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Fir tree fastening of turbomachinery blades—II
Int. I. Mech. Sci. Vol. 24, No. 6. pp. 385-391. 1982 Printed in Great Britain.
FIR TREE
0020--74031821060385-07503.0010 Pergamon Press Ltd.
FASTENING OF TURBOMACHINERY BLADES--II STEP LOAD ANALYSIS G. D. SINGH and S. RAWTANI M. A. College of Technology, Bhopal-462007, India
(Received 7 July 1981; in revised form 12 October 1981) Summary--Blade-root fastening of the fir-treeshape, used in securing turbomachinery blades to the rotor has been analysed by considering it as an assembly of the step-segments, the stiffness characteristics of which are already obtained in Part-l. The general equation of equilibrium as well as the compatability of deformation between the steps of the blade and the disc give the necessary 'n' linear algebraic equations for the evaluation of the step loads in the n-step root. The method can accommodate the centrifugal body forces of the root itself,the friction between the blade and disc-steps and the difference in the material of the blade and the disc. NOTATION
Ai, Ji value of (pdN) and (qdN) for ith blade-step B~,Ki value of (P2/F) and (q,/F) for ith blade-step
C~, Lj value of (P3) and (q3) for ith blade-step Di, M~
value of (pJT) and (qJT) for ith blade-step
I:, neck-section load at the bottom of the ith blade-step I=o centrifugal force of the semi-blade value defined by equation (4) number of steps s, centrifugal body force due to the mass of the ith blade-step 8 relative displacements defined by equations (6) and (8) Iz coefficient of friction friction angle (3 the quantity refers to disc step
G
n
INTRODUCTION
The fir-tree type of root fastening is one of the most commonly employed methods used for securing the turbomachinery blades onto the rotor disc. One of the important design consideration is to accurately evaluate the load carried by each step, at the surface of contact between the blade and the disc. Photoelastic analysis or application of the finite element method to the entire blade root and disc portion, are the methods commonly used to analyse blade roots. The photoelastic analysis requires the preparation of the plastic model of the fir-tree root to close tolerances so that simultaneous contact could be achieved at all the contact surfaces between the blade-root and the disc. The finite element method requires the preparation of a gridwork of elements. For the required accuracy over five hundred nodes m a y have to be employed. The input data to the computer would consist of coordinates of all these nodes as well as the node numbers joining each element. This process of inputting the data is both time consuming and error prone. The computational effort consists of evaluation of individual stiffness matrices of all the elements and the solution of about a thousand linear equations. Thus, these methods are highly time consuming and also necessitate the repetition of the entire analysis, if the designer wishes to alter any parameter of the root shape. The purpose of the present investigation is to establish a sound theoretical procedure for the evaluation of step loads, which would retain almost all the advantages of a rigorous solution and at the same time involve very much less computation by the designer. In fact, the solution of a n-step root would require solution of only 'n' linear algebraic equations, which m a y be carried out without recourse to a digital computer. 385
386
G . D . SINGH and S, RAWTANI
THEORY (a) Load-deflection characteristic o f step Fig. I shows the semi-step of a blade root indicating the geometrical parameters a, b, h, W, x , 3 and y. It also shows the four loadings, namely, (i) normal step load N, (ii) neck-section load F, (iii) distributed centrifugal load due to the mass of the root itself and (iv) tangential friction load T, acting on the semi-step. The vertical deflections of point P for each of the loading cases are denoted as pt, P2, P3 and P4, while those of the point Q as qt, q2, q3, and q, respectively. The expressions for these deflections have been obtained and are given as equations (3.1)-(3.8) in Part I. The following coefficients are introduced: A = -Pl ~,B = ~,C
= p3, D - _- ~P4 ,
ql K = ~ , L = q~, M -_ -~. q4 J = -'~,
(1)
An enlarged view of the disc and the blade steps in contact is shown in Fig. 2. The steps are numbered serially from 1 to n, where n is the total number of steps. The numbering order of these steps increase towards the centre of the disc. For the ith blade-root step the coefficients in equation (1) are denoted as A~, B , - - - , M~. For the ith disc-step they are denoted as A[, BI, - . . . . , M[. (b) Equilibrium equations Referring to Fig. 2 the mid point of the contact surface between the ith blade step and the ith disc step is denoted as P , the central neck section point at the bottom of the ith blade step is denoted as Q~. Also, the central necksection point at the top of the ith disc step is denoted as Q~. The resultant of normal and tangential loads are denoted as N~ and T~ respectively. These are identical for the blade as well as the disc step. The normal load at the bottom neck section of the ith step of the blade is denoted as F~. Similarly the normal load at the top neck section of the ith step of the disc is denoted as FI. Due to their own mass, the steps of the blade as well as the disc would be subjected to centrifugal body forces. The direction of this force at any point would be along the radial line joining the point with the centre of the disc. Since the radius of the disc is very much larger than the semi width of the blade/disc, the distributed centrifugal forces can be considered as acting along the Y-axis. For the ith blade-step this force is denoted as S~, and for the ith disc-step as SI. Fig. 3(a) is the free body diagram of the semi-blade root from the nth to the ith step subjected to normal and tangential step loads Nj and T~ and the centrifugal body forces Sj(j '-- i,- . . . . n). The load at the top neck section is F~_I. Assuming Coloumb friction between the contact surfaces of the disc and the blade, Tj = v, • N~ = t a n ,b ' N j
(2)
where, p, = tan $ is the coefficient of friction. The normal load Nj is inclined at an angle (a + B) to the Y-axis and hence the resultant of N i and Tj is N~ • s e c $ , and this is inclined at an angle (a + 3 - $ ) to the Y-axis. The equilibrium equation for the vertical forces is,
j=tS i it n FI=Gj=~,÷N~- ~ Si =
(3a)
):1
or,
l llt
B
l I i TT e-
I lT .I
ll a
o
z
i
F FIG. I. Geometry of an individual step of a fir-tree root showing the forces acting on it.
