APM-56-8

Design and Calculation of S team -T urbine Disk W heels

By I. MALKIN,1 PHILADELPHIA, PA.

A sim ple procedure for ca lcu la tion o f s te am -tu rb in e disk wheels toge ther w ith a discussion of som e questions concerning th e ir elastic resistance in various conditions is offered in th e following artic le . New solu tions of th e problem of ro ta tin g disks are developed an d th e ir applica­tio n in disk design is show n by p rac tica l exam ples. T he resu lting m ethod in designing tu rb in e disks is represen ted by a general schem e w ith a s tan d ard tab le.

I n t r o d u c t io n

TH E calculation of steam-turbine disk wheels often forms an actual problem of great im portance in steam -turbine design, although m any methods for the solution of the

problem of rotating disks are known and used. To be of prac­tical use in design and development, a satisfactory solution of the problems involved is expected to yield simple procedures and practical standards and to reduce the present cumbersome methods of stress calculation to a minimum of m athematical work. To develop new solutions of this kind is the main purpose of the present contribution.

By introducing certain new profile curves into the analytical form of the problem of ro tating disks, formulas for th e stresses are obtained, which are much simpler than those yielded by any other analytical solution of the problem. These curves are suggested by certain conditions of integrability of linear differ­ential equations and are designated by the author as “exponen­tial profiles.” D ue to the mathem atical properties of the new formulas they easily adm it of a simple and complete numerical representation by means of a “ standard table,” which when once calculated can be used in disk computations for any special given conditions. Such a table is presented in th is paper for disks of the “first exponential profile.” This will be followed in a later article by a similar table for the still more im portant “ second exponential profile.” Examples of disks of the new profiles w ith the corresponding stress distributions are given in Figs. 3 to 6.

Taking into consideration the varied wheel proportions resulting from modem blade dimensions, an additional investi­gation was necessary. This consisted of a revision of the m athe­matical fundamentals of disk design and a check of their validity under the new conditions. The approximate theory of rotating disks, developed by Stodola and used in this paper as in all calculations in disk design throughout the technical literature,

1 Westinghouse Electric & Manufacturing Company, Dr. Malkin was the winner of the prize medal of the Teclmische Hochschule Berlin-Charlottenburg. His theoretical studies with Professor Dr. Max Planck and Professor Dr. R. von Mises at the University of Berlin were preceded by four years of practical work in mechanical engineering with industrial companies. Dr. Malkin was connected with the AEG-Turbinenfabrik, Berlin, in charge of research work in elasticity, especially in disk vibrations, and with the Institute of Applied Mathematics and Mechanics of Dr. R. von Mises at the University of Berlin. Since January, 1932, he has been connected with the Westinghouse Company, South Philadelphia Works.

Contributed by the Applied Mechanics Division and presented at the Annual Meeting, New York, N. Y., December 4 to 8, 1933, of T h e A m e r ic a n S o c ie t y or M e c h a n ic a l E n o i n e e b s .

N o t e : Statements and opinions advanced in papers are to be understood as individual expressions of their authors, and not those of the Society.

is based upon the results obtained from the exact methods of the m athem atical theory of elasticity. These results are ex­amined in Appendix No. 1.

The stresses in a turbine disk are usually calculated for over­speed conditions. A natural question arising in design is th a t concerning the stresses in th e disk due to th e fit pressure when the wheel is a t rest; the design having been calculated originally for overspeed conditions. This particular problem adm its of a simple general solution, as will be shown, by reducing the condi­tions to those of a disk of constant thickness a t rest, the behavior of which is known. In th is way we easily find th a t the profile curve has practically no influence on the stress distribution under static conditions; the tangential as well as the radial stresses a t the bore being about 60 and 40 per cent, respectively, of the tangential stress a t the bore in overspeed conditions. This result was checked on a system of disks of the “first exponen­tia l profile” and graphically represented by the curves in Fig. 8. These curves show the change in th e radial and the tangential stresses a t the bore corresponding to variations in the ratio of th e disk thickness a t the bore to th a t a t the rim, under static conditions.

The discussion treating w ith th e stress distribution in a disk a t rest is completed by some general remarks concerning the problem of the stress variation w ith varying speed. This is in the interest of a be tter understanding of the elastic behavior of the disk under various conditions occurring in practical service.

The last section of this contribution deals w ith the influence of a hub relief, such as shown in Fig. 9, upon the stress distribu­tion in the disk. The solution of th is additional problem is graphically represented in Figs. 9 to 14 and m ay be expressed briefly as follows. In overspeed conditions no change of any practical im portance is caused by a hub relief. Under static conditions, the stresses w ithin th e disk are smaller than in a disk w ithout hub relief. Im m ediately a t the bore the stresses undergo a certain modification, too, b u t th e strength, as defined by M ohr’s theory, is not affected.

T h e F i b s t E x p o n e n t i a l P r o f i l e

Consider the differential equation of rotating disks in the form [37],5

2 See Appendix No. 2.585

wherein E is Young’s modulus of elasticity; v, Poisson’s ratio; a, the angular velocity of the ro tating disk given by the formula 30a = irn, n being the num ber of revolutions per m inute; pi, the specific mass of the disk m aterial; r, the radius; 2y, the thick­ness of the disk; and u the radial displacement. The coefficients of the differential equation, namely

586 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

being free of the function u, which is to be determined, a new profile y = f(r) shall be introduced in using the following pro­cedure suggested by elem entary methods of integration of linear differential equations.3

By integrating the differential equation, term by term , ac­cording to th e rules of partia l integration we obtain

corresponding to the differential Equation [42], for the stress <rr.6 We may restrict ourselves to the consideration of the homogeneous equation, because the particular integral corre­sponding to w2 is determined by [3].

By introducing our new profile [1] into Equation [4] and in using z = ftr4/,J as a new independent variable, we have, because of

the differential equation

instead of Equation [4].By following the m ethod indicated by theory,5 we find th a t

th e last integral can be represented by an infinite series of the form

Furtherm ore i t is known from the teachings of the theory of functions of a complex variable, th a t the domain of convergency of this series is determined by th a t point (singular point) nearest to z = 0, for which the coefficient of dVr/dz2 in [5] vanishes. This coefficient being z2, the radius of convergency will be equal to <s°; the Series [6] will converge for any finite value of z =

Indeed; by introducing Series [6] into Equation [5] wefind

This shows th a t our Series [6] converges faster than th a t for th e exponential function e', the ratio an/a n - \ in the la tte r case being equal to

As the series for el converges in the entire complex plane, ou r Series [6] converges even faster. From [7] we obtain by intro­ducing n = 1, 2, 3, 4, . . . . ,

Correspondingly, th e th ird integral for a, appears in the form

In substituting th is series into Equation [35] we find the expres­sion

as the general term of the series representing the th ird integral, for at, which is, accordingly, given by

6 See Appendix No. 2.

3 A. Forsyth, “Differential Equations,” German edition, Braun­schweig, 1912, p. 101.

4 See Appendix No. 2.6 A. R. Forsyth, “Differential Equations,” German edition,

Braunschweig, 1912, p. 573.

In using Equations [36]4 we find the corresponding expressions for the stresses. The last integral corresponding to the constant C, om itted before, can be determ ined as follows.5

Consider the homogeneous differential equation

D being again a constant of integration.The particular integral corresponding to the term w ith u 2 in

original E quation [37] can now be determined by variation of the constant D occurring in the last integral. If v is assumed to be equal to 1/3, the particular integral appears in a finite form, namely

and pu tting tem porarily C = 0 we find one of the two integrals of the reduced E quation [37]. T his first integral is easily found to be

C being an arb itra ry constant of integration. In considering the reduced equation

or

where a and — 0 are the two constants of integration. Only positive values of the constant /3 will be considered in studying the first exponential profile, for reasons which will be indicated.

For the profile [1] the original differential Equation [37] assumes the form

This is a differential equation for th e profile function y. I ts solution is

where Pi = dPi/dr, and so on. Hence the order of Equation [37] is lowered if F 0 •— P i' + P 2" = 0, or

APPLIED MECHANICS APM-56-8 587

where n = mass density and u = angular velocity, K and L being constants determined by two conditions concerning the boundary stresses (see operation 4). The functions f{z), g(z), <pi(z)i ¥>2(2) , 'P1(2 ) , tote) finally follow from the “standard table.”

