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Thermo-elastic finite element fatigue failure analysis
EngineeringFailureAnalysis, Vol 2, No. 3 pp. 197-207, 1995 Copyright© 1995ElsevierScienceLtd Printedin Great Britain. All rightsreserved 1350-6307/95 $9.50 + 0.00
Pergamon
1350-6307(95)00016-X
THERMO-ELASTIC
FINITE ELEMENT ANALYSIS
FATIGUE FAILURE
G R E G O R Y L. F E R G U S O N and S. R. GULLAPALLI Department of Mechanical Engineering, The University of Alabama, Box 870276, Tuscaloosa, AL 35487-0276, U.S.A.
(Received 22 May 1995) Abstract--Finite element analysis software has given not only the design engineer, but forensic engineers as well, a tool for analyzing complex situations. However, even with such advanced analysis tools, the engineer is often required to adapt the results to accommodate a specific situation. For instance, finite element analysis is excellent at examining static or dynamic loads against a failure criteria. However, in situations such as fatigue multiple loadings must be combined to develop the failure stresses. This is rather easy to do for one loading case that is completely reversed, However, for multiple loadings that are not reversed, other methods must be developed to augment the finite element analyses. Using such an augmented approach, a thermal fatigue failure of a gas-fired heat exchanger was accomplished. This paper details the procedure followed.
1. INTRODUCTION local firm produces a gas-fired heat exchanger used to heat oil. The exchanger was ,~xperiencing weld cracking in its submerged portion after a relatively short field ;ervice. It was originally hypothesized the failure was due to dissimiliar materials laving different coefficients of thermal expansion. The exchanger was fabricated from :wo varieties of thin (-0.0516 and 0.0615 in.) stainless steel (SS) formed by rolling nto an annular ring shape. The design called for the outer ring to be made from H0 SS while the inner ring and transition region between the inner and outer rings vere made from 304 SS. The inner annular ring had small fins protruding on the ~urface facing the outer ring to provide extra heat transfer surface area. The two lalves were welded together with a 430 SS filler wire. The combustion gases enter the mnular ring through a rectangular entrance port made from 310 SS and welded to the rater ring with the 430 SS filler wire. The heating gases flowed around the inside of he annulus, circumscribing an arc of approximately 315 ° before exiting through a port nade from a 1.75 in. diameter 304 SS pipe welded to the outer ring. The problem area was where the rectangular inlet joined the outer annular ring at n acute angle. At this welded connection, cracks were developing after approxiaately 18 months of actual field use. A metallurgist conducted a material and weld uality study on a damaged unit to ascertain if poor welding techniques were the ulprit for premature failure. After examining micrographs of many welds, it was oncluded that the welds were of a high quality, but the complexity of the design and ssembly process prevented full weld penetration. It was further recommended that a nite element analysis be conducted to quantify stress levels and potential failure lechanisms that might exist in regions where large stress gradients were expected. After examining a failed specimen, several failure modes were hypothesized: different thermal expansion coefficients, residual thermal stresses from welding, thermal stress failure, thermal fatigue failure, reduced high-temperature strength, and combinations of the above.
G. L. FERGUSON and S. R. GULLAPALL1
198
Based on the hypothesized failure modes, a finite element analysis series was identified to ascertain if any of these failure modes were the root cause of the problem. The success of any finite element analysis depends on the modeling proficiency and assumptions made by the analyst. To properly load the model, the actual environmental loads and conditions encountered in use by the exchanger for the various thermal cycles had to be identified. Concurrent to the model development was a test phase to map the worst-case thermal distribution in the exchanger. This profile was used to validate model geometry, material properties and boundary conditions for the same situation. Such a step was necessary to insure fidelity of the model which was used for the other thermal load cycles. The thermal temperature mapping for the most extreme case was done to insure the analytical model correctly distributed temperatures for the most disadvantageous case. Without a correct thermal distribution mapping, the thermally induced displacements, strains and stresses could not be considered valid.
2. D E V E L O P M E N T OF T H E R M A L L O A D CYCLES One of the first requirements was to quantify the expected and/or actual usage patterns of the heat exchanger when in field operations. This was important in quantifying whether thermal fatigue is a failure mechanism, and, if so, what is the projected life. To quantify the "typical" thermal load environment for the exchanger, three cycles were defined. These are shown in Fig, 1 and described later. 2.1. Once per day (load case 2) Every day the unit is started up to begin operations for that day. During this cycle, the system goes from shutdown equilibrium to full up operational temperatures. This means the system bad to cool down and reach equilibrium overnight. It was assumed the temperature of all materials, oil and air in the exchanger was stable at 70 °F at this condition plus there were not residual stresses present in the exchanger. Then, at startup, the unit begins heating up to its normal operating conditions, which are the most extreme conditions. Once the operating condition is reached and stabilized, the following temperatures are known: Temperature of the metal at inlet: 1284 °F Temperature of the metal at exhaust: 450 °F Temperature of the oil: 340 °F What was not known at this time was the hot air temperature distribution within the exchanger. This was quantified through a trial-and-error iteration process involving finite element analysis until the predicted thermal distribution matched the thermal distribution from testing. Therefore, load case 1 is defined as the equilibrium Tern
i
M ~
L~
/ " cycle
Vday
cycle
..~ [~> time
Fig. 1. Typical thermal profile vs time.
Thermo-elastic finite element fatigue failure analysis
199
state existing at startup where no stresses exist. Consequently, load case 2 is the full up operational conditions listed.
2.2. Cook cycle (load case 3) Once the unit reaches normal operationing parameters it is put into standby mode until an actual cook cycle begins. Typically, there are 72 cook cycles per day. A cook cycle starts at a lowered temperature and continues until operational maxima are again reached and maintained for the cook cycle, approximately 10min. The conditions at the beginning of the cook cycle depend on how long the system has been in standby mode. Assuming an average standby time of about 5 min results in the following parameters for the starting conditions for case 3: Temperature of the metal at inlet: 750 °F l'emperature of the metal at exhaust: 375 °F Temperature of the oil: 330 °F l'his thermal cycle then ranges from case 3 just listed to the case 2 maximum.
