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Fatigue life estimation of fillet-welded tubular T-joints subjected to multiaxial loading
Accepted Manuscript Fatigue life estimation of fillet-welded tubular T-joints subjected to multiaxial loading Andrea Carpinteri, Joel Boaretto, Giovanni Fortese, Felipe Giordani, Ignacio Iturrioz, Camilla Ronchei, Daniela Scorza, Sabrina Vantadori PII: DOI: Reference:
S0142-1123(16)30330-9 http://dx.doi.org/10.1016/j.ijfatigue.2016.10.012 JIJF 4096
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
22 August 2016 9 October 2016 12 October 2016
Please cite this article as: Carpinteri, A., Boaretto, J., Fortese, G., Giordani, F., Iturrioz, I., Ronchei, C., Scorza, D., Vantadori, S., Fatigue life estimation of fillet-welded tubular T-joints subjected to multiaxial loading, International Journal of Fatigue (2016), doi: http://dx.doi.org/10.1016/j.ijfatigue.2016.10.012
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SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil
Corresponding Author: Sabrina Vantadori,
email: [email protected]
1
ABSTRACT This
study
approach
to
investigates multiaxial
the
applicability
fatigue
strength
of
a
stress-based
assessment
of
fillet-
welded tubular T-joints, used in the so-called “H” component of an agricultural sprayer.
Experimental strain measurements on such a
structural component under sprayer service conditions have been performed.
A finite element analysis of the component is herein
developed under linear elastic hypothesis. based
critical
plane
criterion
is
applied
Finally, a stressto
several
material
points located in the proximity of the intersection between the end of a brace and a chord of one above-mentioned T-joint, where crack nucleation is experimentally observed.
KEYWORDS:
agricultural
sprayer,
critical
fatigue, multiaxial loading, T-joint
2
plane-based
criterion,
NOMENCLATURE C
shear stress vector acting in the critical plane
Ca
shear stress amplitude
m
slope of the S-N curve for fully reversed normal stress
m*
slope of the S-N curve for fully reversed shear stress
N
normal stress vector perpendicular to the critical plane
Na
normal stress amplitude
N a ,eq
equivalent normal stress amplitude
Nm
normal stress mean value
Nf
finite
life
fatigue
strength,
expressed
in
loading
cycles
t
time
T
period
Tcal w
finite life fatigue strength, expressed in seconds normal unit vector perpendicular to the critical plane
angle between the averaged direction 1ˆ and the normal w to the critical plane
, ,
Euler angles
af ,1
fully reversed normal stress fatigue limit
eq,a
equivalent uniaxial normal stress amplitude
u
af ,1
ultimate tensile strength fully reversed shear stress fatigue limit pulsation
3
1. INTRODUCTION In agriculture, herbicides and fungicides are commonly used to avoid harmful insects and herbs which can damage the crops. apply
such
products,
the
pulverisation
process,
by
To
which
agricultural defensive drops are evenly divided and distributed on the plants, is performed by machines named agricultural sprayers. The
most
common
one
is the
arm
sprayer,
which
is
a metal
bearing structure (named bar) equipped of spray nozzles (Figure 1(a)).
The
structural
bar
is
guided
in
the
component,
called
“H”
vertical
component
movements
due
to
its
by
a
shape
(Figure 1(b)), whose function is to raise and lower the sprayer bar. Figure 1.
Such a component is constituted from welded tubular elements, the intersections of which represent geometrical discontinuities like T-joints.
This work examines a T-joint, because it is the
weakest link of the component with respect to the failure (Figure 2).
Such a T-joint consists of a chord with a rectangular hollow
cross-section and a brace with a cylindrical hollow cross-section, both made of C25E steel.
The end of the brace is welded on the
chord by a fillet weld.
Figure 2.
4
This T-joint concentrates high stresses in the vicinity of the weld
[1,2].
These
zones
represent
locations
of fatigue crack
initiation and propagation due to cyclic loading [2-5].
The crack
initiation points are indicated by the arrows in Figure 2.
Hence,
a reliable fatigue assessment is needed [6,7]. Under uniaxial cyclic loading, fatigue assessment can be made through different approaches available in the literature, some of them [8,9] being recognised by Standard Codes and Recommendations: the
nominal
stress
approach
(global
approach)
[6,10-13],
the
structural stress approach (intermediate approach between global and local one) [6,11,14,15], and the notch stress approach (local approach) [6,11].
Whereas a global approach directly proceeds
from the external force and moment or from nominal stresses in the critical cross-section computed according the classical continuum mechanics, without explicitly modelling the stress concentration effect
[6,9],
strain
parameters
approach, fictitious
the
a
approach
[6,9].
stress
notch
condition.
local
For
state
rounding
Therefore,
from
example,
in
is
as
the
proceeds
determined
an
local the
by
idealisation
structural
stress
stress
notch
stress
introducing of
the
method,
or
a
actual
which
is
based on the fundamental idea to take into account the stress component perpendicular to the weld line and to reduce it to a linearised
distribution,
can
be
considered
an
intermediate
approach between the afore-mentioned ones [6,16,17]. The
local
stress
parameters
used
summarised in Figure 3 [14]. 5
in
such
approaches
are
Figure 3.
Many alternative methods have been proposed in the literature in recent years by taking full advantage from local parameters.
In
this context, it is worth mentioning the following methods: the notch stress-intensity factor approach [18-22], the Strain Energy Density
(SED)
approach
[23-29],
the
critical
distance
approach
[30-32], the high stress volume approach [33], the Peak Stress Method
(PSM)
[34-36].
Crack
propagation
approaches
are
also
methods
are
available [6,37-39]. Under
multiaxial among
fatigue them,
loading,
those
based
different
proposed
and,
on
the
critical
plane
concept.
In this scenario, the aim of the present work is to show
that the critical plane-based criterion proposed by Carpinteri et al. [40-44] can be applied in terms of notch stresses (by using one of the afore-mentioned stress analyses suggested by Standard Codes and Recommendations) to perform the fatigue assessment of a steel T-joint in the agricultural sprayer “H” component.
Similar
to the critical distance approach by Taylor [30-32], the above criterion
is
applied
to
some
material
certain distances from the weld toe.
points
characterised
by
Such distances are measured
along the experimental crack paths (Figure 2) developed in the “H” component after 2000 hours of sprayer operation under the typical service condition.
6
The
paper
experimental
is
organised
campaign
as
carried
follows.
out
for
sprayer service condition is described.
the
In “H”
Section component
2,
an
under
Section 3 is dedicated to
the finite element analysis of the “H” component.
In Section 4,
the stress-based critical plane criterion is employed to perform the fatigue assessment for a T-joint of the “H” component, and the final conclusions are provided in Section 5.
2. EXPERIMENTAL CAMPAIGN
The
experimental
campaign
aimed
to
determine
the
strain/stress
field in the “H” component of an agricultural sprayer under a typical service condition: application of herbicides in crops of a Brazilian
city
(Jaboticaba,
São Paulo)
[45].
The
bar
of the
agricultural sprayer was 28m long, and its motions consisted of six
independent
coordinates
(Figure
4):
three
coordinates
controlled the longitudinal, vertical, and lateral displacements, whereas the other three referred to rotational movements about the l-axis (called scroll motion), m-axis (called yaw motion), and naxis (called pitching motion), respectively.
Figure 4.
Such a “H” component consists of fillet-welded tubular T-joints in as-welded condition, characterised by a leg length equal to 5mm. 7
2.1 Experimental procedure The strain field in the “H” component was measured at points W1, W2, and W3, shown in Figure 5 [45]. rosettas
mounted
on
a
More precisely, two tee-
complete-bridge
Wheatstone
circuit
were
arranged on each chord (circuits W1 and W2), whereas two fish-bone strain gages mounted on a complete-bridge Wheatstone circuit were arranged on the brace (circuit W3).
Figure 5.
The service condition investigated consists of the maneuvers listed in Table 1.
In more detail:
(a) shifting the tractor, dragging the agricultural sprayer from the farmhouse to the crops on unpaved road.
Such a travel was
made twice a day, since the tractor left the farmhouse with the herbicide-tank full and came back with the tank empty; (b)
application
of
uncultivated field.
the
herbicide
on
the
perimeter
of
the
In such an operation, only the spray nozzles
located on one side of the bar with respect to the “H” component were opened; (c) application of the herbicide on the cultivated area.
