REPORT 952

DIRECT METHOD (II? DESIGN AND STRESS ANALYSIS OF ROTATINGDISKS WITH TEMPERATURE GRADIENT

By S. S. MAXSOX

SUMMARY

.4 methodis presentedfor the.determination oj the contour ojdi~ks, typijied by those of aircrajl gas turbines, to incorporatearbitrarye[astic-stressdist~”butionsrecsulting.frome{ther centrif-ugal or combinedcentrifugal and thermalefects. The ~peci~ed&resse8may be radial, tangential, or any combinationof the ttco.For exa.m.pie,the equivalent8tre8smaybe ~pen~ed a.teachradiu8ha8much a8 thi8 stre88i8 genera[[y used a8 the criterion of th?occurrence of plasticjlow. We is made of thefinitedifferenceapproach in 8oltiingthe stressequationalthe amount of com.puta--tiomnece8sary in the eco[ution of a de8ign being greatly reducedby the judicious 8election of point @ation8 and by the aid of ade8ign chart. Use of the chartaand of a preselectedschedule ofpoint stations i8 also applied to the direct problem of~nding thee[a8tic- and p[astic-8tre88di8W3ution.in di8k8of a g?len design.,thereby eJecting a great reduction in the amount of ca[cu[ationcompared w-”th pret<ou8ig publi8hed method8. IUu8tratireezan@e8 are pre8en.tedto show computationalprocedures in thedetermination of a new design and in analyzing an exi8tingdesign for elastic ~tre88and for stre8se8resultingfrom plastic&uL

INTRODUCTION

The important role of the disk in the gas turbine hasdirected much attention to the ana.Iysis of stresses in rotatingdisks with temperature gradienk Early in the development.of stress analysis of steam turbines, StodoIa (reference 1)presented the basic equation for stresses in such disks, aswell as a limited approach toward the solution. Morerecent ly, in connection with the development of the gasturbine for aircraft propukion, the problem of determiningtherms.I stresses resulting from temperature gradient hasbeen actively in-wstiga.ted. Several methods have beendeveloped (references 2 to 5) that can be used to calculatethe stresses in a given disk due to different ia.1 expans-ion and to wriation in physical properties of variousparts of the disk operating at diHerent temperatures. Thus,for disks already designed, it. is readiIy possible to determinewith a high degree of accuracy the centrifugal and thermt-dstresses resulting from a given speed and temperaturedistribution.

Little attention has, however, been directed towards theproblem of designing the profile of a disk, -which, when oper-ated at a given speed and temperature distribution, -iviIIincorporate a given total-stress system-centrifugal plus

thermal. A direct method of design for prescribed totaJ-stress systems was de~eIopecI at the lNACA Lewis laboratoryduring 1948. AIthough the design method can readily beextended to include plastic flow-, in the interest of simplicityonly eIastic stresses are considered.

The specified stresses may be either radiaI or tangentialw-dues, m is common in engineering practice, or the speci-fication may refer to any combination of these stresses. Forexa.mpIe, specification of the equivalent stress ah each racliusmay be desired inasmuch as this stress is frequently used asa criterion of the occurrence of pIastic flow. The method isbased on the finite-difference equations derived in reference 3and makes use of several simpIe design charts, presentedherein, that facilitate the calculations. An illustration ispresented for tracing t-he actual steps that may be followedin evoIving a design.

A simplified method of analyzing a.n e.sieting disk forcentrifugal and thermaI stress is also described. The methodis especially useful in checking the stresses in a disk designedfor prescribed stress distributions, particularly if minordeviations from the design profile are introduced for practicalreasons, or if creep or plastic flow, not considered during the _____design stage, are IikeIy to occur. Applications to arbitrarydisks are equa.Uy -did. EssentiaIIy, the method is anadaptation of the finite-difference approach of references 3and 4, the principaI modifications lying in the use of a pre-selected distribution of point stations and the addition ofse=reraI alinement charts to faciIit ate the computations.

AMhough the methods presented are of special advantagein cslctiations invoIving thermal stresses, they are suitabIefor disks subjected solely to centrifugal effects. Simpleequations may be obtained both for design or for stresstwdysis of rotating disks by setting all terms involv@ tem-perature equaI to zero. The subject of design for prescribedcentrifugal-stress distribution has likewise received onIyIimited attention (examphx of vdich maybe found in refer-ences 6 and 7); the presenb method may therefore satisfy nneed in this fieId of design.

DIRECT METHOD OF DISK DESIGN

Equations,-The equations for strees analysis of rotatingdisks are (reference 3)

$ (rhuJ–hu,+-P&hr2=0 (1)

103

-.. i

104 REPORT 952—hTATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

and

$(3)-W%)+: ‘“AT’-(l+PIY)=” ‘2)

(All symbols are defined in .th~..appendix.)Equation (2) is a statement.of the relation that must exist

between the radial and tangential stress dktributions,regardless of the shape of the disk, and therefore does. ..motco~ain the disk thickness .&A.Once the st.re.sses,.lmve beenchosen to be compatible in accordance with equation (2), hcan be determined from equation (1).