2.3, Batch cycle (load case 4) A thermostat has been placed in the oil to insure that the oil is maintained at the appropriate temperature during the cook cycle. The thermostat has been set to keep Ihe oil between 330 and 340 °F. The inlet and exit metal temperatures as well as the expected hot air entrance and exit temperatures are not quantified for this condition as they are extremely difficult to capture accurately. At one extreme they will be ;lightly less than the conditions for load case 2, except that the oil is at 330 °F. As ;uch, it was assumed that load case 4 was identical to load case 2 except the oil was at 330 °F instead of 340 °F. The batch cycle then ranges between load case 4 and load zase 2, and is repeated 5 times during the cook cycle.
3. F A I L U R E C R I T E R I A The finite element analysis method can be used to determine whether the ~xchanger has been overstrained or overstressed. The maximum stress or strain from oad case 2 can be compared against the allowable strength/strain for the operating emperature at a given location. If the allowable strength/strain is exceeded, failure :an be assumed to occur. However, allowable strengths are based on uniaxial tensile est data, whereas the complex geometry of the exchanger creates a three-dimensional tress state in the material. The complex stress state must be resolved into an ,.quivalent uniaxial stress condition before comparison against allowables can be nade. Usually, the distortion-energy theory (or more commonly the yon Mises stress) s used to obtain an equivalent uniaxial stress state in the elastic domain. MSC/ ~ASTRAN is used to calculate the von Mises stress based on the stress tensor found :t each node. Therefore, static yield or static ultimate failure can be directly letermined. On the other hand, determining fatigue failure is more difficult. Fatigue ailure is based on there being an oscillating stress envelope. That is, stresses range rom a minimum to a maximum value that repeat cyclically for a given number of epetitions equivalent to "infinite" life. There are two areas that can create problems a defining fatigue failure. The first is the fatigue criterion. The second is how to ccount for cumulative damage inflicted from different thermal cycles. The usual ,rocess in quantifying fatigue failure criterion is to establish the modified endurance imit [1]. Shigley defines the modified endurance limit as S e = kakbkckdkeate,
(1)
200
G . L . FERGUSON and S. R. G U L L A P A L L I
where: Se k~ kb kc ke ke S'e
is is is is is is is
the modified endurance limit in stress units, a surface finish factor, a size factor, a load factor, a temperature effects factor, a miscellaneous factor which is usual for stress concentrations, and the idealized endurance limit in stress units at infinite life.
For materials having an ultimate strength less than 220 ksi, S'~ is 50.4% of the ultimate strength. Each of these coefficients must be quantified to determine the modified endurance limit. The surface finish factor, k,, is given as k, = aS~,
(2)
where a and b are coefficients based on the finish, and S, is the ultimate strength in stress units (kpsi for this case). Assuming the finish for the exchanger is equivalent to hot rolled steel, then a = 14.4 and b = - 0 . 7 1 8 . According to Shigley, the weld finishes should always be the as-forged conditions unless a better finish is specified and achieved. For as-forged conditions, a = 39.9 and b = -0.995. Therefore ka = 14.4S~ ~7~s
for the exchanger
or
(3) ka = 39.9Su °995
for the welds.
Since the loading is caused by thermal expansion and not actual forces, moments or torques, it is difficult to quantify k b and kc. Since the structural deformation is basically axial, it is reasonable to assume that kb and k~ can be based on axial loadings. Hence, kb is 1 and k~ is 0.923. The next coefficient is for strength degradation at temperatures other than the reference temperature on which strengths are based, usually 70 °F. In this case, strength vs temperature curves for the materials involved were known; therefore, all calculations can be based on the strength at that temperature, so k d = 1. Finally, the miscellaneous effects' factor is used to quantify discontinuities or cross-sectional geometry changes in the elements that amplify stresses above what the normal elastic theory predicts. As the exchanger is modeled using finite element analysis, the stresses predicted will reasonably include the effects of geometric discontinuities as a fine mesh was used. Hence, the miscellaneous effects' factor can be set to unity and its effect seen in the actual calculated stresses. Thus, Eqn (1) becomes Se = 6.6988S~ 28;
for the exchanger (4)
or
Se = 8.5612S~ °°5
for the welds.
It is important to note that Se will be different at each location on the exchanger. This is due to the temperature varying around the geometry of the ring, and, since S u is a function of the temperature, Se becomes a function of the geometry. Experimental results have suggested that the fully modifed endurance limit is the lowest allowable stress limit that can be expected in a cyclic stress situation and occurs at the definition of infinite life, usually 1 million cycles for steel. For a static load, that is, no stress cycling, the stress limit is the ultimate strength. Tests have shown that, for the first 1000 cycles, the strength decreases slightly to about 90% of the ultimate strength. From this point, the strength decreases rapidly (logarithmically) until the endurance limit is reached. Based on this knowledge, the expected life can be found for a given stress state. Figure 2 shows a typical stress-life ( S - N ) curve. If the stress fluctuation is completely reversed, then the magnitude, or amplitude,
Thermo-elastic finite element fatigue failure analysis
201
sZ S u .
0.9S u"
Se -
10 3
10 4
10 5
10 6
N cycles
Fig. 2. Typical S - N diagram for steel.
9f the stress can be compared directly with the modified endurance limit for a safety 7actor, or with the S - N diagram to determine the expected life. One problem occurs zehere the stresses are not completely reversed as there is a fluctuation amplitude on :op of the mean stress value. This leads to defining the mean and amplitude variances ~f stress as a a --
Ormax -- O'min
and
(7m =
O'max Jr" O'min
2
(5)
2
Several failure theories have been proposed to quantify safety based on the tmplitude and mean values of the fluctuating stress. The most widely used is the nodified Goodman approach which is graphically shown in Fig. 3. Mean stresses are plotted on the horizontal axis, and include the actual mean and dtimate stresses. On the vertical axis, the actual amplitude and modified endurance imit stresses are plotted. The modified endurance limit and ultimate strengths are oined by a straight line indicating the safe line. If the intersection of the amplitude tnd mean stresses is below the safe line, the combination of stresses is acceptable. If t is above the line, the situation is unacceptable. Knowing the relationship between he ultimate strength and the modifed endurance limit is linear allows an expression o be developed that expresses a safety factor for the combination stress state, or a__~a+ am _ 1 , Se Su n
(6)
S a
S
e
Se a
in
S a
Fig. 3. Modified G o o d m a n diagram.
