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Al-khwarizmi Engineering

Journal

Al-Khwarizmi Engineering Journal, Vol.3, No2,pp32 -48 (2007)

Effect of the Mechanical and Thermal Stresses of Rotating Blades

Assist. Prof. Dr. Nabeel K. AL-sahib Malik M. Ali Baghdad University / AL-Khwarizmi College of Engineering

((( R R R eee ccc eee iii vvveee d d d 111 111 D D D eee ccc eee mmmbbb eee r r r 222000 000 6 6 6 ;;; aaa ccc ccc eee p p p t t t eee d d d 222 555 A A A p p p r r r iii lll 222 000 000 7 7 7 )))

Abstract: Rotating blades are the important parts in gas turbines. Hence, an accuratemathematical estimation (F.E.M) of the stresses and deformations characteristics wasrequired in the design applications to avoid failure. In recent years there are researchersinterest in the effect of temperature on solid bodies has greatly increased, The main of thisstudy investigated the thermal and rotational effects. So, the thermal stresses due to high

pressure and temperature are studies, also determine the steady state stresses anddeformations of rotating blades due to mechanical effect. Many parameters such asthickness and centre of rotating are investigated in this paper. The study results can ensuregood recommendation for the effect of the mechanical and thermal stresses of rotary

blades.

Keywords: Thermal Stresses, Finite Element Method, Mechanical Stresses , BladeMaterial , Design .

IntroductionRotating blades was one of the

common structural elements used inseveral practical machines such asturbomachinery system, turbojet engines,turbofan, and helicopter rotors. Becauseturbomachinary blades are assumed as ashell, the numerical technique should becapable of representing this complicatedstructure based upon the shell theory. Inthis paper the type of rotating blades were

used is rectangular blade, the geometry for this type was show in figure (1).Ramamurti and Sreenivasamurthy [1]studied the dynamic stress analysis of rotating twisted and tapered blades. The

finite element method was used to determine thestresses and deformations. Three-dimensional,twenty-node isoparametric elements have beenused for the analysis. Extensive analysis has

been done for various pre-twist angles, skewangles, breadth to length ratios, and breadth tothickness ratios of the blades. Experiments werecarried out to determine the stresses for theverification of the numerical results. Lee [2]determined the frequencies and mode shapes of

turbomachinery blade having both camber and twist.The body forces were considered to be centrifugalforces generated within the shell due to rotation of the blade. Omprakash and Ramamurti

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Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)33

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metal. But for high temperature gasturbines, Nimonic alloy series arerecommended. They are basically

Nickel-Chromium alloys, whichwithstand oxidation, mechanicalstresses, and creep at workingtemperatures in excess of 1000 o C. in themiddle part of the turbine, thetemperature and stress conditions aremoderate, but corrosion is likely. Brassis probably the most suitable material.

In the final stages, and especially inlarge machines, the blades are long andheavily stressed. Fatigue action due tovibration may exist. Further than this,

anti corrosion properties are alsodesirable. Brass is a suitable materialwhere the stress conditions allow of itsuse, but when the stresses are high, 5%

Nickel Steel, Stainless Steel, Monelmetal or Phosphor Bronze may be used.

Turbine Blade Design Turbine blades operate at 1400 0C andare required to have a service life of 10000- 20000 hours. The techniquesused to meet these requirements are:

1) The use of nickel basedsuperalloys (containing many

precipitates, solid solution atomsand having high oxidationresistance)

2) The blades are cast a singlecrystal. This means that there areno grain boundaries within thestructure and thus minimizescreep see Fig. (2)

3) The blades are internally cooled

to allow increased operatingtemperatures. Typical coolingdesign features are shown in Fig.(3)

4) The blades are coated to increasethe oxidation resistance as shownin Fig. (4). A timeline of techniques to raise engine-operating temperatures is shownin Fig. (5 &6)