In such
an operation, the tractor wheels remained between the planting rows
and,
when
the
machine
reached
the
end
of
the
row,
it
performed a U-curve crossing the planting rows; (d) braking: this maneuver occurred about five times each travel. 8
Table 1 shows the duration of each maneuver above, referred to both one day (corresponding to 10 hours of sprayer operation) and 2,000 hours of sprayer operation.
Note that we assumed that each
of such maneuvers was performed twice a day: one when the tractor fuel tank is full and the other, when the fuel tank is empty. Table 1
2.2 Results This section describes the collected data and their treatment. By using the signals coming from the above three circuits, the maximum principal stress sequence, related to the maneuvers listed in Table 1, was determined at each material point where the strain gauges were arranged.
The acquisition frequency was equal to 1kHz
and the cutoff frequency, in the low pass filter used, was equal to 20Hz. The sequences obtained were two [45]: one was determined by averaging the strain signals coming from the W1 and W2 circuits (such a sequence was named 1,c in the following, where c stands for chord) and the other determined by the strain signals coming from the W3 circuit (named 1,b in the following, where b stands for brace).
An example of such load histories is shown in Figure
6, where both 1,c
and 1,b
are plotted over a time interval of
about 210.0sec.
9
Figure 6
Then, the rain-flow counting procedure was applied to both 1,c and 1,b time history, and the value of damage over a time interval equal to 2,000 hours was calculated according to the PalmgrenMiner rule [45] for each load sequence by employing the fatigue properties of the “H” component material (C25E steel), which are listed in Table 2 together with the mechanical ones .
Table 2.
3. NUMERICAL MODEL A linear elastic finite element analysis is performed on the “H”
components
(Workbench
15.0)
prismatic
(8
through
the
[46],
using
nodes)
and
Commercial SOLID185 tetrahedral
Package finite (10
Ansys
14.5
elements,
both
nodes).
The
discretization employed is shown in Figure 7, where the finite element size is derived from a convergence analysis, being the minimum size employed equal to about 0.7mm [45]. Figure 7.
The forces transferred by the sprayer bar to the “H” component can be schematised by those named F1 and F2 in Figure 8. It was numerically proved [45] that, when only the force F1 is applied to 10
the
model,
the
maximum
principal
stresses
in
the
braces
are
significantly lower than those in the chords (up to about 110 times lower) and, therefore, such stresses in the chords can be assumed to only be linked to the force F1.
On the other hand, it
was numerically proved [45] that, when only the forces F2 are applied to the model, the maximum principal stresses in the braces are
greater
greater)
than
and,
those
in
therefore,
the
such
chords
(up
to
stresses
in
the
about
10
braces
times
can
be
assumed to only be linked to the forces F2. Figure 8.
Under
such
assumptions,
deterministic
full
histories for both F1 and F2 can be determined.
reverse
time
Such loadings
were characterised by the fact that the damage values computed in correspondence
to
the
points
of
W1
(or
W2)
and
W3
circuit
arrangements (deduced by considering the maximum principal stress time histories in such points) were equal to those obtained by directly processing the 1,c [45].
and 1,b
experimental load sequences
Note that the duration of the force time histories, shown
in Figure 9, was arbitrarily chosen equal to 12sec. Figure 9.
11
4. FATIGUE STRENGTH ASSESSMENT
4.1 The critical plane criterion A multiaxial fatigue criterion
based on the so-called critical
plane approach has been proposed by Carpinteri et al. [40-44] to estimate the fatigue strength (either endurance limit or fatigue lifetime)
of
smooth
metallic
periodic
multiaxial
loading
proportional).
structural (both
components
proportional
under and
any non-
The main steps of the criterion are hereafter
summarised. Let us consider the stress state at a material point P on the body surface (Figure 10) and the corresponding principal stress directions. fatigue
Such
loading:
directions
are
therefore,
generally
the
averaged
time-varying
under
principal
stress
directions can be determined by using suitable weight functions [47] and the averaged values ( ˆ, ˆ, ˆ ) of the principal Euler angles (Figure 10).
Figure 10.
By
employing
the
weight
function
proposed
in
Ref.[41],
the
averaged principal Euler angles coincide with the instantaneous ones at the time instant when the maximum principal stress (being
1t 2 t 3 t )
achieves
its
12
maximum
value
during
1 the
loading
cycle.
By
means
of
the
angles
ˆ, ˆ, ˆ ,
the
averaged
principal stress directions ( 1ˆ, 2ˆ , 3ˆ ) are identified. Then, the normal
w
to the critical plane is linked to the
direction 1ˆ through an off-angle , defined as follows ( w belongs to the principal plane 1ˆ 3ˆ , and the rotation is performed from 1ˆ to
3ˆ ):
The
multiaxial
2 3 1 af ,1 / af ,1 45 2
fatigue
limit
condition
(1)
is
expressed
by
the
following non-linear combination of an equivalent normal stress amplitude, N a ,eq , and the shear stress amplitude, Ca , acting in the critical plane [43]:
N a,eq / af ,1 2 Ca / af ,1 2 1
(2)
N a,eq N a af ,1 N m / u
(3)
with:
where
Nm
and
Na
are the mean value and the amplitude of the
normal stress, respectively,
af ,1 is the fully reversed normal
stress fatigue limit,
af ,1 is the fully reversed shear stress
u
is the ultimate tensile strength of the
fatigue limit, and material.
13
In order to transform the actual constant amplitude periodic multiaxial stress state into an equivalent uniaxial normal stress amplitude, eq,a , Eq.(2) can be written as follows:
eq,a
N a2,eq
af ,1 / af ,1 2 Ca2
af , 1
(4)
The finite life fatigue strength, N f , is computed (in terms of number
of
loading
cycles)
by
using
Basquin-like
relationships,
that is:
a af ,1 N f N 0
where a and
m
m
(5)
is the amplitude of normal stress at fatigue life N f ,
is the slope of the S-N curve for fully reversed normal
stress (tension or bending).
For fully reversed shear stress,
such a relationship is given by:
a af ,1 N f N 0
(6)
m*
where a is the amplitude of shear stress at fatigue life N f , and m * is the slope of the S-N curve for fully reversed shear stress
(torsion). Therefore, Eq.(4) can be rewritten as follows:
N a2,eq
af ,1 / af ,1 2 N f / N 0 2m N 0 / N f 2m*Ca2
af , 1 N f / N 0 m
(7)
Note that the finite life fatigue strength in terms of time, Tcal , can
be
obtained
by
multiplying
Nf
equivalent uniaxial normal stress. 14
for
the
period
T
of
the
By
following
approach applied
the
proposed to
same
by
philosophy
Taylor
[30-32],
of
the
the
critical
above
distance
criterion
is
some material points (named critical points in the
following), characterised by certain distances from the weld toe, measured along the experimental crack paths shown in Figure 2. Details are presented in next Section.
4.2 Multiaxial lifetime estimation The critical points are chosen in correspondence to those nodes of the finite element model in order to minimise the distances from the experimental crack paths (Figure 2). Firstly, such paths are digitalised from the pictures of the Tjoint
near
the
welded
zone.
Then,
the
critical
points
are
identified on the finite element model, as is shown in Figure 11. Figure 11.
Let us consider the stress state in the brace, obtained from the procedure presented in Section 3, at points B1, B2 and B3 in Figure
11.
For
each
point,
the
stress
state
is
practically
biaxial, characterised by the following stress tensor components:
xx (t ) xx , a sin ( t )
(8a)
yy (t ) yy , a sin ( t )
(8b)
xy (t ) xy , a sin ( t )
(8c)
15
where
is the pulsation, t is the time and
phase shift.
is the angle of
For the examined material points, the amplitudes of
the afore-mentioned components and the corresponding values are listed in Table 3, together with the values of the amplitudes of the stress components in the critical plane ( N a ,eq and Ca ) and the calculated finite life fatigue strength ( Tcal ). Table 3.
It can be noted that the most critical point between the examined ones in the brace is B3, being such a point characterised by a greater value of both
N a ,eq and Ca
(Table 3).
The value of Tcal
increases 3 orders of magnitude by moving from point B3 to point B1 along the experimental crack path in the brace. Now consider the stress state in the chord, obtained from the procedure presented in Section 3, at points C1, C2 and C3 in Figure
11.
