. mens. For a series of specimens, the same density of mesh is preserved, so that the different computations are realizedin the same conditions. A typical mesh for a U-shapednotch is given in Figure 2. The other meshes are similar. One of the objectives of the mesh is to provide, at the endof the computation, the stress gradient on the ligament. For small notch radii, it is very important to have small enough elements, so that that the stress gradient is correctly taken into account. The mesh should maintain a compromise between this last condition and a reasonable number of elements, in order to reduce the time for the computation. It shouldbe notedthat all the meshes were designed automatically by the program in the same way for the large notch radii as for the small notch radii. This results in very dense meshes for a large part of the small notch radii specimens, but this should not affect the result. 3.2 Symmetry and boundary conditions Symmetrical cases were meshedon a quarter of the plate, as the specimen exhibits two perpendicular axes of symmetry. The non-symmetrical cases, consequently, were meshedon a half specimen, as one of the axes of symmetry disappeared. The corresponding meshes are shown in Figure 3, Figure 4 and Figure 5. The boundary conditions were applied, taking into account these conditions of symmetry, so that the behaviour of the total plate is properly represented. The horizontal displacements of the nodes of the ligament were blockedup andthe vertical displacement of one of these nodes was also blocked. The load was applied as an arbitrary uniform horizontal displacement of the opposite side, so that the stress can be considered as uniform far away from the ligament. 3.1 Meshing As most of the configurations deal with partly circular notches, quadratic triangular elements were used to fit, as well as possible, with the shape of the speci- 3.3 Material and behaviour The plate material was considered as elastic and isotropic. It was taken as a standard steel with a modulus of elasticity of 2.1·1011 Pa anda Poisson's ratio of 0.3. Figure 2: Typical mesh for U-shapednotches Slika 2: Tipično zamreženje plošče z U-zarezo 342 MATERIALI IN TEHNOLOGIJE 37 (2003) 6 P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES Figure 3: Typical mesh for semi-circular andU-shapedspecimen Slika 3: Tipično zamreženje plošče s polkrožno in U-zarezo Figure 4: Typical mesh for semi-circular andV-shapedspecimen Slika 4: Tipično zamreženje plošče s polkrožno in V-zarezo Figure 5: Typical mesh for U- andV-shapedspecimen Slika 5: Tipično zamreženje plošče z U- in V-zarezo 3.4 Computations For a given configuration of notches, the program was launched, doing loops on the D/d and r/D ratios. The results are given in terms of the stress-concentration factor versus the ratio r/D, the curves being parame-terisedwith respect to the ratio D/d. The computations were controlledby the visualization of the stress fieldin the plate. Typical examples are given in Figure 6, Figure 7, Figure 8 and Figure 9. 4 RESULTS 4.1 Results on symmetrical notches Figure 10 to Figure 12, show the results of ktn or ktg plottedversus r/d for different D/d ratios. It is easy to compare with literature1, andan example of the literature data is given in Figure 13. The numerical results are very close to those foundin the tables, providedthe r/d Figure 6: Representation of the horizontal stress distribution in a plate with a semi-circular notch Slika 6: Predstavitev horizontalne porazdelitve napetosti v plošči s polkrožno zarezo MATERIALI IN TEHNOLOGIJE 37 (2003) 6 343 P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES Figure 7: Representation of the horizontal stress distribution in a plate with semi-circular andU-shapednotches Slika 7: Predstavitev horizontalne porazdelitve napetosti v plošči s polkrožno in U-zarezo and D/d ranges are the same. This validates the FE computation. 4.2 Results for non-symmetrical notches The results of the computations for the nonsymmetrical plates are presentedin Figure 14 to Figure 16. The global aspect of the curves is similar to that obtainedwith the symmetrical cases. However, it should be notedthat for the lowest values of D/d, in the SC-U andSC-V cases, the computedstress concentration factor decreases with r/d, then increases. This unexpectedbehaviour can be correlatedwith the aspect of the tfxx (horizontal) stress fieldas shown in Figure 7 to Figure 9. The cases with this behaviour are represented by Figure 7 and Figure 8. It is observedthat the stress fieldis highly non-symmetrical when, on the opposite, the stress fieldremains quite symmetrical in the U-V case (Figure 9). To facilitate a comparison, six curves of Ktn versus r/D are plottedon the same graph for a given value of D/d=13 (Figure 17). It is clear that the curves in all the Figure 8: Representation of the horizontal stress distribution in a plate with semi-circular andV-shapednotches Slika 8: Predstavitev horizontalne porazdelitve napetosti v plošči s polkrožno in V-zarezo 344 Figure 9: Representation of the horizontal stress distribution in