[3] carried out the steady state,dynamic stress and deformationanalysis of high pressure stageturbomachinery bladed discs takinginto account all the geometriccomplexities involved and included thecontributions due to initial stress andmembrane behavior, which used atriangular shell element with sixdegrees of freedom per node. Yoo [4]studied the vibration analysis of rotating pre-twisted blade with aconcentrated mass. The blade has anarbitrary orientation with respect to therigid hub to which it is fixed. Theequations of motion are derived basedon a modeling method that employshybrid deformation variables. Theresulting equation for the vibrationanalysis was transformed in thedimensionless parameters on the modelcharacteristics of the rotating bladeswere investigated through numericalanalysis

Blade Materials The elimination of blade corrosion and

designing against creep and fatigue, toa large extent, is a problem of findingsuitable materials. Another factor,especially in high-pressure work, is theworking temperature of the blades.Under high temperatures, somematerials tend to disintegrate, andcertain mechanical properties, e.g.hardness (in the sense of resistance toabrasion), are affected adversely.

The properties, which are desirablefor example in turbine blading, varyaccording to the conditions of operationand according to the location of the

blading in the turbine [5]. At high- pressure end, the temperature is high,and the blades are sometimes subjectedto erosive action. Great strength is notrequired as the blades are usually short.The material should be hard at hightemperatures. Suitable materials are 5%

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Finite Element technique

The difficulties in evaluating the stress anddeformations of rotating blades comes fromthe complex geometry of blades due toasymmetry of cross-section, pre-twist, andmounting the blades at a skew angle to therotating disk. The pre-twist and skew of the

blade cause coupling in both bendingdirections, and the asymmetry of the cross-section causes coupling with the tensionalmotion of the blade. For the abovecomplexities, all previous investigations inthis field had declared that plate or shellanalysis for blades is suitable and reliable.Therefore, the shell theories with finite

element provide a useful tool to analyzesuch structures. A superparametric shellelement will be used to investigate thethermal and mechanical stresses of rotating

blades. This element consists of four corner and four midside nodes as shown in Fig. (7).The degrees of freedom considered at The sare the three translations u, v, w of themidsurface and two rotations and of thenormal to the midsurface. The Cartesiancoordinates of any point of the shell and thecurvilinear coordinates can be written in the

form:

hn

m

l

f

z

y

x

f

z

y

x

i

i

i

i

i

i

i

i

3

3

3

2

(1)

where l 3i, m 3i and n 3i are the directioncosines of the normal to the middle surface.Here f i is a function taking a value of unityat the node i and zero at all other nodes, it iscalled as shape function ( 4).as shown intable.(1).In the kinematics formulation twoassumptions are imposed:

1- Nodal fiber isinextensible.

2- Only small rotationsare considered.

The displacements at any point (, , )can be expressed in terms of the

nodal displacements as

i

ii

i

ii

i

i

i

ii f

h N

w

v

u

f

w

v

u

2

8

1

8

1

... . (2)

In this formula the symbol i denote thefollowing matrix:

ii

ii

ii

i

nn

mm

ll

12

12

12

Column 1 in this array contains negativevalues of the direction cosines of thesecond tangential vector V 2i , andcolumn 2 has the direction cosines for the first tangential vector V 1i. Thesevector are orthogonal to the vector V 3iand to each other.The strain in the direction the normal to

the mid-surface is assumed to be negligible

( z )The normal to the mid-surface of the

shell element will remain normal to the mid-

surface after deformation.The displacement shape functions may becast into the matrix form:

[f i]=[f Ai]+ [f Bi] (i=1,28) (3)

where

[f Ai]=

00100

00010

00001

, f i &

[f Bi]=ii

ii

ii

nn

mmll

12

12

12

000

000000

2ih f i .. (4)

The 3 X 3 Jacobian matrix required in this

formulation is:




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Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)

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[J]=

,,,

,,,

,,,

z y x

z y x

z y x

(5)

where , indicate differentiation withrespected to the symbol following the coma.The derivatives in matrix [J] can be foundfrom equation (1)

ih

ii

ih

iii

ii

ih

iii

ii

l f x

l f x f x

l f x f x

i

i

i

32

8

1,

32

8

1,

8

1,,

32

8

1,

8

1,,

And

so on

For this element, six types of non-zerostrains exit, as follows:

z x

y z

x y

z

y

x

zx

yz

xy

z

y

x

uw

wv

vuw

v

u

,,

,,

,,

,

,

,

.. (6)