Solution of compatibility equation for specified stresses,.—A finite-difference method for the solution of equations.(1)and (2) is presented in reference 3. choosing dkcrete pointstations along R radius permits the stresses fit tha.nthstation to be expressed in terms of the .stressc~ at the (n-l)ststation. Equation (2) in finite-difference form becomes

C’. U,,.–U,, n=FnU,,n_l,Q_U,,’. U,,._l+II’. (3)

The coefficients C’., D’,, and so forth are, in general, com-plicated expressions involving the radii T. and m-l at thent,h and (n-l)st stations as well as the properties of thedisk material at the two stations. A great simplificationin the expressions results if the radii r% and r~_l me notchose-n at random, but are chosen according to a “preselected”schedule, for marnple, according to the relation

‘n–T’-1=o.200rm

If, in addkion, Poisson’s ratio ~ is assumed constant andequal to 0.333, equation (3) reduces .t.o

1.133 r,,,– O.466 C,,n=: (0.833 u,,.-1—”.166 Ur,m-1) —~~~’.n-l

=Zn (4)

Equation (4) is useful in the cenfi~al region of the diskwhere values of rmare relatively low. In orde.r to obtainreasonable spacing of the point stations in the rim regionwhere the radii are large, it ie desirable. to choose the spacingaccording to the relation

‘n–rn-’=o.050T*

Equation (3) then becomes ..__

1.033 a,,m–O.366a,,m=& (0.965 gAm-l-0.298ur.m-1) -

EnH’==Zfi (5)

As an example, equation (4) is considered. If u,,.:,- anda: ,n-l are already established and if the temperatures at t.kenth and (n-l )st station are known, the value of Z, can bedetermined; then the only unknowns are u,,. and u~,n. “Theequation is linear in a~,~ and ui,~ so that whatever are theactual values, a plot of w,,. against Ut,ti results in a straightline defined by equation (4). In order to establish theindividual values of u,,, and al,n, another condition must be

specifiecL In some designs, specification of either t-he radialor the t.angent.ial stress maybe desired; hence, the unspecifiedstress may then be. directly determined from equation (4) orfrom the corresponding straight-line pIot, A disadvantage ofstarting a design by specifying the radkl or tangential st.rcssis that.. the unspcc.ified stress mrty be too Klgh or in somoother way unfavorable. It maybe more desirable to specifythe equiva.Ient stress u,, which is defined by the equation

.—

u~,n= Uf,~~—ur,Bal,n+cr@* (6)

This stress is usually used as the criterion for pIastic-Jlowcalculations (reference 4) and is therefore a logical designparameter. Furthermore, specification of the cquivrdent.stress insures that neither the ra~lal nor the hmgent.ial stresswill be .tiexpectedly high. A plot of a~,n against. at,n for agiven value of Cc,mresults in an ellipse inclined a,~450 to thecoordinate axes. The intersection of the Iinc defuicd byequation (4) and the e.llipsc.of equation (6) defines the specificvalues of. a,,, and r,,.. Although, in general, a straight. lineintersects an ellipse at two points, the. propw point” fordesign use is usually obvious from the physicrd nature of thoproblem.

Design chart .—A design chart based on equations (4) rmd(6) is presented in figure 1(a). The stmight. lines correspondto different values of .Z. in equation (4) and t,lm ellipsescorrespond to different values of equivalent stress r~,ti. ThBchart is used to calculate. the stresses at the nth station t-ift crthe stresses have been ~etermined at the (n-l)st station.The value of .2&is calculated b-y equation (4) from conditionsat the (n-l)st st at.ion and the temperature at the nth station,thereby establishing the line along which u,,. and a~,. mustlie. Intersection of this line with the ellipse correspondingto the assigned value of. equivalent stress at the 71th stint ionestablishes u,,. and a,,.. Thus, starting at u station ncrtr thocenter of the disk (or for a disk with a central hole, at. thoedge of the hole), where the radial and tangent id st.rcsscsb;ar k.iown relations to the equivalent. stress, makes itpossible to proceed with the dct.erminat.ion of stresses atsuccessive sta,t.ions of inc.rensing radius.

The corresponding chart “for equation (5) is figure 1(~), ‘- -which is used in determining the st.rcss conditions in t.ho rimregion; whereas figure 1(a) is used for determining t-he stresscondititins in the central region of the disk.

Deter_@nation of disk profile,-When tho st.rceses lii~rebeen determined, the shape of tho disk can be obtained fromthe finite-difference equivalent of equation (1). For rmndomvabs of r~ and ra-l, the equation assumes the form

—..

For ‘n~~”-’= 0,200 and P.= P.+ =constant density, equa-

tion (7) becomes.

:.=.h,_, lo.oocTr,;– rt,n+pnrnwx= - 8.00 cT,,n_l+c,,n-1—0.640parn2ws....

(8)

METHOD OF DESIGN AND STRESS XNALYSIS OF ROTATLWG DISKS WITH TEiMPEIL4TURE GRADIENT 105

Radial stress,

,103ZnIOOXI03

80

60

40

20

0

20

40

60

80

)0

-200 ~ ‘ I t I I I I I I ,“,I

o 20 40 60 80 IOox 103

0- ~,n , lb/sq in.

(a) r-= 02W. m r-=o.oa

FIGUREl.—Desfgn charts for determining radisf and tangentii stresses.

106 REPORT 952—NATIONAL .ADvISoRy coMMI~EE FoR AERoNAu~cS

Tm—rn_land. for =0.050 and pm=pn-l=constantrn .- -.equation (7) becomes

hn_l 40,00c,,. —u’,.+p.r.2u2x= 38.00 U,,n_I+ Ut,n_l–0.903PnTnsUa

density,

(9)

Bccausc equations (8) and (9) give. only ratios of thick-ness and not, specfic thicknesses, a starting point for c.a.lcu.lating actual thicknesses must be obtained from otherconsiderations, for example, those at the rim. For diskswith the familiar fir-tree blade attachments, one approachis as follows:

With reference to figure 2, if sufEcient clearance is providedin the fastening between the. blades and fihe disk to preventcircunlfe.rential tightening, t.hg radial stress in the dkk at thebase of the se.rmticsns (at radius rJ is the sum of the cen-trifugal stresses produced by the blades. and by the serratedregions of the rim.