For
each
point,
the
stress
state
is
practically
biaxial, characterised by the following stress tensor components:
xx (t ) xx , a sin ( t )
(9a)
zz (t ) zz, a sin ( t )
(9b)
xz (t ) xz , a sin ( t )
(9c)
For the examined material points, the amplitudes of the aforementioned components and the corresponding 16
values are listed in
Table 4, together with the values of the amplitudes of the stress components in the critical plane ( N a ,eq and Ca ) and the calculated finite life fatigue strength ( Tcal ). Table 4.
It can be noted from Table 4 that the damage accumulated in the chord is greater than that in the brace, and the most critical point of the examined ones in the chord is C1, being characterised by a greater value of both N a ,eq and Ca .
A Tcal
value increase of
about 50% is observed by moving from point C1 to point C3 along the experimental crack path in the chord. The
result
of
Tcal
obtained
related
to
the
brace
point
B3
(equal to 4.0474(10)6 sec) is in quite satisfactory agreement with the experimental observation time equal to 2,000 hours (that is, 7.2(10)6
sec),
after
agricultural sprayer.
cracks
shown
in
Figure
2
make
unfit
the
On the other hand, the results related to
the critical points in the chord are too conservative, although the criterion results ensure the safety. Now consider points B3
and C1, that is,
the most critical
points in the brace and in the chord, respectively. analyses what happens to Tcal from 0 to 180°.
This study
when the shift phase is made to vary
Figure 12 illustrates the theoretical results
17
determined through the proposed criterion in terms of Tcal
, whereas Figure 13 shows Tcal
against
against .
Figure 12.
Figure 13.
For both point B3 and point C1, a minimum value of Tcal
is noticed
in correspondence to a phase shift equal to about 0°.
It means
that no particular attention has to be paid regarding a possible non-proportionality of the applied loading, since no decrease in fatigue life is observed to occur. Finally, by comparing the results shown in Figures 12 and 13, it can be concluded that the accumulated damage in the chord is always greater than that in the brace, independent from the phase shift value.
5. CONCLUSIONS
In the present paper, the applicability of a stress-based approach to fatigue strength assessment of fillet-welded tubular T-joints, used in the so-called “H” component of an agricultural sprayer, has
been
examined.
In
particular,
a
critical
plane-based
criterion has been applied to several critical points, located in 18
the proximity of the intersection between the end of a brace and a chord for a T-joint of the “H” component, where crack nucleation is experimentally observed. The analysis performed, in particular that related to point B3 of the brace, is quite satisfactory.
As a matter of fact, the
computed finite life fatigue strength is in quite good agreement with the experimental observation time, after cracks make unfit the agricultural sprayer. Furthermore, analyses have been performed by varying the phase shift angles between the acting stress components.
This study has
highlighted that no particular attention has to be paid in design regarding a possible non-proportionality of the applied loading, since no decrease in fatigue life has been observed to occur.
Acknowledgements The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR), the National
Council
for
Scientific
and
Technological
Development
(CNPq - Brazil) and the Coordination for the Improvement of Higher Education Personnel (CAPES – Brazil).
19
REFERENCES
[1] Blacha Ł, Karolczuk A. Validation of the weakest link approach and the proposed Weibull based probability distribution of failure for fatigue design of steel welded joints. Eng Fail Anal 2016;67: 46-62. [2] N'Diaye A, Hariri S, Pluvinage G, Azari Z. Stress concentration factor analysis for welded, notched tubular T-joints under combined axial, bending and dynamic loading. Int J Fatigue 2009;31:367-74. [3] Stenberg T, Barsoum Z, Balawi SOM. Comparison of local stress based concepts - Effects of low-and high cycle fatigue and weld quality. Eng Fail Anal 2015;57:323-33. [4] Chang E, Dover WD. Parametric equations to predict stress distributions along the intersection of tubular X and DT-joints. Int J Fatigue 1999;21:619-35. [5] Liu G, Zhao X, Huang Y. Prediction of stress distribution along the intersection of tubular T-joints by a novel structural stress approach. Int J Fatigue 2015;80:216-30. [6] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. 2nd ed. Cambridge: Woodhead Publishing; 2006. [7] Fricke W. Fatigue analysis of welded joints: state of development. Mar Struct 2003;16:185–200. [8] Susmel L. Four stress analysis strategies to use the Modified Wöhler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading. Int J Fatigue 2014;67:38–54. [9] Fischer C, Fricke W, Rizzo CM. Fatigue assessment of joints at bulb profiles by local approaches. In: Taylor & Francis Ltd, editors. Proceedings of the 5th International Conference on Marine Structures - Analysis and Design of Marine Structures V, Southampton: CRC Press; 2015, p. 251-59. [10] Eurocode 3: Design of Steel Structres – Part 1-1: General Rules for Buildings, ENV 1993-1-1, European Committee for Standardisation, Brussels; 1992. [11] Japanese Society of Steel Construction (JSSC): Fatigue design recommendations for steel structures, Technical Report No.32, Tokyo; 1995. [12] Lotsberg I, Larsen KP. Fatigue design in the new Norwegian structural design code. In: Proceeding of the Nordic Steel Conference, Bergen; 1998. [13] Recommendations for fatigue strength of welded components. Hobbacher A, editor. Cambridge: Abington Publishers; 2007. [14] Stress determination for fatigue analysis of welded components. Niemi E, editor. Cambridge: Abington Publishers; 1995. [15] Niemi E, Fricke W, Maddox SJ. Structural hot-spot stress approach to fatigue analysis of welded components – designer’s guide. Cambridge: Woodhead Publishing; 2003. 20
[16] Marin T, Nicoletto G. Fatigue design of welded joints using the finite element method and the 2007 ASME Div.2 Master curve. Frattura ed Integrità Strutturale 2009;9:76-84. [17] Poutiainen I, Marquis G. A fatigue assessment method based on weld stresses. Int J Fatigue 2006;28:1037-46. [18] Lazzarin P, Tovo R. A notch intensity factor approach to the stress intensity of welds. Fatigue Fract Eng Mater Struct 1998;21:1089-103. [19] Atzori B, Lazzarin P, Tovo R. From a local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints. Fatigue Fract Eng Mater Struct 1999;22:369-81. [20] Lazzarin P, Tovo R. Relationship between local and structural stress in the evaluation of the weld toe stress distribution. Int J Fatigue 1999;21:1063-78. [21] Lazzarin P, Livieri P. Notch stress intensity factors and fatigue strength of aluminium and steel welded joints. Int J Fatigue 2001;23:225-32. [22] Atzori B, Meneghetti G. Fatigue strength of fillet welded structural steels: finite elements, strain gauges and reality. Int J Fatigue 2001;23:713-21. [23] Livieri P, Lazzarin P. Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int J Fract 2005;133:247– 76. [24] Lazzarin P, Livieri P, Berto F, Zappalorto M. Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading. Eng Fract Mech 2008;75:1875–89. [25] Berto F, Lazzarin P. A review of the volume-based strain energy density approach applied to V-notches and welded structures. Theor Appl Fract Mec 2009;52:183-94. [26] Berto F, Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local apporaches. Mat Sci Eng R 2014;75:1-48. [27] Lazzarin P, Berto F, Zappalorto M. Rapid calculations of notch stress intensity factor based on averaged strain energy density from coarse meshes: theoretical bases and applications. Int J Fatigue 2010;32:1559-67. [28] Lazzarin P, Berto F, Gomez FJ,Zappalorto M. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessments of welded joints. Int J Fatigue 2008;30:1345-57. [29] Lazzarin P, Berto F, Radaj D. Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch. Fatigue Fract Eng Mater Struct 2009;32:713-35. [30] Taylor D, Wang G. A critical distance theory which unifies the prediction of fatigue limits for large and small cracks and notches. In: Wu XR, Wang ZG, editors. Proceedings of the Fatigue’99, vol. 1, Beijing, China: Higher Education Press; 1999, p. 579-84. 21
[31] Taylor D, Barrett N, Lucano G. Some new methods for predicting fatigue in welded joints. Int J Fatigue2002;24:509-18. [32] Crupi G, Crupi V, Guglielmino E, Taylor D. Fatigue assessment of welded joints using critical distance and other methods. Eng Fail Anal 2005;12:129-42. [33] Sonsino CM. Multiaxial fatigue of welded joints under inphase and out-of-phase local strains and stresses. Int J Fatigue 1995;17:55-70. [34] Meneghetti G., Guzzella C., Atzori B. The peak stress method combined with 3D finite element models for fatigue assessment of toe and root cracking in steel welded joints subjected to axial or bending loading. Fatigue Fract Eng Mater Struct 2014;37:722-39. [35] Meneghetti G., De Marchi A., Campagnolo A. Assessment of root failures in tube-to-flange steel welded joints under torsional loading according to the Peak Stress Method. Theoretical and Applied Fracture Mechanics 2016;83:19-30. [36] Meneghetti G., Marini D., Babini V. Fatigue assessment of weld toe and weld root failures in steel welded joints according to the peak stress method. Welding in the World 2016;60:559-72. [37] Skorupa M. Load interaction effects during fatigue crack growth under variable amplitude loading – a literature review. Part I: empirical trends. Fatigue Fract Eng Mater Struct 1998;21:987-1006. [38] Skorupa M. Load intraction effects during crack gorwth under variable amplitude loading - a literature review. Part II: qualitative interpretations. Fatigue Fract Eng Mater Struct 1999;22:905-26. [39] Dijkstra OD, Van Straalen IJ. Fracture mechanics and fatigue of welded structures. In: Proceeding of the International Conference on Performance of Dynamically Loaded Welded Structure, New York: WRC; 1997, p.225-39. [40] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Special Issue on “Welded connections” - Int J Fatigue 2009;31:188-96. [41] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue assessment using a simplified critical plane-based criterion. Int J Fatigue 2011;33:969-76. [42] Carpinteri A, Spagnoli A, Vantadori S, Bagni C. Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution. Fatigue Fract Eng Mater Struct 2013;36:870-83. [43] Araújo J, Carpinteri A, Ronchei C, Spagnoli A, Vantadori S. An alternative definition of the shear stress amplitude based on the Maximum Rectangular Hull method and application to the C-S (Carpinteri-Spagnoli) criterion. Fatigue Fract Eng Mater Struct 2014;37:764-71. [44] Carpinteri A, Ronchei C, Scorza D, Vantadori S. Critical plane orientation influence on multiaxial high-cycle fatigue assessment. Physical Mesomechanics 2015;18:348-54. 22
[45] Giordani FA. Estudo de Metodologias para medir a vida em fadiga Multiaxial nao proporcional. Master thesis 2015. Promec/UFRGS, Brazil. In Portugues. http://hdl.handle.net/10183/118864 [46] Ansys 14.5. www.ansys.com; 2016 [accessed 11.08.16]. [47] Macha E. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Mat Wiss U Werkstofftech 1989;20:132–36 and 153-63.