(3b)
Fir tree fastening of turbomachinery blades--II
v
387
F.
O~)%blodestep
~"•
step
imdisc step i~ bloclesteP
4.,
Oi
o'. -~disc step [q
Qm FIG. 2. Nomenclature for division of blade-root and disc-stub into step elements.
In particular,
where,
G = cos (a +/~ - ~b).sec ~b
(4)
Fig. 3(b) is the free body diagram of the semi-disc root from 1st to ( i - l)th step, subjected to normal and tangential loads Nj and T~ and the centrifugal body forces S~j = I, 2, 3 . . . . . i - 1). The load at the bottom neck section is F~. The equilibrium equation for the vertical forces gives the relation, i-1
i-I
or, i -2
i-2
(c) Relative displacements of consecutive step contact points The relative displacement of P~ and P~-a is evaluated treating them as points on the blade-step surface. Referring to Fig. 2 this may be expressed by the relation: Relative displacement of P~ w.r.t P~-I = Relative displacement of Pi w.r.t Qi-i +Relative displacement of Qi-i w.r.t Q~-2 -Relative displacement of P H w.r.t Q~-2 or,
8 = 81 + ' ~ - ~.
(6)
The relative displacement (Bi) of P~ with respect to Q~-I can be obtained from the load-deflection characteristics of the ith blade-step. This displacement is the sum of the displacements caused by the four load cases as discussed earlier and is given by, 81 = Pl +P2 +P~ +P~, for the ith blade-step, i.e. 8] = AiNi + BiFi + Ci + DiTi.
(7.1)
388
G. D. SINGH and S. RAWTANI
t
ri- -.l
,
Ittlt-tI[ I/n llli lll ~ .
f/~,,Ti
L_..~
( b)
(a)
FIG. 3. Nomenclature for forces acting on a (a) blade-root step and (b) disc-step,
Similarly,
a2
Ni-i + Ki-i • E-I + Li-i + M i - i
• Tl-i
(7.2)
a3 = A i - i " N i - i + Bi-i ' Fi-i + C i - i D i - i " Ti-i.
(7.3)
=
Ji-i
"
and,
The relative displacement of these same points P~ and P~._t is next obtained by treating them as part of disc-steps. This gives the following relations;
where, 8;=
- A~ . N~ - B : . F I + C; - D ~ . T~
8 ~ = - J } . N~ - K I . F} + L ~ - M } .
8 j = -Ai-u ' N i - i - - B ~ - L "
FI_I
T~
(9)
+Ci-i -DI-I " T,-I.
When considering the blade steps, the directions of the loads correspond to those given in Fig. 1. While considering the disc steps, only the centrifugal body force retains the direction considered in Fig. 1. All the other force directions are opposite. This explains the negative sign for the Ist, 2rid and the 4th terms in equation (9). Equating 8 from equations (6) and (8) and substitution from equations (7) and (9) yields the relation: N~{AI + A ~ - J i + I.L(Di + D ~ - NI-,{AH
+ A~_,-Ji-,
+la.(Di_, + D ' i - , - M i
M~)} ,)} + Fi ' Bi + E I'
( K i j - B i - i ) + F ; . ( B ~ - K~) - FI_, • B i - i = C, , - Ci + CI - C~ a - LI - L~_I
(10)
The values of F~ and F,_~ can be substituted from equation (3). Similarly the value for F~ and F~-t can be substituted from equation (5). The equation (10) will then be in terms of the normal step loads N only. When there are ' n ' steps in the blade root, the equation (10) will yield ( n - I ) linear algebraic simultaneous equations corresponding to i = 2, 3 . . . . . n. One more equation is given by the overall equilibrium equation of the semi-blade root as per equation 3(c). Solution of these ' n ' simultaneous equations gives the value of normal step loads N~, N2, N3 . . . . Since the number of steps in a fir-tree blade root is generally less than five, the solution of cor[esponding number of algebraic equations can easily be carried out by the designer.
