D E T E R M I N I N G T H E A D A P T A B I L I T Y OF T U R B I N E B L A D E S

V. N'. S t e l ' m a s h u k a n d A. M. K a m i n s k i i UDC621.438-226.2.001.24:539.319

Statement of the Problem. The high tempera ture s t r e s se s occurr ing in the nozzle and working blades of the f i rs t stages of turbines under nonsteady conditions cause plast ic deformations which, after compen- sation of the tempera ture fields, cause the appearance of residual s t r e s ses . Added to the s t r e s s e s f rom mechanical loading, the axial force, and the bending moments these s t r e s s e s may accelera te fai lure of the blade under steady conditions. In addition, the residual field of s t r e s se s occurr ing in start ing up i n c r e a s e s the s t r e s ses which appear in stopping which, in turn, may cause plast ic deformation of the opposite sign and failure of the blade in alternating plast ic flow. Therefore , in determining the s trength of the nozzle or working blades it is also n e c e s s a r y to take into account phenomena occurr ing in nonsteady conditions, the most r igorous of which are s tar t ing and stopping.

An accurate mathematical analysis of the kinetics of the s t r e s s -de fo rma t ion condition of blades with an increase in the number of s t a r t - stop cycles is a quite complex problem, the solution of which is made more difficult by the absence of the values for the pa r ame te r s of the change in the deformation d iagram for the mater ia ls used, by insufficient study of the rules of totaling the damage in operation under var ious con- ditions, and by a number of other c i rcumstances .

This work makes an attempt to rate the capacity of blades of plast ic mater ia ls under nonsteady condi- tions on the basis of the theory of adaptability. The basic theorem of the theory of adaptability, the static theorem of Melan [1], confirms that if there is such a field of residual s t r e s s e s which totaled with the ope- rating s t r e s ses at each point of the par t does not at any moment in t ime exceed the yield strength of an ideal plastic material , then after a number of initial cycles the behavior of the par t becomes completely elastic. As a resul t the operation of the blade under varying conditions will not be accompanied by plast ic de forma- tion and the number of cycles before failure will exceed the serv ice life of 103-104.

In the data of the repor t an approximate analytical solution is p re fe r red . The advantages of this for analyzing the influence of various factors on the adaptability of blades is especial ly marked in the planning stage in choosing the basic pa rame te r s of the design.

To simplify the calculations the following assumptions are made:

1. The change in profi le thickness along the chord (Fig. la) is determined by the equation !

_ _ - k l It = ( l - - X ? t) "~ + tLtXt , (1) /to

where the relat ive coordinate xt = xi/~i l changes f rom zero to unity depending upon the distance f rom the c ro s s - s ec t i on with the grea tes t profi le thickness toward the front (i = 1) and r e a r (i = 2) edges of the p ro - file (~1 = OA/AB, ~2 = 1-~1, #1 = mr/h0).

The sections of the center line of the profi le are the parabolas z = ( 1 - ~ ) ~ , and their common apex coincid.es with the center of the c ross section of grea tes t thickness.

For hollow blades the contour of the hollow is s imi la r to the outside contour with a coefficient of s i - militude ~v. The geometr ic charac te r i s t i c s of such a profi le are determined by formulas presented in the appendix.

2. The field of t empera tures under nonsteadyconditions is constant ac ross the thickness of the profi le and along the length of the center line is descr ibed with a sufficient degree of accuracy by the exponential relationship (Fig. lb):

Translated f rom Prob lemy Prochnost i , No. I, pp. 17-21, January, 1972. Original ar t icle submitted Februa ry 1, 1971.

�9 1972 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy o[ this article is available [rom the publisher [or $15.00.

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a~

Fig. 1.

m

rs~

I i

t s$1

i t =xz

= -

Geometric parameters of the profile (a) and the tempera ture field (b).

Ttxt) = T O (1 + ~t~rt), i -~ 1, 2, (2)

where T o is the t empera tu re of the c ross sect ion corresponding to the maximum thickness, r i = ( T i - T0)/T0 (T i is the t empera tu re of the front (i = 1) and back (i = 2) edges), and r i are p a r a m e t e r s charac te r iz ing the sharpness of the change in the t empera tu re fields on the front (i = 1) and r e a r (1 = 2) edges.

This descr ipt ion of the tempera ture field coincides sufficiently well with the descr ipt ion of exper imen- tal fields obtained, for example, in [2, 3].

3. The produce of the modulus of elast ici ty E and the coefficient of l inear expansion a changes only slightly within the range of t empera tu res T 1- T o and T 2 - T o and therefore the average values in this range will be used. According to the date of [14] the e r r o r f rom this method of averaging is small.

