Stress concentration factors for pin lever of runner blade mechanism ...
Transcript of 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
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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
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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;
- the mesh version 2 with 174376 finite elements –
Fig.6;
- the mesh version 3 with 258779 finite elements –
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
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- 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.
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.
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[14]*****- Strength and lifetime duration calculus
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400-289, November, 2009.
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