Edge-of-contact stresses in blade attachments in gas turbines

Transcription

1 Surfae Effets and Contat Mehanis XI 87 Edge-of-ontat stresses in blade attahents in gas turbines G. B. Sinlair Departent of Mehanial Engineering, Louisiana State University, USA Abstrat By drawing on analytial solutions to ontat probles in the literature, an approah is developed for obtaining two-diensional, elasti, edge-of-ontat stresses in blade attahents in gas turbines. The approah is validated on a benhark proble. It uses stress resultants fro finite eleent analysis (FEA): stress resultants onverge ore rapidly than stresses theselves, so that suh FEA is less deanding. In addition to reduing FEA, the analytial approah reveals with expliit expressions the nature of edge-of-ontat stresses in blade attahents. In partiular in this regard, it identifies a hoop stress oponent right at the edge of ontat that is potentially daaging to blades. Keywords: two-diensional onforing ontat, ontat with frition, elasti ontat stresses. 1 Introdution At this tie, gas turbine engines are the nor for powering larger planes in oerial and ilitary use. In addition, land-based gas turbines have largely replaed reiproating engines as power generators. Essential to the suess of gas turbines in both roles has been the engineering of suffiient strutural reliability. A key aspet of this engineering effort has been the design of fan blade attahents to fan hubs or rotors. While the strutural integrity of these attahents has iproved with tie, failures do still our. The intent of the work reported here, then, is to offer a eans of gaining inreased understanding of the ritial edge-of-ontat stresses ourring in blade attahents in gas turbines: suh understanding an aid in ahieving yet better strutural integrity of attahents. ISSN (on-line) WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press doi:10.495/secm130081

2 88 Surfae Effets and Contat Mehanis XI A ross setion of a typial dovetail blade attahent used in airraft engines is shown in fig. 1. Therein the base of one blade in a fan is restrained fro oving radially outward (upwards in fig. 1) when rotating by ontat on two flats (C-C' in fig. 1) within its portion of the rotor. Attahents of this ilk are oon in the airraft industry. Fir tree attahents with ultiple pairs of ontating flats are also used in the industry, as well as in power generation. Figure 1: Cross setion of a dovetail attahent of a blade base to a rotor. With slipping on the ontat flats during loading up to peak rp, tensile edgeof-ontat stresses are produed in the blade and the rotor (at C and C', respetively, in fig. 1). With even inor variations in operating rp, these tensile stresses an flutuate signifiantly and thus initiate fatigue raks that ultiately lead to failure. Understanding these tensile edge-of-ontat stresses that are the root auses of these failures is of fundaental iportane in trying to design against failures. Finite eleent analysis (FEA) offers a eans of aking a deterination of the edge-of-ontat stresses in blade attahents under varying operating onditions. One of the attributes of FEA is its ability to siulate entire geoetries like that shown in fig. 1. Nonetheless, FEA faes soe hallenges in aurately resolving the loal edge-of-ontat stresses present. In the first instane this is beause these stresses have high gradients, indeed even infinite WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

3 Surfae Effets and Contat Mehanis XI 89 gradients right at the edge of ontat. In the seond this is beause the expanded ontat region within whih the peak ontat stresses our is itself quite sall (typially about 1% of the ontat flat at axiu rp). Thus FEA ust not only resolve stresses with severe gradients but also ake a suffiiently aurate deterination of the region within whih these stresses at. Over the years a nuber of papers have reported attepts to use FEA to apture edge-of-ontat stresses in blade attahents in gas turbines (see Sinlair et al. [1] for referenes). These finite eleent treatents are all within the ontext of two-diensional elastiity. The ost refined esh eployed has ontat eleents with extents that are of the order of 1/100 of the edge radius, itself a sall diension. While results fro this FEA are onsistent with a nuerial analysis that has onverged to within 4% for soe of the edge-ofontat stresses, for the ruial tensile stress onvergene reains probleati. Here, therefore, we adopt a different approah. We use FEA to its best advantage, naely siulating an entire geoetry like that of fig. 1, but then only require the FEA to deterine stress resultants on ontat flats. Being integrals of stresses, resultants are far less sensitive to severe stress gradients, and are also fairly insensitive to preise extents of ontat. Thereafter we use an analytial approah to obtain edge-of-ontat stresses. This analytial approah is based on an adaptation of the two-diensional, losed-for, elasti solution for a flat punh with rounded edges indenting an elasti half plane given in Shtaeran []. The intent of eploying this analytial approah is not only to iprove the resolution of edge-of-ontat tensile stresses but also to inrease understanding of edge-of-ontat stresses by furnishing soe losed-for expressions for key ontributions. In what follows, we begin in Setion by desribing in greater detail the speifi blade attahent used to deonstrate the appliation of the analytial approah. Thereafter, in Setions 3, 4 and 5, we develop the analytial approah used for the ontat stress then other edge-of-ontat stresses. To validate the approah, in Setion 6 we opare with what we believe to be the ost aurately deterined edge-of-ontat stresses found with FEA. We lose in Setion 7 with rearks on soe onsequenes of the approah in general. Contat onfiguration The ross setion of a dovetail blade attahent shown in fig. 1 is, in fat, the one hosen for deonstrating the analytial approah developed subsequently. This hoie is priarily beause this is the attahent that has been subjeted to the ost refined FEA we are aware of in [1]: in addition, we an obtain the neessary speifiations fro [1] for this attahent. The ross setion in fig. 1 is assued to be in a state of plane strain out of the plane of the figure. Furtherore, the fore F indued by the rotation of the blade is assued to be shared equally by the two ontating flats restraining the blade. Then the ross setion in fig. 1 is syetri about its enter line, and it suffies to onsider but one of the ontating flats. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