The general procedure will consist of the following operations (see Fig. 1):

(1) From the values of r», a and ha, which usually are given in practical design, and h0, which has to be assumed and varied, the constant 0 is to be calculated according to [14].

(2) W ith the constant /3 the values of z a t the bore and a t

the rim are to be calculated from [13], i.e., z0 = (3r04/ 3, Za = 0a(3) For these two values of z the corresponding values of the

functions f(z), g{z), <p i (z ) , ¥>2(2 ) , <h(z), ^ 2(2 ) are to be taken from the “standard table” and introduced into formulas [1 2 ].

(4) From the two boundary conditions, o> = 0 (or a , = — p0, where p0 is a comparatively small am ount; see example in the following section) a t the bore (r = ra), and a r —r a,a, where a,a is a given am ount following from the centrifugal forces of the blading for r = a, Equations [12] yield the corresponding values of the constants K and L.

(5) W ith these values of the constants K and L the stresses o> and at can be calculated easily for any point 2 by using Equa­tions [1 2 ] and the “standard tab le .”

Further practical rules are given in the next section which covers a detailed example of disk design.

Finally it should be noted tha t, for practical purposes, it is of some advantage to introduce r 2 = z3̂ 0 ~ ’/* into the terms A r2/(z) and A r2g(z) in Equations [12]. These terms then appear in the form A 13 s/ 2F(z) and A t3 3̂ !G(z), respectively, where F and G are functions of z only. The stresses, [12], then are independent of r, being functions of z, of 8 determined by [14], and of the constants K and L (boundary conditions).

P r a c t ic a l E x a m p l e o f D i s k D e s i g n — F i r s t E x p o n e n t i a l

P r o f i l e

We present now, following the general scheme given in the last section, a practical example of disk design which involves solving a problem characterized by the following data:

A disk is to be calculated for r0 = 6.75 in., a = 18.75 in., the w idth a t the rim, determined by the dimensions of the blading, being b = 2ha = 4 in., while the working speed is n = 3600 rpm and the pull exerted by the centrifugal forces of the blading a t 20 per cent overspeed is a, = 13,000 lb per sq in. for r — a.

h0 and ha being the boundary values of y, while

the profile curve being given by formula [1 ].The functions /(z), g(z), <p i (z ) , ¥>2(2), ^ 1(2), ^2(2) can be calcu­

lated once for all and pu t together into a “standard table” to be used in practical design. Such a “standard table” is given in the next section and is followed by a detailed example of disk calculations.

G e n e r a l P r o c e d u r e :n D e s i g n i n g D i s k s o f t h e F ir s t

E x p o n e n t i a l P r o f i l e

The results of the foregoing section can be summarized as follows: For a disk of the profile

the radial stress o> and the tangential stress at can be represented by the formulas

n being the mass density and w the angular velocity, and

where K and L are arbitrary constants to be determined from the boundary conditions, wrhile A is given by

By introducing the Integrals [2] and [3] for the displacement u in Equations [36] we find the corresponding integrals for the stresses a r and at, and in using those integrals as well as expressions [8 ] and [9] we obtain the complete solution in the form

588 TRANSACTIONS OF THE AMERICAN SOCIETY OP MECHANICAL ENGINEERS

ST A N D A R D TABLE FOR D ISK S OF T H E FIR ST E X PO N E N T IA L PROFILE

z /(*) 9(e) <n(«) $1(2) ^2(2)0 ----- CO CO — CD CO 1.0000 1.0000

0.01 — 5939100.0000 6100700.0000 989.8500 1016.7878 1.0080 1.00540.02 — 728550.0000 775350.0000 346.2690 365.5067 1.0161 1.01070 .03 — 215255.5555 223566.6666 186.4090 202.2745 1.0243 1.01620 .04 — 89775.0000 100235.0000 — 119.6920 133.5710 1.0326 1.02160 .05 — 45420.0000 52140.0000 — 84.6203 97.1567 1.0409 1.02710 .0 6 — 25961.1111 30671.1111 — 63.5790 75.1387 1.0493 1.03270 .07 —. 16139.6426 19647.5199 — 49.8007 60.6103 1.0577 1.03830 .0 8 —. 10668.7500 13368.7500 — 40.2156 50.4292 1.0662 1.04390 .09 — 7389.7119 9542.7982 — 33.2304 42.9563 1.0748 1.04960.10 — 5310.0000 7070.0000 — 27.9587 37.2783 1.0835 1.05500.11 _ 3930.2012 5497.9696 _ 23.8659 32.8409 1.0923 1.06100.12 — 2980.5555 4225.0000 —- 20.6139 29.2935 1.1011 1.06680.13 — 2306.7290 3376.5495 — 17.9790 26.4017 1.1100 1.07270 .1 4 —. 1816.1832 2746.7961 — 15.8100 24.0079 1.1190 1.07850 .15 _ 1451.1135 2268.8915 — 13.9991 21.9987 1.1280 1.08450 .16 — 1174.2186 1899.2186 — 12.4686 20.2920 1.1372 1.09040 .17 — 960.6935 1608.4434 — 11.1609 18.8269 1.1464 1.09640 .18 — 793.6180 1376.3327 — 10.0334 17.5584 1.1557 1.10250 .19 — 661.1848 1188.6070 — 9.0529 16.4544 1.1651 1.10860.20 — 555.0000 1035.0000 — 8.1935 15.4765 1.1745 1.11470.21 _ 469.1427 907.9636 _ 7.4356 14.6142 1.1841 1.12090.22 — 398.6078 801.9126 — 6.7623 13.8467 1.1937 1.12710 .23 — 340.5840 712.6055 —. 6.1617 13.1603 1.2034 1.13340.24 — 292.3549 636.7936 — 5.6221 12.5417 1.2132 1.13970 .25 — 252.0000 572.0000 — 5.1361 11.9843 1.2231 1.14610 .26 — 218.0009 516.2246 —. 4.6958 11.4787 1.2331 1.15250 .27 — 189.4683 468.2058 — 4.2949 11.0175 1.2431 1.15890 .28 — 164.6505 425.8751 — 3.9294 10.5975 1.2533 1.16550 .29 — 143.6336 389.0549 — 3.5941 10.2120 1.2635 1.17200.30 — 125.5551 356.6661 — 3.2861 9.8580 1.2738 1.17860 .31 _ 109.9346 328.0398 _ 3.0017 9.5318 1.2842 1.18520 .32 — 96.3863 302.6362 — 2.7388 9.2305 1.2947 1.19200 .33 — 84.5894 279.9977 — 2.4947 8.9517 1.3053 1.19870 .34 — 74.1617 259.6280 — 2.2677 8.6931 1.3160 1.20550 .3 5 — 65.2436 241.5737 — 2.0560 8 .4524 1.3268 1.21230 .36 — 57.3038 225.2043 — 1.8580 8.2283 1.3376 1.21920 .3 7 — 50.3008 210.4173 — 1.6725 8.0194 1.3485 1.22620 .38 —- 44.1100 197.0166 — 1.4982 7 .8238 1.3596 1.23320 .39 — 38.6229 184.8422 — 1.3341 7.6408 1.3708 1.24030 .40 — 33.7500 173.7500 — 1.1794 7.4694 1.3820 1.24740 .41 _ 29.4099 163.6137 _ 1.0331 7.3084 1.3933 1.25450 .42 — 25.5424 154.3402 — 0.8946 7.1572 1.4048 1.26170 .43 — 22.0848 145.8272 — 0.7633 7.0147 1.4163 1.26900 .44 — 18.9890 137.9965 — 0.6384 6.8806 1.4280 1.27630 .45 —. 16.2139 130.7818 — 0.5195 6.7539 1.4397 1.28370 .46 —. 13.7213 124.1178 — 0.4062 6.6343 1.4516 1.29110 .47 — 11.4797 117.9529 —. 0.2979 6.5214 1.4635 1.29860 .4 8 — 9.4616 112.2390 — 0.1944 6.4147 1.4756 1.30610 .49 — 7 .6420 106.9335 — 0.0952 6.3136 1.4877 1.31370 .50 — 6.0000 102.0000 — 0.0000 6.2177 1.5000 1.32140 .51 _, 4.5162 97.4031 + 0.0915 6.1299 1.5124 1.32910.52 —. 3.1745 93.1146 0.1794 6.0407 1.5249 1.33680 .53 — 1.9606 89.1087 0.2642 5.9588 1.5374 1.34470 .54 —— 0.8611 85.3600 0.3459 5.8812 1.5502 1.35260 .55 + 0.1353 81.8481 0.4249 5.8074 1.5630 1.36050 .56 1.0388 78.5528 0.5013 5.7371 1.5759 1.36850 .57 1.8582 75.4575 0.5753 5.6703 1.5889 1.37660 .58 2.6016 72.5473 0.6469 5.6068 1.6021 1.38470 .59 3.2765 69.8056 0.7165 5.5464 1.6153 1.39290 .60 3.8888 67.2225 0.7841 5.4888 1.6287 1.40110 .61 4.4419 64.7836 0.8499 5.4339 1.6422 1.40940 .62 4.9496 62.4800 0.9139 5.3817 1.6559 1.41780 .63 5.4074 60.2517 0.9763 5.3319 1.6696 1.42620 .6 4 5.8228 58.2398 1.0371 5.2845 1.6834 1.43470 .65 6.1994 56.2857 1.0966 5.2392 1.6974 1.44330.66 6.5407 54.4327 1.1547 5.1961 1.7115 1.45190 .67 6.8496 52.6740 1.2116 5.1550 1.7258 1.46060.68 7.1292 51.0019 1.2673 5.1159 1.7401 1.46930.69 7.3836 49.4088 1.3218 5.0786 1.7546 1.47810 .70 7.6094 47.9007 1.3754 5.0430 1.7692 1.48700 .7 5 8.4445 41.3331 1.6297 4.8890 1.8442 1.53250 .8 0 8.9062 36.0938 1.8662 4.7691 1.9225 1.57970.85 9.1227 31.8461 2.0899 4.6774 2.0044 1.62870 .9 0 9.1769 28.3539 2.3045 4.6092 2.0899 1.67970 .95 9.1237 25.4468 2.5133 4.5518 2.1792 1.73261.00 9.0000 23.0000 2.7183 4.5305 2.2727 1.78761.05 8.8306 20.9200 2.9216 4.5151 2.3701 1.84471.10 8.6326 19.1359 3.1248 4.5135 2.4722 1.90421.15 8.4178 17.5937 3.3292 4.5242 2.5787 1.96591.20 8.1944 16.2499 3.5360 4.5462 2.6902 2.03011.25 7.9680 15 .0720 3.7462 4.5787 2.8065 2.09691.30 7.7424 14.0327 3.9608 4.6210 2.9283 2.16631.35 7.5202 13.1108 4.1807 4.6725 3.0555 2.23851.40 7.3032 12.2888 4.4065 4.7329 3.1885 2:31351.45 7.0926 11.5520 4.6390 4.8018 3.3275 2.39171.50 6.8889 10.8890 4.8791 4.8791 3.4729 2.47291.55 6.6926 10.2896 5.1272 4.9644 3.6250 2.55751.60 6.5039 9.7461 5.3841 5.0578 3.7839 2.64541 .70 6.1490 8.7993 5.9270 5.2685 4.1238 2.83231.80 5.8231 8.0042 6.5132 5.5112 4.4955 3.03481.90 5.5241 7.3291 7.1481 5.7865 4.9021 3.25462.00 5.2500 6.7500 7.8373 6.0957 5.3468 3.49222.10 4.9983 6.2488 8.5869 6.4402 5.835 3.7502.20 4.7671 5.8113 9.4036 6.8223 6.368 4.0302 .3 0 4.5541 5.4269 10.2941 7.2440 6.951 4.3352 .40 4.3576 5.0867 11.2663 7.7085 7.589 4.6652 .50 4 .1760 4.7840 12.3271 8.2184 8.282 5.024