a specimen with V-shapedandU-shapednotches Slika 9: Predstavitev horizontalne porazdelitve napetosti v plošči z Vin U-zarezo different cases are very close to each together, except at each end, when r/D is very small. Taking into account the particularly non-symmetrical geometry of the cases under consideration, it is assumed that there is a strong bending effect that induces a stress gradient in the ligament, and that this enhances the stress concentration, andthat it is more sensitive for the higher r/D values. However, it shouldbe notedthat, in the SC-U case, the geometry evolves with the D/d and r/d parameters. In other words, beyondthe computedthresholdvalues of the parameters, the U-shapednotch becomes a partially circular notch, andthen it vanishes. This geometrical effect is shown in Figure 18, with vertical lines showing the limit between the U-shapednotches andthe partially 0 0,5 1 (2r/D) Figure 10: Stress-concentration factors for a plate in tension with two semi-circular notches Slika 10: Faktor koncentracije napetosti za natezno obremenjeno ploščo z dvema polkrožnima zarezama MATERIALI IN TEHNOLOGIJE 37 (2003) 6 P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES 1 — i,uČ — 1,05 \ 1,1 — 1,2 1 — 1,3 — 1,4 A j — 1,5 — 1,6 1,7 1,8 \\\ 1,9 2 \ 3 5 Č _ io 0,15 rld 0,3 Figure 11: Stress-concentration factor for a plate in tension with two U-shapednotches Slika 11: Faktor koncentracije napetosti za natezno obremenjeno ploščo z dvema U-zarezama 15,5 13,5 11,5 4 9,5 7,5 5,5 3,5 1,5 0,0 0,3 0,1 0,2 r/d Figure 14: Stress-concentration factors for a plate with semi-circular andU-shapednotches Slika 14: Faktor koncentracije napetosti za natezno obremenjeno ploščo s polkrožno in U-zarezo 8,5 - n 7,5 6,5 h ' li 5,5 4,5 - . Č_ 3,5 FČ 2,5 1,5 ČČČČČ ___1,3 ___1,4 1,5 ___1,6 ___1,66 -----1,7 ___1,8 ___1,9 ___2 3 0,1 0,2 0,3 r/d Figure 12: Stress-concentration factor for a plate in tension with two V-shapednotches (? = 60°) Slika 12: Faktor koncentracije napetosti za natezno obremenjeno ploščo z dvema V-zarezama (? = 60°) '¦ \ ; ,.„, - _ *: X \ \ \ \ :Č:i v ¦d' • Y \\ \ "\ \\ Č j \ \ \\\ \ \ \ j t ) :č ¦ •• s; \ \ \ \ \ \ X X /«.z \ ¦ ¦ "\: . \ ¦\ \ ; \ Č Č \ \ .K xČ \ N N \. :\ \ X\Č- X ČČ \ ' / X "-'Č >>Č :' '- --Č /--v .-.J ',; '¦!¦/¦ / - 1 0i p/h ; : . : ¦ ' '¦: '¦' , , Figure 13: The stress-concentration factor for a plate with two U-shapednotches following the literature1 Slika 13: Faktor koncentracije napetosti za natezno obremenjeno ploščo z dvema U-zarezama1 9,5 Č 8,5 7,5 6,5 5,5 4-4,5 3,5 2,5 1,5 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 3 5 0,00 0,05 0,10 0,15 r/d 0,20 0,25 0,30 Figure 15: Stress-concentration factors for a plate with semi-cricular andV-shapednotches Slika 15: Faktor koncentracije napetosti za natezno obremenjeno ploščo s polkrožno in V-zarezo 9,5 4 8,5 7,5 4 6,5 5,5 4,5 3,5 2,5 4 1,5 0,0 0,5 0,2 0,3 r/d Figure 16: Stress-concentration factor for a plate with U- and V-shapednotches Slika 16: Faktor koncentracije napetosti za ploščo z U- in V-zarezo 0 MATERIALI IN TEHNOLOGIJE 37 (2003) 6 345 P. JODIN, C. CERA: ASYMMETRICAL NOTCHES IN PLATES 10 4 _ .u-u .v-v .sc-v .se-u U-V .SC-se 0,1 0,2 r/D 0,3 0,4 Figure 17: Comparison between stress-concentration factors for the different cases and D/d = 1.3 and a = 60° (V-shapednotches) Slika 17: Primerjava medfaktorji koncentracije napetosti za različne geometrijske primere in D/d = 1,3 in a = 60° (V-zareza) circular notches, andthe curves are limitedto the point where the notch vanishes. Of course, the conclusions that have been derived for the SC-U case are also validfor the SC-V case. 5 CONCLUSIONS A numerical determination of the elastic stress-concentration factors in plates subjectedto tension with non-symmetrical notches has shown that the computed values of Ktn are similar to that foundin the literature for symmetrical cases. However, noticeable differences are reportedfor extreme values of the r/D ratio. Moreover, for the lowest values of D/d, it is observedfor SC-U and SC-V geometries that the stress-concentration factor increases for r/D > 0.15. This unexpectedbehaviour is attributedto the large asymmetry of the specimen. D/d -1,1 _ 1,325 1,55 _ 1,775 '•ČNČČ "¦*—*. 2 -J—Pt- 0,05 0,1 0,15 0,2 0,25 0,3 0,35 r/d Figure 18: Geometrical limits for computation in the SC-U case. Vertical lines indicate transition from U shape to partially circular shape. The curves are limitedto the point where the U-notch vanishes. Slika 18: Geometrične meje za izračun primera plošče s polkrožno in U-zarezo. Vertikalne linije označujejo prehodiz U- v delno polkrožno obliko. Krivulje so omejene do točke, kjer U-zareza izgine. To confirm these computations, it will be necessary in the future to realise experimental measurements of the stress-concentration factor. Moreover, it wouldbe interesting to investigate such geometries for a calculation of the elastoplastic stress- andstrain-concentration factors. 6 REFERENCES 1 R.E. Peterson, Stress andconcentration factors, 2nd ed., Wiley, New York, 1974 14 _ 2 8 6 2 2 0 0 o o 346 MATERIALI IN TEHNOLOGIJE 37 (2003) 6