23
SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil
Corresponding Author: Sabrina Vantadori,
email: [email protected]
24
CAPTIONS Figure 1. (a) Agricultural sprayer; (b) “H” component (sizes in mm). Figure 2.
Fatigue cracks experimentally observed in one T-joint of
the “H” component, highlighted under sprayer service condition at: (a),(b) brace and (c),(d) chord.
The crack initiation points are
indicated by the arrows. Figure
3.
Schematisation
of
the
stress
parameters
used
in
the
global, intermediate global-local and local approaches. Figure 4.
Possible movements of the sprayer bar examined.
Figure 5. Complete-bridge Wheatstone circuits arranged on both the brace and the chords. Table
1.
Service
condition
duration,
related
to
10
investigated:
hours
and
maneuvers
2,000
hours
and of
their sprayer
operations. Figure 6. Load histories over a time interval of 6.0sec: (a) 1,c and (b) 1,b . Table 2. Mechanical and fatigue properties of the C25E steel. Figure 7. Discretisation employed in the finite element analysis. Figure 8. Schematisation of the forces transferred by the sprayer bar to the “H” component in the finite element model. Figure 9. Deterministic full reverse time histories for both F1 and F2 forces. Figure 10. Reference systems: fixed frame
25
PXYZ and averaged frame
P1ˆ 2ˆ 3ˆ . Figure 11. Examined critical points in the brace and in the chord. Table
3.
Amplitudes
amplitudes
of
stress
of
the
applied
components
on
stresses, the
phase
critical
shift,
plane,
and
calculated finite life fatigue strength for the critical points B1, B2, B3 in the brace. Table
4.
Amplitudes
amplitudes
of
stress
of
the
applied
components
on
stresses, the
phase
critical
shift,
plane
and
calculated finite life fatigue strength for the critical points C1, C2, C3 in the chord. Figure 12. Calculated finite life fatigue strength, phase shift,
Tcal , versus
, at point B3 of the brace.
Figure 13. Calculated finite life fatigue strength, phase shift, , at point C1 of the chord.
26
Tcal , versus
( a)
( b)
27
Figure 1.
(
(
c)
d)
Figure 2. 28
Notch stress Hot-spot stress
Structural stresses
Figure 3.
29
Nominal stresses
Figure 4.
X
W3
W1
W2 Y
Figure 5.
30
Z
Table 1.
31
MANOUVERS
FUEL TANK
DURATION 10 h
road Application the
of
herbicide
DURATION 2000 h
[h] Travel on unpaved
-
[h]
Full
0.50
100
Empty
0.50
100
Full
1.00
200
Empty
1.00
200
Full
2.25
450
Empty
2.25
450
Full
0.50
100
Empty
0.50
100
Full
0.50
100
Empty
0.50
100
Full
0.25
50
Empty
0.25
50
(perimeter) Application the
of
herbicide
(cultivated area) U - curves
Perimeter curves
Braking
32
-
STRESS, 1c [MPa]
MAXIMUM PRINCIPAL
20
(a)
15 10 5 0 -5 -10 0
20
40
60
80
100
120
TIME, t [sec]
33
140
160
180
200
STRESS, 1b [MPa]
MAXIMUM PRINCIPAL
60
(b)
40 20 0 -20 -40 -60 -80 0
20
40
60
80
100
120
140
160
180
TIME, t [sec]
Figure 6.
Table 2.
Material
u
E
[MPa]
[GPa]
af ,1
m
af ,1 m * [MP
[cycles
[MPa] a]
C25E
N0
470.
198.
141.
0
0
0
34
1/5
86. 0
] -
1.0
1/8
(10)6
200
Figure 7.
35
Figure 8.
36
30000
F1 F2
FORCE [N]
20000 10000 0 -10000 -20000 -30000 0
1
2
3
4
5
6
7
TIME, t [sec] Figure 9.
37
8
9
10
11
12
Z 3
(a)
P
X 1 Y
(b) 3
2 Z
P 1
2
(c) 3 P 2 1
38
Figure 10.
Figure 11.
Table 3.
POINT No.
xx, a yy, a
xy, a
[M
[MP
Pa] B1
[MP a]
11 .30
a]
10. 17
N a ,eq
Ca
Tcal
[
[MPa
[MPa
[sec]
]
]
13.3
28.8
7
0
°]
28.
0
91
39
2.5145E+ 09
B2
41 .48
B3
27. 63
99 .74
27.
0
59 62.
22
30.
0
11
15.2
31.3
6
5
63.5
57.9
6
7
1.2829E+ 09 4.0474E+ 06
Table 4.
POINT
xx, a
zz, a
xz, a
[MP
[MP
[MP
N a ,eq
Ca
Tcal
[
[MPa
[MPa
[sec]
]
]
124.
113.
4.0656E
46
60
+04
115.
105.
6.8599E
57
48
+04
112.
102.
8.1402E
79
94
+04
No.
a] C1
206 .98
C2
07
0
93
51
50.
0
32 26.
42
°]
62.
38.
195 .99
a]
41.
197 .25
C3
a]
44. 04
40
0
CALCULATED FATIGUE LIFE, Tcal / 107 [sec]
8.0
8.00E+007
Point B3
6.0
6.00E+007
4.0
4.00E+007
2.0
2.00E+007
0.4
0
30
60
90 120 150 180
PHASE SHIFT, [°]
Figure 12.
41
CALCULATED FATIGUE LIFE, Tcal / 105 [sec]
1.0
1.00E+005
Point C1 0.8
8.00E+004
0.6
6.00E+004
0.4
4.00E+004
0
30
60
90 120 150 180
PHASE SHIFT, [°]
Figure 13.