In using our “ first exponential profile” [1 ] we find, ac­cording to operation 1 of the scheme, given in the last section,

By varying the ratio 2h0/4 w ithin the limits 2 and 3, respectively, we obtain the corresponding values of /3 and, according to operation 2 of the scheme, those of z0 = /3r„4/», Z(i = /3a4/,‘ from the following table:

TABLE l2Ao/4.0

2 . 02 .53 .0

00.01870.02470.0296

0.2380.3150.378

0.9301.2301.475

Now we introduce the boundary conditions accord­ing to operations 3 and 4 of the scheme. We first re­quire th a t, a t the overspeed na = 1.2 X 3600 = 4320 rpm, th e pressure between the shaft and the disk should disap­pear, th a t is, the radial stress o> m ust be equal to zero a t the bore (r = r0) for n 0 = 4320 rpm, while a t the rim (r = o) the disk is affected by a radial stress a, = 13,000 lb per sq in., as indicated above. The stresses are given by formulas [12], wherein the constant A is

we find

where

and

Zo = —569/(z0) Za = 13,000 — 4395/(2*)

In using our “ standard table” we find by interpolation the values which are tabu lated in Table 2. (See page 589.)

Correspondingly we will have from our Formula [12], the following approximate values:

No.123

TABLE 3 2 V 4 .0

2.0 2 .5 3 .0

<rto (lb/in.*) 59,00052.50049.500

As a result of our prelim inary calculations, we obtain the curve shown in Fig. 2, representing the tangential stress o-io a t the bore as a function of the ratio h o /h a = 27io/4.0 for given constant values of a, = 0 a t the bore (r = r0) and <Tr = 13,000 lb per sq in. a t the rim (r = a), re-

APPLIED MECHANICS APM-56-8 589

zo /(zo ) Zo <pi(zo) <pi(zo)0 .2 3 8 — 300 171,000 — 5 .7 5 1 .2 20 .3 1 5 — 108 61,500 — 2 .8 5 1 .3 00 .3 7 8 — 46 25,900 — 1 .5 5 1 .36

T A B L E 2

ta f (z a ) 9 .1 5 8 .0 5 7 .0 0

0 .9 3 01 .2 3 01 .4 7 5

Z a <pi(za) — 27,200 2 .4 3 — 22,400 3 .6 5 — 17,800 4 .7 5

<p2(za) g(zo) (zo)2 .1 5 650 1 2 .6 5

323 9 .3 5200 7 .8 5

2 .7 73 .4 0

'Mzr)1 .1 41 .1 91 .2 4

spectively. For 2/i0/4.0 = 1 the ordinate is calculated from the known formulas for the disk of constant thickness inasmuch as the use of those for our exponential profile requires, in this special case, a complicated passage to the lim it 0 — > 0 .

Even by the approximate result represented by Fig. 2 the conclusion is justified, th a t the rate of the decrease in the amount of the tangential stress a t the bore is getting smaller and smaller with the increase in the value of the ratio ho/ha. This result can be assumed to be of general character although it is ob­tained here for a special case.

From the diagram Fig. 2 we have now to decide upon the value of ho/ha = 2/i0/4.0 on the basis of the maximum admissible value of <ri0. The question of the maximum admissible working stress a t the bore in overspeed conditions will not be discussed here generally. For our example we consider m0 = 52,500 lb per sq in. as a permissible am ount for the tangential stress a t the bore for 20 per cent overspeed, corresponding to the value h0/ha = 2.5. Then we will have

0 = 0.0247

from Table 1, giving a profile curve represented by

in Table 2.

Correspondingly, w ith ovo = —500 lb per sq in. for r = r0 a t the overspeed, Z 0 = —500 + 12.5 X 6.502 X 125.56 = 65,900 instead of Z0 = 61,500 in Table 2; Za = 13,000 — 12.5 X 18.941 X 7.968 = —22,750 instead of Za = —22,400

Therefore

z y (in.) r (in.) /(*)

T A B L E 4

o(z) Mz) <pz(z) ^ t ( z )0 .3 0 5 .0 7 6 .5 0 — 125 .555 3 5 6 .6 6 6 — 3 .2 8 6 9 .8 5 8 1 .2 7 4 1 .1 7 90 .6 0 3 .7 6 10 .9 4 3 .8 8 9 6 7 .2 2 3 0 .7 8 4 5 .4 8 9 1 .6 2 9 1 .4010 .9 0 2 .7 8 14 .83 9 .1 7 7 2 8 .3 5 4 2 .3 0 5 4 .6 0 9 2 .0 9 0 1 .6 8 01.10 2 .2 8 17 .21 8 .6 3 3 19 .136 3 .1 2 5 4 .5 1 4 2 .4 7 2 1 .9 0 41 .25 1 .9 6 18 .9 4 7 .9 6 8 15 .072 3 .7 4 6 4 .5 7 9 2 .8 0 7 2 .0 9 7

Instead of the given limiting radii r0 = 6.75 in. and a = 18.75 in., we introduced in this table boundary values of r approxi­mately equal to r0 and a, respectively, which in our “standard table” correspond to tabulated values of z and its functions /(z), g(z), <ei(z), tpi(z), ip i(z), i/-2(z). The reasons for this pro­cedure are as follows.