S in
202
G.L. FERGUSON and S. R. GULLAPALLI
where n represents the desired safety factor. If the safety factor is unity, this equation can be rearranged to solve for a new value of Se (shown in Fig. 3 as S*) which is equivalent to making this a completely reversed stress state, or S~': -
<' ]
(7) (7 m __
--
S U
Therefore, S~~ is the equivalent amplitude stress at a zero mean. This value can be used with Fig. 2 to determine the estimated life at this stress level. So, if S* is less than S~ then infinite life can be expected. If S~ exceeds S~ then the finite life for that stress level can be estimated. To do this, S* must be compared to 90% of the ultimate strength. If greater, the life is less than 1000 cycles and can be found by interpolating between 90 and 100% of the ultimate strength. On the other hand, if S* is less than 90% of the ultimate strength but greater than the modified endurance limit, the expected life is between 1000 and 1 million cycles. In this range, the life can be expressed algebraically as N
Sj~ ~h
where a -
(0.9S, f
and
S~.
1log (0.9Su ] b = -5 \~7-~ l
The last critical item involved in the fatigue analysis is how to combine the three separate loading cycles. The most commonly used cumulative damage analysis is based on the Palmgren-Miner theory [1, 2]. Simply stated, the theorem says the sum of the ratio of the number of actual cycles for each load case vs the expected life at that stress equivalent amplitude must be less than 1. Mathematically, that is ) n~
i~1~ ~ 1,
(9)
where: ni j
is the number of actual cycles at a given stress state, is the number of projected cycles at the given stress state, and is the number of defined cycles forming the cumulative damage set.
For this situation, ] is 3 as three thermal cycles have been identified. Further, the number of actual cycles per load case can be quantified in terms of the number of days the system can be cycled. In essence, n~ is the number of daily cycles, n 2 is the number of cook cycles, which is equals 72n~, and n3 is the number of batch cycles, which is 5n2 or 360nl. Therefore, at the limit of the Palmgren-Minor theory: nl --
NI
n~ +
-
N~
n~ +
~-
N~
=
t =
tl, 72nl 360nl :- + - + - Ni N, N3
(10)
This can be rearranged to solve for n~, or the number of cooking days, as nl =
l
•
(11)
- -1+ - - 72 + - -360 N1 N• N3 One particular drawback to using Miner's theory is that it does not account for the order of load cycling. All lifes are based on the material being considered virgin for each load cycle. In reality, if a loading cycle does cause damage, the next cycle loads a damaged specimen and not virgin material. Manson [2] examined this situation and suggested a theory that examined when damaged nuclei occurred and how crack propagation affected life. His approach resulted in an equation similar to Miner's
Thermo-elastic finite elementfatiguefailureanalysis
203
except the expected life is corrected for damage inflicted from an earlier cycle. For most metallic materials, Manson found the projected life can be expressed as N' = N - 14N °'6,
(12)
where N is the projected life of a virgin specimen in excess of 730 cycles. If N is less than 730 cycles, N' = N. Making use of this, Eqn (11) can be recast as nl =
1
1 ---+N'I
72
360 -t--N'2 N'3
,
(13)
¢¢here N' is either the undamaged or damaged life, N, based on which cumulative ]amage theory is used. Again, it must be stressed that the methodology just described must be performed it each node. Each node is subjected to cyclic stresses but the endurance limit varies ;ince the ultimate strength varies around the circumference of the exchanger, as do he nodal temperatures. The life of the exchanger is then based on the minimum nl ~ncountered for all nodes used to represent the system. Many finite element analysis :odes will not allow combinations of stress states as needed for such a fatigue malysis. Therefore, for this study, a macro routine was programmed to read the 'esults files from a complete set of thermal finite element analysis loadings, to levelop n1 for each node, and then to find the minimum nl.
4. FINITE ELEMENT MODEL MSC/ARIES was used to model the exchanger for subsequent analysis by ¢ISC/NASTRAN. Since the exchanger is roughly symmetrical about a plane located t half the vertical height, symmetry boundary conditions were used to minimize aodel size. The exchanger material is thin, so two-dimensional shell elements of ppropriate thickness were utilized. These allow for thermal and stress distributions in he plane of the element. However, thermal distributions through the thickness annot be calculated. Many modeling iterations were made in an effort to correctly capture the essence of he combustion hot air distribution around the annulus. MSC/ARIES provides for anctional definitions in either linear or quadratic form. However, the expected aermal profile logarithmically decreases from the hot inlet to the cold exit. It is nportant to have an accurate hot-air profile as this is used for the convection heat ~ansfer calculations. To capture the hot-air profile, within MSC/ARIES limitations, equired the model's circumference to be broken into eight regions of approximately 5° arc each. The beginning, end and midpoint temperatures in each region were etermined based on a logarithmic temperature decrease around the full exchanger ircumference. These values were then distributed around the arc segment using ISC/ARIES's quadratic formulation. Many trial-and-error iterations were required efore the predicted thermal profile of the exchanger for load case 2 equalled the leasured thermal profile. The hot-air temperature distributions for other load tuations were extrapolated from load case 2. The model was meshed with eight-node quadrilateral elements as shown in Fig. 4. Ysing higher-order elements containing mid-side nodes preserves numerical accuracy 'ith significantly fewer elements than if only four-node quadrilateral elements had een used. Mesh refinement was incorporated around the inlet and exit portals to etter capture the stress raisers expected to exist in these regions. The modeling ~sulted in 1102 elements and 3481 nodes describing the inlet, circumference and exit ~gions of the half model. Table 1 lists the basic material properties used for the taterials [3-5]. Material properties for the materials are a function of temperature and have been aantified. However, to prevent a non-linear material analysis from being required
204
G . L . F E R G U S O N and S. R. G U L L A P A L L I
Fig. 4. Meshed model.