42
SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy 2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil Corresponding Author: Sabrina Vantadori, email: [email protected]
HIGHLIGHTS The fatigue strength assessment of an agricultural sprayer is performed The weakest links of such a sprayer against fatigue failure are the T-joints A stress-based critical plane criterion is applied The comparison between theoretical and experimental results is quite satisfactory
43
S0142-1123(16)30330-9 http://dx.doi.org/10.1016/j.ijfatigue.2016.10.012 JIJF 4096
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
22 August 2016 9 October 2016 12 October 2016
Please cite this article as: Carpinteri, A., Boaretto, J., Fortese, G., Giordani, F., Iturrioz, I., Ronchei, C., Scorza, D., Vantadori, S., Fatigue life estimation of fillet-welded tubular T-joints subjected to multiaxial loading, International Journal of Fatigue (2016), doi: http://dx.doi.org/10.1016/j.ijfatigue.2016.10.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil
Corresponding Author: Sabrina Vantadori,
email: [email protected]
1
ABSTRACT This
study
approach
to
investigates multiaxial
the
applicability
fatigue
strength
of
a
stress-based
assessment
of
fillet-
welded tubular T-joints, used in the so-called “H” component of an agricultural sprayer.
Experimental strain measurements on such a
structural component under sprayer service conditions have been performed.
A finite element analysis of the component is herein
developed under linear elastic hypothesis. based
critical
plane
criterion
is
applied
Finally, a stressto
several
material
points located in the proximity of the intersection between the end of a brace and a chord of one above-mentioned T-joint, where crack nucleation is experimentally observed.
KEYWORDS:
agricultural
sprayer,
critical
fatigue, multiaxial loading, T-joint
2
plane-based
criterion,
NOMENCLATURE C
shear stress vector acting in the critical plane
Ca
shear stress amplitude
m
slope of the S-N curve for fully reversed normal stress
m*
slope of the S-N curve for fully reversed shear stress
N
normal stress vector perpendicular to the critical plane
Na
normal stress amplitude
N a ,eq
equivalent normal stress amplitude
Nm
normal stress mean value
Nf
finite
life
fatigue
strength,
expressed
in
loading
cycles
t
time
T
period
Tcal w
finite life fatigue strength, expressed in seconds normal unit vector perpendicular to the critical plane
angle between the averaged direction 1ˆ and the normal w to the critical plane
, ,
Euler angles
af ,1
fully reversed normal stress fatigue limit
eq,a
equivalent uniaxial normal stress amplitude
u
af ,1
ultimate tensile strength fully reversed shear stress fatigue limit pulsation
3
1. INTRODUCTION In agriculture, herbicides and fungicides are commonly used to avoid harmful insects and herbs which can damage the crops. apply
such
products,
the
pulverisation
process,
by
To
which
agricultural defensive drops are evenly divided and distributed on the plants, is performed by machines named agricultural sprayers. The
most
common
one
is the
arm
sprayer,
which
is
a metal
bearing structure (named bar) equipped of spray nozzles (Figure 1(a)).
The
structural
bar
is
guided
in
the
component,
called
“H”
vertical
component
movements
due
to
its
by
a
shape
(Figure 1(b)), whose function is to raise and lower the sprayer bar. Figure 1.
Such a component is constituted from welded tubular elements, the intersections of which represent geometrical discontinuities like T-joints.
This work examines a T-joint, because it is the
weakest link of the component with respect to the failure (Figure 2).
Such a T-joint consists of a chord with a rectangular hollow
cross-section and a brace with a cylindrical hollow cross-section, both made of C25E steel.
The end of the brace is welded on the
chord by a fillet weld.
Figure 2.
4
This T-joint concentrates high stresses in the vicinity of the weld
[1,2].
These
zones
represent
locations
of fatigue crack
initiation and propagation due to cyclic loading [2-5].
The crack
initiation points are indicated by the arrows in Figure 2.
Hence,
a reliable fatigue assessment is needed [6,7]. Under uniaxial cyclic loading, fatigue assessment can be made through different approaches available in the literature, some of them [8,9] being recognised by Standard Codes and Recommendations: the
nominal
stress
approach
(global
approach)
[6,10-13],
the
structural stress approach (intermediate approach between global and local one) [6,11,14,15], and the notch stress approach (local approach) [6,11].
Whereas a global approach directly proceeds
from the external force and moment or from nominal stresses in the critical cross-section computed according the classical continuum mechanics, without explicitly modelling the stress concentration effect
[6,9],
strain
parameters
approach, fictitious
the
a
approach
[6,9].
stress
notch
condition.
local
For
state
rounding
Therefore,
from
example,
in
is
as
the
proceeds
determined
an
local the
by
idealisation
structural
stress
stress
notch
stress
introducing of
the
method,
or
a
actual
which
is
based on the fundamental idea to take into account the stress component perpendicular to the weld line and to reduce it to a linearised
distribution,
can
be
considered
an
intermediate
approach between the afore-mentioned ones [6,16,17]. The
local
stress
parameters
used
summarised in Figure 3 [14]. 5
in
such
approaches
are
Figure 3.
Many alternative methods have been proposed in the literature in recent years by taking full advantage from local parameters.
In
this context, it is worth mentioning the following methods: the notch stress-intensity factor approach [18-22], the Strain Energy Density
(SED)
approach
[23-29],
the
critical
distance
approach
[30-32], the high stress volume approach [33], the Peak Stress Method
(PSM)
[34-36].
Crack
propagation
approaches
are
also
methods
are
available [6,37-39]. Under
multiaxial among
fatigue them,
loading,
those
based
different
proposed
and,
on
the
critical
plane
concept.
In this scenario, the aim of the present work is to show
that the critical plane-based criterion proposed by Carpinteri et al. [40-44] can be applied in terms of notch stresses (by using one of the afore-mentioned stress analyses suggested by Standard Codes and Recommendations) to perform the fatigue assessment of a steel T-joint in the agricultural sprayer “H” component.
Similar
to the critical distance approach by Taylor [30-32], the above criterion
is
applied
to
some
material
certain distances from the weld toe.
points
characterised
by
Such distances are measured
along the experimental crack paths (Figure 2) developed in the “H” component after 2000 hours of sprayer operation under the typical service condition.
6
The
paper
experimental
is
organised
campaign
as
carried
follows.
out
for
sprayer service condition is described.
the
In “H”
Section component
2,
an
under
Section 3 is dedicated to
the finite element analysis of the “H” component.
In Section 4,
the stress-based critical plane criterion is employed to perform the fatigue assessment for a T-joint of the “H” component, and the final conclusions are provided in Section 5.
2. EXPERIMENTAL CAMPAIGN
The
experimental
campaign
aimed
to
determine
the
strain/stress
field in the “H” component of an agricultural sprayer under a typical service condition: application of herbicides in crops of a Brazilian
city
(Jaboticaba,
São Paulo)
[45].
The
bar
of the
agricultural sprayer was 28m long, and its motions consisted of six
independent
coordinates
(Figure
4):
three
coordinates
controlled the longitudinal, vertical, and lateral displacements, whereas the other three referred to rotational movements about the l-axis (called scroll motion), m-axis (called yaw motion), and naxis (called pitching motion), respectively.
Figure 4.
Such a “H” component consists of fillet-welded tubular T-joints in as-welded condition, characterised by a leg length equal to 5mm. 7
2.1 Experimental procedure The strain field in the “H” component was measured at points W1, W2, and W3, shown in Figure 5 [45]. rosettas
mounted
on
a
More precisely, two tee-
complete-bridge
Wheatstone
circuit
were
arranged on each chord (circuits W1 and W2), whereas two fish-bone strain gages mounted on a complete-bridge Wheatstone circuit were arranged on the brace (circuit W3).
Figure 5.
The service condition investigated consists of the maneuvers listed in Table 1.
In more detail:
(a) shifting the tractor, dragging the agricultural sprayer from the farmhouse to the crops on unpaved road.
Such a travel was
made twice a day, since the tractor left the farmhouse with the herbicide-tank full and came back with the tank empty; (b)
application
of
uncultivated field.
the
herbicide
on
the
perimeter
of
the
In such an operation, only the spray nozzles
located on one side of the bar with respect to the “H” component were opened; (c) application of the herbicide on the cultivated area.
In such
an operation, the tractor wheels remained between the planting rows
and,
when
the
machine
reached
the
end
of
the
row,
it
performed a U-curve crossing the planting rows; (d) braking: this maneuver occurred about five times each travel. 8
Table 1 shows the duration of each maneuver above, referred to both one day (corresponding to 10 hours of sprayer operation) and 2,000 hours of sprayer operation.
Note that we assumed that each
of such maneuvers was performed twice a day: one when the tractor fuel tank is full and the other, when the fuel tank is empty. Table 1
2.2 Results This section describes the collected data and their treatment. By using the signals coming from the above three circuits, the maximum principal stress sequence, related to the maneuvers listed in Table 1, was determined at each material point where the strain gauges were arranged.
