Finite Element Based Multi-Axial Strain Fatigue Analysis of Compressor Cylinders Abstract: This paper describes the finite element based multi-axial strain fatigue analysis procedure for reciprocating compressor cylinders, and investigates the effect of the fillet radius of cylinder valve shelf corner and the pre-load of cylinder valve cover bolts to the fatigue life at valve shelf corner zones. The analysis results can be used as a technical reference for investigation of cylinder fatigue failure. 1 Introduction XXX Inc has been commissioned to carry out the stress and failure analyses for several reciprocating compressor cylinders which have been found considerable cracks initiating at valve shelf corner zones. Major factors affecting the fatigue failure at valve shelf corner zones have been examined using the modified Goodman method. This paper is intended to perform the fatigue failure analysis at cylinder valve shelf corner zones using the finite element based multi-axial strain fatigue analysis technique, and investigate the effect of the fillet radius of valve shelf corner and the pre-load of valve cover bolts to the fatigue life. The analysis is carried out based on a generalized cylinder model with representative operating conditions. The analysis results are not related to fatigue failure investigation of any real cylinder product, but may be used as a general technical reference for investigating the cylinder fatigue failure. Figure 1 shows the geometrical model of a representative reciprocating compressor cylinder. A significantly long crack was found at its valve shelf corner (Figure 2). It has been recognized that the crack initiates and propagates as a result of cyclic loads applied on the valve shelf. During every working cycle of compressor, the gas pressure applied to passage surfaces keeps at the specified discharge (or suction) pressure for discharge (or suction) valve, where the gas pressure applied to working surfaces changes from the value of suction to that of discharge pressure repeatedly. The resultant force applied to the valve is therefore a cyclic load, causing a cyclic stress at valve shelf corner zones accordingly. The pre-load of bolts used to tight the valve to cylinders increases the mean value of this cyclic stress. Because of the complex geometrical shape of cylinder, Finite Element Analysis (FEA) technique [1] is employed as a power computational tool to investigate the highly localized stress at valve shelf corner zones as well as the stress distribution over the entire cylinder under the combined loads. As the significant plastic deformation was observed at valve shelf corner zones and the stress is of a general three-dimensional state, multi-axial strain fatigue analysis approach [2] is thus applied for this study. 2 Finite Element Stress Analysis 1 (1) FEA model of compressor cylinder Figure 3 shows the FEA model of a quarter of the cylinder including valve and valve accessories. The model is consisted of 3D solid elements. In order to capture the highly localized stress around valve shelf corner zones, the fillet of valve shelf corner was accurately modeled as that in the geometrical model (two different fillet radius values were used, respectively, in the FEA model to investigate the effect of the fillet radius), and the corner zones were meshed with the element size as small as 0.2 mm (Figure 4). The mesh was further verified by checking the stress difference obtained from nodal (average) stress and element (non-averaged) stress less than 5%. (2) Loads and boundary conditions The loads applied to the FEA model include gas pressures applied to the passage and working surfaces of cylinders, and the pre-load of bolts. In this study, we set the discharge and suction gas pressures as 16.72MPa and 5.602MPa, respectively, and use two different pre-load values, 310.17MPa (45 ksi) and 241.24MPa (35 ksi). The clamping force for each bolt was calculated by multiplying the pre-load of bolts and the effective tensile area of bolts. Then, it was applied to stretch the bolts and push on the nut contact area on the valve cover. The calculated bolt stretch (measured with respect to the valve cover) under the clamping force is locked and used as a load for the subsequent stress analysis under gas pressure loads. The total loads are summarized in Table 1. Table 1: Loads for Stress Analysis Loads Pre-load of Bolts (MPa) Passage Gas Pressure (MPa) Working Gas Pressure (MPa) Clamping Force 310.17 - - Clamping Force 241.24 - - Pressure High/ Low Stretch from clamping 16.72 5.602 Pressure High/ High Stretch from clamping 16.72 16.72 Symmetrical plane constraint conditions were applied to 3 mutual-perpendicular symmetrical sections. In addition, a selected node was fixed to prevent the model from rigid movement in 3 coordinate directions. (3) Material properties 2 The mechanical properties of cylinder materials were shown in Table 2 below. The materials are set to be bi-linear type with Von Mises yield criterion. Table 1: Mechanical Properties of Cylinder Components Yielding Strength y (MPa) Tangential Modulus ET (Pa) 1.66E+11 0.31 379.21 33.09E+9 Valve 2.0E+11 0.29 330.67 40.0E+9 Gasket 2.0E+11 0.29 330.67 40.0E+9 Bolts 2.0E+11 0.29 517.11 40.0E+9 Cylinder Component Elasticity Modulus E (Pa) Body Poisson ratio (4) FEA stress results Three