4 90 Surfae Effets and Contat Mehanis XI Figure : Contat geoetry and oordinate; loal stresses. The loal ontat onfiguration for the right ost restraining flat is shown in greater detail in fig.. Therein L is the length of the ontat flat that is oon to both the blade and the rotor, and this flat is inlined at an angle to the horizontal diretion. At the ends of this flat there is a oon edge radius r, on the rotor at the upper end and on the blade at the lower. Under loading, atual ontat expands a sall aount onto these radii, shown sheatially as extending to C and C' in fig. : the horizontal distane between C and C' is denoted as the ontat length L. To desribe variations of stresses within L, a oordinate x aligned with the ontat flat and having origin O at the enter of the flat is introdued. Contat between the blade and the rotor extending over the flat in fig. is produed by a noral pressure ating in onert with a bending stress. For the first of these, the assoiated noinal pressure on the ontat flat is given by p = N L, N = F os sin, (1) where N is the resultant noral fore on a ontat flat and is the oeffiient of frition between the blade and the rotor. In eqn (1), slipping between the blade and the rotor has been assued (we review soe justifiation for this assuption subsequently in Setion 5). For the seond of these, the assoiated noinal stress is = x L, =6 M L, () WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

5 Surfae Effets and Contat Mehanis XI 91 where M is the resultant oent ating and is taken to be positive when adding to N at x = L/. For the dovetail attahent of fig. 1, F follows fro the geoetry and density of the blade, as well as the rotational speed at whih the fan operates. For M, on the other hand, analysis is required beause this oent is statially indeterinate. Sine M is a stress resultant rather than a stress, this analysis an be fairly readily perfored with D or even 3D finite eleents. (For soe other blade attahents, suh as fir tree attahents, F ay also have to be deterined using FEA.) Our general objetive here, then, is to deterine the stresses produed in the blade and the rotor under the ation of N and M that satisfy: the field equations of plane-strain elastiity, athed trations and noral displaeents within the ontat region, Aonton s law within the ontat region, stress-free onditions outside of the ontat region, and the usual ontat onstraints. These last require that the ontat stress between the blade and the rotor be nowhere tensile, and that displaeents of the blade and the rotor do not let the two touh outside of the ontat region. In partiular, we are interested in the loal stresses at or near the edge of ontat. There are three stress oponents of prinipal interest in this viinity: the noral ontat stress, the opanion shear ontat stress, and the noral hoop stress h. These are shown ating in their adopted positive senses in the lose-up in fig.. Thus is positive when opressive, whih it always is, while h has the ore noral sign onvention of being positive when tensile beause it an be both tensile and opressive. The atual dovetail attahent in [1] is one used by General Eletri Airraft Engines and is ade with a titaniu alloy. For this attahent, -1 r L = 7 5, = tan (3) As in [1], we onsider two liiting ases for frition effets: =0.0 and =0.4. The first of these is for the iniu oeffiient for fritionless operation, while the seond is for the axiu oeffiient estiated to our in dovetail attahents when they are ade with the titaniu alloy (see Hady and Waterhouse [3]). Then, as in [1], orresponding loading has for =0.0, 0.4, defined by -3-3 p E = 5.66 x 10, 3.61 x 10, (4) respetively. In eqn (4), E is the ontat odulus and is 1-b 1- r E =, Eb Er -1 (5) where E b and b are the Young s odulus and Poisson s ratio for the blade, E r and r like oduli for the rotor. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

