N. S. Vyas Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208 016 India

J. S. Rao Department of Mechanical Engineering, Indian Institute of Technology, New Delhi 110 016 India

Fatigue Life Estimation Procedure for a Turbine Blade Under Transient Loads Fatigue analysis and consequent life prediction of turbomachine blading requires the stress load history of the blade. A blade designed for safe operation at particular constant rotor speeds may, however, incur damaging stresses during start-up and shut-down operations. During such operations the blade experiences momentary resonant stresses while passing through the criticals, which may lie in the speed range through which the rotor is accelerated. Fatigue due to these transient influences may accumulate to lead to failure. In this paper a technique for fatigue damage assessment during variable-speed operations is presented. Transient resonant stresses for a blade with nonlinear damping have been determined using a numerical procedure. A fatigue damage assessment procedure is described. The fatigue failure surface is generated on the S-N-mean stress axes and Miner's Rule is employed to estimate the accumulation of fatigue.

Introduction Blade fatigue is a multidisciplinary problem and various aspects of the problem have been subjects of extensive research; see articles by Rao (1987), Srinivasan (1984), and Rieger (1985). Blade fatigue is influenced by the static and dynamic stress fields on the blade, fatigue properties of the blade material, loading history, and the environment of operation. Fatigue crack usually initiates in a region of high stress at some metallurgical or structural discontinuity and if critical conditions of operation are sustained, the crack may grow to lead to failure. Various factors may affect the blade behavior in several different ways to make the failure problem very specific. The steps toward fatigue analysis can be, generally, summarized as (Rao and Vyas, 1986): 8 Generating the stress loading biography. • Definition of blade fatigue parameters under operating environment. • Life estimation using fatigue theories. The blade during its operation span, in addition to constant speed operation, undergoes various transient rotor operations like step-up, step-down, partial admission, etc. Attempts have been made by several researchers to obtain mathematical models for accurate determination of blade response under steady as well as transient excitations. Stress response of single rotating blades was obtained by Rao et al. (1986). Transient vibration analysis has been undertaken by Irretier (1986) and Vyas et al. (1987). Models for multiple blade systems have also been developed. Some recent works include Jones and Muszynska

(1983), MacBain and Whaley (1984), and Sinha and Griffin (1984). The problem of stress response determination is made complex by the nonlinearity of damping in the vibratory system. Energy dissipation through Coulomb, hysteretic, and viscous mechanisms of damping is a blade specific problem. Rao et al. (1986) developed an experimental technique for determination of damping properties under the overall influences of the above-described mechanisms for rotating blades and defined damping ratios in various vibratory modes as nonlinear functions of rotor speed and the vibratory strain amplitude. A numerical procedure has been developed by Rao and Vyas (1989, 1990) to account for the nonlinearity in damping for determination of resonant stresses during steady and transient rotor operations. In this study the influence of transient resonant stresses experienced by the blade during step-up/down operations on its fatigue life is investigated. The transient resonant stress, determined for a blade with nonlinear damping using a numerical procedure in conjunction with Reissner's functional, Ritz procedure, and modal analysis, defines the alternating stress level experienced by the blade. The mean stress level is a function of the speed of rotation and the steady component of the steam/gas force. S-N and mean stress diagrams are coupled to define the fatigue failure surface for the blade under assumed conditions of operating environment. Calculations to illustrate accumulation of damage are done using Miner's rule.

Contributed by the International Gas Turbine Institute and presented at the 37th International Gas Turbine and Aeroengine Congress and Exposition, Cologne, Germany, June 1-4, 1992. Manuscript received by the International Gas Turbine Institute February 4, 1992. Paper No. 92-GT-78. Associate Technical Editor: L. S. Langston,

Blade Vibration Model A single free-standing blade can be considered a tapered, twisted asymmetric aerofoil cross-sectional beam mounted on a rotating disk at a stagger angle (Fig. la). The cross section

198/Vol. 116, JANUARY 1994

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For accurate determination of blade stress fields, Reissner's functional has been employed, which simultaneously yields good stress and displacement fields. The Reissner's functional is given by {Tijeu-U*(Tjj)}dv

/* =

Biuidv+ C

N

Fig. 1(a)

Tjiijds

(2)

Using beam theory and stress-strain displacement relationships, the above can be shown to be

Blade mounted on rotating disk

-Myx"

+Mxy"

+TeB' +

2

2C

2

_ M Jylyl+M Ixlxl - 2 MxMy Ixlyl dz (3) 2E(Iylyl Ixlxl ~ / Xiyi The dynamic Reissner functional is set as L=T-IR p

/

^

y\

*

ry]

Using shape functions — X

Blade cross section Fig. 1(6)

x=ZAi{t)Mz)