Table I. Material properties
Material 304 SS 310 SS 430 SS 18SR SS
Thermal expansion coefficient (in,/in.-°F) 9.2 8.0 5.6 5.9
× × x x
10 -6 10 -6 10-~ 10 -~
Thermal conduction coefficient (BTU/s-in.-°F) 2.176 1.852 2.820 2.750
x × × N
10 -4 10-4 10 4 10 -4
Yield strength
Ultimate strength
(psi at room temperature) 30 45 45 60
x × × N
103 103 103 103
84 95 77 80
× x x ×
103 103 103 103
(as the presently available version of MSC/NASTRAN does not support non-linear material properties) the thermal expansion coefficient, Young's modulus of elasticity, density and thermal conductivity coefficient were assumed to be constant at room temperature values. The strengths at that temperature were only utilized in the macro written to calculate the expected fatigue life. In this case, the properties varied per nodal temperature. The macro reads the temperatures at the node and calculates the appropriate properties for that temperature based on a regression analysis derived to express strength vs temperature. For instance, Fig. 5 shows ultimate tensile strength vs temperature for 304 SS. To model steady-state thermal equilibrium, all thermal producing gradients must be included. The simplest way to define the heat input is through holding the inlet and exit metal temperatures constant at their respective levels corresponding to the case being analyzed (see the earlier discussion on thermal load cycles). The temperature
Thermo-elastic finite element fatigue failure analysis
205
90-
~.o
70
60-
g ~ 2
50-
~
4113020 0
I 500
1000
1500
Temperature
Fig. 5. Ultimate strength for 304 sS vs temperature.
~rofile is then affected by free convection on either the top or the bottom surface of he plates as well as through conduction across the planar span of the elements. The allowing convection coefficients were used: aetal to air--outside ring letal to air--inner rings aetal to oil--all surfaces
4.82 × 10 -6 BTU/s-in.2-°F, 6.75 × 10 -6 BTU/s-in.2-°F, 11.57 × 10 -6 BTU/s-in.2-°F,
'he reason for the different convection coefficients for metal to air is due to the fins n the inner ring being more efficient. However, the fins were not modeled but are pproximated in the finite element model by a higher convection coefficient of the mer ring.
5. R E S U L T S The predicted temperature distribution for the worst case is shown in Fig. 6 for ,ad case 2. It matches the measured thermal distribution very closely. Based on this ose match, the other load case thermal distributions were assumed to realistically :flect the actual situation. The attendant stresses caused by the worst-case thermal distribution is shown tperimposed on the deflected shape in Fig. 7. It is important to point out that the :ale of deflection has been grossly exaggerated to demonstrate where maximum splacements are occurring. It is readily apparent that the hot-air inlet is the most storted. The life expectancy equations from the fatigue failure discussion were programmed the C language. The M S C / N A S T R A N generated nodal stresses and nodal mperature that have been reformatted for the M S C / A R I E S display are read in by e program. The program then performs all calculations based on the nodal mperature. It should be pointed out that nodal stresses are the average of all the ~n Mises stresses from all elements connected to that node. Therefore, in areas of :treme stress gradients, the average nodal stress may be less than the elemental :esses. The minimum life for the model was found at node 2888. This node is at the tersection (weld joint) of the inlet, outer ring and transition piece from the inner to e outer ring on the side nearest the exit port. This location is where the weld failure is occurring on service units. All nodes acting along the weld line between the inlet d the outer ring were analyzed against properties for the parent material(s) as well
200
G, L. FERGUSON and S. R. GULLAPALLI
Fig. 0. M S C / N A S T R A N
predicted thermal distribution.
!i ;;
,:
}
i!
~5
Fig. 7. Stress distribution for load c a s e 2.
as for the weld material. It turned out that the weld material properties controlled the life. In other words, the life expectancy at node 2888 for the parent material (310, 430, or 18SR SS) is significantly greater than for the weld material. The results of the C program calculations for node 2888 are shown in Table 2.
Thermo-elastic finite element fatigue failure analysis
207
Table 2. Summaryof node 2888 results Reference temperature (°F)
;tress (kpsi) diner's life* (cycles) danson's life* (cycles) diner's lifet (cycles) ¢lanson's lifet (cycles)
diner's damaged life (days) danson's damaged life (days)
669.88 Load case 1
Load case 2
Load case 3
Load case 4
0.00
47.04 13,004 8887 9753 6289
21.53 93,757 80,288 56,409 46,480
33.69 695,386 650,567
SS sheet*
Weldt
1184 991
527 442
*Based on hot rolled surface finish. tBased on as-forged surface finish. 6. C O N C L U S I O N S This work suggests that thermal fatigue, exacerbated by mismatched thermal xpansion coefficients, is the probable cause of failure. The predicted life for the xisting exchanger, which is controlled by the 430 weld material, accurately predicts le actual failure assuming the weld material is the as-forged surface finish, i.e. 42-527 days c o m p a r e d to an actual failure life of approximately 500 days. The nalysis was based on the model staying in the linear-elastic regime, and material roperties such as thermal conductivity and thermal expansion remaining constant as ~e t e m p e r a t u r e increases. The thermal expansion coefficient for the inner and outer rings for the baseline esign are very nearly identical, which suggests identical growth/deformation. Howver, the circumferential growth of the outer ring will be greater than the inner ring mply due to the ring radius. This means a larger radial deflection will result, thereby ,creasing stresses at the intersection. The weld material in these areas must then be 91e to withstand this increased stress state. It is also observed that there is a roughly )% reduction in tensile strength at operating temperatures for the 430 SS. The weld Laterial in these areas must be able to withstand the increased stress state. In conclusion, the finite element m e t h o d is an extremely useful tool to help analyse totems prior to implementation or as a forensic engineering tool to determine why a ilure occurred. H o w e v e r , as with any tool, engineering judgement must be used and ',suits m a y have to be augmented with extra work to obtain the final answer. Such a tuation was demonstrated in determining the cause of a thermal fatigue failure. In Lis case, the original hypothesis of thermal growth mismatch causing the problem as found to be partially correct. The weld material used was the real culprit as it had sufficient strength at the operating t e m p e r a t u r e to withstand the thermal cyclic adings to which it was exposed. A simple solution to correcting the problem would to replace the 430 SS weld rod with a better matched material, such as 18SR SS eld wire.
REFERENCES J. E. Shigley and C. R. Mischke, Mechanical Engineering Design (5th edn), McGraw-Hill, New York (1989). J. A. Collins, Failure of Materials in Mechanical Design (2nd edn), Wiley Interscience, New York (1993). D. Peckncr and I. M. Bernstein, Handbook of Stainless Steels, Chap. 19, McGraw-Hill, New York (1959). Allegheny Ludlum Steel Corp., Stainless Steel Fabrication (1958). TCT Stainless Steel of Nashville, Inc., Material Specifications for Various Stainless Steels, vendor specifications.