In using interpolations as indicated above, we multiply by large numbers the errors introduced by these interpolations in the course of our calculations. Considerable inaccuracy can be caused hereby in the results. I t is much better, therefore, to proceed in the way shown in our Table 4, namely, to use numbers tabulated in the “standard table” w ithout interpola­tions. Naturally, some deviations will be caused by the fact tha t, in the special example treated , the boundary conditions actually refer to z0 = 0.315 and za = 1.230, according to the Tables 1 and 2, and not to z0 = 0.30 and z« = 1.25, according to Table 4. B ut these deviations will be smaller generally than the errors produced by interpolations. And, besides, these deviations due to inaccuracy in fulfilling the boundary conditions are of negligible order, because in practical cases those boundary values cannot be sta ted very accurately anyway.

In using Table 4, with the approximate values of z0 and za as given by the first and the last lines of th a t table, respectively, we find th a t

F i g s . 3 a n d 4

stress distribution in using the stress Formulas [12], according to operation 5 of the scheme given in the last section:

T A B L E 5r ( in .) V (in.) <rr ( lb / in .2) at ( lb /in .a)

6 .5 0 5 .0 7 — 500 52,4001 0 .9 4 3 .7 6 13,850 34,20014 .8 3 2 .7 8 15,700 28,20017 .21 2 .2 8 14,600 25,20018 .9 4 1 .9 6 13,000 23,400

The profile curve as given by the first two columns of this table is represented by Fig. 3. The stress distribution according to the th ird and the fourth columns of Table 5 is shown in Fig. 4.

According to the general conclusions drawn in Appendix No. 1 the maximum tangential stress a t the bore will be about 5 per cent larger than the average am ount given in our Table 5. The maximum stress, therefore, in our example will be about 55,000 lb per sq in.

As to the profile curve, it does not differ appreciably from a straight line. This, obviously, will be still more nearly the case for disks w ith a ratio h0/h a < 2.5, the radial dimensions being of the same order as in the example treated.

T h e S e c o n d E x p o n e n t i a l P r o f i l e

If the foregoing method of developing the first exponential profile be applied to Equation [41 ],7 a new solution is obtained

7 See Appendix No. 2.

W ith these values of K and L we obtain from Table 4 the following

according to the same table. The constant a is given by

Now we use again our “standard table” of the first exponential profile and obtain the following table:

590 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

characterized by a profile curve for which

the values of y a t the bore (r = r0) and a t the rim (r = a), re­spectively.

This solution is rem arkable for the fact th a t both of the stresses as well as the profile are given by finite expressions; as to the exponential function e‘, occurring in those expressions, i t is tabulated very extensively. Therefore the formulas of the “second exponential profile” can be used for steep profile curves as well; th a t is, for curves characterized by large values of 0. This is of some practical advantage for calculation of the disk p art connecting the wheel w ith the rim. For such a part 0 has a large negative value, in either of both profiles, and the formulas of the “ first exponential profile” cannot be used, be­cause in the vicinity of such 0 values the series occurring in the stress expressions converge very slowly. Formulas [16] and [17] are, of course, free of such objections. This explains why positive values of (—0) were no t considered in our Formulas [1], [10], and [11], I t m ust be remarked, however, th a t the stress formulas are the less accurate, the steeper the profile curve becomes.

By tabulating the functions

W ith these values of the constants A and B and with th a t of 0 the following table is obtained by using some intermediate values of z:

T A B L E 6

z a r ( l b / i n . 2) a t ( l b / i n . 2) r ( in .) V ( in .)0 .9 3 — 250 53,000 6 .6 5 5 .0 71.00 4,650 47,400 7 .4 2 4 .7 51 .2 5 13,700 36,300 10 .37 3 .6 81.50 17,000 30,800 13 .6 0 2 .8 71 .7 5 14,800 26,600 17 .1 5 2 .2 31.86 13,200 24,500 18 .8 5 2.00

The profile and the stress distribution according to this table are shown in Figs. 5 and 6, respectively. For a comparison with the results obtained above for the “ first exponential profile” (see the first example) the profile curve and the stress distribu­tion, represented by Figs. 3 and 4, respectively, are shown dotted in Figs. 5 and 6.

A “standard table” for the “second exponential profile” will be available later.

and which m ay be designated as the “ second exponential pro­file.” The stresses are, w ith v = 1/3, given by the expressions

where A and B are the two arbitrary constants of integration, while 0 is determined by the ratio h0/h a; h„ and ha being again

We introduce the values z0 = 0.93; z„ = 1.86 into our further calculations, according to the practical rules given in the last section. By substituting

<r, = —250 lb per sq in. for z = zo = 0.93 o> = 13,200 lb per sq in. for z = Za = 1.86

we find from Equations [16] and [17]

This equation shows th a t yn is always smaller than yi, provided th a t a t the boundaries (r = r0 and r = a) yi and yu are, respec­tively, equal to each other. The profile curve y\ has a hori­zontal, and the profile curve yn a vertical tangent a t the point r — 0 (see Fig. 5).

P r a c t ic a l E x a m p l e o f D i s k D e s i g n — S e c o n d E x p o n e n t i a l

P r o f i l e

As an example we trea t here the same problem, as in the last section, in using our “second exponential profile.” The pro­cedure is essentially the same as indicated in the scheme given above for the “first exponential profile” and we find, for h0/h a = 2.5,

From these two equations we find

are the profile curves joining the point r = r0, y = y0 with the point r — a, y = ya. Then we will have

all calculations can be accomplished in the manner outlined for the “first exponential profile.”

C o m p a r i s o n o f t h e Two P r o f i l e s

I t is, of course, of practical interest to compare the two ex­ponential profiles. Suppose

APPLIED MECHANICS APM-56-8 591

S t r e s s e s D u e t o F it P r e s s u r e — E l a s t ic R e s i s t a n c e W i t h

V a r y i n g S p e e d

The boundary conditions for the problem of stresses (p, , pi) in a disk due to fit pressure in resting conditions are8

where ato is the tangential stress a t the bore for « = wo (over­speed). W ith these boundary conditions the problem is to be confined to special profiles. I t is, however, possible to draw general conclusions from the following consideration.

For a disk of constant thickness the general solution for stresses due to fit pressure under static conditions is given by the for­mulas9

(Ci = const, and C 2 = const.)

By introducing the boundary conditions given above, we find

L and M are arb itrary constants of integration and /i(r), fi{r), gi(r), gz{r), h\(r), h2(r) are the integral functions for the profile considered, the boundary conditions are to be stated from which the constants L and M should follow as functions of the speed «.

Designate by pr(w) and pt(u) the disk stresses as functions of «. They usually are calculated for the overspeed « = w0 and appear then as functions of r, so th a t, w ith our former designations,

The boundary conditions for a, and at are, as we know,

where r0 and a are the radii of the bore and the rim respectively wrhile aa designates the radial tension a t the rim exerted by the centrifugal forces of the blading a t the overspeed. These boundary conditions for ar result in a corresponding value of at a t the bore, equal to ato, and from the values of aro and ato a t the bore, for « = w0, the boundary conditions [43] and [44] for p, = pr(w = 0) and pt = pt{oi = 0) under static conditions are obtained, as showTi in Appendix No. 2. Now, by using the same manner of reasoning we find the condition

where K is a constant independent of the speed u, to be true for any speed w. This condition can be w ritten, with close approximation, in the form11

11 See Appendix No. 2.

where 2r0 and 2a are the bore and the rim diameter, respectively. The stress distribution given by [18] is shown graphically in Fig. 7. The formulas [18] together with Fig. 7 adm it of a simple interpretation. As soon as the ratio a /r0 exceeds a cer­tain value, say a fro ^ 3, the stress distribution according to the curves pr and pt in Fig. 7 does not vary essentially with varying a. The maximum values of pr and pt always take place a t the bore, and these maximum values are always about 40 to 60 per cent respectively of the maximum tangential stress ato a t the bore under overspeed conditions. In other words, only the parts of the disk in the vicinity of the bore are essen­tially carrying the fit loading under static conditions, an increase of the outer radius a being of little influence on the functions [18].10

This result can be easily generalized for disks of other profiles. We have only to realize th a t the disk of constant thickness investigated above can be subdivided into several separate rings; the larger the diam eter of any of them, the less its influence on the stress distribution of the whole disk under static conditions, as shown above, and the less, consequently, the stress variations in the disk due to the reduced thickness of th a t ring as compared with the original thickness of the parallel sided disk. This means th a t the statem ent developed above for disks of constant thick­ness is qualitatively and, with a certain approximation, also quantitatively true for disks of any profile.

