# Stress concentration factors for pin lever of runner blade

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Stress concentration factors for pin lever of runner blade

Latest Trends on Engineering Mechanics, Structures, Engineering Geology Stress concentration factors for pin lever of runner blade mechanism from Kaplan turbines ANA-MARIA PITTNER, CONSTANTIN VIOREL CAMPIAN, DORIAN NEDELCU, DOINA FRUNZAVERDE, VASILE COJOCARU Faculty of Engineering “Eftimie Murgu”University of Resita No. 1-4, P-ta Traian Vuia, 320085, Resita ROMANIA [email protected], [email protected], [email protected], [email protected], [email protected], http://www.uem.ro Abstract: This paper presents a comparative study between different values of stress concentration factor Kt and fatigue notch factor Kf determined with different methods for a specific part. The study was specially created for the pin lever of runner blade mechanism from a Kaplan turbine. The paper tries to offer the most adequate choice, from the author’s point of view, which can be made between the obtained values after processing the needed data issue from geometrical details and service conditions. The methods used to obtain the look-up values are graphical, analytical and numerical. For this analyze it will be taken in consideration from the oldest to the recent methods used by engineering community. Key-Words: stress concentration factor, fatigue notch factor, graphical, analytical, numerical. methodologies to be apply for quantification of this influence. It was considered that this were the problem that can be incriminated for structures taken out of service, before the initial estimated time. The phenomenon is known such as fatigue failure. Finally, the measurement of effects induced by stress concentrators on the stress values, in a specified location, can be made multiplying the calculated stress value with a stress concentration factors. The engineering community usually uses, in order to predict the real load, the so called fatigue notch factor Kf . 1 Introduction In the last four decades there have been issued a lot of studies having like principal concerns designing against fatigue failures. The attention of design engineer is focused on the overall structure as well as its components when exposure to service conditions assume numerous fluctuating loads and attendant stress and strain histories which may result in fatigue failure. Previously, large factors of safety were used into design components because of the lack of knowledge and understanding of interactive effects. These safety factors are no longer needed since the development of extensive computer software packages. Using adequate software can make a realistic estimation about the real values of local stresses in the interested points of a structure. In absence of one of that specialized software may still be used an analytic method to determined the value of stress concentration factors needed to be taken in to consideration for a real estimation of strength in service. It is known that the presence of one or more stress concentrators (abrupt variation of section, material defects, improper work of surfaces, etc) provoke the appearance of unexpected values for local stresses. These values were proved to be, many times, much higher that the one obtained from strength calculations. Starting from these points various theories were developed concerning the main factors that induce such rising and the ISSN: 1792-4294 2 Problem Formulation In order to find the value of fatigue notch factor, we firstly must determine de stress concentration factor Kt. 2.1 The theoretical stress concentration factor Kt The value of stress concentration factor can be determined using three methods: - from diagrams experimentally raised; - using analytical algorithms; - from finite element analysis. In Fig.1 there are illustrated an experimentally diagram raised for bending of a 181 ISBN: 978-960-474-203-5 Latest Trends on Engineering Mechanics, Structures, Engineering Geology The most common definition, used to describe de interdependence between Kf and Kt is: K f = 1 + q ⋅ (K t − 1) (2) stepped round bar with a shoulder fillet (based on photoelastic tests of Leven & Hartman, Wilson & White). where: q – fatigue notch sensitivity. The fatigue notch sensitivity q is the measure of the degree of agreement between Kt and Kf. In specialized literatures two relations are mostly used, to define the q's value: - according to Neuber [2] 1 q= (3) a′ 1+ r where: r - the concentrator’s radius; a ′ - material constant depending on the mechanical properties of the analyzed material. - according to Peterson [3] 1 q= (4) a 1+ r where: r – the concentrator’s radius; a – material