The acquisition frequency was equal to 1kHz
and the cutoff frequency, in the low pass filter used, was equal to 20Hz. The sequences obtained were two [45]: one was determined by averaging the strain signals coming from the W1 and W2 circuits (such a sequence was named 1,c in the following, where c stands for chord) and the other determined by the strain signals coming from the W3 circuit (named 1,b in the following, where b stands for brace).
An example of such load histories is shown in Figure
6, where both 1,c
and 1,b
are plotted over a time interval of
about 210.0sec.
9
Figure 6
Then, the rain-flow counting procedure was applied to both 1,c and 1,b time history, and the value of damage over a time interval equal to 2,000 hours was calculated according to the PalmgrenMiner rule [45] for each load sequence by employing the fatigue properties of the “H” component material (C25E steel), which are listed in Table 2 together with the mechanical ones .
Table 2.
3. NUMERICAL MODEL A linear elastic finite element analysis is performed on the “H”
components
(Workbench
15.0)
prismatic
(8
through
the
[46],
using
nodes)
and
Commercial SOLID185 tetrahedral
Package finite (10
Ansys
14.5
elements,
both
nodes).
The
discretization employed is shown in Figure 7, where the finite element size is derived from a convergence analysis, being the minimum size employed equal to about 0.7mm [45]. Figure 7.
The forces transferred by the sprayer bar to the “H” component can be schematised by those named F1 and F2 in Figure 8. It was numerically proved [45] that, when only the force F1 is applied to 10
the
model,
the
maximum
principal
stresses
in
the
braces
are
significantly lower than those in the chords (up to about 110 times lower) and, therefore, such stresses in the chords can be assumed to only be linked to the force F1.
On the other hand, it
was numerically proved [45] that, when only the forces F2 are applied to the model, the maximum principal stresses in the braces are
greater
greater)
than
and,
those
in
therefore,
the
such
chords
(up
to
stresses
in
the
about
10
braces
times
can
be
assumed to only be linked to the forces F2. Figure 8.
Under
such
assumptions,
deterministic
full
histories for both F1 and F2 can be determined.
reverse
time
Such loadings
were characterised by the fact that the damage values computed in correspondence
to
the
points
of
W1
(or
W2)
and
W3
circuit
arrangements (deduced by considering the maximum principal stress time histories in such points) were equal to those obtained by directly processing the 1,c [45].
and 1,b
experimental load sequences
Note that the duration of the force time histories, shown
in Figure 9, was arbitrarily chosen equal to 12sec. Figure 9.
11
4. FATIGUE STRENGTH ASSESSMENT
4.1 The critical plane criterion A multiaxial fatigue criterion
based on the so-called critical
plane approach has been proposed by Carpinteri et al. [40-44] to estimate the fatigue strength (either endurance limit or fatigue lifetime)
of
smooth
metallic
periodic
multiaxial
loading
proportional).
structural (both
components
proportional
under and
any non-
The main steps of the criterion are hereafter
summarised. Let us consider the stress state at a material point P on the body surface (Figure 10) and the corresponding principal stress directions. fatigue
Such
loading:
directions
are
therefore,
generally
the
averaged
time-varying
under
principal
stress
directions can be determined by using suitable weight functions [47] and the averaged values ( ˆ, ˆ, ˆ ) of the principal Euler angles (Figure 10).
Figure 10.
By
employing
the
weight
function
proposed
in
Ref.[41],
the
averaged principal Euler angles coincide with the instantaneous ones at the time instant when the maximum principal stress (being
1t 2 t 3 t )
achieves
its
12
maximum
value
during
1 the
loading
cycle.
By
means
of
the
angles
ˆ, ˆ, ˆ ,
the
averaged
principal stress directions ( 1ˆ, 2ˆ , 3ˆ ) are identified. Then, the normal
w
to the critical plane is linked to the
direction 1ˆ through an off-angle , defined as follows ( w belongs to the principal plane 1ˆ 3ˆ , and the rotation is performed from 1ˆ to
3ˆ ):
The
multiaxial
2 3 1 af ,1 / af ,1 45 2
fatigue
limit
condition
(1)
is
expressed
by
the
following non-linear combination of an equivalent normal stress amplitude, N a ,eq , and the shear stress amplitude, Ca , acting in the critical plane [43]:
N a,eq / af ,1 2 Ca / af ,1 2 1
(2)
N a,eq N a af ,1 N m / u
(3)
with:
where
Nm
and
Na
are the mean value and the amplitude of the
normal stress, respectively,
af ,1 is the fully reversed normal
stress fatigue limit,
af ,1 is the fully reversed shear stress
u
is the ultimate tensile strength of the
fatigue limit, and material.
13
In order to transform the actual constant amplitude periodic multiaxial stress state into an equivalent uniaxial normal stress amplitude, eq,a , Eq.(2) can be written as follows:
eq,a
N a2,eq
af ,1 / af ,1 2 Ca2
af , 1
(4)
The finite life fatigue strength, N f , is computed (in terms of number
of
loading
cycles)
by
using
Basquin-like
relationships,
that is:
a af ,1 N f N 0
where a and
m
m
(5)
is the amplitude of normal stress at fatigue life N f ,
is the slope of the S-N curve for fully reversed normal
stress (tension or bending).
For fully reversed shear stress,
such a relationship is given by:
a af ,1 N f N 0
(6)
m*
where a is the amplitude of shear stress at fatigue life N f , and m * is the slope of the S-N curve for fully reversed shear stress
(torsion). Therefore, Eq.(4) can be rewritten as follows:
N a2,eq
af ,1 / af ,1 2 N f / N 0 2m N 0 / N f 2m*Ca2
af , 1 N f / N 0 m
(7)
Note that the finite life fatigue strength in terms of time, Tcal , can
be
obtained
by
multiplying
Nf
equivalent uniaxial normal stress. 14
for
the
period
T
of
the
By
following
approach applied
the
proposed to
same
by
philosophy
Taylor
[30-32],
of
the
the
critical
above
distance
criterion
is
some material points (named critical points in the
following), characterised by certain distances from the weld toe, measured along the experimental crack paths shown in Figure 2. Details are presented in next Section.
4.2 Multiaxial lifetime estimation The critical points are chosen in correspondence to those nodes of the finite element model in order to minimise the distances from the experimental crack paths (Figure 2). Firstly, such paths are digitalised from the pictures of the Tjoint
near
the
welded
zone.
Then,
the
critical
points
are
identified on the finite element model, as is shown in Figure 11. Figure 11.
Let us consider the stress state in the brace, obtained from the procedure presented in Section 3, at points B1, B2 and B3 in Figure
11.
For
each
point,
the
stress
state
is
practically
biaxial, characterised by the following stress tensor components:
xx (t ) xx , a sin ( t )
(8a)
yy (t ) yy , a sin ( t )
(8b)
xy (t ) xy , a sin ( t )
(8c)
15
where
is the pulsation, t is the time and
phase shift.
is the angle of
For the examined material points, the amplitudes of
the afore-mentioned components and the corresponding values are listed in Table 3, together with the values of the amplitudes of the stress components in the critical plane ( N a ,eq and Ca ) and the calculated finite life fatigue strength ( Tcal ). Table 3.
It can be noted that the most critical point between the examined ones in the brace is B3, being such a point characterised by a greater value of both
N a ,eq and Ca
(Table 3).
The value of Tcal
increases 3 orders of magnitude by moving from point B3 to point B1 along the experimental crack path in the brace. Now consider the stress state in the chord, obtained from the procedure presented in Section 3, at points C1, C2 and C3 in Figure
11.
For
each
point,
the
stress
state
is
practically
biaxial, characterised by the following stress tensor components:
xx (t ) xx , a sin ( t )
(9a)
zz (t ) zz, a sin ( t )
(9b)
xz (t ) xz , a sin ( t )
(9c)
For the examined material points, the amplitudes of the aforementioned components and the corresponding 16
values are listed in
Table 4, together with the values of the amplitudes of the stress components in the critical plane ( N a ,eq and Ca ) and the calculated finite life fatigue strength ( Tcal ). Table 4.
It can be noted from Table 4 that the damage accumulated in the chord is greater than that in the brace, and the most critical point of the examined ones in the chord is C1, being characterised by a greater value of both N a ,eq and Ca .