Pergamon
1350-6307(95)00016-X
THERMO-ELASTIC
FINITE ELEMENT ANALYSIS
FATIGUE FAILURE
G R E G O R Y L. F E R G U S O N and S. R. GULLAPALLI Department of Mechanical Engineering, The University of Alabama, Box 870276, Tuscaloosa, AL 35487-0276, U.S.A.
(Received 22 May 1995) Abstract--Finite element analysis software has given not only the design engineer, but forensic engineers as well, a tool for analyzing complex situations. However, even with such advanced analysis tools, the engineer is often required to adapt the results to accommodate a specific situation. For instance, finite element analysis is excellent at examining static or dynamic loads against a failure criteria. However, in situations such as fatigue multiple loadings must be combined to develop the failure stresses. This is rather easy to do for one loading case that is completely reversed, However, for multiple loadings that are not reversed, other methods must be developed to augment the finite element analyses. Using such an augmented approach, a thermal fatigue failure of a gas-fired heat exchanger was accomplished. This paper details the procedure followed.
1. INTRODUCTION local firm produces a gas-fired heat exchanger used to heat oil. The exchanger was ,~xperiencing weld cracking in its submerged portion after a relatively short field ;ervice. It was originally hypothesized the failure was due to dissimiliar materials laving different coefficients of thermal expansion. The exchanger was fabricated from :wo varieties of thin (-0.0516 and 0.0615 in.) stainless steel (SS) formed by rolling nto an annular ring shape. The design called for the outer ring to be made from H0 SS while the inner ring and transition region between the inner and outer rings vere made from 304 SS. The inner annular ring had small fins protruding on the ~urface facing the outer ring to provide extra heat transfer surface area. The two lalves were welded together with a 430 SS filler wire. The combustion gases enter the mnular ring through a rectangular entrance port made from 310 SS and welded to the rater ring with the 430 SS filler wire. The heating gases flowed around the inside of he annulus, circumscribing an arc of approximately 315 ° before exiting through a port nade from a 1.75 in. diameter 304 SS pipe welded to the outer ring. The problem area was where the rectangular inlet joined the outer annular ring at n acute angle. At this welded connection, cracks were developing after approxiaately 18 months of actual field use. A metallurgist conducted a material and weld uality study on a damaged unit to ascertain if poor welding techniques were the ulprit for premature failure. After examining micrographs of many welds, it was oncluded that the welds were of a high quality, but the complexity of the design and ssembly process prevented full weld penetration. It was further recommended that a nite element analysis be conducted to quantify stress levels and potential failure lechanisms that might exist in regions where large stress gradients were expected. After examining a failed specimen, several failure modes were hypothesized: different thermal expansion coefficients, residual thermal stresses from welding, thermal stress failure, thermal fatigue failure, reduced high-temperature strength, and combinations of the above.
G. L. FERGUSON and S. R. GULLAPALL1
198
Based on the hypothesized failure modes, a finite element analysis series was identified to ascertain if any of these failure modes were the root cause of the problem. The success of any finite element analysis depends on the modeling proficiency and assumptions made by the analyst. To properly load the model, the actual environmental loads and conditions encountered in use by the exchanger for the various thermal cycles had to be identified. Concurrent to the model development was a test phase to map the worst-case thermal distribution in the exchanger. This profile was used to validate model geometry, material properties and boundary conditions for the same situation. Such a step was necessary to insure fidelity of the model which was used for the other thermal load cycles. The thermal temperature mapping for the most extreme case was done to insure the analytical model correctly distributed temperatures for the most disadvantageous case. Without a correct thermal distribution mapping, the thermally induced displacements, strains and stresses could not be considered valid.
2. D E V E L O P M E N T OF T H E R M A L L O A D CYCLES One of the first requirements was to quantify the expected and/or actual usage patterns of the heat exchanger when in field operations. This was important in quantifying whether thermal fatigue is a failure mechanism, and, if so, what is the projected life. To quantify the "typical" thermal load environment for the exchanger, three cycles were defined. These are shown in Fig, 1 and described later. 2.1. Once per day (load case 2) Every day the unit is started up to begin operations for that day. During this cycle, the system goes from shutdown equilibrium to full up operational temperatures. This means the system bad to cool down and reach equilibrium overnight. It was assumed the temperature of all materials, oil and air in the exchanger was stable at 70 °F at this condition plus there were not residual stresses present in the exchanger. Then, at startup, the unit begins heating up to its normal operating conditions, which are the most extreme conditions. Once the operating condition is reached and stabilized, the following temperatures are known: Temperature of the metal at inlet: 1284 °F Temperature of the metal at exhaust: 450 °F Temperature of the oil: 340 °F What was not known at this time was the hot air temperature distribution within the exchanger. This was quantified through a trial-and-error iteration process involving finite element analysis until the predicted thermal distribution matched the thermal distribution from testing. Therefore, load case 1 is defined as the equilibrium Tern
i
M ~
L~
/ " cycle
Vday
cycle
..~ [~> time
Fig. 1. Typical thermal profile vs time.
Thermo-elastic finite element fatigue failure analysis
199
state existing at startup where no stresses exist. Consequently, load case 2 is the full up operational conditions listed.
2.2. Cook cycle (load case 3) Once the unit reaches normal operationing parameters it is put into standby mode until an actual cook cycle begins. Typically, there are 72 cook cycles per day. A cook cycle starts at a lowered temperature and continues until operational maxima are again reached and maintained for the cook cycle, approximately 10min. The conditions at the beginning of the cook cycle depend on how long the system has been in standby mode. Assuming an average standby time of about 5 min results in the following parameters for the starting conditions for case 3: Temperature of the metal at inlet: 750 °F l'emperature of the metal at exhaust: 375 °F Temperature of the oil: 330 °F l'his thermal cycle then ranges from case 3 just listed to the case 2 maximum.