constant depending on the material strength. Also Peterson gives us an analytical relationship for relatively high strength steels subjected to axial or bending fatigue: 5.0 4.5 r 4.0 D M M d 3.5 Kt = σmax / σnom σnom = 32M / πd3 D/d=3 3.0 2 2.5 1.5 1.2 2.0 1.1 1.05 1.5 1.0 1.01 0.01 1.02 0.05 0.10 0.15 0.20 0.25 0.3 Fig. 1 As is shown in Fig.1, the value of Kt can be chose from the experimentally raised diagrams depending on three geometrical characteristics: value of large diameter D, value of small diameter d and value of fillet radius r. In that case, the usual problem that appears is to not find the specific diagram for the specified material of analyzed structure. In that case must be chose a similar diagram raised for another material with appropriate mechanical characteristics tested for the same load. Analytical method assumes calculation of Kt, using a mathematical equation like [1]: 1 (1) K = 1+ t,inc 2 1,8 2070 mm a = 0,0254 ⋅ (5) σr where: σ r - ultimate tensile strength of material. The analytical relation to determinate value of Kf, proposed by expert group from FKM, is [1]: K t ,inc K f ,inc = (6) nσ (r ) ⋅ nσ (d ) where: nσ (r ) - Kt-Kf ratio of the component for normal stress or for shear stress according to r; nσ (d ) - Kt-Kf ratio of the component for normal stress or for shear stress according to d; The Kt-Kf ratio for normal stress nσ (r ) is [1]: 3 r r r r d 0,62⋅ + 11,6⋅ ⋅ 1 + 2 ⋅ + 0,2⋅ ⋅ t d d t D where: r, t, d, D – geometrical dimensions in accordance with execution drawing of lever. Mathematical equation (1) is valide only for the cases in witch r ⊳ 0 and d D ⊲ 1 , case that is represented also by ours. To determinate Kt by finite element analysis it was made a linearly static analysis using Cosmos Design Star software. Rm − aG − 0 , 5+ bG ⋅MPa (7) nσ = 1 + Gσ ⋅ mm ⋅ 10 where: aG, bG – material constants; Gσ - the related stress gradient; Rm – tensile strength of material. The analytical relation to determine the related stress gradient to value of r is [1]: 2,3 Gσ (r )int,inc = ⋅ (1 + ϕ ) (8) r 2.2 The fatigue notch factor Kf The fatigue notch factor will be determined from theoretical stress concentration factor Kt using an analytical relation. The connecting relations between the two mathematical equations have different form in accordance with different vision of the one that studied the problem of fatigue behavior of structures. ISSN: 1792-4294 182 ISBN: 978-960-474-203-5 Latest Trends on Engineering Mechanics, Structures, Engineering Geology where: ϕ - coefficient depending by ratio t d . The Kt-Kf ratio for normal stress nσ (d ) is [1]: the centrifugal force FC for the runner speed 71.43 rot/min; the axial thrust on the blade FAH , resulted from the measurements on model; Rm − aG + bG ⋅MPa nσ = 1 + Gσ ⋅ mm ⋅ 10 (9) The analytical relation to determine the related stress gradient to value of d is [1]: Gσ (d )int,inc = 2 d the tangential force on the blade FT . Loads applied to the lever are presented in Fig.4. (10) 3 Problem Solution In Fig.2 there are presented the load block after which the calculation is made. Fig. 4 In the present analysed case, the main component is the lever and the blade & trunnion will be replaced by remote loads [4]. In Table 1 there are presented the loads applied to lever for static analyses and the other parameters necessary to be use for a complete analyze. Fig. 2 In accordance with Fig.2, and taking into consideration the tensile strength of material, it was chosen the experimentally diagram for our case [3]: Table 1 Lever loading Centrifugal force [N] 4069495 Blade & trunnion & lever mass force [N] 286900 Runner Head blade Case H angle [m] [grade] Fig. 3 The entry data for calculations of Kt , after the first two methods are: r = 10 mm d = 338 mm D = 375 mm t = 18,5 mm The linearly static analysis it was made by specialist of CCHAPT [4]. For linearly static analysis, firstly it is necessary to define the loads applied to the lever: the gravity force of the runner blade – lever Tangential force F link, max FT [N] [N] 1 +17.5 25 1775041 1336898 2 +10 25 1771437 1024955 3 +10 31.4 1887115 1292381 3100000 The static analysis was made for different values of global element size GMS for 36....20 mm to lead to a great precision. For the study, there were made analyses for four dimensions of meshes: - the mesh version 1 with 145772 finite elements – Fig.5; - the mesh version 2 with 174376 finite elements – Fig.6; - the mesh version 3 with 258779 finite elements – Fig.7; – trunnion assembly G =286900 N; ISSN: 1792-4294 Thrust force Fax [N] 183 ISBN: 978-960-474-203-5 Latest Trends on Engineering Mechanics, Structures, Engineering Geology Table 3 - the mesh version 4 with 400750 finite