389
Fir tree fastening of turbomachinery blades--ll
DESIGN PROCEDURE The design procedure for evaluating the normal step loads is summarised below: (i) The steps in the blade root and the disc are numbered as illustrated in Fig. 2 and for each step the values of parameters h, W, a, b, a,/3 and 3' are noted. (ii) The values of A~, Bi, Ci, D. J. K, L~ and Mi (i = I to n) are calculated from equation (1) for blade as well as disc steps. (iii) The centrifugal body force (Si) is calculated for each step of the blade and the disc. The centrifugal force of the blade (excluding the root) is denoted as 2F0. (iv) The values obtained above are substituted in equations (10) and 3(c). The solution of the resulting ' n ' equations gives the step loads Nw, N2, N 3 , - . . . . •
Numerical computations and discussion The procedure outlined above is applied to the case of a three-step fir-tree root subjected to centrifugal force (2F0) of the blade alone. The geometrical parameters and elastic constants are taken to be identical for the disc and the blade. The coefficient of friction is taken as zero. The actual computations are carried out as Case I in the Appendix. The above data has been selected as the computations for this case have already been carried out[l] using the theory of elasticity approach. The results for the value of F0 = 11500 kg. are given in Table l column 3. The figures in parenthesis are the values of the vertical components of step-loads as obtained in Ref. [l]. It is seen that the maximum difference in results is about 4% which proves the accuracy of the present method of analysis. In the method detailed in Ref. [I] the effect of the centrifugal body force due to the root mass has been neglected. In the present analysis the effect of these body forces can be taken into consideration. In Case 2 the computations of step loads has been carried out taking into account this body force only. The vertical component of the step loads work out as V~ = 30.0230, V: = 37.8509 and V3 = 45.5592 kg per mm. While these values will change with blade geometry it is significant to note that the last blade step (nearest to the disc-centre) takes the greatest part of this load and the first step the least part. In computations carried out in Case 1 the coefficient of friction was taken as zero to be able to compare the results with those of Ref. [1]. Since the other data considered correspond to the values of steel, the value of/.t should be approx. 0.42. The loads for this case have been computed in Case 3, and are given in column 4 of Table I. In the present case the coefficient of friction has not significantly affected the vertical component of the loads carried by the steps. However, it has significantly altered the normal and tangential components. The results could be different for different geometries. The effect of friction and root mass on the normal and tangential components of step load could be compared by taking a particular value for F0 and a suitable value for root mass. In a typical intermediate stage blade of a steam turbine the blade mass may be of the same order as the root mass, since the root is generally of uniform thickness and the blade is of a narrow aerofoil section tapering towards the tip. The centre of gravity of the blade is at a greater radius than that of the centre of gravity of the root mass. It may be reasonable to take, as an illustrative example, the centrifugal force of the blade as twice the centrifugal force due to the root mass. For the blade having Fo = 11,500 kg and centrifugal force due to semi-root mass as 5620 kg (for a blade root of 50 mm thickness), the step loads are compared in Table 1, for the different cases.
TABLE l. COMPARISON OF STEP-LOADS Step No.
Component
1
Normal Tangential Vertical Normal Tangential Vertical Normal Tangential Vertical
2 3
Root-mass neglected /z = 0.0 # = 0.42 4424 0 3830(3910) 4430 0 3836(3680) 4424 0 3830(3910)
3567 1498 3827 3551 1491 3821 3567 1498 3827
Root-mass included ~t = 0.0 /z = 0.42 6157 0 5331 6616 0 5728 7054 0 6108
4988 2095 5367 5302 2227 5705 5619 2360 6047
The step loads are considerably higher when the centrifugal force of the root mass is also taken into account. The inclusion of friction affects the normal and tangential components of the step-loads significantly. For the geometry under consideration the friction has not significantly altered the vertical component of step loads. CONCLUDING REMARKS A fir-tree type of root fastening has been considered as made up of several distinct steps, the shapes of which are defined by seven different parameters. Two reference points have been shown to be relevant in assembling the different steps to form a complete fastening.
390
G.D. SINGHand S. RAWTAN1
T h e e q u a t i o n s b a s e d o n e q u i l i b r i u m a n d e q u a l i t y of r e l a t i v e d e f o r m a t i o n s h a v e b e e n g i v e n for a g e n e r a l ' n ' step root. T h e m a j o r a d v a n t a g e s of the p r e s e n t m e t h o d are: (i) T h e c o m p u t a t i o n s i n v o l v e d in c a l c u l a t i n g the step loads are c o m p a r a t i v e l y s i m p l e , r e q u i r i n g the s o l u t i o n of ' n ' l i n e a r a l g e b r a i c e q u a t i o n s for a n - s t e p root. (ii) It is n o t n e c e s s a r y to a s s u m e t h a t the elastic c o n s t a n t s as well as the step w i d t h are i d e n t i c a l for b l a d e r o o t a n d the disc. (iii) Coefficient of sliding f r i c t i o n b e t w e e n the b l a d e a n d disc step c a n be t a k e n into a c c o u n t . (iv) T h e c e n t r i f u g a l b o d y f o r c e s d u e to the r o o t m a s s itself n e e d n o t b e n e g l e c t e d . (v) D i f f e r e n t steps of the b l a d e m a y h a v e d i f f e r e n t v a l u e s of g e o m e t r i c parameters. I n the p r e s e n t a n a l y s i s the d i s t r i b u t i o n of n e c k - s e c t i o n load has b e e n t r e a t e d as u n i f o r m for d e f l e c t i o n a n a l y s i s , w h i c h m a y n o t be e x a c t l y t r u e . O n c e the load c a r r i e d b y e a c h step is o b t a i n e d the m a x i m u m stress at the fillets c a n b e o b t a i n e d b y t r e a t i n g the step as a s h o r t c a n t i l e v e r a n d u s i n g a n a p p r o p r i a t e stress c o n c e n t r a t i o n f a c t o r . Acknowledgement--The valuable suggestions given by Dr. R. Kitching, of the University of Manchester Institute of Science and Technology, for the extension and improvement of the manuscripts of the papers are gratefully acknowledged.