FEA running models were developed by combining the valve shelf corner fillet radius and bolt pre-load load in different way. The first model used the valve shelf corner radius r of 0.015 inch and the pre-load Fpre of 310.17MPa, the second model used r of 0.020 inch and Fpre of 310.17MPa, and the third model used r of 0.015 inch and Fpre of 241.24MPa. Each model was run in two steps. The first step is to calculate the cylinder stress especially the displacements for each bolt stretch under the clamping force. The second step is to use the bolt stretches as constraint loads, together with the operating loads of gas “Pressure High/ Low” and “Pressure High/ High” shown in Table 2, to calculate the cylinder stress distribution, especially the highly localized stress at valve shelf corner zones. The maximum stresses calculated at valve shelf corner zones are shown in Table 3. Figure 5 to Figure 11 show the stress and deformation distributions for parts of analysis results. Table 3: Maximum Von Mises Stresses at Valve Shelf Corner Zones Model r = 0.015” Fpre = 310.17MPa r = 0.020” Fpre = 310.17MPa Clamping Stress (MPa) High/ Low Stress (MPa) High /High Stress (MPa) 458.53 421.47 412.16 455.56 388.08 376.07 3 r = 0.015” Fpre = 241.24MPa 371.91 385.65 373.91 3. Multi-axial Strain Fatigue Analysis Fatigue has been recognized as one of main failure forms for mechanical components subjected to cyclic loads. Fatigue failure analysis is generally performed by using stress-life or strain-life approach [2]. The stress-life approach is suitable for high-cycle fatigue problems where all stresses, even local ones, remain elastic, and the number of cycles to failure is large (typically, > 106 cycles). The fatigue life is calculated using S N curve (stress vs number of cycles curve). In contrast, the strain-life approach is mainly used for low-cycle fatigue problems where the material endures significant plastic straining leading to short life (<106 cycles). The fatigue life is calculated using N curve (strain vs number of cycles curve). However, the strain-life approach is also amenable to the treatment of the long life high-cycle fatigue problems [2]. The use of a consistent quantity, strain, in dealing both low- and high- cycle fatigue failure demonstrates considerable advantages. In this study, the FEA results indicate the maximum stress at valve shelf corner zones goes beyond the material yielding limit, the strain-life approach is thus a suitable choice. For application of the strain-life approach, the first step is to determine the total strain range caused by cyclic loads, and then determine the number of load cycles to failure, i.e. the fatigue life, using strain amplitude and fatigue life relationship, i.e. the - N curve [3]. Figure 12 illustrates a typical cyclic stress-strain material behavior which is used to characterize the cyclic strain for a given cyclic load. It actually defines a single fatigue cycle caused by a complete loading and unloading process in stress-strain space. This cyclic curve is generally generated experimentally, but may be also characterized by the following relationship [3]: 1 n' 2 ' E 2K (1) Where is the total strain range consisting of elastic e and plastic p components, is the total stress range, E is the elasticity modulus, K’ is the cyclic strength coefficient, and n’ is the cyclic strain hardening exponent. Once the cyclic strain range is obtained, a relationship between the total strain range and the fatigue life, in terms of number of cycles Nf to failure, becomes a key for strain based fatigue analysis. Manson [4], Coffin [5], Morrow [6], Smith, Watson, Topper [7], etc. have expended considerable efforts to develop practically applicable 4 mathematical relationship. At present, a well-established relationship accounting for the effect of mean stress is the Morrow Equation [2] which is expressed as follows: ' f o 2N f 2 E b 'f 2 N f c (2) Where 'f is the fatigue strength coefficient, o is the mean stress, b is fatigue strength exponent, 'f is the fatigue ductility coefficient, and c is the fatigue ductility exponent. Figure 13 illustrates the strain-life curves based on the Morrow Equation (2) in which, one of straight lines is for the elastic strain, and the other is for the plastic strain. The strain-life relationship based on equation (5) is directly applicable for the onedimensional stress and strain state, i.e. the uni-axial stress state. In this study, the cylinder stress at any location is of three-dimensional state, that is, the multi-axial stress state. An equivalent stress-strain approach [2, 8] for dealing with this multi-axial stress state fatigue problem is thus employed. This is where an equivalent stress and strain are calculated under multi-axial loading and then applied to uni-axial data. In the equivalent stress-strain approach using Von Mises method, the prediction of yield in terms of (x, y, and z) component stress is [2, 8]: 0.7071 x y 2 y z 2 z x 2 6 xy2 yz2 zx2 And the yielding will occur when exceeds the monotonic yielding limit. equivalent strain parameter that is used for fatigue analysis is as follows [2, 8]: 1 1 2 2 2 3 2 3 1 2 1 1 2 (3) The (4) Where 1, i (i=1, 2, 3) is the principal stress and strain, respectively. Although there are other approaches available for dealing with multi-axial stress fatigue problems, such as the critical plane approach [8], the equivalent stress-strain approach has gained widest acceptance [2, 3]. 