4. The t ime of steady operation f rom star t up to stopping (especially, with increased speeds) is small , which allows us to not take into account the phenomenon of relaxation of residual s t r e s s e s at the t e m p e r a - ture of steady operation [5].

The conditions of adaptability are considered for three basic conditions: s ta r t up, s teady operat ion (at the nominal speed), and stopping. However, the resul ts obtained may be general ized for severa l s teady and variable conditions as also for severa l cr i t ical rat ios of t empera tures acting in different t i m e intervals within the l imits of a single nonsteady condition.

ApproximateFormulas for the Tempera ture Stresses. Fo r blade profi les it is convenient to use a cen- t ra l sys tem of coordinates developed relat ive to the main axes at some small angle 0 so that the axis x0 is d i rected paral le l to the chord of the profi le and the axis Y0 perpendicular to it (Fig. la).

In this sys tem of coordinates (x 0, Y0) the known formula for tempera ture s t r e s se s [6] taking into ac- count the s t r e s se s f rom mechanical loads acquires the form

t t tF+ e a T , ~ I x - - gtdF F.aro ) I=~

zt = EczTo " F + x \ r l=ltl__12xg

+ Y l x l ~ - - Ix. ~

15

~,o

46

o,z

o

, , l e o !-~176 ,q,, m 7 x dx

/ /

/ /

i i / g~

40 e7.~ a,~ ~z5 o ~ a

' , / e l ,~,Jb-r)~ d ~ . .

-~;~/-~~ ~j /

zey ~,o ez: o,: a,z~ a ~- b

Fig. 2. Values of the aux i l i a ry funct ions .

F o r " t e m p e r a t u r e loads" the fol lowing e x p r e s s i o n s a r e obtained:

2

F 1)" k t + r I + 1 '

2

F i=I

x (1 - - ~n+3)/A,~+, (n~) + k~ + q. +

2 r t, I~, ~__~Ii (4)

F l = 1

where ~0 = x0/l; and 0 = Y0 / Aa re the re la t ive coord ina t e s of the c e n t e r of g r a v i t y (Fig. la) .

F o r a sol id p rof i l e ~p = 0. The values en te r ing t he se e x p r e s s i o n s of the funct ions of A u (n) a r e g iven in Fig. 2a.

The coord ina t e s of the a r b i t r a r y point of the p ro f i l e m a y be p r e s e n t e d in the f o r m

l

x = ( _ _ l ) , ~ t l _ _ X o ; Y ( l _ x ~ A _ _ Y o + e [ ( 1 _ - , ~ - - ho = x~ ) ~i + ~ ~Z/] T - (5)

(-- 1 ~<e~<- -% ~p~< e..< 1)

Having subst i tu ted Eq. (2) and (5) in Eq. (3) we obtain I

o t = E a T o [a + b,; t + cx.-2 + d,x~' + e,x~' + f (1 - - x, . , ]. (6)

where

a = o) - - I~x o + -~, ( A - - Yo) - - 1;

b t = (-- 1)t~fll~;

c = - - "l'A;

ho d t = e ~ y -~- ;

e l = - - T l ;

'(! co = - f tdF +

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~" = l=I u -- l#,v

? = IxI u - - 12xu

The Conditions of Adaptability. A blade adapts to the action of cycl ical ly changing loads and t e m p e r a - tures if there exists such a field of residual s t r e s se s that totaled with the operating s t r e s se s for each of the most rigid conditions under considerat ion (start ing up, s teady operation, stopping) it does not exceed the corresponding limiting charac te r i s t i c of the material . In accordance with this we have three groups of conditions of adaptability, which must be fulfilled for all points of the profi le

• ] • a t + a ~< o~; (7)

+ a~ • %

The f i rs t component in the left por t ion of the inequalities is the s t r e s ses f rom tempera tu re and mechanical loads, the second is the residual s t r e s ses , and the values on the right hand side are the limiting cha rac - t e r i s t i cs of the mater ia l . One pr ime mark indicates the values for the most dangerous moment of s tar t ing up, two pr ime marks the most dangerous moment of stopping, and the values without p r ime marks r e f e r to s teady operation. The most dangerous is assumed to be the moment of s teady operat ion corresponding to the grea tes t gradients in the t empera tu re field or a moment somewhat removed f rom it in t ime and direct ion corresponding to the highest general t empera ture level. As limiting charac te r i s t i cs it is proposed to use the yield s t rength ~0 of the mater ia l for the conditions of start ing up and stopping and the creep s t rength a0* for an equivalent operating t ime at the nominal condition for conditions of long t ime operation.

Assuming that the s t r e s ses at various port ions of the profile may have different signs, all possible combinations of signs must be utilized in the sys tem obtained.