The stress-resultant vector in the localcoordinate system is,

{f }={f xf yf xy Q y Q x M x M y M xy }T..(7)

Where Q x was the shear stress per unit length inthe x and y direction. The relationship betweenthe stress resultants and the generalized strainscan be stated as follows,

{f }=[D]{} (8)where [D] is the rigidity matrix. A typicalrigidity matrix for flat plate shell is given

24)1(

1212

1212

21

21

21

2

2

22

22

0000000

000000

000000

0000000

0000000

0000000

0000001

0000001

1

h

hh

hh

k

k Eh D ..(9)

k =shear shape factor (assumed k=1.2) ( 5).

Numerical Solution Two cases were studied, steady state and thermal

stresses as follows:Steady -State Analysis

In this section the steady state stresses anddeformations were calculated, The blades consideredwere made of steel; and the material properties are:modules of elasticity = 207 * 10 3 Mpa, Poissonsratio = 0.3, Density = 7850 kg/m 3. Other informationare: Speed of rotation = 5000 r.p.m, Disk radius/width = 4, width = 0.05m. In this respect, anumerical study for a rectangular cross-section bladeis given in details because it is a common model.

Comparison of the results

A comparison with experimental published resultsis achieved between the present work and theexperimental work [1].

In this comparison, a rectangular steel blade isconsidered, having the following dimensions andmaterial properties: Length = 100 mm, Width =25mm, Thickness =3mm, Skew angle = 90 o , Radiusto root = 125mm, and maximum speed of rotation =2500 r.p.m, Modulus of elasticity = 207 * 10 3 Mpa,Poissons ratio =0.3, Density = 7850 kg/m 3 . Straingauges were used as transducers and the results areshown in Table 2Effect of Rotational speed

The variation of stresses and deformations witheight speeds of rotation (250, 500, 750, 1000,1250 1750 and 2000 r. .m for three as ect




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Fig. (2) Turbine blades cast to promote different grain structures [9]

Fig. (3) Blade cooling techniques [10]

Fig. (4) History of turbine blade coating systems [11]




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Fig. (5) Developments to increase engine-operating temperatures [10]

Fig. (6) Developments to increase turbine metal capability [8]

=1

=-1

=-1

=1=1

=-1

Fig. (7) Eight noded shell element




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Serendipity 8-node element:

Corner nodes: )1)(1)(1(41

iiiii f

Midside nodes: )1)(1(21

)1)(1(21 2222 iiiii f

Table (1) Shape Function for Midsurface Interpolation of Shell Elements

Table 2 Comparison with experimental results. (radial stress)

o Distance Exp. (Mpa) Presentwork (Mpa)

(r.p.m)0 0.1 L 9.1154 9.0086 2500

15 0.25 L 5.7115 5.5613 2000

z, w

i, v

Fig. (8) Degrees of freedom at a node

i




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0

1

2

3

4

5

6

0 250 500 750 1000 1250 1500 1750 2000 2250

speed of rotation (r.p.m)

v_

d e

f l e c

t i o n

{ m m

} x 1 0 ^ - 2

l/b=3l/b=4l/b=5

0

1

2

3

4

56

7

8

9

0 250 500 750 1000 1250 1500 1750 2000 2250


x x_

s t r e s s e s

{ M p a

}

l/b=3l/b=4l/b=5

Fig. (9) Variation of v-deflection with speed of rotation Fig. (10) Variation of xx-stresses with speed of rotation

Fig. (11) Variation of yy-stresses with speed of rotation

0

3

6

9

12

15

18

21

24

27

30

33

0 250 500 750 1000 1250 1500 1750 2000 2250


y y_

s t r e s s e s {

M p a

}

l/b=3l/b=4l/b=5

R=0.225 m, h=0.0018 mSkew angle =20 o

Pre-twist angle = 12 oAl-alloy (5052)