A Tcal
value increase of
about 50% is observed by moving from point C1 to point C3 along the experimental crack path in the chord. The
result
of
Tcal
obtained
related
to
the
brace
point
B3
(equal to 4.0474(10)6 sec) is in quite satisfactory agreement with the experimental observation time equal to 2,000 hours (that is, 7.2(10)6
sec),
after
agricultural sprayer.
cracks
shown
in
Figure
2
make
unfit
the
On the other hand, the results related to
the critical points in the chord are too conservative, although the criterion results ensure the safety. Now consider points B3
and C1, that is,
the most critical
points in the brace and in the chord, respectively. analyses what happens to Tcal from 0 to 180°.
This study
when the shift phase is made to vary
Figure 12 illustrates the theoretical results
17
determined through the proposed criterion in terms of Tcal
, whereas Figure 13 shows Tcal
against
against .
Figure 12.
Figure 13.
For both point B3 and point C1, a minimum value of Tcal
is noticed
in correspondence to a phase shift equal to about 0°.
It means
that no particular attention has to be paid regarding a possible non-proportionality of the applied loading, since no decrease in fatigue life is observed to occur. Finally, by comparing the results shown in Figures 12 and 13, it can be concluded that the accumulated damage in the chord is always greater than that in the brace, independent from the phase shift value.
5. CONCLUSIONS
In the present paper, the applicability of a stress-based approach to fatigue strength assessment of fillet-welded tubular T-joints, used in the so-called “H” component of an agricultural sprayer, has
been
examined.
In
particular,
a
critical
plane-based
criterion has been applied to several critical points, located in 18
the proximity of the intersection between the end of a brace and a chord for a T-joint of the “H” component, where crack nucleation is experimentally observed. The analysis performed, in particular that related to point B3 of the brace, is quite satisfactory.
As a matter of fact, the
computed finite life fatigue strength is in quite good agreement with the experimental observation time, after cracks make unfit the agricultural sprayer. Furthermore, analyses have been performed by varying the phase shift angles between the acting stress components.
This study has
highlighted that no particular attention has to be paid in design regarding a possible non-proportionality of the applied loading, since no decrease in fatigue life has been observed to occur.
Acknowledgements The authors gratefully acknowledge the financial support of the Italian Ministry of Education, University and Research (MIUR), the National
Council
for
Scientific
and
Technological
Development
(CNPq - Brazil) and the Coordination for the Improvement of Higher Education Personnel (CAPES – Brazil).
19
REFERENCES
[1] Blacha Ł, Karolczuk A. Validation of the weakest link approach and the proposed Weibull based probability distribution of failure for fatigue design of steel welded joints. Eng Fail Anal 2016;67: 46-62. [2] N'Diaye A, Hariri S, Pluvinage G, Azari Z. Stress concentration factor analysis for welded, notched tubular T-joints under combined axial, bending and dynamic loading. Int J Fatigue 2009;31:367-74. [3] Stenberg T, Barsoum Z, Balawi SOM. Comparison of local stress based concepts - Effects of low-and high cycle fatigue and weld quality. Eng Fail Anal 2015;57:323-33. [4] Chang E, Dover WD. Parametric equations to predict stress distributions along the intersection of tubular X and DT-joints. Int J Fatigue 1999;21:619-35. [5] Liu G, Zhao X, Huang Y. Prediction of stress distribution along the intersection of tubular T-joints by a novel structural stress approach. Int J Fatigue 2015;80:216-30. [6] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. 2nd ed. Cambridge: Woodhead Publishing; 2006. [7] Fricke W. Fatigue analysis of welded joints: state of development. Mar Struct 2003;16:185–200. [8] Susmel L. Four stress analysis strategies to use the Modified Wöhler Curve Method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading. Int J Fatigue 2014;67:38–54. [9] Fischer C, Fricke W, Rizzo CM. Fatigue assessment of joints at bulb profiles by local approaches. In: Taylor & Francis Ltd, editors. Proceedings of the 5th International Conference on Marine Structures - Analysis and Design of Marine Structures V, Southampton: CRC Press; 2015, p. 251-59. [10] Eurocode 3: Design of Steel Structres – Part 1-1: General Rules for Buildings, ENV 1993-1-1, European Committee for Standardisation, Brussels; 1992. [11] Japanese Society of Steel Construction (JSSC): Fatigue design recommendations for steel structures, Technical Report No.32, Tokyo; 1995. [12] Lotsberg I, Larsen KP. Fatigue design in the new Norwegian structural design code. In: Proceeding of the Nordic Steel Conference, Bergen; 1998. [13] Recommendations for fatigue strength of welded components. Hobbacher A, editor. Cambridge: Abington Publishers; 2007. [14] Stress determination for fatigue analysis of welded components. Niemi E, editor. Cambridge: Abington Publishers; 1995. [15] Niemi E, Fricke W, Maddox SJ. Structural hot-spot stress approach to fatigue analysis of welded components – designer’s guide. Cambridge: Woodhead Publishing; 2003. 20
[16] Marin T, Nicoletto G. Fatigue design of welded joints using the finite element method and the 2007 ASME Div.2 Master curve. Frattura ed Integrità Strutturale 2009;9:76-84. [17] Poutiainen I, Marquis G. A fatigue assessment method based on weld stresses. Int J Fatigue 2006;28:1037-46. [18] Lazzarin P, Tovo R. A notch intensity factor approach to the stress intensity of welds. Fatigue Fract Eng Mater Struct 1998;21:1089-103. [19] Atzori B, Lazzarin P, Tovo R. From a local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints. Fatigue Fract Eng Mater Struct 1999;22:369-81. [20] Lazzarin P, Tovo R. Relationship between local and structural stress in the evaluation of the weld toe stress distribution. Int J Fatigue 1999;21:1063-78. [21] Lazzarin P, Livieri P. Notch stress intensity factors and fatigue strength of aluminium and steel welded joints. Int J Fatigue 2001;23:225-32. [22] Atzori B, Meneghetti G. Fatigue strength of fillet welded structural steels: finite elements, strain gauges and reality. Int J Fatigue 2001;23:713-21. [23] Livieri P, Lazzarin P. Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int J Fract 2005;133:247– 76. [24] Lazzarin P, Livieri P, Berto F, Zappalorto M. Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading. Eng Fract Mech 2008;75:1875–89. [25] Berto F, Lazzarin P. A review of the volume-based strain energy density approach applied to V-notches and welded structures. Theor Appl Fract Mec 2009;52:183-94. [26] Berto F, Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local apporaches. Mat Sci Eng R 2014;75:1-48. [27] Lazzarin P, Berto F, Zappalorto M. Rapid calculations of notch stress intensity factor based on averaged strain energy density from coarse meshes: theoretical bases and applications. Int J Fatigue 2010;32:1559-67. [28] Lazzarin P, Berto F, Gomez FJ,Zappalorto M. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessments of welded joints. Int J Fatigue 2008;30:1345-57. [29] Lazzarin P, Berto F, Radaj D. Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch. Fatigue Fract Eng Mater Struct 2009;32:713-35. [30] Taylor D, Wang G. A critical distance theory which unifies the prediction of fatigue limits for large and small cracks and notches. In: Wu XR, Wang ZG, editors. Proceedings of the Fatigue’99, vol. 1, Beijing, China: Higher Education Press; 1999, p. 579-84. 21
[31] Taylor D, Barrett N, Lucano G. Some new methods for predicting fatigue in welded joints. Int J Fatigue2002;24:509-18. [32] Crupi G, Crupi V, Guglielmino E, Taylor D. Fatigue assessment of welded joints using critical distance and other methods. Eng Fail Anal 2005;12:129-42. [33] Sonsino CM. Multiaxial fatigue of welded joints under inphase and out-of-phase local strains and stresses. Int J Fatigue 1995;17:55-70. [34] Meneghetti G., Guzzella C., Atzori B. The peak stress method combined with 3D finite element models for fatigue assessment of toe and root cracking in steel welded joints subjected to axial or bending loading. Fatigue Fract Eng Mater Struct 2014;37:722-39. [35] Meneghetti G., De Marchi A., Campagnolo A. Assessment of root failures in tube-to-flange steel welded joints under torsional loading according to the Peak Stress Method. Theoretical and Applied Fracture Mechanics 2016;83:19-30. [36] Meneghetti G., Marini D., Babini V. Fatigue assessment of weld toe and weld root failures in steel welded joints according to the peak stress method. Welding in the World 2016;60:559-72. [37] Skorupa M. Load interaction effects during fatigue crack growth under variable amplitude loading – a literature review. Part I: empirical trends. Fatigue Fract Eng Mater Struct 1998;21:987-1006. [38] Skorupa M. Load intraction effects during crack gorwth under variable amplitude loading - a literature review. Part II: qualitative interpretations. Fatigue Fract Eng Mater Struct 1999;22:905-26. [39] Dijkstra OD, Van Straalen IJ. Fracture mechanics and fatigue of welded structures. In: Proceeding of the International Conference on Performance of Dynamically Loaded Welded Structure, New York: WRC; 1997, p.225-39. [40] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Special Issue on “Welded connections” - Int J Fatigue 2009;31:188-96. [41] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue assessment using a simplified critical plane-based criterion. Int J Fatigue 2011;33:969-76. [42] Carpinteri A, Spagnoli A, Vantadori S, Bagni C. Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution. Fatigue Fract Eng Mater Struct 2013;36:870-83. [43] Araújo J, Carpinteri A, Ronchei C, Spagnoli A, Vantadori S. An alternative definition of the shear stress amplitude based on the Maximum Rectangular Hull method and application to the C-S (Carpinteri-Spagnoli) criterion. Fatigue Fract Eng Mater Struct 2014;37:764-71. [44] Carpinteri A, Ronchei C, Scorza D, Vantadori S. Critical plane orientation influence on multiaxial high-cycle fatigue assessment. Physical Mesomechanics 2015;18:348-54. 22
[45] Giordani FA. Estudo de Metodologias para medir a vida em fadiga Multiaxial nao proporcional. Master thesis 2015. Promec/UFRGS, Brazil. In Portugues. http://hdl.handle.net/10183/118864 [46] Ansys 14.5. www.ansys.com; 2016 [accessed 11.08.16]. [47] Macha E. Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Mat Wiss U Werkstofftech 1989;20:132–36 and 153-63.