6 9 Surfae Effets and Contat Mehanis XI 3 An analytial approah for the ontat stress absent bending effets Here we adapt the analytial solution of Shtaeran [] so as to furnish estiates of the ontat stress in fig. produed by the noinal pressure p alone. Initially we do this for the fritionless ase: later, in Setion 5, we disuss the effets of frition. A ross setion of the plane-strain ontat onfiguration analyzed in Shtaeran [] is shown in fig. 3(a). Therein a rigid sooth punh is pressed by a noral fore into an elasti half plane. The punh has a flat base with rounded edges: we take the extent of the flat to equal L and the edge radii to equal r in oon with the ontat onfiguration of fig.. Then we take the agnitude of the reote fores pushing the two together to be N in order to have a noinal pressure of p on the ontat flat. The half plane has a Young s odulus E and a Poisson s ratio. In fig. 3(a), we ontinue to eploy an x oordinate arranged as earlier in fig.. By superiposing displaeents fro the Flaant line load on an elasti half plane, the ontat stress ating in fig. 3(a) an be shown to satisfy the integral equation 1- I x= up x, (6) E for x < L, where L is now the ontat extent in Shtaeran s proble (L > L). In eqn (6), I is the integral resulting fro the superposition of line loads, and u P is the displaeent produed in the half plane by the punh profile. Thus fro, for exaple, Johnson [4], p. 17, I x L - x- = ln d, (7) L -L and u P, using the approxiation of Hertz [5], is given by for x L, up x = -x- L 8r for L < x L, (8) wherein is the depth of penetration of the punh flat. In eqn (6), it is understood that ust be positive to be in opliane with the ontat onstraint requiring ontat stresses be opressive. Further in eqn (6), L is to be adjusted so that the half-plane displaeents outside of the ontat region do not touh the punh and so are in opliane with the seond ontat onstraint. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

7 Surfae Effets and Contat Mehanis XI 93 Figure 3: Shtaeran ontat onfigurations (a) original onfiguration, (b) odified equivalent onfiguration. To adapt this integral equation so that it applies to the ontat onfiguration of fig., we first let the half plane repliate the blade and hene replae E and by orresponding values for the blade, E b and b. Next we let the punh be WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

8 94 Surfae Effets and Contat Mehanis XI deforable with oduli E r and r. Then the displaeents produed in punh are to be subtrated off u P on the right-hand side of eqn (6). Following Steuerann [6], we let these displaeents also be represented by those attributable to the superposition of half-plane line loads. Aordingly, with both of these odifiations, eqn (6) beoes 1-1- r I x= u x- Ix, E b P b Er for x < L. That I(x) on the right-hand side of eqn (9) reains the sae as in eqn (7) is a onsequene of assuing the sae Green s funtion for the half plane and the punh, together with the fat that both are ated on by the sae ontat stress. The assuption of a oon Green s funtion, as well as the Hertz approxiation used in eqn (8), are only reasonable provided the expansion of the ontat area does not progress too far around the edge radii. In Persson [7], an assessent is ade of the extent that ontat an extend and have to two siplifiations reains reasonable. Soewhat surprisingly, [7] finds this to be the ase provided ontat regions do not extend horizontally ore than 40% of ontat radii. Beause we expet lateral ontat extensions to be less than this for the present lass of ontat probles, the two siplifiations an be expeted to be reasonable here. We do, though, hek that this is indeed the ase subsequently. Returning to eqn (9) and rearranging it by plaing both I ters on the lefthand side, we see that the only hange to the original integral equation for Shtaeran s proble, eqn (6), is that the oeffiient of I, 1- E, is replaed -1 by E where E is the ontat odulus of eqn (5). It follows that the only hange needed to be ade to Shtaeran s solution for eqn (6) so that it applies to a deforable punh and half plane respetively having the oduli of the rotor and the blade is the aforeentioned exhange. Finally in adapting the onfiguration of fig. 3(a) to that of fig., we note that the punh profile u p an equally well be interpreted as the aount the indentor would overlap or interpenetrate the half plane were it not for the opensating displaeents in eqn (9). With this interpretation, it is of no onsequene whether u p oes fro interpenetration of the punh into the half plane or vie versa. For that atter, the punh ould interpenetrate the half plane for a portion of the ontat region and the half plane interpenetrate the punh for the reainder and eqn (9) would reain unhanged. Hene eqn (9) holds in effet for the onfiguration of fig. 3(b), and therefore the Shtaeran s solution for eqn (6) applies for the ontat stress in fig. provided the noted oduli exhange is ade. Aordingly for this ontat stress, fro Shtaeran [], os os EL sin sin -sin = sin ln, (10) r sin - sin sin for x L, where (9) WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