FIGURE2.—DIak nomznclaturs for derivation of rim tIrickness.

The total centrifugal force due to airfoil sections of blades is

ArmzAbI (lo)

The serrated bases of the blades and the base-retainingsections of the dkk maybe considered as a discontinuous. ringof material subjected to centrifugal effect, the serrat.ionsserving to eliminate tangential stresses. When it is assumedthat at the rim the thickness h~ (fig. 2) is spe.cfied from con-siderat.ione of blade design and that the thkkness taperslineady to a vahm hb (as yet, to be determined) at the base ofthe serrated section of the disk, then approximately:

Total cent.rifugaI force on serrated sections of blades anddisk is

(speed’) (volume) (average density) (average radius)or

The radial stress at the base of the slots is equal to the radialstress U,,SIat the rim of the continuous section of the disk and

presumably has already been cletermined from the diskdesign. Hence:

. .

Total resisting force at rim of continuous-disk

= (stress) (area)

. = Ur, b .%rr~ha (12)

W%en the sum of equations (10) and (11) is equutcd toequat.io>. (12),

When h~has been determined, the thicknesses at the remain-ing statjons in the disk may be. obtained by succcssivo

muhip]~cation by the ratio &. An application Of the

method is ilhstrated by the following example.

ILLUSTRATIVE DESIGN

Requirements.-The assumed problem is to design a diskof 9-inch radius to carry blades that. require a 1.Winch bmcthickness (h~ = 1.9). In order to retain the blades, a serrat-cdregion of l-inch ra.cii?ddepth is necessary (rb = 9–1 = S in.).Based on measurements of similar disks, the tempcmturodistribution is assumed to be as shown in figure 3, and as aninitial design, the equivalent stress at. each location is assumedto be equal to the .elastic limit of the r@e.rial at the opc.ratingtemperature. The disk is to operate at. a speed of 11,850 rpmand the density of the disk and blade material is 0.29 poundper cubic inch.

lUGUKE?.—Variation of temperature with disk radius assumedfor Il]ustrutivc dcslgn.

Establ@hment of stresses satisfying the compatibilityequation,—The calculations are shown in table 1. The diskis divided into 18 stations. Column 1 indicates the radius ateach station as a fraction of the radius of the continuoussection of the disl{. These values have been chosen to yield

—r~_lnine stations with” ~n =0.200 and nine stations with

ra—ra_l=0,050. The actual value of ra at euch station is

~n _

METHOD OF DESIGN AND STRESS A2TALYSIS OF

obtained by multiplying the values of column 1 by the diskradius, -ivhich in this case is 8 inches. The radii are listed inCohlmn 2.

The difference between the act ual temperature ~nd thetemperature at -ivhich there is no t.hermal stress is listed incoIumn 3. Because in this case the stress-free condition is ata room temperature of 70° F, the Mn term of cohnnn 3 isobtained by subtracting 70° F from the actual temperaturesin figure 3.

The elastic moduli and the coefficients of tlermal txpa.n-sion at the operating temperatures of the various st aticms melisted in columns 4 and 5, respectively.

The vahes ctmM’m, It= CY.AT.-Ci,_,AZ’F_,, and EaH~are then calculated as shown in columns 6, 7, and 8, respec-tively.

The quantity rX2pX&at. each station is then caIcuIated, asshown in column 9. Column 10 lists dues of the ratio

E.— needed for subsequent csdcuhdions.E.-,

Cohmm.s 11 and 12 are temporarily bypassed, and in col-umn 13 the desired equivalent stress at each station isentered. In this design, the equivalent. stressw are takenequal to the elastic limits of the material at the operatingtemperatures, which for the temperatures of figure 3 areshown in column 13.

The essential stress determinations are then effected bysimultaneous operation on columns 11 to 15. At station a,the radial and tangent ial stresses are both equal to u. and aretherefore both entered as 76,500 pounds per square inch incolumns 14 and 15. These stresses are used to det.m”nethe ~alue of Z at station 2 equal to 49,600, as listed in cohnn.n12. Reference to figure 1(a) then shows that the line Z=49,600 intersects the ellipse u.=76,300 at a,,.= 76,000 andat a,,.= 75,000, which are then Iist.ed in columns 14 and 15.In the same manner, these wdues are used to determine thestresses at. station 3 and progressively at. subsequent stations.For example, at station 8, 2=32,200 and ac=75,000. Inter-section of the corresponding straight Iine and eUipse yields,in figure 1(a), a,,.= 83,000 and C(,m=62,000.

At station 10, there is a change to “~~-1=0.050. The

only effect is to change the coefficients in column 1I and tointroduce the use of figure 1(b) instead of I(a). At elation10, for example, Z= 13,800 and rc=’i3,000. Intersection ofthe corresponding straight Iine and eIIipse in figure 1(b)yields cr,,m=83,500 and u~,n=42,500, which are listed incolumns 14 and 15, and are subsequently used to obtainthe value for Z at station 11. Thus, by successive andsimultaneous operation on the various cohmms, values ofu7,*and u~,nmay be obtained for each station that are con-sistent with the compatibility equation and combine to yield

ROTATIXG DISKS WITH TEMPERATURE GRJ4DIEXT 107

an equivalent stress at each station of any desired value.(If the imposed requirements are impossible to attain, thenecessary straighh Iine and eUipse WW not intersect. Thus,for example, were it arbitrarily required to achieve a.n equiv-a.hmt stress at the rim of 40,000 lb/sq in. while maintainingthe equiva.lent stresses at dl other stations at the values listedin cohunn 13, the line Z= 60,900 would not intersect theellipse c.=40,000. The indication is merely thafi no physicaldisk can satisfy the requirements and that the requirementsmust be changed)