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0

1

2

3

4

5

6

7

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Thickness (mm)

v_ d e

f l e c t

i o n

{ m m

} 1 0 _ 2

l/b=3l/b=4l/b=5

0

1

2

3

4

5

6

7

8

9

10

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Thickness (mm)

x x_

s t r e s s e s {

M p a

}

l/b=3l/b=4l/b=5

0

5

10

15

20

25

30

35

40

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Thickness (mm)

y y_

s t r e s s e

s { M p a

}

l/b=3l/b=4l/b=5

Fig. (12) Variation of v-deflection with thickness Fig. (13) Variation of xx-stresses with thickness

Fig. (14) Variation of yy-stresses with thickness

R=0.225 m, =2000r.p.m

Skew angle =20 o

Pre-twist angle = 12 o

Al-alloy (5052)




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0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3

x - a x is { c m }

T e m p e r a

t u r e

{ C

0 .0 0 E + 0 0

2 .0 0 E + 0 2

4 .0 0 E + 0 2

6 .0 0 E + 0 2

8 .0 0 E + 0 2

1 .0 0 E + 0 3

1 .2 0 E + 0 3

1 .4 0 E + 0 3

1 .6 0 E + 0 3

0 0 . 0 1 0 . 0 2 0 . 0 3x - a x is { c m }

x x

_ s

t r e s s e s

{ M P a }

Fig. (15 Variation of temperature with x-axis

Fig. (18) Variation of yy-stresses with x-axis

0 . 0 0 E + 0 0

1 . 0 0 E + 0 2

2 . 0 0 E + 0 2

3 . 0 0 E + 0 2

4 . 0 0 E + 0 2

5 . 0 0 E + 0 2

6 . 0 0 E + 0 2

7 . 0 0 E + 0 2

8 . 0 0 E + 0 2

9 . 0 0 E + 0 2

1 . 0 0 E + 0 3

0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3

x - a x is { c m }

y y

_ s

t r e s s e s

{ M P a

}

0 . 3 5

0 . 3 9

0 . 4 3

0 . 4 7

0 . 5 1

0 . 5 5

0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3

x - a x is { c m }

u_

d e

f l e c

t i o n

{ m m

}

Fig. (17) Variation of xx-stresses with x-axis

Fig. (16) Variation of u-deflection with x-axis




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Fig. (19) Contour of temperature

Fig. (20) Contour of u-deflection




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Fig. (21) Contour of xx-stresses

Fig. (22) Contour of yy-stresses




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Reference:

1- Ramamurti, and Sreenivasamurthy;Dynamic stress analysis of rotatingtwisted and tapered blades, J. StrainAnalysis, Vol. 15, No. 3, 1980

2- J. K. Lee, A. W. Leissa and A. J.Wang, Vibration of blades withvariable thickness and curvature byshell theory, J. of Eng. For GasTurbine and Power, Vol. 106, pp. 11-16, 1984.

3- V. Omprakash and V. Ramamurti,Dynamic stress analysis of rotatingturbomachinery blade-disc systems, J.of Computers and structures, Vol. 32,

No. 2, pp. 477-488, 1989.

4- H. H. Yoo, J. Ykwak and J. Ghung,

Vibration analysis of rotating pre-twist blades with a concentrated mass,J. of sound and vibration, Vol. 240,

No. 5, pp. 891-908, 2001

5- Kearton, W. J. ,Steam turbine

operation, Sir Petman & Sons, Ltd.,4th ed. 19456. Flower, H.M. ed., HighPerformance Materials in Aerospace ,Chapman &Hall, London,

7. Sims, C.T., Stoloff, N.S. & Hagel,W.C. eds., Superalloys II , John Wiley&Sons, New York, 1987

8. Askeland, D.R., The Science and Engineering of Materials: Third SI Edition ,Chapman and Hall, London, 1996

9. Callister, W.D., Jr, MaterialsScience and Engineering an

Introduction: 5 th Edition , John Wiley and Sons Inc, New York, 2000

10. Ashby, M.F. & Jones, D.R.H., Engineering Materials 1: An Introduction totheir Properties and Applications, 2 nd

Edition , Butterworth Heinemann,Oxford, 1997

11. Sims, C.T., Stoloff, N.S. & Hagel,W.C. eds., Superalloys II , John Wiley

&Sons, New York, 1987

12- Aeronautical Materials Teacher Reference, School of Material Scienceand Engineering, University of NewSouth Wales 2001




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