Stress concentration factors for pin lever of runner blade

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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
am.pittner@uem.ro, v.campian@uem.ro, d.nedelcu@uem.ro, d.frunzaverde@uem.ro,
v.cojocaru@uem.ro, 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.
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 .
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
ISSN: 1792-4294
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
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The most common definition, used to describe de
interdependence between Kf and Kt is:
K f = 1 + q ⋅ (K t − 1)
(2)
stepped round bar with a shoulder fillet (based on
photoelastic tests of Leven & Hartman, Wilson &
White).
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]
1
q=
(3)
a′
1+
r
where: r - the concentrator’s radius;
a ′ - material constant depending on the
mechanical properties of the analyzed
material.
- according to Peterson [3]
1
q=
(4)
a
1+
r
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:
5.0
4.5
r
4.0
D
M
M
d
3.5
Kt = σmax / σnom
σnom = 32M / πd3
D/d=3
3.0
2
2.5
1.5
1.2
2.0
1.1
1.05
1.5
1.0
1.01
0.01
1.02
0.05
0.10
0.15
0.20
0.25
0.3
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]:
1
(1)
K = 1+
t,inc
2
1,8
 2070 
 mm
a = 0,0254 ⋅ 
(5)
 σr 
where: σ r - ultimate tensile strength of material.
The analytical relation to determinate value of
Kf, proposed by expert group from FKM, is [1]:
K t ,inc
K f ,inc =
(6)
nσ (r ) ⋅ nσ (d )
where: nσ (r ) - Kt-Kf ratio of the component for
normal stress or for shear stress
according to r;
nσ (d ) - Kt-Kf ratio of the component for
normal stress or for shear stress
according to d;
The Kt-Kf ratio for normal stress nσ (r ) is [1]:
3
r
r 
r
r d
0,62⋅ + 11,6⋅ ⋅ 1 + 2 ⋅  + 0,2⋅   ⋅
t
d 
d
t D
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 r ⊳ 0 and d D ⊲ 1 , 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.

Rm 

− aG − 0 , 5+
bG ⋅MPa 

(7)
nσ = 1 + Gσ ⋅ mm ⋅ 10
where: aG, bG – material constants;
Gσ - the related stress gradient;
Rm – tensile strength of material.
The analytical relation to determine the related
stress gradient to value of r is [1]:
2,3
Gσ (r )int,inc =
⋅ (1 + ϕ )
(8)
r
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.
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where: ϕ - coefficient depending by ratio t d .
The Kt-Kf ratio for normal stress nσ (d ) is [1]:
the centrifugal force FC for the runner
speed 71.43 rot/min;
the axial thrust on the blade FAH , resulted
from the measurements on model;

Rm 

− aG +
bG ⋅MPa 

nσ = 1 + Gσ ⋅ mm ⋅ 10
(9)
The analytical relation to determine the related
stress gradient to value of d is [1]:
Gσ (d )int,inc =
2
d
the tangential force on the blade FT .
Loads applied to the lever are presented in Fig.4.
(10)
3 Problem Solution
In Fig.2 there are presented the load block after
which the calculation is made.
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.
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]:
Table 1
Lever loading
Centrifugal force [N]
4069495
Blade & trunnion & lever mass force [N]
286900
Runner
Head
blade
Case
H
angle
[m]
[grade]
Fig. 3
The entry data for calculations of Kt , after the
first two methods are:
r = 10 mm
d = 338 mm
D = 375 mm
t = 18,5 mm
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
Tangential
force
F link, max
FT
[N]
[N]
1
+17.5
25
1775041
1336898
2
+10
25
1771437
1024955
3
+10
31.4 1887115
1292381
3100000
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;
- the mesh version 2 with 174376 finite elements –
Fig.6;
- the mesh version 3 with 258779 finite elements –
Fig.7;
– trunnion assembly G =286900 N;
ISSN: 1792-4294
Thrust
force
Fax
[N]
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Table 3
- the mesh version 4 with 400750 finite elements –
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.
GMS
Case 1
Case 2
Case 3
Finite
Elements
number VonMises VonMises VonMises
max
[MPa]
36
30
25
20
145772
174376
258779
400750
max
[MPa]
395,50
413,02
413,70
411,50
max
[MPa]
382,7
401,8
398,6
394,3
387,6
407,0
403,1
394,1
The values of stress concentration factors and
fatigue notch factors, determinate with all methods
reminded previously are presented in Table 4.
Fig.5
Table 4
Stress concentration
factor Kt
Fig.6
from
Graphics
after
FKM
after
FEM*
Fatigue notch
factor Kf
after
after
after
Neuber Peterson FKM
1,3 ÷
2,13
1,6
*Finite element methods
2,2
Fig.7
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.
Stresses values σînc [N/mm2]
φ=0º,
φ=+10º,
φ=+10º,
Work
H=25 m
H=25 m
H=31,4 m
regime
Case 1
Case 2
Case 3
- closing
course
232,008
249,678
297,490
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).
ISSN: 1792-4294
1,97
4 Conclusion
Table 2
course
2,18
Fig.8
Table 2 present the value of stresses obtained
through classical methods.
- opening
2,15
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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-09400-289, November, 2009.
ISSN: 1792-4294
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ISBN: 978-960-474-203-5