Determination of disk profile.-Estab1ishment of stressesby means of the compatibility equation makes it possible toobtain thickness ratios in accordance with equations (7) and(8). Before proceeding, however, examination of the radialstress at the rim rr,b shouId be made in order to determinewhether or not it will yield a reasonable value of thickness hbin accordance with equation (13). Substitution of the values

iV=54 ;,,=27,000

A,,= O.625 p&!~=I150

ra=8.00 rd=9.00

ii.=1.9

ih equation (13) rewdts in the rdation between hb and rrzshow-n in figure 4. Obviously, from this figure, values ofUr,bbelow 18,000 pounds per square inch will yield inordi-nately Iarge thickness values of fib, whereas vfdues of C,Jabove 40,000 pounds per square inch wiIl yield inordinatelylow vzdues while overstressing the rim material, which isopera.t~w at a high temperature. The allowable radial rimstress can be ascertained from a detaiIed analysis of creepstrength and stress-rupture va.lues of the rim materiaI forthe desired life of the wheel; but in any case, figure 4 indi-cates that the reasonable range for r,J is betiiveen 18,000and 40,000 pounds per square inch.

FIGCF.E4.—ReIstfon between Chfcknessat rfm of eontffuous sectionof disksnd rsdiaI streseat that Ioeatfon.

108 REPORT 952—NATIONAL ADVISORY

Obviously, the value of u,,i in ta.ble”I obtained in the firstdetermination (u,,~= approximately 0) does not lie wit,hlnthe aforementioned reasonable tilts. The reason for thevery low wdue of r~,b,however, is the designation of the lowvalue of u&a= 58,000 pounds per square inch at rim of con-tinuous section of disk. The compatibility conditiondemands ordy that ~t,b and a, ,b lie along the line 2=60,900(column 12, table I). By modification of the demand ..onae,~, the individual values of. c,,, and u,,~ can be moved toother locations along 2=60,900. Thus, for example, by achange in the requirements on CT,,,to 63,500, 64,000, 67,700,and 72,900 pounds per square inch, the values of C7,IJcan besuccessively moved to 18,00.0, 20,000, 30,000, and ~0,000pounds per square inch with...corresponding changes in ~t,o;all these values are given in table 1. Physical] y, the signifi-cance of this result is the impracticability y of attempting tofind a disk for which stresses~t the rim. will .uot exe,?.ed...he.elastic limit if all the other. regions are to operate exactly atthe elastic limit. Allowing the rim to operate somewhatabove the elastic limit permit~ the re:rna.inde.r.of the ~.k.. toba kept at the elastic limit. Under. ~nusu.a.l. conchtlone,modifying several stations near the ri~. may be necessary.

The values of ‘“-’~ are now computed according to equa-

tions (8) and (9), the necessary calculations being shown incolumns 16, 17, and 18. Actual values of hfi are obtained

hn-,in column 19 by successive multiplication by the ratio ~

n

starting with the rim value ho determined from figure 4- foreach value of m,,a and proceeding toward the center of thedisk. The disk profiles and the stress distributions areshown in figure 5. The slenderness of the profiles relativeto the rim is the result of operation at the elastic limit inmost of tho disk. More prtiitical disks can be obtained byassuming r, values yielding-greater factors of safety. Thesubject of allowable design stresses, however, requires con-siderable experimental investt~gation.

tbdiai dia iance, r, in.

FIGURE5.—Stressdistributionssnd disk profllcs for illustrative designswith sevorslsssigmdvaiues of radial rim strsss.

COMMIT>EE

SIMPLIFIED

FOR .AERONA?TICS

METHOD OF STRESS ANALYSIS OF GIVEN DISK

ELASTIC STRESS

A method for calculating the elast,ic-stress distribution ina disk of given design is presented in reference 3. Althoughthis method can be directly used to checlc the stress dist.ri-but.ion in any disk de.signed by the procedure describedherein, simplifications can be made that appreciably rcdiccthe amount of labor involved in calculation.

Simplification of equations,—Briefly, the method of refer-ence 3 &_n.sists in rewriting stress equations (1) and (2) infinit.e~difle.re.nce form and solving these equations to obtain

u,,n=A,,nct,=+l?,,n

1(14)

u,,n=At,.uz,a+Bt,.

where Ut,dis the tangential stress r-d.the first poinb station in _the disk and remains unknown until the end of tho Calcuh- .tion. The coefficients A,,, and so forth are determined fromthe equations

A r,n =KnA,,.-l+L,Agrn-* 1

Thus, if the coefficients A,, A,, B,, and B, are known at the(n-l) st”station, they are readily determined at the nt.h stittio.nfrom equations (15). The coefficients at the first station(r=a) are known. For a solid disk

. A,,.=A,,.= 1

Br,==BtBa=O I

and for a disk with a central hole

Ar,==B,,a=B,,ti=o

At,a=l 1

(16)

(17)

Hence: successive applications of equations (15) yield thecoefficients for subsequent substitution in equations (14) foractual stress determinations. Th~ va.luo of .ul,c is determinedfrom the condition at the rim where the. radial stress is known.Thus, from the fist of equations (14)

(18)

where a,, o is the radial rim stress and A,, Oand B,, b are thecoefficients as determined in the last of the successive applica-tions of.equations (15).

The”bulk of the computational work in a stress analysis iscontained in the dcterminat.ion of the coefficients K,, Lx, andso forth, in equations (15). When the stations are randomlychosen, the equations for these coefficients arc complicated.