REFERENCE 1. E. SHIRATORI,M. HOSHIYAand N. SAKURAI,Bull. T o k y o Inst. o f Tech. 80, 25 (1967). APPENDIX 1 The procedure of computation is illustrated for the case of a three step fir-tree root for which a solution based on the theory of elasticity is available j. Equations (10) and (3) when applied to a three-step root take the form; [ H I { N } = {R}
(AI)
where, H,t = - { A i + A { - Jt + # ( D t + D { - M r ) - G ( B ~ -
K~)}
Hi2 = {Az + A~ - J ~ + t~( D2 + D~ - M ~) + G( K j - BI)} HI3 = G(B~ + K i - BO Hz, = G ( B ; - K~ - B~)
/422= - {A2 + A~ - .12 + I2(D~ + D~ - M2) - G ( B ; - K~)} /423 = {A3 + A~ -J~ +/z(D~ + D~ - M;) + G ( K z - B2)} //3, = H32 = H33 = G
R, = C , - C z + C { - C ~ - L ~ - L ,
+ S~ . B2 +($2 + $3).
(K, - B , ) - SI • (B~ - K{)
R2 = Cz - C3 + C~ - C~ - L~ - L2 + S3" (K~ - B2) - ( S ; +St). ( B ; - K ; ) + S;. B{ R3 = (Fo + S, + $2 + S3). Case 1
The following data is taken for the blade and disc-steps: E = 2 x 104 kg/mm:, t = I ram, a = I0°, /3 -- 20°, a = 6.9 mm, b = 11.50 ram, Wt = 35.13 mm, W2= 18.32 mm, WI = 1.5 ram, W~ = 18.32 ram,
/J. = 0.0 y = 39.5° h =23.1 mm W3= 1.5 mm W~ = 35.13 mm.
(Centrifugal body force of the root steps is neglected.) The values of the coefficients for the blade-step and for the disc-step are calculated and given in Table AI.
391
Fir tree fastening of turbomachinery blades--ll TABLE A1. COEFFICIENTS FOR BLADE AND DISC-STEPS ( X I0 +4) Coeff. A B C D J K L M S
Step I Blade Disc -3.1318 -0.0430 0.0 - 1.9590 -0.0180 -0.4411 0.0 +0.0664 0.0
-2.9510 -0.2978 0.0 -1.9547 -0.3696 - 1.4507 0.0 +0.0977 0.0
Step 2
Step 3
Blade
Disc
Blade
Disc
-3.0416 -0.1704 0.0 -1.9568 -0.1938 -0.9457 0.0 +0.0821 0.0
-3.0416 -0.1704 0.0 -I.9568 -0.1938 -0.9457 0.0 +0.0821 0.0
-2.9510 -0.2978 0.0 -1.9547 -0.3696 -1.4507 0.0 +0.0977 0.0
-3.1318 -0.0430 0.0 -1.9590 -0.0180 -0.4411 0.0 +0.0664 0.0
From equation (4), G = cos (a +/3 - &)' sec 4~ = 0.866.
Equation (A1) when applied to this three-step root takes the following form, 6.736 × 10 -4 10.492 × 10 -4 [ 0.866
-6.234 × 10-4 6.234 × 10-4 0.866
-0.492 x 10-4 -6.736 x 10-4 0.866
N N2 N3
= F0
This gives N, = 0.3847 Fo, N: = 0.3853 Fo and N3 = 0.3847 F0 as step-loads acting normal to the step surface of the blade. The vertical components are V~ = 0.3331 Fo, V2 = 0.3336 Fo and V3 = 0.3331 Fo. Case 2 In the above case the effect of pull of the blade on the root was considered and the effect of centrifugal force due to the mass of the root was neglected. Here, only the effect of the latter is considered. Let, p = 7.85 × 104/9810, r = 1000 mm and, w = 314 rad/sec. The values of the coefficients C, L, and S only are now different from those given in Case-1 and are given below in the Table A2. TABLE A2. VALUES OF COEFFICIENTS (x 10 +4)
Coeff. C L S
Step 1 Blade Disc
Step 2 Blade Disc
Step 3 Blade Disc
25.386 10.078 531330
24.623 11.0163 378112
23.859 11.954 224892
23.859 11.954 224892
24.623 11.016 378112
25.386 10.078 531330
The values of the remaining coefficients remain the same. Equation (AI) gives, 6.736x10-4-6.234x10-4-0.492x10~']{N, 0.492 × 1 0 - 4 6.234 × 10-~ -6.736 × 10-' / 0.866 0.866 0.866 J
I
N: ?43
=
- 6 4 . 8 4 0 x 1 0 -~] -64.840 x 10-'~ • 113.433
The above on solving gives N~ = 34.6687 kg, N: = 43.7078 kg and N~ = 52.6088 kg. The vertical components of these forces are V~ = 30.0230 kg, I/2 = 37.8509 kg and V3 --- 45.5592 kg per mm thickness of the root. Case 3 In this case the effect of the coefficient of friction, #, at blade/disc contact surface is considered when the root is subjected to only the blade pull, the effect of root mass being neglected, as was so in Case 1. Taking g = 0.42 the value of G, from equation (4) is 1.076. From the equation (A1) the following relation is obtained:
I
8.570 x 1 0 - 4 - 7 . 9 9 6 x 10~ 0.611 × 10-4 7.996 ×10-' 1.076 1.076
-0.611 x 10-41 { N I l {00} -8.570 ×10-' / N2 = . 1.076 J N; F0
The above on solving gives N, = 0.3102 Fo, N: = 0.3088 Fo and N3 = 0.3102 Fo as normal step-loads. Their vertical components are V~ = 0.3338 F0, V2 = 0.3323 F0 and V3 = 0.3338 Fo.