MX = L DM

Mz)

y = LBi(t)fi(z)

My = L Ei(t) hi(z)

0 = EC,-(O/;U)

r e = S E,(t) hi(z)

(5)

where

Elemental displacement relations

of the blade at a height z is shown in Fig. \(b). The centroid of the cross section is located at C, while O is the center of flexure. The kinetic energy of the blade can be shown to be (Raoet al., 1986)

-H-

(4)

y.(Z) =

(i+2)(i+3)z;+l_/q±3)zf+2 6 3

fi(Z) = Zi-

+

/(m)z;+3 6

i+\

2

A{(x + ryey + (y-rx6) } dz {(x' +ry6+ry6)2

+ - ycg62dz + -po>2 \ ^ +

(1-Z) ; (5a) i+l which satisfy the boundary conditions of a cantilever, Ritz process

{y'~r'xd-rx6')2}{RIx+h)dz

2

2

2

dL = 0, etc. dAi

2

+ sin ^ A (x + ryd) dz + cos A (y-rx 6) dz Jo •'o - sin 2(j>\A(x + ryd) (y - rx d)dz

(6)

is applied to the dynamic Reissner functional above to obtain the equations of motion

(1)

p2[M\{q}-[K]{q\=Q

(7)

Nomenclature A = area of cross section Bj = body force distribution a, b = coefficients in trigonometric series of forcing functions C = torsional stiffness E = modulus of elasticity Fx, Fy = forcing functions / = shape function for bending deflections 7 = shape function for angular deflections Hmk = wth harmonic response in the kth mode h = shape function for bending and twisting moments /, = A0(l~z)+l/2Ai(l -zi)+... +^ r ( / „

z"+i)

+1

= l/2A0(l2-z2) +

J

n

+ l/3Al(lI~z2)+. l

..

+l

\(l + -z" ) n+2 = second moment of area about x\ x\ axis Journal of Engineering for Gas Turbines and Power

Iyxy\ /^i^i Icg IR L / M Mx, My m N ns pK Q R Rf

= = = = = = = = = = = = = = =

second moment of area about y\ y\ axis product moment of area about x\ y\ axis moment of inertia per unit length Reissner's functional Reissner's dynamic functional blade length moment moments in x and y direction harmonic number cycles to failure number of nozzles natural frequency in Arth mode forcing vector disk radius endurance limit modifying factor = kakbkckdkekf rx, ry = coordinates of the center of flexure Se = endurance limit for 103 cycles S'e = endurance limit for N cycles JANUARY 1994, Vol. 116 /199

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Table 1

Geometric and material properties I

Dist cm

cm

0.0

cm

2.127 .3447

I

yy cm

4

0.150

C x 10

r r „ x N-m y— cm rad

xy 4 cm

.9489

.4705

.9702

0.075

-.213 .075

2.5

1.738 .3651

.6717

.4158

.7776

- . 0 9 4 .135

5.0

1.647 .4927

.5174

.4146

.7576

- . 0 8 2 .073

7.5

1.263 .5054

.3097

.2943

.6113 - . 1 7 8

10.0

1.008 .4356

.1543

.2243

.4424

-.282

-.222

11.0

0.736

.0858

.1572

.3087

-.303

-.453

.3256

L = 11.1 cm, E = 2 0 0 GPa, R = 18 cm, G = 9 0 ° , p = 0 . 0 8 N / c m 3 , n

Table 2

-

-.113

CL075 1 -

Distance f r o m zero f o r c e p o s i t i o n (mm)

0.04

Fig. 2 Forcing functions

the blade in the x, y , 6 directions, respectively, can be expressed in Fourier form as

75 GPa

= 12.

FX(Z,

t) = a0X(Z)

c o s {mns(u>0t + - a t2)}

+ ^] amx(Z)

Fourier component amplitudes

+ Yjbmx(Z)

sin \mns(ui0t + -at2)}

m F (N/cm) mx 0

0.0622

mx 0.0000

F (N/cm) y a my my -0.0064

0.0000

M (N-cm/cm) b

mM 0.0280

Fy(Z,

+ ^jamy{Z)

cos {mns(oi0t + ^]bmy(Z)

0.0000

1

-0.0642

-0.0169

0.0153

-0.0392

-0.0269

-0.0173

2

0.0110

0.0140

-0.0063

0.0095

0.0037

0.0089

3

-0.0024

0.0005

0.0024

0.0005

-0.0006

0.0004

4

-0.0027

0.0013

0.0012

0.0019

-0.0010

0.0011

5

-0.0001

-0.0020

0.0007

-0.0012

0.0001

-0.0012

6

-0.0005

0.0028

-0.0018

0.0015

-0.0007

0.0017

where [A/] a n d [K] are mass a n d stiffness matrices a n d the vector (