9 Surfae Effets and Contat Mehanis XI 95 x L -1-1 =os, =os, = L L. (11) To quantify using eqn (10), the ontat length L needs to be deterined. Also fro Shtaeran [], this an be effeted by solving 8 pr = E L -sin (1) for, hene and L. This transendental equation an be readily solved nuerially. By way of exaple, for the edge radius of eqn (3) and the fritionless loading of eqn (4), solving eqn (1) with the seant ethod yields = (13) This orresponds to L r L = L -L is the lateral ontat extension for a single end of the ontat region. This is arkedly less than the perentage assessed as aeptable in Persson [7]. With of eqn (13) and hene of eqn (11), of eqn (10) an be evaluated when noralized by p. This leads to the ontat stress distribution shown in fig. 4(a) for 0 x L (the distribution for x <0 is siply a refletion beause of eqn (10) is syetri about x = 0). This stress distribution features a sharp peak within the ontat region extension onto the rounded edge. By differentiating eqn (10) with respet to x and setting d dx =0, the loation of the axiu ontat stress,, an be found. Thus = =, where results fro solving equal to 8%, where sin sin - ot = ln, (14) for <. This transendental equation an also be readily solved nuerially. For the sae fritionless ase as for of eqn (13), this leads to = Using eqn (13), this orresponds to x L = , (15) where x is the loation of. That is, the axiu ontat stress ours but 1/3 of a perent of the length of the ontat flat outside of this flat. At this loation, eqn (10) has p = (16) WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

10 96 Surfae Effets and Contat Mehanis XI Figure 4: Representative ontat stress =0 half of the ontat region; (b) edge-of ontat stress. : (a) stress distribution over The stress distribution near this axiu is shown ore learly in the lose up of this viinity in fig. 4(b). WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

11 Surfae Effets and Contat Mehanis XI 97 Inluded in fig. 4(b), starting at the edge of ontat x = L /, is a loal Hertzian ontat stress distribution sharing the sae peak stress as. After Hertz [5] for a ylindrial roller, this stress is given by x where = - for 1, x x x x = 1- H x, (17) and x = L - x. The Hertzian stress of eqn (17) agrees with for 0 x 1 orresponding to x x L to within 1%. This is not surprising sine and H share both the sae stress values and the sae stress gradients at x = x and L. We take advantage of this agreeent between the two later when we asseble ontributions to the hoop stress, h. 4 An adjustent to the ontat stress to reflet bending effets At the outset in attepting to inorporate the effets of bending, we assue that these are not of a suffiient agnitude to negate the opressive nature of the ontat stresses produed by unifor pressures and thereby lead to lift off and separation in the ontat region. For flat punhes with sharp edges Gladwell [8], p. 74, has that suh will be the ase if of eqn () is liited by p <3. (18) In lieu of any ore appropriate liit, we adopt that of eqn (18) in what follows. Fortunately for the dovetail attahent in [1], while noinal bending stresses are appreiable, they are not lose to the liit iposed by eqn (18). For this attahent, Sinlair and Corier [9] reports that p = 0.36, 0.58, (19) for =0.0, 0.4, respetively. Furtherore, for fir tree attahents, noinal bending stresses are relatively odest, and so also in opliane with eqn (18). Unfortunately for punhes with rounded edges as in fig. 3(a), the bending ounterpart to eqn (10) would not appear to be available in the literature. There are, however, analytial solutions for for both p and when the punh is opletely flat with sharp edges. For p, is given in Sadowsky [10], while for, is given in Gladwell [8]. Aordingly we next try to take advantage of these flat punh solutions to arrive at an estiate of the effets of additional bending ontributions on ontat stresses for punhes with sall radii rounding their edges rl 15. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

12 98 Surfae Effets and Contat Mehanis XI Fro Sadowsky [10], when a sooth flat punh of width L is pressed by a unifor pressure p into an elasti half plane, the ontat stress produed is given by p f =, 1- xl f (0) for x <L, where x ontinues to be as in fig. 3(a). Now L = L beause of the sharp orners, and stress singularities our at the edges of ontat. Fro Gladwell [8], the orresponding result for the sae onfiguration but now with ating on the punh is f where f 4 x =, 3 1- L xl (1) for x <L. Now it is assued that ats in onert with suffiient p that ontat is preserved over x <L, and again L = L. For this to be the ase, f should be less that f : this is so when eqn (18) holds. The ontat stress of eqn (1) shares the sae stress singularity as its predeessor in eqn (0). Hene the quotient f f has a finite liit as x L. If we let f denote the peak stresses in the viinity of x = L for p alone and f the orresponding stress for p in onert with, eqns (0) and (1) give f f =1. 3p () Thus the right-hand side of eqn () is the fator that peak ontat stresses for flat punhes are inreased by as the result of bending ontributions. The ontat stress of eqn (0) for the punh with sharp orners agrees with its ounterpart of eqn (10) for the punh with orners rounded in aord with eqn (3) to within 3% over the entral 90% of the ontat region. Outside of this entral region, signifiant differenes in ontat stresses our for the two punhes. However, beause the stresses agree for the entral 90%, the two punhes ust share oon fores ating thereon. Hene, sine they share oon total fores, they ust also share oon fores ating on their outer regions. It follows, therefore, that the fator in eqn () reflets the inrease in applied fores ating on the outer ontat regions as a result of bending effets. For the Hertzian ontat of a ylinder, inreasing the applied fore or p has the peak ontat stresses, H, inrease proportional to p 1/ (see, e.g., [4], p. 47). This is beause inreases in p are itigated to a degree by expansions in the WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