This general result shall be checked now in calculating, for a special disk, the Stresses pr and pi due to fit pressure under static conditions. The question may be treated on a somewhat more general basis by considering the disk as rotating with varying speeds. This is useful for a better understanding of the elastic behavior of the disk in various conditions of the actual service.

Having the general solution for rotating disks in the form

where A is a certain constant determined by the profile, while

8 See Appendix No. 2.9 A. Stodola, “ S team T u rb in es ,” sec. 76, E q . 22, w ith w = 0 and

A = 0.10 Cf. A. Foeppl, Fest.igkeitslehre, Sect. 56 (“ D ickw andige R oeh-

ren ” ).

592 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS

where <rto is the tangential stress a t the bore (r = ro) for co = coo (overspeed). A t the outer boundary, r = a, we will have

from which we obtain

If we substitute co = 0 in the general Equations [22] and calculate the corresponding values of L and M , the following expressions for the stresses due to fit pressure under static conditions are obtained from [21 ]:

where <ra is the value indicated above. The boundary condi­tions [19] and [20] are to be introduced into

According to the general statem ent developed above these stresses m ust reach amounts of about

respectively, a t the bore, for any profile, if o ^ 3r0. By substi­tu ting r = ro into [23] we find

These are two equations for the two constants L and M . These constants of integration, therefore, appear as functions of the speed co and in consequence the problem of stress distribution in terms of the speed is solved.

For the disk of constant thickness, for instance, we have

for a special example. Consider a disk w ith a = 21 in., r0 = 7 in., while the ratio yo/ya may vary between 1 and 3. For yo/ya = 1 the formulas for disks of constant thickness are to be used. Generally we will have

W ith the given values of a and r0 we find the values of Zo and Za according to the formulas

and then, by using the “ standard table,” w ith the symbols <pi, <p2f i£i, and fa, occurring in th a t table, the corresponding values of fi , fiy gi, and g<i are obtained as follows:

v * /y o

2.0

3 .0

TABLE 7 / i * <pi ft

!20 IZa

0.2080.9020.3301.430

—7 .52 .3

—2 .54 .5

14.64 .69 .04 .8

ffi ■» <ps fft m 'pi1.19 1.122 .10 1.691.31 1.203 .28 2.36

By substituting these values in Formulas [24] we find the ratios Pro(0)/<ri0 and pta(0)/ata for the exponential profile w ith yo/ya = 2 and yo/ya = 3, respectively. For the disk of constant thickness the aforementioned ratios are to be found from the Formulas [18]. The results obtained in this way are set forth in the following table:

y o /y a123

TABLE 8Pro (0)/<Tto —0.444 —0.397 — 0.369

Pio (0)/<r 0.556 0.603 0.631

These results are graphically represented in Fig. 8. From these it is evident th a t, even for ro/a = 1/3, the stresses due to fit pressure under static conditions do not depend essentially on the form of the disk; which is in full accordance w ith the state­m ent developed above. The quantitative agreement with the result yielded by the general consideration is very satisfactory.

An im portant result is expressed, furthermore, by Equation [19] in its original physical meaning. The strength of an elastic

These formulas shall be applied to our “first exponential profile”

and the constants of integration are

So we finally have for the disk of constant thickness

representing the stresses for any speed between co = 0 and a = wo, provided th a t ato and aa are the tangential stress a t the bore and the radial stress a t the rim, respectively, a t the over­speed. If w = 0 is substituted, the solution as developed above applies to stresses due to fit pressure under static conditions.

Referring to the general Equations [21 ] for pT (a) and pt(a) and Equations [22] for the constants L and M as functions of the speed o> for any profile, it is easily seen th a t both stresses always can be represented in the form

where Fi(r) and Ft(r) are certain functions of r. In other words, the stresses vary proportionally to cos, the proportionality factor being a function of r.

12 See Appendix No. 2.

APPLIED MECHANICS APM-56-8 593

In other words, the stress curves are completely determined if two conditions are to be fulfilled corresponding to the two con­stants of integration occurring in the general expressions for the stresses.

Now consider the same disk with a hub relief according to Fig. 9. We require again the conditions covered by Equations [26] to be fulfilled. As these two relations determine the stress curves, we will have the same stress diagram (see Fig. 12) in the new conditions, as in Fig. 11, as long as the disk profile is the same, i.e., from r = a to r = r0 + A, where r0 + A is the radius corresponding to the point where ar vanishes in Fig. 11. This restriction (the vanishing of o> for r = r0 + A) will be removed below.

As to the stresses in the disk, or rather ring, represented by the section ABCD, it is easy to see th a t the radial stress is equal to zero a t the outer edge BC. Indeed, the radial stress being zero a t the edge B E of the disk BGFE, the equilibrium requires th a t the same be the case along BC in the disk ABCD. The tangential stress along BC in the same small disk is deter­mined, as well known, by the condition th a t the tangential elongation u a t the cylindrical section BC m ust be the same for both disks. This elongation being given by the formula

the stress at m ust be the same on both sides of BC, since ar is the same on both sides. In other words, neither a, nor at can jump a t the outer edge BC of the disk ABCD . So both of the boundary values of ar and at are known a t the outer edge BC of the small disk, and the stresses are completely determined, therefore, for the part between r = r0 and r = r0 + A according to previous statements. Furthermore, these boundary values being the same in Fig. 9 (for the ring ABCD ) and Fig. 10 (for the ring A 'B 'C 'D ') the stresses within the part between r = ra and r = r0 + A must be the same in both cases. Therefore

F ig s . 9 t o 14

the stress distribution for Fig. 12 is identical with th a t for Fig. 11.This result is developed on the basis th a t ov = 0 for r = To + A

in Fig. 11, i.e., th a t the depth of the hub relief is equal to the distance A a t the end of which ar vanishes in the disk without hub relief. If ar = S for r = r0 + A in Fig. 11, where S is a small amount, positive or negative, then or jumps in Fig. 12, and we will have

body is characterized, according to the well-known theory of Otto Mohr, by the difference of the largest and the smallest of the three principal stresses. The axial stress at, i.e., the principal stress along the axial direction, being approximately equal to zero, the difference a t the bore will be always equal to pt — pr, because a t the bore pro(u) ^ 0 and pio(w) > 0. According to Equation [19] this difference in stress values a t the bore is always the same a t any speed. Therefore, we have, w ith close approximation, the general result, th a t the strength of the disk at the bore is always the same at any speed « within the limits0 ^ u < OJO*

D i s k W i t h H u b R e l i e f

An instructive example for application of the results developed in the last section will be obtained by studying stress conditions in disks with hub relief as shown in Fig. 9.

First consider a disk designed w ithout hub relief (Fig. 10). The corresponding stresses may be represented by Fig. 11, the boundary conditions being given, for w = wo, by the relations

where cra is the tension due to the centrifugal forces of the blading. Of course, the stress curves in Fig. 11 are also com­pletely determined, if the conditions [25] are replaced by

594 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

to the left of BC. The corresponding value of at follows from the jum p Bat of at according to our Equation [27]:

As 5 is a small am ount anyway, the stresses do no t jum p essen­tially a t the section BC and the whole consideration above is practically no t affected a t all. This means th a t under rotating conditions no change in stress distribution is caused by the hub relief.