FIR TREE
0020--74031821060385-07503.0010 Pergamon Press Ltd.
FASTENING OF TURBOMACHINERY BLADES--II STEP LOAD ANALYSIS G. D. SINGH and S. RAWTANI M. A. College of Technology, Bhopal-462007, India
(Received 7 July 1981; in revised form 12 October 1981) Summary--Blade-root fastening of the fir-treeshape, used in securing turbomachinery blades to the rotor has been analysed by considering it as an assembly of the step-segments, the stiffness characteristics of which are already obtained in Part-l. The general equation of equilibrium as well as the compatability of deformation between the steps of the blade and the disc give the necessary 'n' linear algebraic equations for the evaluation of the step loads in the n-step root. The method can accommodate the centrifugal body forces of the root itself,the friction between the blade and disc-steps and the difference in the material of the blade and the disc. NOTATION
Ai, Ji value of (pdN) and (qdN) for ith blade-step B~,Ki value of (P2/F) and (q,/F) for ith blade-step
C~, Lj value of (P3) and (q3) for ith blade-step Di, M~
value of (pJT) and (qJT) for ith blade-step
I:, neck-section load at the bottom of the ith blade-step I=o centrifugal force of the semi-blade value defined by equation (4) number of steps s, centrifugal body force due to the mass of the ith blade-step 8 relative displacements defined by equations (6) and (8) Iz coefficient of friction friction angle (3 the quantity refers to disc step
G
n
INTRODUCTION
The fir-tree type of root fastening is one of the most commonly employed methods used for securing the turbomachinery blades onto the rotor disc. One of the important design consideration is to accurately evaluate the load carried by each step, at the surface of contact between the blade and the disc. Photoelastic analysis or application of the finite element method to the entire blade root and disc portion, are the methods commonly used to analyse blade roots. The photoelastic analysis requires the preparation of the plastic model of the fir-tree root to close tolerances so that simultaneous contact could be achieved at all the contact surfaces between the blade-root and the disc. The finite element method requires the preparation of a gridwork of elements. For the required accuracy over five hundred nodes m a y have to be employed. The input data to the computer would consist of coordinates of all these nodes as well as the node numbers joining each element. This process of inputting the data is both time consuming and error prone. The computational effort consists of evaluation of individual stiffness matrices of all the elements and the solution of about a thousand linear equations. Thus, these methods are highly time consuming and also necessitate the repetition of the entire analysis, if the designer wishes to alter any parameter of the root shape. The purpose of the present investigation is to establish a sound theoretical procedure for the evaluation of step loads, which would retain almost all the advantages of a rigorous solution and at the same time involve very much less computation by the designer. In fact, the solution of a n-step root would require solution of only 'n' linear algebraic equations, which m a y be carried out without recourse to a digital computer. 385
386
G . D . SINGH and S, RAWTANI
THEORY (a) Load-deflection characteristic o f step Fig. I shows the semi-step of a blade root indicating the geometrical parameters a, b, h, W, x , 3 and y. It also shows the four loadings, namely, (i) normal step load N, (ii) neck-section load F, (iii) distributed centrifugal load due to the mass of the root itself and (iv) tangential friction load T, acting on the semi-step. The vertical deflections of point P for each of the loading cases are denoted as pt, P2, P3 and P4, while those of the point Q as qt, q2, q3, and q, respectively. The expressions for these deflections have been obtained and are given as equations (3.1)-(3.8) in Part I. The following coefficients are introduced: A = -Pl ~,B = ~,C
= p3, D - _- ~P4 ,
ql K = ~ , L = q~, M -_ -~. q4 J = -'~,
(1)
An enlarged view of the disc and the blade steps in contact is shown in Fig. 2. The steps are numbered serially from 1 to n, where n is the total number of steps. The numbering order of these steps increase towards the centre of the disc. For the ith blade-root step the coefficients in equation (1) are denoted as A~, B , - - - , M~. For the ith disc-step they are denoted as A[, BI, - . . . . , M[. (b) Equilibrium equations Referring to Fig. 2 the mid point of the contact surface between the ith blade step and the ith disc step is denoted as P , the central neck section point at the bottom of the ith blade step is denoted as Q~. Also, the central necksection point at the top of the ith disc step is denoted as Q~. The resultant of normal and tangential loads are denoted as N~ and T~ respectively. These are identical for the blade as well as the disc step. The normal load at the bottom neck section of the ith step of the blade is denoted as F~. Similarly the normal load at the top neck section of the ith step of the disc is denoted as FI. Due to their own mass, the steps of the blade as well as the disc would be subjected to centrifugal body forces. The direction of this force at any point would be along the radial line joining the point with the centre of the disc. Since the radius of the disc is very much larger than the semi width of the blade/disc, the distributed centrifugal forces can be considered as acting along the Y-axis. For the ith blade-step this force is denoted as S~, and for the ith disc-step as SI. Fig. 3(a) is the free body diagram of the semi-blade root from the nth to the ith step subjected to normal and tangential step loads Nj and T~ and the centrifugal body forces Sj(j '-- i,- . . . . n). The load at the top neck section is F~_I. Assuming Coloumb friction between the contact surfaces of the disc and the blade, Tj = v, • N~ = t a n ,b ' N j
(2)
where, p, = tan $ is the coefficient of friction. The normal load Nj is inclined at an angle (a + B) to the Y-axis and hence the resultant of N i and Tj is N~ • s e c $ , and this is inclined at an angle (a + 3 - $ ) to the Y-axis. The equilibrium equation for the vertical forces is,
j=tS i it n FI=Gj=~,÷N~- ~ Si =
(3a)
):1
or,
l llt
B
l I i TT e-
I lT .I
ll a
o
z
i
F FIG. I. Geometry of an individual step of a fir-tree root showing the forces acting on it.