METHOD OF DESICW AND STRESS ANALYSIS OF

.A great reduction in the a.mounk of labor is et?ected if the

rn—rR_l.stations are chosen according to an assigned va.lue of rnIn addition, if Poisson’s ratio is assumed constant as p= 0.333the expressions assume the simphfled form given in thefollom”ng table:

TABLll .4

1

~--mo

-(0.02++!+0.043,=) r*w-

0.9!20EnH’.

Alinement charts.-Evalua.tion of the coefficients K.,A?., and so forth from the expressions given in ta.ble A ispossible. A fwther reduction in labor, however, can beeffected by the use of the alinement charts in figure 6. Forexa.mple, in figure 6(a), joining with a,straight Ii.ne the points

&=o.9s7 and ‘~– —1.185 yields intersections on the suit-

able sales for valu~s ~.= 0.199, L’m= 0.808, K.=0.972, andK’n=O.256, which check closely with calculations based onthe expressions in table .4. If the detiity can be consideredconstant between two stations (p.= pn_J, the values of Mnand M’. can also be obtained from the charts in figure 6.

hj+,As an exa.mple, if h,—= 1.185, rn*P&=20.5X10S, and -E@’a=

hn-15.5 x 10S, the corresponding -raIues of –h: and r=spd are

joined by a straight line t.o determine the supplementarypoint P, which is then joined to 13’JY’.=5.5x 10s to yieldintersections M,= —4.3 X103 and M’== —6.6X103. Be-tween any two stations where a change in density occurs,the expressions for Mm and M’s in the table must be used.

Illustrative example.—The necessary c.alculationa areshown in table II for the determination of the elastic stressesin a disk of given design, which are the disks formerly de-signed for the various assumed radial rim stresses. Thecoeillcients K., L., and so forth listed in columns 20 t-o 25are determined from values listed in table I used with theaIinement charts of figure 6. Values beyond the range ofthe charts given in figure 6 may be computed from theequations given in table A. Cohunns 26 to 29 list the A andB coefficients as determined from equations (15) in which the

ROTATLWG DISKS WITH TIWPERATURE GRADIENT 109

values listed in equations (16) are used as the initial condi-tions. The value of r,,= is then determined with the aid ofequation (18), and the radial and tangentitd stresses at allstations are obtained from equations (14) and listed in col-umns 30 and 31, respect-iveIy. The equivalent stresses atvarious stations are listed in column 32, and the yield stressesat each sta.t.ion, whkh were the design criterions, are listedin column 33. A correlation of satisfactory engineeringaccuracy exists between the computed stresses and the designstresses shown in columns 14 and 15 of table I.

PLASTIC STRESSES

Modification of elastic-stress eq~ations.—.{ method for

calculating the stress distribution in a dislr subjected to plastic

flow- is presented in reference & The form of the calculationis simfla.r to that of an elastic-stress deterfiat.ion, except

that the term lY’min the expression for Mm and M’* (table .4)is replaced by IFn-P’nl where Pf, depends on the plasticstrains in the radial and tangential directions. In practice,the plastic. strains are estimated and a stress clist.ribution isdetermined. Comparison of the plastic. strains based on thecalculated stress values with the plastic strains assumed inorder to determine these stresses permits better subsequentestimates. When resultant strain vrdues agree closely withassumed strain values, the strains are assumed to be correct.

The values of P’n to be used in the application of the sire- .pIified method are given in the following table:

TABLE B

, 1r--r=+

r. I IO.XI) I 0.100&,m+O.125&,r,+-u33&..+O.SZi ihm-1 I

.0.050 O.O%-&..+ O.O25&m-1-I.0!25&,.-l-O.974At.a-11

where A,,* is the plastic radial strain at the nth station, andthe other A terms represent other plastic strains accordingto the notation of this report. The detailed procedure forestimating the plastic strains and checking -with computedvahms is given in reference 4.

SimiIarly for creep calculations, ~’w terms are added to theP’n terms, the form of the expressions being identical tothose for P’. in table B except that the A terms are replacedby creep terms 8, as discussed in reference 4.

Illustrative example.— As an ilhstration of practical pro-cedure, the disk previously designed for a radial stress of30,000 pounds per square inch at the rim is considered. Aspointed out in the design calculations, any radiaI stress aboveOpounds per square inch at the rim will cause the equivalentstress to exceed the eIastic Iimit; hence, plastic ffow mustoccur. The amount of plastic flow in this case is small andis con6.ned to the single station at the rim, but the methodinvolved is essentially the same for the more complicatedcases involving plastic flow at several stations and previouslyaccumulated plastic flow and creep. Examination of refer-ence 4 wiII indicate the procedure in the more complicatedcases.

110

EnEn-l

1.00

.991

REPORT 952—NATIONAL ADvISORY (!OMM~’’l’EE FOR AERONAUTICS

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454

50

“i

..=

I55

-..-

.@

r, —r.+(s4 y=o,goo.

:-[

L-2

:-3

---4.

.

E‘. .

I ‘.

t-5 ‘\, -6

E

E-k -.

-~

-8

-9

-lo

-11

-12

;-13

;-[4

+15

‘-16

~ L.L.t

k

),0t

—

[ 13

.-

.—

FIGURE6.—Alinirnent charts for deterrnkdrrgconstantsfor finite-differenceequationsof stressesin turbine dkks. $m=~.-:.

.