Now consider both disks, w ith and th a t w ithout hub relief, under static conditions. The stress distribution in the la tte r is represented by Fig. 13 in accordance w ith our previous con­siderations, and we know, th a t

according to [28] and [29], we will have

For r = r0 this difference m ust be equal to S 'T ' in the case of Fig. 10 and to P 'Q ' in the case of Fig. 9; so we have

and

where c is an arbitrary constant of the same kind as b. There­fore, for r = r0 + A

while a t the bore (r = ro) the difference of both stresses pi and p , m ust be the same a t any speed w. Therefore, under static conditions, this difference m ust be, w ith reference to Fig. 12

Now we have

and, according to [31]

and the question is, to determine, in an approximate way, the stress distribution satisfying the last two boundary conditions in the disk shown in Fig. 9. S tarting w ith the outer edge r = a in Fig. 14 we state, th a t for the disk p art BEFO (Fig. 9) the stress distribution is completely determined if we assume a certain value of pt for r = a, p r being equal to zero a t th a t point. Assuming p i to be of the same am ount s as in Fig. 13 a t the outer edge r = a, we find, of course, th e same stress curves p , and pt in Fig. 14, as in Fig. 13, for r0 + A ^ r ^ a. At the point r = ra + A the stresses a, and <ri jum p, according to Equations [28] and [29], from the points S and T to the points P and <2, respectively. Since

This shall be proved as follows.For r0 + A ^ r r0 th e profile represents a disk of constant

thickness in both Fig. 9 and Fig. 10. Therefore th e general expressions for the stresses in bo th of them , for static conditions, are represented by th e relations

which is nothing other th an Equation [30].We have found, therefore, th a t if, for r 0 + A ^ r jC a, the

stresses pt and p, in Fig. 14 are the same as in Fig. 13, the dif­ference of the stress values pm and pro a t the bore, represented by P 'Q ’, will be larger than the same difference under rotating conditions as represented in Fig. 12. This being impossible, our assumption, according to which the stress curves are the same for r0 + A ^ r ^ a in both disks with and without hub relief in resting conditions, m ust be altered. In other words, for r — a the tangential stress m ust be less than s in Fig. 13. We then obtain stress curves pt and pr, determined by both boundary conditions for r = a. I t is easily seen th a t the curves pt and pr qualitatively correspond to each other physically. If the radial stress, which is compression, is diminishing from pr to pr (Fig. 14), the tangential stress, which is tension, m ust diminish from pt to pt. A t a certain definite decrease of both stress functions the state will be reached a t which the previously mentioned stress difference a t the bore will assume the required value, independent of the speed «. This means th a t by using a disk w ith hub relief, the stresses are the same under rotating conditions, as in a disk without hub relief, and they are smaller, throughout the disk except the hub, under static conditions, while a t the bore they remain within permissible limits deter­mined by the relation p«> — pro = const, a t any speed a.

Appendix N o. 1T h e P r o b l e m o f R o t a t in g D i s k s i n t h e M a t h e m a t ic a l

T h e o r y o f E l a s t ic i t y

The rigorous mathem atical basis for the design of steam- turbine disk wheels consists in a solution of the following problem w ith boundary conditions.

Consider the state of strain and stress in an isotropic elastic body of revolution rotating about its axis z-z w ith the angular velocity w. The displacement will be symmetrical about the same axis. The components of the displacement being u in the radial and w in the axial direction, the conditions of elastic equilibrium are expressed by the differential equations18

13 A. E . H . Love, “ M athem atica l T heory of E la stic ity ,” 4 th ed., C am bridge, 1927, p. 146, an d p. 104.

or approximately

o-ro being of negligible amount. The stress distribution m the disk w ith hub relief under static conditions is again determined by the boundary conditions given above. These are th a t a t the outer edge we m ust have

APPLIED MECHANICS APM-56-8 595

The boundary conditions express the fact th a t the lateral surfaces of the disk, represented by the equation y — f(r), where f ( r) is a known function of r, are free of stresses, while the edges r = ro and r = a (Fig. 15) can be affected by given forces. In special problems the inner boundary surface can vanish (disk without bore). The stresses in the rotating disk are to be determined.

In the form [32] the problem has been solved for a few special forms of the meridian curve (profile) only, par­ticularly for the cylinder14 and for the ellipsoid15 under certain boundary conditions.16

I t is of practical interest and importance for compari­son with the procedure used in technical applications as reproduced below to know the results yielded by the methods of the theory of elasticity. Therefore a concise report concerning the solutions just mentioned shall be given.

We designate by a , , a t, and a* the normal stresses along the radius r, the tangent, t, to the circle of the radius r, and the direction, z, of the axis of rotation, respectively. As to the shear stresses, the components rrt and Tzt vanish by reasons of symmetry about the axis, while t „ 0 generally.

Consider now the stress distribution determined by the equations

This stress distribution is produced by a certain elastic displacement satisfying Equations [32].17 The solution [33], if applied to the case of a cylindrical disk with­out bore, fulfils the boundary conditions a t the plane surfaces z = ± I. Indeed, the stress component at and the shear stresses vanishing identically, they vanish also a t the surfaces z = =*= I. B ut a t the cylindrical surface r = a both the radial stress oy and the tangential stress <7i follow a special law of distribution along the axial

11 A. E. H. Love, loc. cit.; and F. Purser, Trans. Royal Irish Acad., vol. 32, 1902.

16 C. Chree, Royal Soe. Proc., London, vol. 58, 1895; treated the general ellipsoid.

16 See also Zeitschr. f. Ang. Math. u. Mech., vol. 3, 1923, p. 319.

17 A. E. H. Love, loc. cit., Eq. [70] and [71], and p. 56.

596 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

direction z, as shown by Equation [33]. According to this law the average am ount of o> along th a t direction vanishes for r = a. The additive correction of Equation [33], necessary to produce boundary stresses a, — 0 for r = a, is discussed in the article of Purser, previously mentioned. This correction is of importance only for calculation of stresses in the vicinity of r = a. For parts which are no t too near to the edge, the stress sta te of the disk is represented by Equation [33] w ith sufficient accuracy according to the well-known principle of de Saint V enant,18 provided th a t the thickness 21 is small compared with the diam eter 2a. The axial distribution of the stresses a t the edge r = a is here of little interest. T hat a t the inner edge in disks w ith central bore- is much more im portant. For such disks the solution [33] is to be replaced by the following formulas:

According to the solutions [33] and [34] the stress variation along the axial direction, z, is of an almost vanishing amount, if I, equal to a half the thickness of the disk, does no t exceed a certain lim it as compared w ith the outer radius of the disk.

Take, for instance, a solid cylindrical disk for which a = 51. Suppose, this ratio justifies the application of the principle of de Saint Venant. In Fig. 16 the curves a, and at show the stress distribution, according to Equations [33], for the middle plane, (z = 0), of the disk. The stress variation along the axial direc­tion z is represented by the parabolical curves Z-Z , analytically expressed by the formula A a = Cz2, C being a constant and — I ^ z ^ I. The am ount of this stress variation is about 2.5 per cent of the maximum stress a t the axis r = 0.

The lower parts of the curves in Fig. 16 are dotted, because in the vicinity of r = a they do no t represent the actual stress conditions, as explained above.

Fig. 17 shows the stress distribution according to Equations [34], in an analogous way, for the same disk as in Fig. 16, bu t having a bore, the radius of which is equal to I. The maximum stress appears again in the vicinity of the inner edge of the disk; bu t it is almost twice as large as in the case of the solid disk of Fig. 16. The stress variability along the axial direction is the same as before, or about 1.5 per cent of the maximum stress in the varied conditions.

The stress distribution in an ellipsoid according to the exact solution cited above is discussed in detail by Stodola.20 We do no t reproduce the complicated formulas for the stresses; only the general results shall be mentioned.

For a solid disk of elliptical meridian curve having a diam eter 2a and a maximum thickness 2c a t the axis, Stodola finds a stress variability along the axis (r = 0) as represented by the following table:

TABLE 9c /a l/s l/ t l/s

A<r/<r (%) 5 13 45

** A. E. H. Love, loc. cit., p. 132.•• A. E. H. Love, ibid., p. 148.20 A. Stodola, “Steam Turbines,” New York, 1927, vol. II, sec.