(3b)
Fir tree fastening of turbomachinery blades--II
v
387
F.
O~)%blodestep
~"•
step
imdisc step i~ bloclesteP
4.,
Oi
o'. -~disc step [q
Qm FIG. 2. Nomenclature for division of blade-root and disc-stub into step elements.
In particular,
where,
G = cos (a +/~ - ~b).sec ~b
(4)
Fig. 3(b) is the free body diagram of the semi-disc root from 1st to ( i - l)th step, subjected to normal and tangential loads Nj and T~ and the centrifugal body forces S~j = I, 2, 3 . . . . . i - 1). The load at the bottom neck section is F~. The equilibrium equation for the vertical forces gives the relation, i-1
i-I
or, i -2
i-2
(c) Relative displacements of consecutive step contact points The relative displacement of P~ and P~-a is evaluated treating them as points on the blade-step surface. Referring to Fig. 2 this may be expressed by the relation: Relative displacement of P~ w.r.t P~-I = Relative displacement of Pi w.r.t Qi-i +Relative displacement of Qi-i w.r.t Q~-2 -Relative displacement of P H w.r.t Q~-2 or,
8 = 81 + ' ~ - ~.
(6)
The relative displacement (Bi) of P~ with respect to Q~-I can be obtained from the load-deflection characteristics of the ith blade-step. This displacement is the sum of the displacements caused by the four load cases as discussed earlier and is given by, 81 = Pl +P2 +P~ +P~, for the ith blade-step, i.e. 8] = AiNi + BiFi + Ci + DiTi.
(7.1)
388
G. D. SINGH and S. RAWTANI
t
ri- -.l
,
Ittlt-tI[ I/n llli lll ~ .
f/~,,Ti
L_..~
( b)
(a)
FIG. 3. Nomenclature for forces acting on a (a) blade-root step and (b) disc-step,
Similarly,
a2
Ni-i + Ki-i • E-I + Li-i + M i - i
• Tl-i
(7.2)
a3 = A i - i " N i - i + Bi-i ' Fi-i + C i - i D i - i " Ti-i.
(7.3)
=
Ji-i
"
and,
The relative displacement of these same points P~ and P~._t is next obtained by treating them as part of disc-steps. This gives the following relations;
where, 8;=
- A~ . N~ - B : . F I + C; - D ~ . T~
8 ~ = - J } . N~ - K I . F} + L ~ - M } .
8 j = -Ai-u ' N i - i - - B ~ - L "
FI_I
T~
(9)
+Ci-i -DI-I " T,-I.
When considering the blade steps, the directions of the loads correspond to those given in Fig. 1. While considering the disc steps, only the centrifugal body force retains the direction considered in Fig. 1. All the other force directions are opposite. This explains the negative sign for the Ist, 2rid and the 4th terms in equation (9). Equating 8 from equations (6) and (8) and substitution from equations (7) and (9) yields the relation: N~{AI + A ~ - J i + I.L(Di + D ~ - NI-,{AH
+ A~_,-Ji-,
+la.(Di_, + D ' i - , - M i
M~)} ,)} + Fi ' Bi + E I'
( K i j - B i - i ) + F ; . ( B ~ - K~) - FI_, • B i - i = C, , - Ci + CI - C~ a - LI - L~_I
(10)
The values of F~ and F,_~ can be substituted from equation (3). Similarly the value for F~ and F~-t can be substituted from equation (5). The equation (10) will then be in terms of the normal step loads N only. When there are ' n ' steps in the blade root, the equation (10) will yield ( n - I ) linear algebraic simultaneous equations corresponding to i = 2, 3 . . . . . n. One more equation is given by the overall equilibrium equation of the semi-blade root as per equation 3(c). Solution of these ' n ' simultaneous equations gives the value of normal step loads N~, N2, N3 . . . . Since the number of steps in a fir-tree blade root is generally less than five, the solution of cor[esponding number of algebraic equations can easily be carried out by the designer.