MEITHOD OF DESIGN &?%!lSTRESS ANMXSIS OF ROTATLNG DISKS WITH TEMPEFL4TURE GRADIELNT

‘En L’“ Ln ~ K~ K‘~ rn2poJz M“ M’nG

EnH’n

1.3~~ lo3~

I.00:

, ~[ f ~~ ~\ , ~

o c1 o 0.17

.955

.99: .16-2 2

.94 1013

.98: 45 -4

’51 :-l4

.055.93 ++.2

720

.97– J4 f-6

T6

25–.92

“i :.13

[, ~1 ;: ~ /

-8.96: 1,2— 30 8

–.9[ 35-2

.12 -10

.95– - - 101.[ 40

-.90.11 45 -12

.94– . : 12

.050 –

3 -

50

:.89 1.1—.10 -3 -14

55.93– - 3

:14

60

u.88 :.09—[.0 ~ -16

.92– : 16

-.67 -18

.91- - : 1.0 ~18

-.86-20:

.90 –- :20

.045 – —.9-22:

T.85 d

.89– . :22

:.05:9 — .

100

;1 ‘

-24:

.8afo5

/ ~ ~1 1 ‘ ] i

:24

.04 110 -26

.83 115

.87 .81

26

.03 120-6 -28

.62

.86 .8 28.02

“040 -30

.81

.65 (b].01

r.—r~x(b) y-o.050-

FIGGEE6.-ConcIuded. AIfnernent chsrts for detcmnfnhg constantsfor Brdtedfference e~ustfon!sof stressesin tcrbfne dfsk-. “. -‘. -L.

112

It?a

t40

/20

100

c.-.

$

* 80“:

;m

60

40

20

0

REPORT 952—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

Stmin, in.jiq.

FIGURE7.–Stress.strain curve for disk material at 1070”F.

The stress-strain curve for the disk material at the oper-ating temperature of 10700.I is given in figure 7. Point A

is the calculated elastic equivalent stress at the rim, B theyield point-, and C the intersection with the stress-straincurve of a vertical line drawn through A. In accordancewith the method of reference 4, the strain CD is the firstapproximation of the plastic strain. The value is listed incolumns 34 and 35 of table 111 for “station b. Columns 36and 37 list the approximated radit-d. and tangential plasticstrains obtained from equations (20) of reference 4, Thevalue of P’. as determinedfrom. table B, is listed in column38. C?olumn 39 corresponds to colugm 8 of table I used inthe elastic-stress calculations; the -onIy difference is thatH~-Pi has replaced Hi. l.hom this point., the calculations

are identical to elastic-stress calculations and are shown in

cohmms 24, 25, and 28 to 32 (table 111), which are similarto the corresponding columns of table II. Further approxi-mations, made in accordance with the method of reference4j are &o shown

approximation.

in table III to obtain greater accuracy of

COMMENT ON SIGNIFICANCE OF THERMAL STRESSES

The design method described assigns equal importanceto the thermal and centrifugal stresses. Tho true relativeimportance of these two types of stress is, however, as yetto be. evaluated. Whereas the centrifugal stresses areinhercn~ly necessary to balance tho centrifugal forces ofrotation-so that yielding due to overstrcssing in onc rcghmshifts the load and thereby increases the stress in anotherregion-the thermal stresses constitute an int-crnally bakmccdsysteni. Tension in one region balances compression inothers; hence, plastic flow in one region reduces the thermalstresses in another region. When considering burstingstrength, for example, centrifugal stresses me probably ofgreatej- importance than thermal stresses. In the case ofrim cracking (reference 8), which is in part due to cyclicplastig flow at the riml the thermaI stresses am, howm.r,fully as important as the centrifugal stresses. ~ikewise,when disk deformations are considered in designing for C1OSOmechanical tolerances, t.hermal stresses must receive atten-tion equal to that given to centrifugal stresses.

The relative importance of centrifugal and thermalstresses depends, to a great extent, on the ductility of {ho -““disk material. As yet, experinmtd data aro very meageron the significance of ductility. In any case, however,

because the distinction between the two Jypes of stress canonly be realized after the occurrence of plastic flow andbecause the behavior of a material in subsequent loadingdepen@ on its previous stress-strain history, thermal st.resscsshould receive detailed attention early in the design in orderto minimize any detrimental effects that may result f~omtheir omission.

LEWIS” FLIGHT PROPULSION LABORATORY,

NATIONAL ADVISORY COMMITTEE TOR “AERONAUTICS;

C!JEVEL.4ND, OHIO, -April 4,1948.

REFERENCES

Stodola, A.: Stcarn and Gas Turbines, vol. L McGraw-Dill BoolcCo., Inc., 1927. (Reprinted, Peter Smith (New Yorlc), 1045.)

Thompson, A. Stanley: Stresses in Rotating Disks at lligh Temper-atures. Jour. Appl. Mech., vol. 13, no. 1, March 1946, pp. A45-A52,

Manson, S. S.: Determination of EIastic Stresses in Ga&TtirbineDisks. NAC.4 Rep. 871, 1947.

MilIenson, M. B., and Manson, S. S.: Determination of Stresses InGas-Turbine Disks Subjected to Phistic I?low and Creep. N’ACARep. 906, 1948.

Leopold, W. R.”: Centrifugal and Thermal Stresses in RotatingDisks. Jour. Appl. Mech., VOI.15, no. 4, Dec. 1948, pp. 322-32&

iWcDowell, C. M.: Prescribed-Cent.rifugaI-Stress Design of RotatingDiscs. Paper presented before SAI?J Ann. Meeting (Detroit),Jan. 12-16, 1948.

‘lhmarlrln, S.: Methods of Stress Calculation in Rotating Dish,NACA TM 1064, 1944.

Manson, S. S,: Stress Investigations in Gas Turbine Discs andB1ades, SAE Quarterly Trans., vol. 3, no. 2, April 1949, pp.229-239.

.