Let us r epresen t the relationship of the mechanical proper t ies of the mater ia l to t empera tu re in the form

% = So(1 ~- ~rP): a ~ = So(l + )~TP'); (8)

then according to Eq.'(2) their relationship to the coordinates is determined by the equations

% , = S O [1 n u X' (To) ~ (1 -b ~f ' )~

%, ----- S O [1 -k X" (To)" (1 -t- ~,x~')'l; (9)

a0~ ---- So (1 + ~.*Tg') .

Since for adaptability the ve ry fact of the existence of an advantageous field of residual s t r e s s e s is important, let us eliminate the s t r e s se s ~ f rom Eq. (7) and convert to the following sys tem:

o; + . ; :L ~; + o, >Io; 1 !

" ~ o;[ (IO) ~ + ~; + "; + ~; ! ~; + G o : k . , • ~; >t 0.

After substituting Eq. (6) and (9) in the sys tem (10) the problem becomes one of investigating expres - sions in the form

!

L----A-I-B~g-I-Cx~-~-D,-~'-I-H.~C'-I- F(I x"q"'-FGi(l-k x;x~')~ q-G;(l ,,-,~o -- nL xur. i ) , (ii) t t i �9

the coefficients of which are obtained f rom the coefficients of the relat ionships of the t empera tu re s t r e s s e s and the proper t ies of the mater ia l to the coordinates in accordance with the operations specified by the con- ditions of adaptability (Eq. (10)).

If over the whole range of values of ~i and e Eq. (11) is positive, then the blade will adapt to the action of unstable tempera ture fields and mechanical loads. The limit of adaptability is reached if Eq. (11) reaches zero even at a single danger point of (~i, e).

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The s a f e t y f a c t o r fo r a d a p t a b i l i t y i s u n d e r s t o o d as the n u m b e r by which it i s n e c e s s a r y to m u l t i p l y the ax ia l f o r c e and the bend ing m o m e n t ac t ing on the b l a d e in o r d e r to r e a c h the l i m i t of a d a p t a b i l i t y wi th the g iven t e m p e r a t u r e f i e l d s at s t a r t up and s topp ing .

T h e r e f o r e , the r a t i n g of a d a p t a b i l i t y r e q u i r e s c h e c k i n g of the p o s i t i v e n e s s of the e x p r e s s i o n f o r L (x, E) at the d a n g e r po in t s of the p r o f i l e . *

In many p r a c t i c a l c a s e s a s a f i r s t a p p r o x i m a t i o n it is p o s s i b l e to u se r 1 = r 2 = 2, ~l = P2 = 0, p = 1, y - ( 1 - ~ ) A - y 0 + e (1 -~ i /n i )h0 /2 . Then the i n v e s t i g a t i o n t a k e s on a q u a d r a t i c f o r m

L = (A + v + ~ + ~;~ + ( 8 , - --'n, ~)~' +/C + " , + ~i+~ + ~+;1~ (~2)

F o r c h e c k i n g the cond i t ion L -> 0 in th i s c a s e i t i s su f f i c i en t to c o n s i d e r the e x t r e m e po in t s of the r a n g e ~ i = 0, xi = i and the po in t of the a n a l y t i c a l e x t r e m e

I B~ - - - - F

- - ~ - - t ~ i

i f 0 -< Xmin "~ I and at t h i s po in t t h e r e i s a m i n i m u m in (K i = C + H i + G~-] + G ' !~] -> 0). S ince the c oe f f i - c i e n t s of the q u a d r a t i c f o r m , b e s i d e s F, do not depend upon the va lue of e and ~he va lue of F i s a l i n e a r funct ion , it is su f f i c i en t to check the cond i t i on L - 0 fo r the e x t r e m e v a l u e s e = +1, and in the c a s e of the a n a l y t i c a l m i n i m u m , a l s o f o r e m i n a s d e t e r m i n e d f r o m the cond i t i on B i - F / n i = 2K i (if mini n l i e s in the a l - l owab le r ange ) .

Tu rn ing f r o m the s i m p l e s t c a s e to the g e n e r a l one, i t i s a p p a r e n t l y p o s s i b l e to m a i n t a i n the c o o r d i n - a t e s of the d a n g e r po in t s r e q u i r i n g check ing as d e t e r m i n e d fo r the s i m p l e s t c a s e . In us ing such an a p p r o a c h i t is p o s s i b l e to g ive r e a s o n s fo r the low c h a n g e a b i l i t y of the func t ion in the r e g i o n of the a n a l y t i c a l e x t r e m e . In addi t ion , e x p e r i m e n t a l d a t a have shown that t h e r m a l fa t igue c r a c k s a r e n o r m a l l y f o r m e d on the e d g e s ~ i = 1) and in the c a s e of l a r g e m e c h a n i c a l l o a d s the po in t wi th the c o o r d i n a t e s x i = 0, e = 1 m a y a l s o be d a n - g e r o u s . T h e r e f o r e , in r a t i n g a d a p t a b i l i t y t h e s e t h r e e p o i n t s on ly a r e of i n t e r e s t fo r a f i r s t a p p r o x i m a t i o n .