389
Fir tree fastening of turbomachinery blades--ll
DESIGN PROCEDURE The design procedure for evaluating the normal step loads is summarised below: (i) The steps in the blade root and the disc are numbered as illustrated in Fig. 2 and for each step the values of parameters h, W, a, b, a,/3 and 3' are noted. (ii) The values of A~, Bi, Ci, D. J. K, L~ and Mi (i = I to n) are calculated from equation (1) for blade as well as disc steps. (iii) The centrifugal body force (Si) is calculated for each step of the blade and the disc. The centrifugal force of the blade (excluding the root) is denoted as 2F0. (iv) The values obtained above are substituted in equations (10) and 3(c). The solution of the resulting ' n ' equations gives the step loads Nw, N2, N 3 , - . . . . •
Numerical computations and discussion The procedure outlined above is applied to the case of a three-step fir-tree root subjected to centrifugal force (2F0) of the blade alone. The geometrical parameters and elastic constants are taken to be identical for the disc and the blade. The coefficient of friction is taken as zero. The actual computations are carried out as Case I in the Appendix. The above data has been selected as the computations for this case have already been carried out[l] using the theory of elasticity approach. The results for the value of F0 = 11500 kg. are given in Table l column 3. The figures in parenthesis are the values of the vertical components of step-loads as obtained in Ref. [l]. It is seen that the maximum difference in results is about 4% which proves the accuracy of the present method of analysis. In the method detailed in Ref. [I] the effect of the centrifugal body force due to the root mass has been neglected. In the present analysis the effect of these body forces can be taken into consideration. In Case 2 the computations of step loads has been carried out taking into account this body force only. The vertical component of the step loads work out as V~ = 30.0230, V: = 37.8509 and V3 = 45.5592 kg per mm. While these values will change with blade geometry it is significant to note that the last blade step (nearest to the disc-centre) takes the greatest part of this load and the first step the least part. In computations carried out in Case 1 the coefficient of friction was taken as zero to be able to compare the results with those of Ref. [1]. Since the other data considered correspond to the values of steel, the value of/.t should be approx. 0.42. The loads for this case have been computed in Case 3, and are given in column 4 of Table I. In the present case the coefficient of friction has not significantly affected the vertical component of the loads carried by the steps. However, it has significantly altered the normal and tangential components. The results could be different for different geometries. The effect of friction and root mass on the normal and tangential components of step load could be compared by taking a particular value for F0 and a suitable value for root mass. In a typical intermediate stage blade of a steam turbine the blade mass may be of the same order as the root mass, since the root is generally of uniform thickness and the blade is of a narrow aerofoil section tapering towards the tip. The centre of gravity of the blade is at a greater radius than that of the centre of gravity of the root mass. It may be reasonable to take, as an illustrative example, the centrifugal force of the blade as twice the centrifugal force due to the root mass. For the blade having Fo = 11,500 kg and centrifugal force due to semi-root mass as 5620 kg (for a blade root of 50 mm thickness), the step loads are compared in Table 1, for the different cases.
TABLE l. COMPARISON OF STEP-LOADS Step No.
Component
1
Normal Tangential Vertical Normal Tangential Vertical Normal Tangential Vertical
2 3
Root-mass neglected /z = 0.0 # = 0.42 4424 0 3830(3910) 4430 0 3836(3680) 4424 0 3830(3910)
3567 1498 3827 3551 1491 3821 3567 1498 3827
Root-mass included ~t = 0.0 /z = 0.42 6157 0 5331 6616 0 5728 7054 0 6108
4988 2095 5367 5302 2227 5705 5619 2360 6047
The step loads are considerably higher when the centrifugal force of the root mass is also taken into account. The inclusion of friction affects the normal and tangential components of the step-loads significantly. For the geometry under consideration the friction has not significantly altered the vertical component of step loads. CONCLUDING REMARKS A fir-tree type of root fastening has been considered as made up of several distinct steps, the shapes of which are defined by seven different parameters. Two reference points have been shown to be relevant in assembling the different steps to form a complete fastening.
390
G.D. SINGHand S. RAWTAN1
T h e e q u a t i o n s b a s e d o n e q u i l i b r i u m a n d e q u a l i t y of r e l a t i v e d e f o r m a t i o n s h a v e b e e n g i v e n for a g e n e r a l ' n ' step root. T h e m a j o r a d v a n t a g e s of the p r e s e n t m e t h o d are: (i) T h e c o m p u t a t i o n s i n v o l v e d in c a l c u l a t i n g the step loads are c o m p a r a t i v e l y s i m p l e , r e q u i r i n g the s o l u t i o n of ' n ' l i n e a r a l g e b r a i c e q u a t i o n s for a n - s t e p root. (ii) It is n o t n e c e s s a r y to a s s u m e t h a t the elastic c o n s t a n t s as well as the step w i d t h are i d e n t i c a l for b l a d e r o o t a n d the disc. (iii) Coefficient of sliding f r i c t i o n b e t w e e n the b l a d e a n d disc step c a n be t a k e n into a c c o u n t . (iv) T h e c e n t r i f u g a l b o d y f o r c e s d u e to the r o o t m a s s itself n e e d n o t b e n e g l e c t e d . (v) D i f f e r e n t steps of the b l a d e m a y h a v e d i f f e r e n t v a l u e s of g e o m e t r i c parameters. I n the p r e s e n t a n a l y s i s the d i s t r i b u t i o n of n e c k - s e c t i o n load has b e e n t r e a t e d as u n i f o r m for d e f l e c t i o n a n a l y s i s , w h i c h m a y n o t be e x a c t l y t r u e . O n c e the load c a r r i e d b y e a c h step is o b t a i n e d the m a x i m u m stress at the fillets c a n b e o b t a i n e d b y t r e a t i n g the step as a s h o r t c a n t i l e v e r a n d u s i n g a n a p p r o p r i a t e stress c o n c e n t r a t i o n f a c t o r . Acknowledgement--The valuable suggestions given by Dr. R. Kitching, of the University of Manchester Institute of Science and Technology, for the extension and improvement of the manuscripts of the papers are gratefully acknowledged.