APPENDIX

SYMBOLS

In addition to the symboJs used herein, the symbolsof references 3 and 4 of which application is made in thisreport are defined. The various coefficients are given intheir general form for random spacing of the point stations.From these coeflkients, the simpbfled coefficients can be

obtained by direct substitution of ~-l=constant andT*

~=conetant.

base area of airfoiI section of blade, sq in.

elastic modulus of disk material, lb/sq in.

axial thickness of disk, in.

number of blades supported by diskradkd dlsta.nce, in.temperature, ‘F

coe%lcient of thermal expansion between actualtemperature and temperature at zero thermal stress,in./(ii.) (“F)plastic increment of strain, in./ii.plastic increment of strain in radial direction, in./in.plastic increment of strain in t.angent.ial direction,

in.fin.temperature increment- above temperature of zero

t.hermal stress, ‘Fcreep increment “in radial direction, in.fii.creep increment in tangential direct-ion, in./ii.plastic strain corresponding to stress crc in tensile

specimen, in./in.Poisson’s ratiomass density of dkk material, (lb) (sec2)/ii.4stress at. base of airfoiI section of blade, lb/sq in.equiwdent stress, lb~sq in.radial stress, lb/sq in.ta.ngential stress, lb/sq in.proportiomd elastic limit, lb/sq in.angular velocity, radians/see

The following supplementary subscripts are used for de-noting values of the preceding symbols in connection withthe finite-difference solution:

n n.th point stat-ionn-l (n-l)st point station

a station at smallest disk radius considered (For disk

with a centra.1 hole, this station is taken at the

radius of the central hole; for a solicI disk, this

station is taken at a radius appro.xirmtely 10

percent of the rim radius, which is larger than that

(5 percent) recommended in reference 3. The

value of 10 percent yields results of ‘satisfactory

accuracy and permits the use of fewer stations

when constant value of ‘“~~-1 is used.)

b station at rim of continuous section of d~k or base

of blades

d station at rim of disk or base of airfoil section ofblades

The following supplementary symbols denote combina-tions of the previous symboIs:

Istress coefficierds defined by equations

~r,n=4.a~i,a+Br,n

at,m=.&n~t.a+Bc,n

= rmh%

=rn_lhz_l

p.-, (1 +Pn-1) (~..T%J.—— —En-1 22J._lr._l

.=; (r~—r._Jhq_l

=a~ATn—an-lAT=_l

C’.G.-I- C. G’..— fn

=C’.HS+GJH’C-P’S-Q’JC’nDs-CxD’n

113

114 REPORT 952—NATIONAL ADVISOR%’ COMMITTEE FOR AERONAUTICS

\“

For ‘njfn-’=0.200,

#

115METHOD OF DESIGN AND STRESS A.N.4LI”SIS OF ROTATING DISKS WITH TEMPEILITURE GRADIXA’T

TABLE L-CALCULATION OF DISK PROFILES FOR ILLUSTRATIVE DESIGN

I 1 3 “8 9

r. - rm-in — r.r. 7s

aAT., W*.Em c%

m w) (6) -(0 .-1—

Ems,

(4)X(71

I-S,

rkx[l)AT.

a a% I O:;::

: .220 .1654 .=ml .2065 .220 .2586 . 2CI0 .3227 .20Q8 : ;E

:% .6301: .050 .66311 .050 .69812 .73513 :R .7i314 ;&o .814

.8571: .05Q .90217 .050 .950h .050 LMnJ

~ :Xlw

23.526.528.423.323.2%. oXi. 427.327.026.926.5%.125.6w. 823.521.4

~ 8Jao+ 3-3-:X1O+

a 90 3.3528.91 3.420s.93 3.5378.95 3.6709.00 3.915

4254;5 5.1529.22 : S&u.2%9.33 &01s9.!0 6.4399.48 8.920S!56 7.5Q59.66 3.1639.ii 8.S919.92 9.g~o

0:yJxlLw

.0!s

.097

.142

.245

.339

.89S

. 2JIL

.243

.357

.421

.481

.585

.656-.i$

L 40XW1.251.37z 75405&919.49

24.6L5.498.320.67

11.16125514.9816.3217. H2202

1.28XIO:2.003.12;WJ

119518.7029.1932.3535.8839.7643.9s4s.z54.0559.36

R%

_.. .

I 11 1912 13 15—.

n,%

18

0.s33X(15)n-lh -0.188X(14)W1

h,,(NJS+lX(EI).+1h.+—. ,

h.

:17)+(1s——

0.997~ yl

L 0221.052L 079L 123L 164

L CBsL0421.0421.018L 019.979.93s.877. 0s7.551.602

l:!!l

8.00X(14)n-!~(15)e, –0.64X(9)

10.COXC14)-(1.9 +(’2)

.%:11)X( IO)-(3)

u.,, u,, .

u,, b=lS,00Q

G,. b= U*. b =20,0)0 20,0i)o

G,. h=40,000

a5LOXNY

: 49.94 49.05 43.0

47.0: 45;s9 37.9

76.5xICF75.074.073.0is. o70.560.562048.0

:;

2L o11.5

–Lo-16.0-31.0-60.0-53.0-620-4s.o-44.5

L S61.87L 851.30L 73

;E1.35L 16

1.76L 76L 74L 69

2:L44L23L IO

1.06L02.98.96;:;

L021.16

-..---

1.50

?ZL 451.40L 33L22L 09.94

.91

.S7

.23

.82

%.87.99

------------.- .-. .L 15

L 35L 361.35L 3L1261.201::

.s5

.as

.79it!.75.74.76.81.92

-------.--.-------. -----

.32

J&:xlrF

&J.;

ti838.532.2125

qsxm

g

691695622707

636xIOZ49s700713727750737823

o.965x(151-1-OXSX(14)*1

10 19.4X10416.1

ii 10.51314 –;!