23
SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy
2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil
Corresponding Author: Sabrina Vantadori,
email: [email protected]
24
CAPTIONS Figure 1. (a) Agricultural sprayer; (b) “H” component (sizes in mm). Figure 2.
Fatigue cracks experimentally observed in one T-joint of
the “H” component, highlighted under sprayer service condition at: (a),(b) brace and (c),(d) chord.
The crack initiation points are
indicated by the arrows. Figure
3.
Schematisation
of
the
stress
parameters
used
in
the
global, intermediate global-local and local approaches. Figure 4.
Possible movements of the sprayer bar examined.
Figure 5. Complete-bridge Wheatstone circuits arranged on both the brace and the chords. Table
1.
Service
condition
duration,
related
to
10
investigated:
hours
and
maneuvers
2,000
hours
and of
their sprayer
operations. Figure 6. Load histories over a time interval of 6.0sec: (a) 1,c and (b) 1,b . Table 2. Mechanical and fatigue properties of the C25E steel. Figure 7. Discretisation employed in the finite element analysis. Figure 8. Schematisation of the forces transferred by the sprayer bar to the “H” component in the finite element model. Figure 9. Deterministic full reverse time histories for both F1 and F2 forces. Figure 10. Reference systems: fixed frame
25
PXYZ and averaged frame
P1ˆ 2ˆ 3ˆ . Figure 11. Examined critical points in the brace and in the chord. Table
3.
Amplitudes
amplitudes
of
stress
of
the
applied
components
on
stresses, the
phase
critical
shift,
plane,
and
calculated finite life fatigue strength for the critical points B1, B2, B3 in the brace. Table
4.
Amplitudes
amplitudes
of
stress
of
the
applied
components
on
stresses, the
phase
critical
shift,
plane
and
calculated finite life fatigue strength for the critical points C1, C2, C3 in the chord. Figure 12. Calculated finite life fatigue strength, phase shift,
Tcal , versus
, at point B3 of the brace.
Figure 13. Calculated finite life fatigue strength, phase shift, , at point C1 of the chord.
26
Tcal , versus
( a)
( b)
27
Figure 1.
(
(
c)
d)
Figure 2. 28
Notch stress Hot-spot stress
Structural stresses
Figure 3.
29
Nominal stresses
Figure 4.
X
W3
W1
W2 Y
Figure 5.
30
Z
Table 1.
31
MANOUVERS
FUEL TANK
DURATION 10 h
road Application the
of
herbicide
DURATION 2000 h
[h] Travel on unpaved
-
[h]
Full
0.50
100
Empty
0.50
100
Full
1.00
200
Empty
1.00
200
Full
2.25
450
Empty
2.25
450
Full
0.50
100
Empty
0.50
100
Full
0.50
100
Empty
0.50
100
Full
0.25
50
Empty
0.25
50
(perimeter) Application the
of
herbicide
(cultivated area) U - curves
Perimeter curves
Braking
32
-
STRESS, 1c [MPa]
MAXIMUM PRINCIPAL
20
(a)
15 10 5 0 -5 -10 0
20
40
60
80
100
120
TIME, t [sec]
33
140
160
180
200
STRESS, 1b [MPa]
MAXIMUM PRINCIPAL
60
(b)
40 20 0 -20 -40 -60 -80 0
20
40
60
80
100
120
140
160
180
TIME, t [sec]
Figure 6.
Table 2.
Material
u
E
[MPa]
[GPa]
af ,1
m
af ,1 m * [MP
[cycles
[MPa] a]
C25E
N0
470.
198.
141.
0
0
0
34
1/5
86. 0
] -
1.0
1/8
(10)6
200
Figure 7.
35
Figure 8.
36
30000
F1 F2
FORCE [N]
20000 10000 0 -10000 -20000 -30000 0
1
2
3
4
5
6
7
TIME, t [sec] Figure 9.
37
8
9
10
11
12
Z 3
(a)
P
X 1 Y
(b) 3
2 Z
P 1
2
(c) 3 P 2 1
38
Figure 10.
Figure 11.
Table 3.
POINT No.
xx, a yy, a
xy, a
[M
[MP
Pa] B1
[MP a]
11 .30
a]
10. 17
N a ,eq
Ca
Tcal
[
[MPa
[MPa
[sec]
]
]
13.3
28.8
7
0
°]
28.
0
91
39
2.5145E+ 09
B2
41 .48
B3
27. 63
99 .74
27.
0
59 62.
22
30.
0
11
15.2
31.3
6
5
63.5
57.9
6
7
1.2829E+ 09 4.0474E+ 06
Table 4.
POINT
xx, a
zz, a
xz, a
[MP
[MP
[MP
N a ,eq
Ca
Tcal
[
[MPa
[MPa
[sec]
]
]
124.
113.
4.0656E
46
60
+04
115.
105.
6.8599E
57
48
+04
112.
102.
8.1402E
79
94
+04
No.
a] C1
206 .98
C2
07
0
93
51
50.
0
32 26.
42
°]
62.
38.
195 .99
a]
41.
197 .25
C3
a]
44. 04
40
0
CALCULATED FATIGUE LIFE, Tcal / 107 [sec]
8.0
8.00E+007
Point B3
6.0
6.00E+007
4.0
4.00E+007
2.0
2.00E+007
0.4
0
30
60
90 120 150 180
PHASE SHIFT, [°]
Figure 12.
41
CALCULATED FATIGUE LIFE, Tcal / 105 [sec]
1.0
1.00E+005
Point C1 0.8
8.00E+004
0.6
6.00E+004
0.4
4.00E+004
0
30
60
90 120 150 180
PHASE SHIFT, [°]
Figure 13.
42
SUBMITTED TO INTERNATIONAL JOURNAL OF FATIGUE SPECIAL ISSUE ON: “PROGRESS IN FATIGUE LIFE ESTIMATION OF WELDMENTS” August 2016 Revised version - October 2016
FATIGUE LIFE ESTIMATION OF FILLET-WELDED TUBULAR T-JOINTS SUBJECTED TO MULTIAXIAL LOADING Andrea Carpinteri1, Joel Boaretto2, Giovanni Fortese1, Felipe Giordani2, Ignacio Iturrioz2, Camilla Ronchei1, Daniela Scorza1, Sabrina Vantadori1
1
Department of Civil-Environmental Engineering & Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy 2
Mechanical Post- Graduate Program,
Federal University of Rio Grande do Sul, Sarmento Leite 425, CEP 90050-170, Porto Alegre, Brazil Corresponding Author: Sabrina Vantadori, email: [email protected]
HIGHLIGHTS The fatigue strength assessment of an agricultural sprayer is performed The weakest links of such a sprayer against fatigue failure are the T-joints A stress-based critical plane criterion is applied The comparison between theoretical and experimental results is quite satisfactory
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