stress concentration factors for pin lever of runner blade mechanism

Stress concentration factors for pin lever of runner blade mechanism

from Kaplan turbines

ANA-MARIA PITTNER, CONSTANTIN VIOREL CAMPIAN, DORIAN NEDELCU,

DOINA FRUNZAVERDE, VASILE COJOCARU

Faculty of Engineering

“Eftimie Murgu”University of Resita

No. 1-4, P-ta Traian Vuia, 320085, Resita

ROMANIA

[email protected], [email protected], [email protected], [email protected],

[email protected], http://www.uem.ro

Abstract: This paper presents a comparative study between different values of stress concentration factor Kt

and fatigue notch factor Kf determined with different methods for a specific part. The study was specially

created for the pin lever of runner blade mechanism from a Kaplan turbine. The paper tries to offer the most

adequate choice, from the author’s point of view, which can be made between the obtained values after

processing the needed data issue from geometrical details and service conditions. The methods used to obtain

the look-up values are graphical, analytical and numerical. For this analyze it will be taken in consideration

from the oldest to the recent methods used by engineering community.

Key-Words: stress concentration factor, fatigue notch factor, graphical, analytical, numerical.

1 Introduction In the last four decades there have been issued a lot

of studies having like principal concerns designing

against fatigue failures. The attention of design

engineer is focused on the overall structure as well

as its components when exposure to service

conditions assume numerous fluctuating loads and

attendant stress and strain histories which may result

in fatigue failure. Previously, large factors of safety

were used into design components because of the

lack of knowledge and understanding of interactive

effects. These safety factors are no longer needed

since the development of extensive computer

software packages. Using adequate software can

make a realistic estimation about the real values of

local stresses in the interested points of a structure.

In absence of one of that specialized software

may still be used an analytic method to determined

the value of stress concentration factors needed to be

taken in to consideration for a real estimation of

strength in service. It is known that the presence of

one or more stress concentrators (abrupt variation of

section, material defects, improper work of surfaces,

etc) provoke the appearance of unexpected values

for local stresses. These values were proved to be,

many times, much higher that the one obtained from

strength calculations. Starting from these points

various theories were developed concerning the

main factors that induce such rising and the

methodologies to be apply for quantification of this

influence.

It was considered that this were the problem that

can be incriminated for structures taken out of

service, before the initial estimated time. The

phenomenon is known such as fatigue failure.

Finally, the measurement of effects induced by

stress concentrators on the stress values, in a

specified location, can be made multiplying the

calculated stress value with a stress concentration

factors. The engineering community usually uses, in

order to predict the real load, the so called fatigue

notch factor Kf .

2 Problem Formulation In order to find the value of fatigue notch factor, we

firstly must determine de stress concentration factor

Kt.

2.1 The theoretical stress concentration

factor Kt The value of stress concentration factor can be

determined using three methods:

- from diagrams experimentally raised;

- using analytical algorithms;

- from finite element analysis.

In Fig.1 there are illustrated an

experimentally diagram raised for bending of a

Latest Trends on Engineering Mechanics, Structures, Engineering Geology

ISSN: 1792-4294 181 ISBN: 978-960-474-203-5

stepped round bar with a shoulder fillet (based on

photoelastic tests of Leven & Hartman, Wilson &

White).

Fig. 1

As is shown in Fig.1, the value of Kt can be chose

from the experimentally raised diagrams depending

on three geometrical characteristics: value of large

diameter D, value of small diameter d and value of

fillet radius r. In that case, the usual problem that

appears is to not find the specific diagram for the

specified material of analyzed structure. In that case

must be chose a similar diagram raised for another

material with appropriate mechanical characteristics

tested for the same load.

Analytical method assumes calculation of Kt,

using a mathematical equation like [1]:

D

d

t

r0,2

d

r21

d

r11,6

t

r0,62

11K

32inct,

⋅

⋅+

⋅+⋅⋅+⋅

+= (1)

where: r, t, d, D – geometrical dimensions in

accordance with execution drawing

of lever.

Mathematical equation (1) is valide only for the

cases in witch 0⊳r and 1⊲Dd , case that is

represented also by ours.

To determinate Kt by finite element analysis it

was made a linearly static analysis using Cosmos

Design Star software.

2.2 The fatigue notch factor Kf The fatigue notch factor will be determined from

theoretical stress concentration factor Kt using an

analytical relation. The connecting relations

between the two mathematical equations have

different form in accordance with different vision of

the one that studied the problem of fatigue behavior

of structures.

The most common definition, used to describe de

interdependence between Kf and Kt is:

( )11 −⋅+= tf KqK (2)

where: q – fatigue notch sensitivity.