elements – Fig.8. After the analysis was processed, the value of theoretical stress concentration factor can be calculated madding the ratio between values of VonMises stresses, resulting from finite element analyses, and the values of stresses obtained according to classical strength calculations. GMS Case 1 Case 2 Case 3 Finite Elements number VonMises VonMises VonMises max [MPa] 36 30 25 20 145772 174376 258779 400750 max [MPa] 395,50 413,02 413,70 411,50 max [MPa] 382,7 401,8 398,6 394,3 387,6 407,0 403,1 394,1 The values of stress concentration factors and fatigue notch factors, determinate with all methods reminded previously are presented in Table 4. Fig.5 Table 4 Stress concentration factor Kt Fig.6 from Graphics after FKM after FEM* Fatigue notch factor Kf after after after Neuber Peterson FKM 1,3 ÷ 2,13 1,6 *Finite element methods 2,2 Fig.7 Analyzing the values revealed in Table 4 we can see that the highest values for stress concentrations factor are obtained using the analytical algorithms proposed by FKM-Guideline. As it is expected, the value of fatigue notch factor Kf is smaller than value of stress concentration factor Kf, no matter what method we apply. In industry, when we speak about big and expensive machines, such as Kaplan turbines, it is justified to chose, for dimensioning, the highest values for multiplication factors, even if it raises supplementary costs. These costs will always be smaller than the ones necessary to repair systematically the structures affected by fatigue. The paper reveals the fact that an analytical method can be use successfully to determine values for stress concentration factors, which can be used to make calculations to estimate fatigue lifetime duration. The values analytically obtained definitely are cover for all security working problems that must be solved through designing process. The chosen of fatigue notch factor became the personal option of designing engineer, the accuracy of results being strictly dependent by his experience. Stresses values σînc [N/mm2] φ=0º, φ=+10º, φ=+10º, Work H=25 m H=25 m H=31,4 m regime Case 1 Case 2 Case 3 - closing course 232,008 249,678 297,490 216,580 233,074 277,752 The numerical analysis confirms the fact that the area with maximum stress value is the fillet area between pin lever and body lever. Table 3 shows the values of VonMises stress obtained through finite element analyses. Having the values of VonMises stress it is easy to find the value of fatigue notch factor. This can be done very simple only by dividing the value of VonMises stress to stress values obtained from classical strength calculations (for the same work conditions). ISSN: 1792-4294 1,97 4 Conclusion Table 2 course 2,18 Fig.8 Table 2 present the value of stresses obtained through classical methods. - opening 2,15 184 ISBN: 978-960-474-203-5 Latest Trends on Engineering Mechanics, Structures, Engineering Geology References: [1]Conle F.A., Chu C.-C., Fatigue analysis and the local stress-strain approach in complex vehicular Structures, International Journal of Fatigue, Vol.19, Supp.No.1, 1997 [2]FKM-Guideline, Analytical strength assessment of components in Mechanical Engineering, Forschungskuratorium Maschinenbau” Frankfurt /Main, 2003 [3]Hertzberg R.W., Deformation and Fracture Mechanics of Engineering Materials, 4th edition, John Wiley&Sons, Inc., New York, 1995 [4]Mercer M., Malton G., Draper J., Investigating fatigue failures using analysis and testing – some do's and don'st, Proceedings of the ABAQUS International User's Conference, Germany, 2003 [5]Muhs D., Wittel H., Jannasch D., VoBiek J., Roloff/Matek .Machine Parts, Vol.I, Ed. Matrix Rom, Bucuresti, Romania, 2008 [6]Neuber H., Theory of Notch Stresses, J. W. Edwards Publ. Ann Arbor, 1946 [7]Peterson R.E., Stress Concentration Design Factors, Ed. J. Wiley & Sons, New York, 1953 [8]Pilkey Walter D., Peterson's Stress Concentration factor, second edition, John Wiley&Sons, Inc., New York, 1997 [9]Rice R.C., Fatigue Design Handbook, 3rd ed., SAE Pub.No.AE-22, Society of Automotive Engineers, Warrendale, PA, 1997 [10]Rusu O., Teodorescu M., Lascu-Simion N., Fatigue of Metals, Vol.I, Calculation's base, Ed. Tehnica, Bucuresti, Romania, 1992 [11]Slocum A.H., Precision Machine Design, Prentice-Hall, Englewood Cliffs, New York, 1992 [12]Socie D.F., Marquis G.B., Multiaxial Fatigue, Society of Automotive Engineers, Inc., Warrendale, PA, 2000 [13]Suresh S., Fatigue of materials, Second Edition, University Press, Cambridge, United Kingdom, 1998 [14]*****- Strength and lifetime duration calculus for runner blade lever of CHE Portile de Fier I turbine, CCHAPT, Technical Report No.U-09400-289, November, 2009. ISSN: 1792-4294 185 ISBN: 978-960-474-203-5