184. See also his article in Zeit. V.D.I., 1907, p. 1259.

where a designates the tangential or radial stress for r = 0, Z = 0 , while A a = o-(r = 0, z = o)— ®'(r-=o, z = l)-

I t is interesting to observe, th a t the stress variability along the axis is of a larger am ount for the elliptical disk than for the cylindrical one. This result is quite natural, if the uniform mass distribution along the axial direction in the case of a cylin­drical disk, causing a uniform distribution of the centrifugal forces along the same direction, is taken into account. B ut even for a disk of elliptical profile, having a maximum thickness equal to 1/ i of the diameter, the absolute value of the excess of the stress am ount over the average stress along the axis remains w ithin the limits of about 7 per cent of the maximum stress.

A remarkable detail is characterized by appearance of normal stresses a , in the axial direction. These stresses a , represent compression in the disk parts of larger, and tension in those of smaller thickness. They reach the amount of only a few per cent of the maximum norm al stress.

From analogy w ith the cylindrical disk we m ay conclude, that, if the elliptical disk is provided w ith a bore, the non-uniformity in the stress distribution along the axial direction a t the bore, as defined by the expression (aimsx — + o-fmin),will be about a half of the am ount indicated by the last table, so th a t the absolute value of the excess over the average stress along the axial direction remains within limits of about 3Vj per cent of the maximum stress for c/a = 1/4.

In summarizing we arrive a t the following conclusion with certain approximation, special regard having been given to the increased proportions in modern disk design: By increasing the thickness of the disk from 1/» to V* of the diameter the absolute value of the excess of the stress values over the average stress along the axial direction increases in disks w ith bore from the order of about 1 per cent to th a t of about 4 per cent of the maximum stress, while the axial stresses a z still remain within negligible limits.

The rigorous treatm ent of the problem of rotating disks in general form, i.e., for given meridian curves, by using methods of mathem atical physics, would be, a t the present stage, in certain branches of the m athematical analysis, a t least exceed­ingly complicated.21 A very valuable new method of analyzing plates and disks22 is invented by G. D. Birkhoff ;2* in its further development th is method m ay acquire practical importance for disk design. As far as technical requirements are concerned the approximate method, introduced by Stodola,24 is very satis­factory. His procedure is based upon the fundamental con­clusion drawn from the exact solutions as previously reported. The non-uniformity of the stress distribution along the axial direction in rotating disks can be neglected within certain limits determined approximately by Table 9 and emphasized in the foregoing conclusion. This basis of strength calculations in disk design does no t lose its validity in the new conditions characterized by increased disk proportions.

Appendix N o. 2D i f f e r e n t i a l E q u a t io n s i n A p p r o x im a t e s F o r m o f

R o t a t in g D is k s

Since the variability of the stresses o> and at along the axial direction is, to all practical purposes, of negligible amount, they

21 See St. Bergmann, Mathem. Ann., vol. 98, 1927, p. 248. A detailed report is given by I. Malkin, Zeit. f. Ang. Math. u. Mech., vol. 10, 1930, p. 182.

22 The mathematical analogy between circular plates under bending and rotating disks is established by L. Foeppl, Zeit. f. Ang. Math, u. Mech., 1922.

28 Phil. Mag., vol. 43, 1922, p. 953; also C. A. Garabedian, Amer. Math. Soc. Trans., vol. 25, 1923, p. 343.

21 Stodola, “Steam Turbines,” sec. 74.

APPLIED MECHANICS APM-56-8 597

use to be considered as functions of r only. W ith this simpli­fication, Stodola obtains from the conditions of equilibrium [see Fig. 18] the equation24

and by introducing [36] into [35] the equation

In elastic conditions the stresses follow from the radial displace­ment u according to the relations

is obtained. The elimination of u from Equations [36] yields the condition

The differential equations reproduced here are used in pre­ceding sections for developing new solutions of the problem of rotating disks. Furtherm ore, in using Equations [36], the boundary conditions for the problem of stresses due to fit pres­sure in the disk under static conditions can be obtained as follows.

Designate byd the bore diam eter of the disk D the diam eter of the shaft

both under undeformed static conditions, and by «o the radial displacement of the disk Mo' the radial displacement of the shaft

both for r = ro and for any speed w between 0 and the over­speed wo- W ith these designations we obviously m ust have

where

28 A. Foeppl, “Technische Mechanik,” vol. V, 1922, p. 87. 26 A. Stodola, loc. cit., sec. 181b.

of compatibility of both stresses.In assuming a certain curve, y — f(r), as the meridian curve

(profile) of the disk a differential equation for u is obtained from Equation [37]. If the solution of this differential equation is known, the stresses can be determined by Equations [36] in connection with the boundary conditions. Should the stresses, as following from these calculations, exceed permissible limits, the assumed profile m ust be modified and the calculations repeated until satisfactory results are obtained.

Sometimes another equivalent procedure is used in solving the systems [35] and [38] of differential equations. A certain kind of stress function S is introduced by the formula25

In using th e well-known formulas for a disk of constant thick­ness27 w ithout bore we will have

Correspondingly

Now. bv introducing

By using these expressions Equation [35] is satisfied identically, while Equation [38] requires

an equation which does not differ much from Equation [37] for the displacement u.

Still another analytical expression for our problem can be obtained by eliminating a, or at from Equations [35] and [38]. By eliminating at, Stodola finds the following differential equa­tion for o>:26

Herein aTo is equal or approximately equal to zero. As to the quantity V«(l — v)nw,2r02, it is a small value which m ay be neglected in the last equation. Indeed, with reference to the example treated in the preceding section we have

27 See Stodola, ibid., sec. 76.

where

so th a t the differential equation of equilibrium in approximate form can be written as follows:

where at and ar are the tangential and the radial stresses, re­spectively, for r — r0 a t th e speed w, in th e disk, while at' and <»' = a, are the corresponding stresses in the shaft for r = r0. The shrink fit is, therefore,

provided th a t, a t any speed o> < wo, there is a certain pressure between the disk and the shaft. The “shrink fit” is corre­spondingly given by

The displacements Uo and ito' are according to Equations [36]

598 TRANSACTIONS OF TH E AMERICAN SOCIETY OF MECHANICAL ENGINEERS

We see th a t if y becomes smaller, p will increase, i.e., we have more terms and the calculations are somewhat longer. However a study of the general solution shows th a t the series have a better convergence for small values of y therefore fewer term s in the series need to be considered.

The method used by the author is practical only if tables for the values of the series are calculated beforehand. Such tables are given in this paper for the case y — V 3, and the author also proposes to have tables calculated for his second case where y = 2/„. Since the calculations are rather long, it m ight be advisable to determine first which profile is better adapted for practical purposes.

Due to the more rapid convergence of the series for small values of y , a larger range for the variable z = f f r y m ay be taken, therefore disk profiles differing more from the conical shape can be approximated w ith y = x/ ( or Vs for instance, than w ith y = 4/ 3 or 2/ 3. I t seems then, th a t tables for a value of y smaller than 2/3 would offer more advantages to the designer of turbine disk wheels th an the second profile proposed by the author.

A r& um 6 of the general solution will be given here: the differential equation30

28 See Stodola, ibid., sec. 81d.29 University of Michigan, Ann Arbor, Mich.30 Author’s Equation [37],

M i is a function of y and 0.The condition [48] for 7 is obtained through the analysis by

making the assumption [47] and [49].

C. R. S o d e r b e r g .32 Dr. M alkin’s contribution to the disk problem is a very real one and I feel certain th a t it will find its place among the classics of the subject.

I t is of importance to the designer to have available a method of stress calculation which is sufficiently rapid to perm it evalua­tion of the stresses for m any combinations in a short time and yet of sufficient accuracy to make the results reliable. During the period of its use, we have had ample opportunity to demon­stra te the great merits of the new method.

Concerning the problem of design limits, i t seems th a t the following questions m ust be discussed: (a) tangential stress a t the inner bore a t overspeed; (6) permanence of the shrink fit on the shaft a t the normal speed, and (c) normal pressure on the shaft a t standstill. These aspects of the failure problem m ust be weighed in the order mentioned. The importance of the last item is difficult to evaluate. Failure in the ordinary sense of the word does not occur under normal pressure until its intensity m aterially exceeds the yield strength. This state is

31 Author’s Equation [5].32 Westinghouse Research Laboratories, East Pittsburgh, Pa.

Mem. A.S.M.E.

or about 2 per cent of m0 = 52,400 lb per sq in. So the boundary condition in question can be w ritten as follows

This is the inner boundary condition for the stresses p t and pr due to fit pressure under static conditions. The other boundary condition expresses the vanishing of p r a t the outer disk edge r = a :

The boundary condition [43] is here obtained in a way some­w hat different from th a t used by Stodola28 in order to show the limits of accuracy of this condition.