The fatigue notch sensitivity q is the measure of the

degree of agreement between Kt and Kf. In

specialized literatures two relations are mostly used,

to define the q's value:

- according to Neuber [2]

r

aq

′+

=

1

1 (3)

where: r - the concentrator’s radius;

a′ - material constant depending on the

mechanical properties of the analyzed

material.

- according to Peterson [3]

r

aq

+=

1

1 (4)

where: r – the concentrator’s radius;

a – material constant depending on the

material strength.

Also Peterson gives us an analytical relationship for

relatively high strength steels subjected to axial or

bending fatigue:

mmar

8,1

20700254,0

⋅=

σ (5)

where: rσ - ultimate tensile strength of material.

The analytical relation to determinate value of

Kf, proposed by expert group from FKM, is [1]:

( ) ( )dnrn

KK

inct

incf

σσ ⋅= ,

, (6)

where: ( )rnσ - Kt-Kf ratio of the component for

normal stress or for shear stress

according to r;

( )dnσ - Kt-Kf ratio of the component for

normal stress or for shear stress

according to d;

The Kt-Kf ratio for normal stress ( )rnσ is [1]:

⋅+−−

⋅⋅+= MPab

Ra

G

mG

mmGn5,0

101 σσ (7)

where: aG, bG – material constants;

σG - the related stress gradient;

Rm – tensile strength of material.

The analytical relation to determine the related

stress gradient to value of r is [1]:

( ) ( )ϕσ +⋅= 13,2

int,r

rG inc (8)

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0 0.01 0.05 0.10 0.15 0.20 0.25 0.3

Kt = σmax / σnom

σnom = 32M / πd3

M D M d

D/d=3

2 1.5

1.2

1.1

1.01

1.05

1.02

r


ISSN: 1792-4294 182 ISBN: 978-960-474-203-5

where: ϕ - coefficient depending by ratio dt .

The Kt-Kf ratio for normal stress ( )dnσ is [1]:

⋅+−

⋅⋅+= MPab

Ra

G

mG

mmGn 101 σσ (9)

The analytical relation to determine the related

stress gradient to value of d is [1]:

( )d

dG inc

2int, =σ (10)

3 Problem Solution In Fig.2 there are presented the load block after

which the calculation is made.

Fig. 2

In accordance with Fig.2, and taking into

consideration the tensile strength of material, it was

chosen the experimentally diagram for our case [3]:

Fig. 3

The entry data for calculations of Kt , after the

first two methods are:

mmt

mmD

mmd

mmr

5,18

375

338

10

=

=

=

=

The linearly static analysis it was made by

specialist of CCHAPT [4].

For linearly static analysis, firstly it is necessary

to define the loads applied to the lever:

� the gravity force of the runner blade – lever

– trunnion assembly G =286900 N;

� the centrifugal force CF for the runner

speed 71.43 rot/min;

� the axial thrust on the blade AHF , resulted

from the measurements on model;

� the tangential force on the blade TF .

Loads applied to the lever are presented in Fig.4.

Fig. 4

In the present analysed case, the main

component is the lever and the blade & trunnion

will be replaced by remote loads [4].

In Table 1 there are presented the loads applied

to lever for static analyses and the other parameters

necessary to be use for a complete analyze.

Table 1

The static analysis was made for different values

of global element size GMS for 36....20 mm to lead

to a great precision. For the study, there were made

analyses for four dimensions of meshes:

- the mesh version 1 with 145772 finite elements –

Fig.5;


Fig.6;


Fig.7;

Lever loading

Centrifugal force [N] 4069495

Blade & trunnion & lever mass force [N] 286900

Case

Runner

blade

angle

[grade]

Head

H

[m]

Thrust

force

Fax

[N]

Tangential

force

FT

[N]

F link, max

[N]

1 +17.5 25 1775041 1336898

3100000 2 +10 25 1771437 1024955

3 +10 31.4 1887115 1292381


ISSN: 1792-4294 183 ISBN: 978-960-474-203-5


Fig.8.

After the analysis was processed, the value of

theoretical stress concentration factor can be

calculated madding the ratio between values of

VonMises stresses, resulting from finite element

analyses, and the values of stresses obtained

according to classical strength calculations.

Fig.5 Fig.6

Fig.7 Fig.8

Table 2 present the value of stresses obtained

through classical methods.