13 Surfae Effets and Contat Mehanis XI 99 ontat region. Thus if we aept the fator in eqn () as being that for inreasing load with bending, we would have H H 1 = 1, 3p (3) where H is a notional peak Hertzian ontat stress with bending present. While the outer edges of a punh with rounded orners do have a siilar stress distribution to that for Hertzian ontat of a ylinder, inside the oon peak value of this stress the stresses for the two differ appreiably. In part this is beause the punh with rounded orners only enjoys stress relief fro expanding ontat fro expansions on one side of the peak ontat stress. Aordingly we an expet the fator for inreasing stresses due to bending for a punh with rounded orners to lie soewhere between that of eqn () for punhes with sharp orners and that of eqn (3) for ylinders. In an attept to obtain this intervening fator, we trak fro eqns (10), (11), (1) and (14) as p varies throughout the range given in eqn (4). We find that is onsistently proportional to p /3 over this range. Consequently for punhes with rounded orners we take 3 = 1, 3p (4) where is when ats as well as p. Then, for the noinal stress ratios of eqn (19), we have = 1.154, 1.44, (5) for = 0.0, 0.4, respetively. Thus with bending the axiu ontat stress of eqn (16) is inreased to p =6.30, (6) when there is no frition. We give the orresponding inreases for when frition ats after we disuss ontat shear stress influenes in the next setion. One is deterined, paralleling eqn (1), for the entire ontat stress distribution with bending present, we take x = 1-1, L (7) for x L, with reaining as given by eqn (10). WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

14 100 Surfae Effets and Contat Mehanis XI 5 An analytial approah for the other edge-of-ontat stresses Here we first ake a deterination of the shear ontat stress,, for the onfiguration of fig.. Thereafter we obine the effets of with those of and F in aking a deterination of the hoop stress, h. In Sinlair and Corier [9], a siple ork odel indiates that attahents like that of fig. 1 an be expeted to slip as loading up oenes if tan >. (8) Equation (8) is ertainly oplied with here (see eqn (3)), and this is typially the ase for other blade attahents. In [1], the FEA finds the blade in the attahent of fig. 1 not only to start to slip during loading up, but also to ontinue slipping throughout loading. Here, then, we assue this to be the ase. However, this is an assuption that really needs to be onfired for other attahents, soething that is fairly readily done with FEA. (Griffin and Cushan [11] reports the sae finding of slipping throughout loading for a firtree attahent.) Under the assuption of slipping, Aonton s law applied at the stress level has =, (9) for x L when there is no bending present. With bending, eqn (9) siply beoes =, (30) for x L, where is the shear stress with ating as well as p. Soe justifiation for the appliation of Aonton s law at the stress level rather than for fores is given in Johnson [4], on p. 04. Unfortunately there would not appear to be a solution in the literature for for punhes with rounded orners when eqn (9) holds. There is, though, a solution for a punh with sharp orners when eqn (9) holds given in Muskhelishvili [1] on p Beause of the sharp orners, this solution has stress singularities at the edges of ontat. Moreover, beause of the different boundary onditions involved, these singularities do not share the sae singularity exponent as punhes with sharp orners absent frition (see eqn (0)). Hene it is only appropriate to opare ontat stresses for punhes with sharp orners and with and without frition well away fro the edges of ontat. With this restrition, it is found that the ontat stresses with and without frition are quite siilar. More preisely for = 0.0 and 0.4 when x < L 4, differenes in are less than 4%. Aordingly here we hoose to ignore any fritional WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

15 Surfae Effets and Contat Mehanis XI 101 effets on ontat stress distributions and take of eqn (10) to apply in eqn (9), of eqn (7) in eqn (30). On the other hand, with frition there are differenes in applied noinal pressures beause soe of the fore F is now balaned by shear stress resultants. These enter through eqn (1) and result in the redued p with frition given in eqn (4). For this p, eqn (1) leads to = and an aeptable L r of 6%. Then eqns (14), (10) and (5) give p =6.35, p =7.90, (31) for =0.4. For the hoop stress h, there are three ontributions: noinal stresses fro the fore F of fig. 1, stresses produed by the ontat stress, and stresses generated by the shear stress. We treat eah of these in turn. Balaning vertial fores for unifor noral stresses in retangular and ylindrial polar oordinates, when the two oordinate systes share a oon origin at the intersetion of the projetions of the ontat flats in fig. 1, diretly yields an estiate of the noinal hoop stress, hn, at the edge of ontat. Thus = F w, 3) hn for x = L /, where w is the width of the blade ross setion at C (fig. 1). For loations iediately above the edge of ontat, w in eqn (3) is redued and the noinal hoop stress beoes F hn =, (33) w L -x os wherein now the plus sign indiates x slightly in exess of L /. Conversely, within the ontat area, w is inreased. In addition, F is redued by virtue of being inreasingly balaned by fores on the ontat flats. Beause these noinal stresses transpire to be an order of agnitude less than the oplete hoop stress, we just odel these redutions in F with a linear fit, and take - F L x hn =, L w L -x os (34) for x < L, as the inus sign on hn indiates. Contat stresses siply produe exatly the sae lateral surfae stresses in D elastiity. This is a onsequene of Mihell s solution in [13] for the strip loading of an elasti half plane that has the sae equality of noral stresses in WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