REFERENCE 1. E. SHIRATORI,M. HOSHIYAand N. SAKURAI,Bull. T o k y o Inst. o f Tech. 80, 25 (1967). APPENDIX 1 The procedure of computation is illustrated for the case of a three step fir-tree root for which a solution based on the theory of elasticity is available j. Equations (10) and (3) when applied to a three-step root take the form; [ H I { N } = {R}
(AI)
where, H,t = - { A i + A { - Jt + # ( D t + D { - M r ) - G ( B ~ -
K~)}
Hi2 = {Az + A~ - J ~ + t~( D2 + D~ - M ~) + G( K j - BI)} HI3 = G(B~ + K i - BO Hz, = G ( B ; - K~ - B~)
/422= - {A2 + A~ - .12 + I2(D~ + D~ - M2) - G ( B ; - K~)} /423 = {A3 + A~ -J~ +/z(D~ + D~ - M;) + G ( K z - B2)} //3, = H32 = H33 = G
R, = C , - C z + C { - C ~ - L ~ - L ,
+ S~ . B2 +($2 + $3).
(K, - B , ) - SI • (B~ - K{)
R2 = Cz - C3 + C~ - C~ - L~ - L2 + S3" (K~ - B2) - ( S ; +St). ( B ; - K ; ) + S;. B{ R3 = (Fo + S, + $2 + S3). Case 1
The following data is taken for the blade and disc-steps: E = 2 x 104 kg/mm:, t = I ram, a = I0°, /3 -- 20°, a = 6.9 mm, b = 11.50 ram, Wt = 35.13 mm, W2= 18.32 mm, WI = 1.5 ram, W~ = 18.32 ram,
/J. = 0.0 y = 39.5° h =23.1 mm W3= 1.5 mm W~ = 35.13 mm.
(Centrifugal body force of the root steps is neglected.) The values of the coefficients for the blade-step and for the disc-step are calculated and given in Table AI.
391
Fir tree fastening of turbomachinery blades--ll TABLE A1. COEFFICIENTS FOR BLADE AND DISC-STEPS ( X I0 +4) Coeff. A B C D J K L M S
Step I Blade Disc -3.1318 -0.0430 0.0 - 1.9590 -0.0180 -0.4411 0.0 +0.0664 0.0
-2.9510 -0.2978 0.0 -1.9547 -0.3696 - 1.4507 0.0 +0.0977 0.0
Step 2
Step 3
Blade
Disc
Blade
Disc
-3.0416 -0.1704 0.0 -1.9568 -0.1938 -0.9457 0.0 +0.0821 0.0
-3.0416 -0.1704 0.0 -I.9568 -0.1938 -0.9457 0.0 +0.0821 0.0
-2.9510 -0.2978 0.0 -1.9547 -0.3696 -1.4507 0.0 +0.0977 0.0
-3.1318 -0.0430 0.0 -1.9590 -0.0180 -0.4411 0.0 +0.0664 0.0
From equation (4), G = cos (a +/3 - &)' sec 4~ = 0.866.
Equation (A1) when applied to this three-step root takes the following form, 6.736 × 10 -4 10.492 × 10 -4 [ 0.866
-6.234 × 10-4 6.234 × 10-4 0.866
-0.492 x 10-4 -6.736 x 10-4 0.866
N N2 N3
= F0
This gives N, = 0.3847 Fo, N: = 0.3853 Fo and N3 = 0.3847 F0 as step-loads acting normal to the step surface of the blade. The vertical components are V~ = 0.3331 Fo, V2 = 0.3336 Fo and V3 = 0.3331 Fo. Case 2 In the above case the effect of pull of the blade on the root was considered and the effect of centrifugal force due to the mass of the root was neglected. Here, only the effect of the latter is considered. Let, p = 7.85 × 104/9810, r = 1000 mm and, w = 314 rad/sec. The values of the coefficients C, L, and S only are now different from those given in Case-1 and are given below in the Table A2. TABLE A2. VALUES OF COEFFICIENTS (x 10 +4)
Coeff. C L S
Step 1 Blade Disc
Step 2 Blade Disc
Step 3 Blade Disc
25.386 10.078 531330
24.623 11.0163 378112
23.859 11.954 224892
23.859 11.954 224892
24.623 11.016 378112
25.386 10.078 531330
The values of the remaining coefficients remain the same. Equation (AI) gives, 6.736x10-4-6.234x10-4-0.492x10~']{N, 0.492 × 1 0 - 4 6.234 × 10-~ -6.736 × 10-' / 0.866 0.866 0.866 J
I
N: ?43
=
- 6 4 . 8 4 0 x 1 0 -~] -64.840 x 10-'~ • 113.433
The above on solving gives N~ = 34.6687 kg, N: = 43.7078 kg and N~ = 52.6088 kg. The vertical components of these forces are V~ = 30.0230 kg, I/2 = 37.8509 kg and V3 --- 45.5592 kg per mm thickness of the root. Case 3 In this case the effect of the coefficient of friction, #, at blade/disc contact surface is considered when the root is subjected to only the blade pull, the effect of root mass being neglected, as was so in Case 1. Taking g = 0.42 the value of G, from equation (4) is 1.076. From the equation (A1) the following relation is obtained:
I
8.570 x 1 0 - 4 - 7 . 9 9 6 x 10~ 0.611 × 10-4 7.996 ×10-' 1.076 1.076
-0.611 x 10-41 { N I l {00} -8.570 ×10-' / N2 = . 1.076 J N; F0
The above on solving gives N, = 0.3102 Fo, N: = 0.3088 Fo and N3 = 0.3102 Fo as normal step-loads. Their vertical components are V~ = 0.3338 F0, V2 = 0.3323 F0 and V3 = 0.3338 Fo.