D iscussionJ. L. M a u l b e t s c h .29 I t can be shown th a t the two profiles

given by the author belong to a series of profiles of the type

where

anda — any positive integer

i.e.,

and th a t any profile of this series will allow a similar solution to the one presented by the author.

The solution for any value of 7 consists of two infinite series and of an expression containing a few term s only. For 7 = 4/a and 7 = 2/ 3, this last expression has three term s. In general, for any value of 7 , the num ber of term s p is:

where

and

This limiting condition for 7 is due to the fact th a t we want to obtain a particular solution of Equation [45] so th a t a homo­geneous equation m ay be obtained.

The particular solution is of the type

and limit 7 to the values

where

and

if we assume

is not homogeneous. I t is reduced to the following homogeneous equation31 which can be solved with series:

APPLIED MECHANICS APM-56-8 599

seldom reached, but the pressures are usually so high th a t the designer does not feel he can ignore them.

A u t h o r ’s C l o s u r e

Mr. Maulbetsch’s generalization of my solution involves two infinite series in each of the two stress expressions. In recom­mending 7 = 1/ i he introduces two further expressions with

as compared with za = 1.25 and za = 1.86, respectively, in the above examples. The increase in za caused by decrease of 7 is a factor of opposite effect to th a t of more rapid convergence.

The necessity of calculating tables cannot be considered as a disadvantage of the au thor’s m ethod, because such tables are necessary in any analytical solution, and it is contended by the writer that, for two reasons, the method of solution as set forth is simpler than any other analytical solution so far developed for the problem. First, it is much more convenient to calculate t he tables in question by using simple finite expressions, or solu­tions with only two infinite series, than by dealing with six in­finite series, as in the case of conical disks, or even with finite expressions of considerable complicacy such as encountered in the case of hyperbolic disks. The other im portant advantage is the fact th a t the w riter’s method of solution is based upon the principle of similarity which, in the case considered, assumes the following form: If the profile curve is given by an expression y — / (r, 0), where 0 is the essential constant param eter characteriz­ing the individual disk, the solution will be obtained generally in terms of two varying essential quantities, r and 0. In using f. i. tables calculated for conical disks we have to multiply all values, for any special disk, by a constant varying w ith th a t disk. The calculations are, in other words, two-dimensional. In the case of the exponential profiles set forth by the writer the solution is given essentially in terms of a certain combination of the variable r and the param eter 0, namely in terms of z — 0r't* and z = 0r ^ , respectively. The main calculations, conse­quently, include only one variable. These are the two advan­tages which are of practical value in tabulating the solutions and in using the tables for design purposes.

As to the question of technical adaptability of the writer’s method of solution, our attention is called to Fig. 19 in conjunc­tion with the following remarks.

The problem of practical disk design is not a problem merely of avoiding the conical profile. I t is rather a problem of profile variation between two limiting profile curves, the conical and the hyperbolic. Fig. 19 is a typical example of the practical possi­

bilities of choice offered to the disk designer by the two expo­nential profiles. They not only represent definite special solu­tions, bu t practically fill the range between the two limiting solutions, inasmuch as, if completed by the latter, they approxi­mate, w ith sufficient accuracy, any interm ediate solution as well. Practical experience shows th a t a slight variation of the profile curve between the two fixed limiting points a t the bore and the rim is of a practically negligible influence on the boundary stresses. After having used, for a certain time, the new method

F ig . 19 T y p i c a l E x a m p l e s o f C h o i c e O f f e r e d b y t h e T w o E x p o n e n t i a l P r o f i l e s

represented by the two exponential profiles the practical designer will, therefore, be able to modify more or less freely the theo­retical profile curves, if this is advisable from the standpoint of weight reduction or required by the necessity of avoiding possible dangerous vibrations. The profiles y = ae~z, z = 0ry, with 7 = 1/ i or 7 = 1/ 6, being of sharper curvature a t the bore than the second exponential profile' and, consequently, situated in the vicinity of the hyperbolic profile, appear quite unnecessary in th is connection because they do not throw any further light on the problem of stress distribution and there would be no justification for the tedious calculations involved.

This is a case of special practical interest. Inasmuch as the hyperbolic disk is regarded as a limiting profile characterized by smaller weight bu t for the most part no t adapted to technical- design purposes because of the sharp curvature in the vicinity of the bore, considerable modification of the profile curve is neces­sary, thereby reducing the degree of accuracy of the stress calcu­lations. Fig. 19 shows th a t, by using the second exponential profile, the designer practically obtains the “ modified” hyper­bolic disk w ithout being forced to abandon the basis of a reliable

terms in each. These two expressions, however, may be worse t han any of the proposed infinite series, because there is no indica­tion as to the rapidity of their convergence. The solutions, thus represented by six infinite series, cannot compete with the exponential profiles, the first of which involves only two infinite series, while the second entails the use of simple finite expres­sions. The more rapid convergence in the case of smaller values of 7 in the first four of Mr. M aulbetsch’s six series would not appear to be a factor of great weight.

There is, furthermore, another factor which is not to be over­looked in connection w ith the question of convergence. If 7 becomes smaller, the independent variable z becomes larger, especially in the vicinity of the rim (see Tables 4 and 6 ). In using 7 = Vi we easily find, for the disk analyzed in Figs. 3 to 6 , that

600 TRANSACTIONS OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

stress analysis. This consideration once more demonstrates the technical importance of the second exponential profile. I t leads to the conclusion tha t, while the hyperbolic disk represents a theoretical limiting profile w ith regard to the im portant question of uniformity of the stress distribution, the second exponential profile very often will represent the corresponding actual limiting curve. This applies especially to the case of smaller bore diame­ters (see Fig. 19) where the exponential profiles differ considerably from each other. B ut even in the case of large bore diameters (see Figs. 3 to 6), the second exponential profile can be considered a t least as a good guide w ith respect to the question of a satis­factory stress distribution a t the bore.

There will be perhaps another occasion later to discuss the problem of axial disk vibrations in connection w ith the problem of profile choice.

In summarizing so far, the following characteristic features of the new solutions should be emphasized: (a) simple m athe­matical representation, (fc) compliance with the principle of similarity, (c) practical use in profile-curve variation within the limits represented by the conical profile and the hyperbolical disk, (d) reliable representation of the “actual” hyperbolical disk, and (e) tendency tow ard uniformity of stress distribution a t the bore. These features make the new solutions well adapted for purposes of practical disk design.

Referring to the problem of design limits as stated by C. R. Soderberg, certain indications concerning the distribution of the

tangential stresses along the bore and their maximum amount at the middle plane (inner bore) of the disk have been given already in Appendix No. 1. If this maximum stress exceeds a certain limit, deformation processes of plastic character m ay be caused w ith the result th a t gradual changes will take place in the shrink fit. This is a question of entirely different nature and beyond the scope of this contribution.

T he problem of normal pressure on the shaft under static conditions adm its of a simple solution. We have seen (Fig. 8) th a t the norm al pressure p , between disk and shaft under these conditions is from 40 to 45 per cent of the tangential stress, <r«>, a t the bore in overspeed conditions. The principal stresses in the shaft under uniform compression are, according to the solution of a well-known special case of the elastic problem .38 <r, = 0 in the axial direction and = <rv — —p, in any two directions in the cross-section of th e shaft. If the problem of failure of the shaft under th is uniform compression is judged from the standpoint of M ohr’s theory, we would have to consider the characteristic am ount of l/ t (ox — <r,) — 1/ i (ay — a,) or, absolutely, V 2pr, as compared w ith th e am ount of Vi^io in the case of the disk. In other words, according to M ohr’s theory, the shaft would be in twice as favorable a condition as the disk, provided the corre­sponding cases of failure are comparable w ith each other .84

33 Loo. cit., A. E. H. Love, p. 144.34 “Festigkeitsversuche,” by Th. von. KArmdn, Zeit. V.D.I., 1911.