Table 2

Stresses values σînc [N/mm2]

Work

regime

φ=0º,

H=25 m

Case 1

φ=+10º,

H=25 m

Case 2

φ=+10º,

H=31,4 m

Case 3

- opening

course 232,008 249,678 297,490

- closing

course 216,580 233,074 277,752

The numerical analysis confirms the fact that the

area with maximum stress value is the fillet area

between pin lever and body lever. Table 3 shows the

values of VonMises stress obtained through finite

element analyses.

Having the values of VonMises stress it is easy

to find the value of fatigue notch factor. This can be

done very simple only by dividing the value of

VonMises stress to stress values obtained from

classical strength calculations (for the same work

conditions).

Table 3

GMS

Finite

Elements

number

Case 1

Case 2 Case 3

VonMises

max

[MPa]

VonMises

max

[MPa]

VonMises

max

[MPa]

36 145772 395,50 382,7 387,6

30 174376 413,02 401,8 407,0

25 258779 413,70 398,6 403,1

20 400750 411,50 394,3 394,1

The values of stress concentration factors and

fatigue notch factors, determinate with all methods

reminded previously are presented in Table 4.

Table 4

Stress concentration

factor Kt

Fatigue notch

factor Kf from

Graphics

after

FKM

after

FEM*

after

Neuber

after

Peterson

after

FKM

2,2 2,15 1,3 ÷

1,6 2,13 2,18 1,97

*Finite element methods

4 Conclusion Analyzing the values revealed in Table 4 we can see

that the highest values for stress concentrations

factor are obtained using the analytical algorithms

proposed by FKM-Guideline.

As it is expected, the value of fatigue notch

factor Kf is smaller than value of stress

concentration factor Kf, no matter what method we

apply.

In industry, when we speak about big and

expensive machines, such as Kaplan turbines, it is

justified to chose, for dimensioning, the highest

values for multiplication factors, even if it raises

supplementary costs. These costs will always be

smaller than the ones necessary to repair

systematically the structures affected by fatigue.

The paper reveals the fact that an analytical

method can be use successfully to determine values

for stress concentration factors, which can be used to

make calculations to estimate fatigue lifetime

duration. The values analytically obtained definitely

are cover for all security working problems that

must be solved through designing process.

The chosen of fatigue notch factor became the

personal option of designing engineer, the accuracy

of results being strictly dependent by his experience.


ISSN: 1792-4294 184 ISBN: 978-960-474-203-5

References:

[1]Conle F.A., Chu C.-C., Fatigue analysis and the

local stress-strain approach in complex

vehicular Structures, International Journal of

Fatigue, Vol.19, Supp.No.1, 1997

[2]FKM-Guideline, Analytical strength assessment

of components in Mechanical Engineering,

Forschungskuratorium Maschinenbau” Frankfurt

/Main, 2003

[3]Hertzberg R.W., Deformation and Fracture

Mechanics of Engineering Materials, 4th edition,

John Wiley&Sons, Inc., New York, 1995

[4]Mercer M., Malton G., Draper J., Investigating

fatigue failures using analysis and testing – some

do's and don'st, Proceedings of the ABAQUS

International User's Conference, Germany, 2003

[5]Muhs D., Wittel H., Jannasch D., VoBiek J.,

Roloff/Matek .Machine Parts, Vol.I, Ed. Matrix

Rom, Bucuresti, Romania, 2008

[6]Neuber H., Theory of Notch Stresses, J. W.

Edwards Publ. Ann Arbor, 1946

[7]Peterson R.E., Stress Concentration Design

Factors, Ed. J. Wiley & Sons, New York, 1953

[8]Pilkey Walter D., Peterson's Stress

Concentration factor, second edition, John

Wiley&Sons, Inc., New York, 1997

[9]Rice R.C., Fatigue Design Handbook, 3rd ed.,

SAE Pub.No.AE-22, Society of Automotive

Engineers, Warrendale, PA, 1997

[10]Rusu O., Teodorescu M., Lascu-Simion N.,

Fatigue of Metals, Vol.I, Calculation's base, Ed.

Tehnica, Bucuresti, Romania, 1992

[11]Slocum A.H., Precision Machine Design,

Prentice-Hall, Englewood Cliffs, New York,

1992

[12]Socie D.F., Marquis G.B., Multiaxial Fatigue,

Society of Automotive Engineers, Inc.,

Warrendale, PA, 2000

[13]Suresh S., Fatigue of materials, Second Edition,

University Press, Cambridge, United Kingdom,

1998

[14]*****- Strength and lifetime duration calculus

for runner blade lever of CHE Portile de Fier I

turbine, CCHAPT, Technical Report No.U-09-

400-289, November, 2009.


ISSN: 1792-4294 185 ISBN: 978-960-474-203-5

stress concentration factors for pin lever of runner blade mechanism

Documents