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