4. Fatigue Analysis Results and Discussions Based on the FEA results and following - N curve parameters [2]: 'f = 896.051MPa (assuming 'f (ksi) f ult +50, f is the true stress at final fracture, and ult is the ultimate tensile strength), 'f f = 0.57 ( f is the true strain at final fracture), b = - 5 0.09, and c = -0.6, the fatigue life of the material at valve shelf corner zones is calculated as shown in Table 4. The number of years to failure is calculated based on the Nf value and assumption of compressor operating speed at 1000 RPM. Table 4: Fatigue Life Analysis Results Model r = 0.015” Fpre = 310.17MPa r = 0.020” Fpre = 310.17MPa r = 0.015” Fpre = 241.24MPa Strain Amplitude /2 (%) Mean Stress (MPa) Fatigue Life (Nf) Number of Years 0.0377 416.82 3.359E+9 6.5 0.0354 382.08 1.605E+10 30.9 0.0339 379.78 2.466E+10 47.6 It can be found from Table 4 that increasing the fillet radius of valve shelf corner or decreasing the pre-load of bolts can effectively increase the fatigue life of materials at valve shelf corner zones. This is because that both ways can reduce the total strain range in a cycle and the mean stress at these critical locations. However, a certain pre-load of bolts is necessary for keeping the gas inside the cylinder without leakage as well as improving the fatigue failure resistance of bolts themselves. An optimum compromise for pre-load value must be tailored to the real case investigated. The fatigue life shown in Table 4 was calculated from the cyclic load resulting from the varying gas pressure. The cyclic loads of bolting and unbolting of the valve cover, and compressor shut down operation (i.e. the pressurization and de-pressurization process) are not included in the analysis. If all these loads are taken into account, the cumulative fatigue damage theory [2] must be employed, and the fatigue life is expected to be less than what was shown in Table 4. 5. Conclusions The fatigue analysis results in this study indicate that the strain-life fatigue analysis is an applicable approach to investigate the cylinder fatigue failure, and both the fillet radius of valve shelf corner and the pre-load of bolts have significant effect on the fatigue life of materials at valve shelf corner zones. As the fatigue life calculation results are sensitive to FEA results, as well as the - N curve parameters, it is strongly recommended that for fatigue life analysis of real 6 engineering products, all material properties used in the FEA model and the - N curve should be obtained experimentally whenever possible. In addition, a well-verified FEA model is also critical to achieve practically useful results. References 1. FEMAP – Finite Element Modeling and Post processing, Version 9.2, UGS, 2006 2. ASM Volume 19, Fatigue and Fracture, 1996 3. Bishop, N. W. M. and Sherratt, F., Finite Element Based Fatigue Calculations, NAFEMS, Int. Assoc. for the Engineering Analysis Community, 2000 4. Manson, S. S., Fatigue: A Complex Subject – Some Simple Approximations, Experimental Mechanics, July 1975, pp. 1-35 5. Coffin, L. F. and Tavernelli, J. F., The Cyclic Straining and Fatigue of Metals, Trans. Metallurgical Society, AIME, Vol. 215, Oct 1959, pp. 794-806 6. Morrow, J., Fatigue Properties of Metals, Fatigue Design Handbook, SAE, 1968, pp. 21-30 7. Smith, K. N., Watson, P., and Topper, T. H., A Stress-Strain Function for the Fatigue of Metals, SMD Report 21, University of Waterloo, Oct. 1969. 8. Krempl, E., The Influence of State of Stress on Low Cycle Fatigue of Structural Materials: A Literature Survey and Interpretive Report, STP 549, ASTM, 1974 7 Valve shelf corner Figure 1: Geometrical Model of a Generalized Cylinder Model Crack Figure 2: Crack Found at Valve Shelf Corner Zones 8 Figure 3: Finite Element Model of a Quarter of Cylinder with Valve and Valve Accessories Fine mesh at valve shelf corner Figure 4: Location of the Ring for Showing Locally Concentrated Stress and Fine Mesh at Valve Shelf Corner Zones 9 Figure 5: Cylinder Von Mises Stress [Pa] Produced by Clamping Stress Load (r = 0.015”, Fpre=310.17MPa ) Figure 6: Cylinder Deformation [m] Produced by Clamping Stress Load (r = 0.015”, Fpre=310.17MPa ) 10 Figure 7: Close-up View of Locally Concentrated Von Mises Stress along Valve Shelf Corner Produced by Clamping Stress Load (r = 0.015”, Fpre=310.17MPa ) Figure 8: Cylinder Von Mises Stress [Pa] Produced by Pressure High / Low Load (r = 0.015”, Fpre=310.17MPa ) 11 Figure 9: Cylinder Deformation [m] Produced by Pressure High / Low Load (r = 0.015”, Fpre=310.17MPa ) Figure 10: Close-up View of Locally Concentrated Von Mises Stress along Valve Shelf Corner Produced by Pressure High / Low Load (r = 0.015”, Fpre=310.17MPa ) 12 (a) (b) © (d) Figure 11: Close-up View of Locally Concentrated Von Mises Stress along Valve Shelf Corner Produced by (a): Pressure High / High Load (r=0.015”, Fpre=310.17MPa ); (b): Pressure High / High Load (r=0.02”, Fpre=310.17MPa ); (c): Clamping Stress Load (r=0.015”, Fpre=241.24MPa); and (d): Pressure High / High Load (r=0.015”, Fpre=241.24MPa) 13 Strain Amplitude, /2 (Log Scale) Figure 12: Material Cyclic Stress – Strain Behaviour Reversals, 2Nf, to Failure (Log Scale) Figure 13: Material Strain – Life Curve 14