-11.1;: -20.917 +J. ;bb —d~ib –42. 7b -42.7b –42 7

38.66X(14).-1+:15).-1-O.W3X(Q

3269X1(F31s331553106!&@2i9424013a7115s7;53;

15671537

1.12L 07

HJ

L 011.08L 23

------2.23 ------

L 93. ----- I

---..------------- ..- , ------

.

_REPORT 952—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TABLE II.—CALCULATION OF ELASTIC STRESSES FOR ILLUSTRATIVE DESIGN

27 23—- . -.

29 30131132 2326

A,,.,

.—u,,., 91 ,“, s,,.,At,m, I B,,.,

.Bt,.,

Fmm table I,(18) and (10) I (la) (9),

b,,’.(-.76,104

\J-(311):+(31p-(301 x(al)

C20)X(23).-1

(?3x(~) -1 +(2s) x(2g) *.+(22) X(27)m-l “+(24)

(22) X(28) .-1+(23) x(29) .-

●(25J(xl) X(26) .-.,p(’x))((!m)r

n

—

a2

;

6

;

1!111213

#161;

bbbb—

Coeflicknts detemnfnedfrom figure 6or from tabIs A LW*X243)

+( )UL. X(27)

+(29)

xl. I xlo~ 70.lXIOI74.6 76.078.6 74.672.3 74.8

74.7% I 74.500.6 14.162.5 74.E46.8 72.442.7 72.0a7.o 71.830.4 71.b21.0 69.9n. a 00.1

-L o 07.a-15. a 05.4-30.2 63.7

.. I

aalal

. w

.809

.864

.ato

.810

.sol

.947

.941

.945

.941

.937

.eaa

.921

.901... -----

.662

.a84

.866

.86a

1: pcJ-:. 4X1OI

L 000 -1.1L 012 -2.11.028 ==8.61.052 -6.0L 0’34 -9.91.162 -1& 61.255 –28. 61.281 –a2. 3L 307 –36. 81.239 –41. 9L 254 –46. 51.367 -61.8L a53 -55.61.306 -57.7L 210 -56.6

-! 4X1W-25-8.7-6.3

-10.1–16. 7-26.9~:~ ;

–6~ 5–71. 5–al. 4–92. a

-104.0-114.7-122.4

;} ;Xlol76;876.377.278.379.9

%:83.182.7al. 778.274.266.856.442.9

;~ :xiol

70.a76.176.076,076.676.073,.578.072.671.670.6

%:: ““”05.803.0

1.000.696

L 004:%

1:loa1.1801.301L 4721.517L 5701.6241.639L 656L 6091.499L aoa

0.816.828.243.851.862.885.624.956.088.993.993.054.970.922.892.824

. . . . . . ..622.571.81a

1.063

0:;g

. 1s4; #

.189

.192

.196

.050

.030

.050

.049

.040

.046,047.044

. ..- ..-..025.037.048.050

-ufxlo~ -L 4XIU-. –L 3

-I. 5–T I -z 9-1.7 -4. a-2.7 -7.2-4.2 -10.1~;; –24. a

-6.9-io -8. a-2.2 -10.0-2.4 -11.7-2.7 -13.1-2.9 –16. 6–3. 2 –16. 9–a. 4 -17.8........... ........-----a. a –22.4-3.4 -22.5-8.9 –22. 7–4. 3 -22.9

..... ...... ..- -------- .....-.942 -87.1.967 –40. 2

:% –55. 5-70.8

.... .........,-123.6–124.9–130.2-125.7

.........18.0

;:

.... ........ ..-..--.=. .....-61.9 62.9–51. 3–47. 8 M-44.3 7a. o

......-.58.058.038.068.0

...... .....-..725.7Q2

1.1221.455

TABLE 111.—CALCULATION OF PLA.STIC STRESSlj&3 FOR ILLUSTRATIVE DESIGN

-.

[!W-30,000]

7T.-7.8-1

.+ ,< ”,-,,!34 35 36 37 38 28 39

-L

24 25 23 29 30— — -— —— ———- —

●#,* ~ml AM, At,., P’., P’”, Em(H’m- M* M’. B,,., BLrq %.,P.?,

0.160(36)~ 0.025(36)-l--x 0.125(36)s-i 0.026(36).-1

0.50X25)X o 56X26)

!?

F~or& “. 6;9$3+P$lM!kL f%%%: “~/$g.g

‘fig. 7) ‘mcimate) r2x(30)-(aO] [2X(31)-(30)] ;:;:ay;: ;g:2~:: [(7’M8)1 & (39) +(24) +(W

— — -— —.

0 0.2OX1O+‘.24)XNY

0, 159X1O+

‘------”---- T ‘-:------- ‘------------

–. 185)(10-J. . . . . . . . . . . . 0.194X104 17.9X1O+ -K7X108 -18.7X1OS. . . . . . . . .

-55. 3X1O: -126. 2X10~ 30.Oxloo 0 0 0 . . . . . . . ..- -. . . . . . . . .._- . . . . . . . . . . . . . . . ..-. . . . . . . . . . . . .. . ---------------- -,22 .21 .169 –. 192 . .. . . . . . . . . . .201 17.7

.. . . . . . . ----–a.7 -la.6

. . . . . . . . .-65.3 -123.0 30.0.—.- —

31 aa

Ui ,“,

wax(n)+-(!W}

.............04.4x 10J

.. . . . . . . . . . . .64,2

..... .....-4a.9xlo.. . . . . . . . .,-4a.7

*