16 10 Surfae Effets and Contat Mehanis XI the surfae. Hene, absent frition, the hoop stress ating in fig. for the dovetail attahent of fig. 1 is given by for x > L, = hn h - - for <, hn x L (35) and it is understood that x only slightly exeeds L / for the upper result. When the shear stresses attending frition at in the positive sense shown in fig., they generate tensile hoop stresses at C, opressive at C'. These stresses an be obtained by superiposing stresses fro the Boussinesq horizontal line load on an elasti half plane. Thus fro, for exaple, Johnson [4], p. 19, and eqn (30), the additional hoop stress, h, generated by fritional shear stresses is given by L h = d, x - (36) is -L where of eqn (7) with replaing x. With a view to larifying the key ontribution steing fro the integral in eqn (36), we exhange for H - H, where H is H of eqn (17) with replaed by. Then the integral involving H an be evaluated analytially: see Poritsky [14]. Denoting the hoop stress fro this integral by we have, H x x x H = x for x < 1, - -1 for > 1, (37) where x reains as in eqn (17). Hene in onjuntion with eqn (35) we have, for the edge-of-ontat hoop stress distribution with frition present, h = - x h - hn for x 1, - -1 x x h hn for 1 x, (38) where L d = - H - x x, (39) x - h H -L and H is the Heaviside unit step funtion. What eqn (38) then shows is that h has an infinite positive gradient as x 1, x > 1, fro the first ter in the upper line, as well as an infinite negative gradient as x 1, x < 1 orresponding to WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

17 Surfae Effets and Contat Mehanis XI 103 x L, x < L, fro of eqn (10) whih ontains sin = 1- xl. It follows that the axiu value of the hoop stress ours right at the edge of ontat and is given by h h hn = x = L, (40) where hn is as in eqn (3). For the frition ase of eqn (4), we opute these various ontributions to h (using a idordinate rule in onjuntion with singularity prograing to alulate the Cauhy prinipal value of the integral for hn in eqn (39)). Noralized by p, the following results are given for the ters in eqn (40) in the order whih they our there: p = = (41) h As expeted fro the nature of the expressions in eqn (38), the first ter in eqn (41) is the ajor ontributor to the peak hoop stress. 6 Validation of the analytial approah The preeding analytial approah akes a nuber of siplifying assuptions. Aordingly we seek to evaluate the validity of this approah by oparing with an aurate finite eleent deterination of orresponding stresses. To this end we hoose the FEA of Sinlair et al. [1] beause there, by refining eshes suffiiently, onverged ontat stresses for the dovetail attahent of fig. 1 are reported. The FEA in [1] eploys four-node quadrilateral eleents in onert with point-to-surfae ontat eleents (PLANE4 and CONTAC48, ANSYS [15]). For these eleents, one ontat extents have been deterined, errors in stresses, e, behave like e = Oh as h 0, (4) where h is the eleent size in the viinity of where is being deterined. The initial esh ( = 1) used in [1] features signifiant esh gradation away fro the ontat region, but nearly onstant eleent sizes within the viinity of the ontat region with h equaling about 16% of the edge radius r. This esh is refined by halving eleent extents to produe two further eshes ( =, 3). Thereafter, to redue oputational effort, the subodelling tehnique of Corier et al. [16] is eployed in the neighborhood of C in fig. 1 wherein the axiu ontat stresses our. This results in three additional eshes ( = 4-6) with eleent sizes ontinuing to be halved. For the first three eshes, ontat extents vary, as do the loations of peak ontat stresses. Consequently ontat stresses are not expeted to have WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