T h e

1.

2.

A P P E N D I X

G e o m e t r i c C h a r a c t e r i s t i c s o f t h e B l a d e P r o f i l e

Su r f ace

2

]+, l= t

Sta t ic m o m e n t s r e l a t i v e to the axes x and y 2

+,7]'+'- "-'-+', { A,C',' + ; '.+`

2

- - I x+ • [A1 (hi) -~- P'l S,, = Llh o (1 ~3) E ( - J ~ l = l

3. C o o r d i n a t e s of the c e n t e r of g r a v i t y

s~ s x . = + ;

4. M o m e n t s of i n e r t i a r e l a t i v e to the axes Y0 and x 0 and the c e n t r i f u g a l m o m e n t of i n e r t i a

2

Iu=Ll2ho(1--cp 4) \ " ~ [A~ ~l _ - k - 7 - ~ ] - - (Xo)+OF;

i= l

i x Lh ~ 9. 2. = [holxmat) + A tx(varJ,

*The l i m i t s fo r the c o e f f i c i e n t s of Eq. ( i i ) r e s u l t i n g f r o m the cond i t ion L - 0 at the d a n g e r po in t s a r e s u p p l e m e n t e d by l i m i t s r e s u l t i n g f r o m the cond i t ion of c a p a c i t y f o r s e l f - a d j u s t m e n t of the f i e ld of r e s i d u a l s t r e s s e s .

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where 2 l [R (n~) + 3~Nki (n~) + 3~t~A2k t (n~) d- 3k i + t ] ; ix(nat)-=-- (1 - - ~p4) ~ ~ ~f .

_ _ ~ 2 ) A o ( n i ) _ o �9 2(1 ,p,) [ A, (,,,) +. . i ~ I

' 2 J [ I~, l + { ' I - - , 3\r V', II

In these formulas ~ is the length of the chord of the profile, L is the length of the center line of the profile, L ~ l [l + 2/3(A/l) 2 1 / ~ ] ; h 0 is the maximum thickness of the profile, A is the maximum pitch of the center line, 5 is the wall thickness of the sloping blade, and ~ is the coefficient of similitude of the profi le (for a solid profi le q~ = 0).

The pa rame te r s n i, k i, ~i = mi/h0, K =A/h 0 and ~i (~l = OA/AB, ~2 = 1 - ~1) determine the shape of the profile. The values of the functions A u (n), Nki (n), R in) are given in Fig. 2a and b.

C O N C L U S I O N S

1. The problem of rating the adaptability of turbine blades (solid or hollow) under the action of t e m p e r a - ture fields and mechanical loads acting at s tar t up, s tat ionary operation, and stopping was considered.

2. On the basis of an analytical concept of the shape of the profile and the tempera ture field express ions for the s t r e s se s in the blade were obtained.

3. It was shown that ra t ingthe adaptability consis ts of checking at the danger points of the profi le the sign of Eq. (11), which is composed of the coefficients of the expressions of s t r e s se s and strength cha rac - t e r i s t i cs of the material .

4. The solution obtained makes it possible to analyzethe influence on the adaptability of bIades of the cha rac te r of the tempera ture field, the shape of the profile, the level and ratio of mechanical and thermal loads, the relationship of the proper t ies of the mater ia l to temperature , and other factors .

5. Approximate analytical express ions for the geometr ic charac ter i s t ics of the solid and hollow blade profi les used in the calculations are given.

I .

2.

3. 4.

5.

6.

L I T E R A T U R E C I T E D

V. T. Koiter, General Theorems of the Theory of E l a s t i c - P l a s t i c Media [Russian translat ion], IL, Moscow (L96D. E. R. Plotkin and E. I. Molchanov, in: Thermal Stresses in Design Elements [in Russian], Vol. 2, Naukova Dumka, Kiev (1963). R. I. Kuriat and Yu. D. Miroshnichenko, Energomashinostroenie, No. 10 (1966). R. I. Kuriat, G. N. Tret 'yachenko, et al., in. The High Temperature Strength of Materials and Design Elements [in Russian], Vol. 4, Naukova Dumka, Kiev (1967). R. F. Khalimon, R. I. Kuriat, et al., in. The High Temperature Strength of Materials and Design Elements [in Russian], Naukova Dumka, Kiev (1967). N. N. Malinin, Izv. Akad. Nauk SSSR, OTN, No.4 (1954).

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