2.3, Batch cycle (load case 4) A thermostat has been placed in the oil to insure that the oil is maintained at the appropriate temperature during the cook cycle. The thermostat has been set to keep Ihe oil between 330 and 340 °F. The inlet and exit metal temperatures as well as the expected hot air entrance and exit temperatures are not quantified for this condition as they are extremely difficult to capture accurately. At one extreme they will be ;lightly less than the conditions for load case 2, except that the oil is at 330 °F. As ;uch, it was assumed that load case 4 was identical to load case 2 except the oil was at 330 °F instead of 340 °F. The batch cycle then ranges between load case 4 and load zase 2, and is repeated 5 times during the cook cycle.
3. F A I L U R E C R I T E R I A The finite element analysis method can be used to determine whether the ~xchanger has been overstrained or overstressed. The maximum stress or strain from oad case 2 can be compared against the allowable strength/strain for the operating emperature at a given location. If the allowable strength/strain is exceeded, failure :an be assumed to occur. However, allowable strengths are based on uniaxial tensile est data, whereas the complex geometry of the exchanger creates a three-dimensional tress state in the material. The complex stress state must be resolved into an ,.quivalent uniaxial stress condition before comparison against allowables can be nade. Usually, the distortion-energy theory (or more commonly the yon Mises stress) s used to obtain an equivalent uniaxial stress state in the elastic domain. MSC/ ~ASTRAN is used to calculate the von Mises stress based on the stress tensor found :t each node. Therefore, static yield or static ultimate failure can be directly letermined. On the other hand, determining fatigue failure is more difficult. Fatigue ailure is based on there being an oscillating stress envelope. That is, stresses range rom a minimum to a maximum value that repeat cyclically for a given number of epetitions equivalent to "infinite" life. There are two areas that can create problems a defining fatigue failure. The first is the fatigue criterion. The second is how to ccount for cumulative damage inflicted from different thermal cycles. The usual ,rocess in quantifying fatigue failure criterion is to establish the modified endurance imit [1]. Shigley defines the modified endurance limit as S e = kakbkckdkeate,
(1)
200
G . L . FERGUSON and S. R. G U L L A P A L L I
where: Se k~ kb kc ke ke S'e
is is is is is is is
the modified endurance limit in stress units, a surface finish factor, a size factor, a load factor, a temperature effects factor, a miscellaneous factor which is usual for stress concentrations, and the idealized endurance limit in stress units at infinite life.
For materials having an ultimate strength less than 220 ksi, S'~ is 50.4% of the ultimate strength. Each of these coefficients must be quantified to determine the modified endurance limit. The surface finish factor, k,, is given as k, = aS~,
(2)
where a and b are coefficients based on the finish, and S, is the ultimate strength in stress units (kpsi for this case). Assuming the finish for the exchanger is equivalent to hot rolled steel, then a = 14.4 and b = - 0 . 7 1 8 . According to Shigley, the weld finishes should always be the as-forged conditions unless a better finish is specified and achieved. For as-forged conditions, a = 39.9 and b = -0.995. Therefore ka = 14.4S~ ~7~s
for the exchanger
or
(3) ka = 39.9Su °995
for the welds.
Since the loading is caused by thermal expansion and not actual forces, moments or torques, it is difficult to quantify k b and kc. Since the structural deformation is basically axial, it is reasonable to assume that kb and k~ can be based on axial loadings. Hence, kb is 1 and k~ is 0.923. The next coefficient is for strength degradation at temperatures other than the reference temperature on which strengths are based, usually 70 °F. In this case, strength vs temperature curves for the materials involved were known; therefore, all calculations can be based on the strength at that temperature, so k d = 1. Finally, the miscellaneous effects' factor is used to quantify discontinuities or cross-sectional geometry changes in the elements that amplify stresses above what the normal elastic theory predicts. As the exchanger is modeled using finite element analysis, the stresses predicted will reasonably include the effects of geometric discontinuities as a fine mesh was used. Hence, the miscellaneous effects' factor can be set to unity and its effect seen in the actual calculated stresses. Thus, Eqn (1) becomes Se = 6.6988S~ 28;
for the exchanger (4)
or
Se = 8.5612S~ °°5
for the welds.
It is important to note that Se will be different at each location on the exchanger. This is due to the temperature varying around the geometry of the ring, and, since S u is a function of the temperature, Se becomes a function of the geometry. Experimental results have suggested that the fully modifed endurance limit is the lowest allowable stress limit that can be expected in a cyclic stress situation and occurs at the definition of infinite life, usually 1 million cycles for steel. For a static load, that is, no stress cycling, the stress limit is the ultimate strength. Tests have shown that, for the first 1000 cycles, the strength decreases slightly to about 90% of the ultimate strength. From this point, the strength decreases rapidly (logarithmically) until the endurance limit is reached. Based on this knowledge, the expected life can be found for a given stress state. Figure 2 shows a typical stress-life ( S - N ) curve. If the stress fluctuation is completely reversed, then the magnitude, or amplitude,
Thermo-elastic finite element fatigue failure analysis
201
sZ S u .
0.9S u"
Se -
10 3
10 4
10 5
10 6
N cycles
Fig. 2. Typical S - N diagram for steel.
9f the stress can be compared directly with the modified endurance limit for a safety 7actor, or with the S - N diagram to determine the expected life. One problem occurs zehere the stresses are not completely reversed as there is a fluctuation amplitude on :op of the mean stress value. This leads to defining the mean and amplitude variances ~f stress as a a --
Ormax -- O'min
and
(7m =
O'max Jr" O'min
2
(5)
2
Several failure theories have been proposed to quantify safety based on the tmplitude and mean values of the fluctuating stress. The most widely used is the nodified Goodman approach which is graphically shown in Fig. 3. Mean stresses are plotted on the horizontal axis, and include the actual mean and dtimate stresses. On the vertical axis, the actual amplitude and modified endurance imit stresses are plotted. The modified endurance limit and ultimate strengths are oined by a straight line indicating the safe line. If the intersection of the amplitude tnd mean stresses is below the safe line, the combination of stresses is acceptable. If t is above the line, the situation is unacceptable. Knowing the relationship between he ultimate strength and the modifed endurance limit is linear allows an expression o be developed that expresses a safety factor for the combination stress state, or a__~a+ am _ 1 , Se Su n
(6)
S a
S
e
Se a
in
S a
Fig. 3. Modified G o o d m a n diagram.