18 104 Surfae Effets and Contat Mehanis XI onverged on this first sequene of eshes, and indeed, as is apparent in what follows, this proves to be the ase. Nonetheless, the finest of these three eshes furnishes the best estiates available of the relative ontributions of bending to peak ontat stresses beause suh estiates are not available fro the subodel eshes sine they fous on the ontat stress at C alone. For this esh ( = 3) then, averaging peak ontat stresses at C and C' and dividing into the peak at C by itself furnishes = 1.,1.3, (43) for = 0.0, 0.4, respetively. Despite underlying ontat stresses for the ratios in eqn (43) not being onverged, the ratios are quite oparable to their ounterparts with the analytial approah in eqn (5) (within 5%). On the three subodel eshes, ontat extents in [1] are adjusted by suessively saller aounts with esh refineent and appear to have onverged with ontat extensions being about 1% of the extent of the ontat flat orresponding to = This is also quite oparable to results for the analytial approah (e.g., eqn (13)). Further, the loations of the peak ontat stresses reain fixed on all three subodel eshes and oinident with that for = 3 (see figs 4(b) and 5(a), [1]). Hene on this entire sequene of four eshes a reasonable deterination of the extent to whih the FEA has onverged is possible. Table 1, therefore, inludes the peak ontat stresses for = 3-6 fro [1] (speifially, fro Table thereof on renoralizing). Table 1: Finite eleent results for edge-of-ontat stresses. Mesh, Mesh size, h/r p =0.0 p =0.4 h p = / / / / (8.64) Corresponding analytial value: WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

19 Surfae Effets and Contat Mehanis XI 105 For the peak ontat stresses in Table 1 for = 0.0, stress inreents suessively halve with esh refineent, or nearly so. Suh nuerial results are onsistent with a linearly onverging analysis. Thus Rihardson extrapolation [17] for linear onvergene an be applied and erely adds the last stress inreent to the last stress ( = 6) to get an extrapolated value. These values loosely orrespond to or h 0, and are inluded in Table 1. The sae situation obtains for eshes = 4, 5, 6 for = 0.4, and extrapolated values are likewise inluded. With linear onvergene, disretization errors in stresses do behave as in eqn (4). Hene perentage errors for esh results,, an be estiated by = x 100 %, (44) where = - -1, being the FEA deterination of the stress of onern on esh. Applying eqn (44) to the results in Table 1 for gives 6 =1.6%, 1.8%, for = 0.0, 0.4, respetively. All told, therefore, it would see reasonable to regard the FEA results of [1] for as being of good auray. For oparison, orresponding analytial results for fro eqns (16) and (31) are also inluded in Table 1. The analytial result for for = 0.0 agrees well with the extrapolated FEA result (within 1%). The analytial result for = 0.4 agrees satisfatorily with the extrapolated FEA result (within 6%). The greater disagreeent for the latter ase is perhaps to be expeted beause the presene of frition inreases the oplexity of the ontat onfiguration and onsequently results in additional siplifiations in the analytial approah (see Setion 5). It also hallenges the FEA ore: despite the FEA indiating that slip ours throughout the ontat region, peak shear ontat stresses diverge fro satisfying Aonton s law by 3.3% in [1], so that perhaps errors in for = 0.4 are soewhat larger than the 1.8% estiate ade using eqn (44). In all, therefore, at least a satisfatory validation of the analytial approah. While not part of the validation of the analytial approah, FEA hoop stresses are in addition inluded in Table 1 to illustrate the hallenges of deterining this stress right at the edge of ontat with finite eleents. In ontrast to ontat stresses, peak hoop stress loations are a never oinident throughout the esh sequene (see fig. 5(b), [1]), and only for = 6 our at the orret loation (x = L ). Consequently stress inreents vary erratially rather than systeatially and are not in agreeent with inreents for linear onvergene. Beause these inreents are dereasing in agnitude, it is possible to apply generalized Rihardson extrapolation as in Roahe [18], on p This leads to the value inluded in parentheses in Table 1: parentheses are used beause experiene has shown that suh large hanges with extrapolation an be unreliable. Rather, instead, we use the inreents to estiate the error with a proedure that WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

20 106 Surfae Effets and Contat Mehanis XI reognizes errati onvergene. In aord with Roahe [18], p. 155, this proedure has, for esh refineent halving eleent extents, where =ln = x 100 %, (45) -1-1 ln is an estiate of the effetive onvergene rate. Applying eqn (45) to the results in Table 1 for h gives 6 =33%. Alternatively, if one were to take the analytial value fro eqn (41) given in Table 1 as being aurate, 6 =4%. Either way, the error level in is not satisfatory. There are reasons why the FEA deterination of an be aurate but that while it is near an infinite stress gradient at x = L, at x for h not be. For, = x the atual stress gradient in is zero. In ontrast, the stress gradients for h at its peak loation, x = L, are infinitely positive then infinitely negative as L is approahed fro above and below (see Setion 5). These severe gradients seriously ipair the deterination of with finite eleents. For peak hoop stresses, therefore, the analytial approah ay offer a signifiantly ore effetive eans of deterining suh stresses than FEA. 7 Conluding rearks h Drawing on the ontat stress solution in Shtaeran [], an analytial approah is developed for blade attahents in gas turbines. The approah enables the FEA of suh attahents to deterine just stress resultants, quantities that onverge ore rapidly than stresses theselves. Given the appropriate resultants, the approah then furnishes key edge-of-ontat stresses. The analytial approah eploys a nuber of siplifying assuptions. To hek the validity of these assuptions, peak ontat stresses are opared with an FEA that has onverged orresponding stresses. Agreeent between the two is quite satisfatory (differenes < 6%). Results for the edge-of-ontat stresses show that signifiant stress onentrations our near the edge of ontat. The ost potentially daaging of these stress raisers is a tensile hoop stress that ours right at the edge of ontat. This stress is likely to be the ost daaging beause, as the analytial approah expliitly shows, it has severe stress gradients when approahed fro either inside or outside of ontat. As a result, sall hanges in operating rp and noinal stresses produe relatively large hanges in this hoop stress as the ontat region shifts. Fousing, therefore, on this key hoop stress at the edge of ontat, there are several options for reduing its severity. Shot peening an be used to redue its agnitude by adding a opressive ontribution. Choosing aterials that are h WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