S in
202
G.L. FERGUSON and S. R. GULLAPALLI
where n represents the desired safety factor. If the safety factor is unity, this equation can be rearranged to solve for a new value of Se (shown in Fig. 3 as S*) which is equivalent to making this a completely reversed stress state, or S~': -
<' ]
(7) (7 m __
--
S U
Therefore, S~~ is the equivalent amplitude stress at a zero mean. This value can be used with Fig. 2 to determine the estimated life at this stress level. So, if S* is less than S~ then infinite life can be expected. If S~ exceeds S~ then the finite life for that stress level can be estimated. To do this, S* must be compared to 90% of the ultimate strength. If greater, the life is less than 1000 cycles and can be found by interpolating between 90 and 100% of the ultimate strength. On the other hand, if S* is less than 90% of the ultimate strength but greater than the modified endurance limit, the expected life is between 1000 and 1 million cycles. In this range, the life can be expressed algebraically as N
Sj~ ~h
where a -
(0.9S, f
and
S~.
1log (0.9Su ] b = -5 \~7-~ l
The last critical item involved in the fatigue analysis is how to combine the three separate loading cycles. The most commonly used cumulative damage analysis is based on the Palmgren-Miner theory [1, 2]. Simply stated, the theorem says the sum of the ratio of the number of actual cycles for each load case vs the expected life at that stress equivalent amplitude must be less than 1. Mathematically, that is ) n~
i~1~ ~ 1,
(9)
where: ni j
is the number of actual cycles at a given stress state, is the number of projected cycles at the given stress state, and is the number of defined cycles forming the cumulative damage set.
For this situation, ] is 3 as three thermal cycles have been identified. Further, the number of actual cycles per load case can be quantified in terms of the number of days the system can be cycled. In essence, n~ is the number of daily cycles, n 2 is the number of cook cycles, which is equals 72n~, and n3 is the number of batch cycles, which is 5n2 or 360nl. Therefore, at the limit of the Palmgren-Minor theory: nl --
NI
n~ +
-
N~
n~ +
~-
N~
=
t =
tl, 72nl 360nl :- + - + - Ni N, N3
(10)
This can be rearranged to solve for n~, or the number of cooking days, as nl =
l
•
(11)
- -1+ - - 72 + - -360 N1 N• N3 One particular drawback to using Miner's theory is that it does not account for the order of load cycling. All lifes are based on the material being considered virgin for each load cycle. In reality, if a loading cycle does cause damage, the next cycle loads a damaged specimen and not virgin material. Manson [2] examined this situation and suggested a theory that examined when damaged nuclei occurred and how crack propagation affected life. His approach resulted in an equation similar to Miner's
Thermo-elastic finite elementfatiguefailureanalysis
203
except the expected life is corrected for damage inflicted from an earlier cycle. For most metallic materials, Manson found the projected life can be expressed as N' = N - 14N °'6,
(12)
where N is the projected life of a virgin specimen in excess of 730 cycles. If N is less than 730 cycles, N' = N. Making use of this, Eqn (11) can be recast as nl =
1
1 ---+N'I
72
360 -t--N'2 N'3
,
(13)
¢¢here N' is either the undamaged or damaged life, N, based on which cumulative ]amage theory is used. Again, it must be stressed that the methodology just described must be performed it each node. Each node is subjected to cyclic stresses but the endurance limit varies ;ince the ultimate strength varies around the circumference of the exchanger, as do he nodal temperatures. The life of the exchanger is then based on the minimum nl ~ncountered for all nodes used to represent the system. Many finite element analysis :odes will not allow combinations of stress states as needed for such a fatigue malysis. Therefore, for this study, a macro routine was programmed to read the 'esults files from a complete set of thermal finite element analysis loadings, to levelop n1 for each node, and then to find the minimum nl.
4. FINITE ELEMENT MODEL MSC/ARIES was used to model the exchanger for subsequent analysis by ¢ISC/NASTRAN. Since the exchanger is roughly symmetrical about a plane located t half the vertical height, symmetry boundary conditions were used to minimize aodel size. The exchanger material is thin, so two-dimensional shell elements of ppropriate thickness were utilized. These allow for thermal and stress distributions in he plane of the element. However, thermal distributions through the thickness annot be calculated. Many modeling iterations were made in an effort to correctly capture the essence of he combustion hot air distribution around the annulus. MSC/ARIES provides for anctional definitions in either linear or quadratic form. However, the expected aermal profile logarithmically decreases from the hot inlet to the cold exit. It is nportant to have an accurate hot-air profile as this is used for the convection heat ~ansfer calculations. To capture the hot-air profile, within MSC/ARIES limitations, equired the model's circumference to be broken into eight regions of approximately 5° arc each. The beginning, end and midpoint temperatures in each region were etermined based on a logarithmic temperature decrease around the full exchanger ircumference. These values were then distributed around the arc segment using ISC/ARIES's quadratic formulation. Many trial-and-error iterations were required efore the predicted thermal profile of the exchanger for load case 2 equalled the leasured thermal profile. The hot-air temperature distributions for other load tuations were extrapolated from load case 2. The model was meshed with eight-node quadrilateral elements as shown in Fig. 4. Ysing higher-order elements containing mid-side nodes preserves numerical accuracy 'ith significantly fewer elements than if only four-node quadrilateral elements had een used. Mesh refinement was incorporated around the inlet and exit portals to etter capture the stress raisers expected to exist in these regions. The modeling ~sulted in 1102 elements and 3481 nodes describing the inlet, circumference and exit ~gions of the half model. Table 1 lists the basic material properties used for the taterials [3-5]. Material properties for the materials are a function of temperature and have been aantified. However, to prevent a non-linear material analysis from being required
204
G . L . F E R G U S O N and S. R. G U L L A P A L L I
Fig. 4. Meshed model.
Table I. Material properties
Material 304 SS 310 SS 430 SS 18SR SS
Thermal expansion coefficient (in,/in.-°F) 9.2 8.0 5.6 5.9
× × x x
10 -6 10 -6 10-~ 10 -~
Thermal conduction coefficient (BTU/s-in.-°F) 2.176 1.852 2.820 2.750
x × × N
10 -4 10-4 10 4 10 -4
Yield strength
Ultimate strength
(psi at room temperature) 30 45 45 60
x × × N
103 103 103 103
84 95 77 80
× x x ×
103 103 103 103
(as the presently available version of MSC/NASTRAN does not support non-linear material properties) the thermal expansion coefficient, Young's modulus of elasticity, density and thermal conductivity coefficient were assumed to be constant at room temperature values. The strengths at that temperature were only utilized in the macro written to calculate the expected fatigue life. In this case, the properties varied per nodal temperature. The macro reads the temperatures at the node and calculates the appropriate properties for that temperature based on a regression analysis derived to express strength vs temperature. For instance, Fig. 5 shows ultimate tensile strength vs temperature for 304 SS. To model steady-state thermal equilibrium, all thermal producing gradients must be included. The simplest way to define the heat input is through holding the inlet and exit metal temperatures constant at their respective levels corresponding to the case being analyzed (see the earlier discussion on thermal load cycles). The temperature
Thermo-elastic finite element fatigue failure analysis
205
90-
~.o
70
60-
g ~ 2
50-
~
4113020 0
I 500
1000
1500
Temperature
Fig. 5. Ultimate strength for 304 sS vs temperature.
~rofile is then affected by free convection on either the top or the bottom surface of he plates as well as through conduction across the planar span of the elements. The allowing convection coefficients were used: aetal to air--outside ring letal to air--inner rings aetal to oil--all surfaces
4.82 × 10 -6 BTU/s-in.2-°F, 6.75 × 10 -6 BTU/s-in.2-°F, 11.57 × 10 -6 BTU/s-in.2-°F,
'he reason for the different convection coefficients for metal to air is due to the fins n the inner ring being more efficient. However, the fins were not modeled but are pproximated in the finite element model by a higher convection coefficient of the mer ring.
5. R E S U L T S The predicted temperature distribution for the worst case is shown in Fig. 6 for ,ad case 2. It matches the measured thermal distribution very closely. Based on this ose match, the other load case thermal distributions were assumed to realistically :flect the actual situation. The attendant stresses caused by the worst-case thermal distribution is shown tperimposed on the deflected shape in Fig. 7. It is important to point out that the :ale of deflection has been grossly exaggerated to demonstrate where maximum splacements are occurring. It is readily apparent that the hot-air inlet is the most storted. The life expectancy equations from the fatigue failure discussion were programmed the C language. The M S C / N A S T R A N generated nodal stresses and nodal mperature that have been reformatted for the M S C / A R I E S display are read in by e program. The program then performs all calculations based on the nodal mperature. It should be pointed out that nodal stresses are the average of all the ~n Mises stresses from all elements connected to that node. Therefore, in areas of :treme stress gradients, the average nodal stress may be less than the elemental :esses. The minimum life for the model was found at node 2888. This node is at the tersection (weld joint) of the inlet, outer ring and transition piece from the inner to e outer ring on the side nearest the exit port. This location is where the weld failure is occurring on service units. All nodes acting along the weld line between the inlet d the outer ring were analyzed against properties for the parent material(s) as well
200
G, L. FERGUSON and S. R. GULLAPALLI
Fig. 0. M S C / N A S T R A N
predicted thermal distribution.
!i ;;
,:
}
i!
~5
Fig. 7. Stress distribution for load c a s e 2.
as for the weld material. It turned out that the weld material properties controlled the life. In other words, the life expectancy at node 2888 for the parent material (310, 430, or 18SR SS) is significantly greater than for the weld material. The results of the C program calculations for node 2888 are shown in Table 2.
Thermo-elastic finite element fatigue failure analysis
207
Table 2. Summaryof node 2888 results Reference temperature (°F)
;tress (kpsi) diner's life* (cycles) danson's life* (cycles) diner's lifet (cycles) ¢lanson's lifet (cycles)
diner's damaged life (days) danson's damaged life (days)
669.88 Load case 1
Load case 2
Load case 3
Load case 4
0.00
47.04 13,004 8887 9753 6289
21.53 93,757 80,288 56,409 46,480
33.69 695,386 650,567
SS sheet*
Weldt
1184 991
527 442
*Based on hot rolled surface finish. tBased on as-forged surface finish. 6. C O N C L U S I O N S This work suggests that thermal fatigue, exacerbated by mismatched thermal xpansion coefficients, is the probable cause of failure. The predicted life for the xisting exchanger, which is controlled by the 430 weld material, accurately predicts le actual failure assuming the weld material is the as-forged surface finish, i.e. 42-527 days c o m p a r e d to an actual failure life of approximately 500 days. The nalysis was based on the model staying in the linear-elastic regime, and material roperties such as thermal conductivity and thermal expansion remaining constant as ~e t e m p e r a t u r e increases. The thermal expansion coefficient for the inner and outer rings for the baseline esign are very nearly identical, which suggests identical growth/deformation. Howver, the circumferential growth of the outer ring will be greater than the inner ring mply due to the ring radius. This means a larger radial deflection will result, thereby ,creasing stresses at the intersection. The weld material in these areas must then be 91e to withstand this increased stress state. It is also observed that there is a roughly )% reduction in tensile strength at operating temperatures for the 430 SS. The weld Laterial in these areas must be able to withstand the increased stress state. In conclusion, the finite element m e t h o d is an extremely useful tool to help analyse totems prior to implementation or as a forensic engineering tool to determine why a ilure occurred. H o w e v e r , as with any tool, engineering judgement must be used and ',suits m a y have to be augmented with extra work to obtain the final answer. Such a tuation was demonstrated in determining the cause of a thermal fatigue failure. In Lis case, the original hypothesis of thermal growth mismatch causing the problem as found to be partially correct. The weld material used was the real culprit as it had sufficient strength at the operating t e m p e r a t u r e to withstand the thermal cyclic adings to which it was exposed. A simple solution to correcting the problem would to replace the 430 SS weld rod with a better matched material, such as 18SR SS eld wire.
REFERENCES J. E. Shigley and C. R. Mischke, Mechanical Engineering Design (5th edn), McGraw-Hill, New York (1989). J. A. Collins, Failure of Materials in Mechanical Design (2nd edn), Wiley Interscience, New York (1993). D. Peckncr and I. M. Bernstein, Handbook of Stainless Steels, Chap. 19, McGraw-Hill, New York (1959). Allegheny Ludlum Steel Corp., Stainless Steel Fabrication (1958). TCT Stainless Steel of Nashville, Inc., Material Specifications for Various Stainless Steels, vendor specifications.