21 Surfae Effets and Contat Mehanis XI 107 otherwise equivalent in ters of strengths in the turbine environent but have a lower oeffiient of frition lower this stress largely proportionally. And preision rowning as in Sinlair and Corier [19] offers a potential eans of opletely reoving peak hoop stresses at the edge of ontat. Aknowledgeent The benefit of tehnial disussions with Dr J. H. Griffin of Blade Diagnostis Corporation during the ourse of preparing this study is gratefully aknowledged. Referenes [1] Sinlair, G.B., Corier, N.G., Griffin, J.H. and Meda, G., Contat stresses in dovetail attahents: finite eleent odeling. Journal of Engineering for Gas Turbines and Power, 14, pp , 00. [] Shtaeran, I.Y., Contat Probles of the Theory of Elastiity, Gostekhizdat Publishing: Mosow, 1949 (in Russian: an English translation is available fro the Foreign Tehnology Division of the British Library, FTD-MT , 1970). [3] Hady, M.M. and Waterhouse, R.B., The fretting wear of Ti-6Al-4V and aged Inonel 718 at elevated teperatures. Wear, 71, pp , [4] Johnson, K.L., Contat Mehanis, Cabridge University Press: Cabridge, U.K., [5] Hertz, H., On the ontat of elasti solids. Journal für die Reine und Angewandlte Matheatik, 9, pp , 188 (in Geran: for an aount in English, see [4], Ch. 4). [6] Steuerann, E., To Hertz s theory of loal deforation in opressed elasti bodies. Coptes Rendus (Doklady) de l Aadéie des Sienes de l URSS, 5, pp , [7] Persson, A., On the stress distribution of ylindrial elasti bodies in ontat. Dotoral dissertation, Chalers Tekniska Hogskola, Göteborg, Sweden, [8] Gladwell, G.M.L., Contat Probles in the Classial Theory of Elastiity, Sijthoff and Noordhoff: Alphen aan den Rijn, The Netherlands, [9] Sinlair, G.B. and Corier, N.G., Contat stresses in dovetail attahents: physial odeling. Journal of Engineering for Gas Turbines and Power, 14, pp , 00. [10] Sadowsky, M.A., Two-diensional probles of elastiity theory. Zeitshrift für Angewandte Matheatik und Mehanik, 8, pp , 198 (in Geran: for an aount in English, see [4], Ch. ). [11] Griffin, J.H. and Cushan, M., Private ouniation, 01. [1] Muskhelishvili, N.I., Soe Basi Probles of the Matheatial Theory of Elastiity, P. Noordhoff Ltd.: Groningen, The Netherlands, [13] Mihell, J.H., The inversion of plane stress. Proeedings of the London Matheatial Soiety, 34, pp , 190. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)

22 108 Surfae Effets and Contat Mehanis XI [14] Poritsky, H., Stresses and defletions of ylindrial bodies in ontat. Journal of Applied Mehanis, 17, pp , [15] ANSYS personnel, ANSYS User s Manual, Revision 5., ANSYS Inorporated: Canonsburg, Pennsylvania, [16] Corier, N.G., Sallwood, B.S., Sinlair, G.B. and Meda, G., Aggressive subodelling of stress onentrations. International Journal for Nuerial Methods in Engineering, 46, pp , [17] Rihardson, L.F., The approxiate arithetial solution by finite differenes of physial probles involving differential equations, with an appliation to the stresses in a asonry da. Philosophial Transations of the Royal Soiety, A10, pp , [18] Roahe, P.J., Fundaentals of Verifiation and Validation, Herosa Publishers: Soorro, New Mexio, 009. [19] Sinlair, G.B. and Corier, N.G., Contat stresses in dovetail attahents: alleviation via preision rowning. Journal of Engineering for Gas Turbines and Power, 15, pp , 003. WIT Transations on Engineering Sienes, Vol 78, 013 WIT Press ISSN (on-line)