ABSTRACT
The process of producing residual
stresses in thick_walled cylinder before
it is putin to usage is called Autofretage, which it means; a suitable large
enough pressureto cause yielding within the wall, is applied toinner surface of
a sylinder and then removed. So that
acompressive residual stresses are generated to acertain radial depth at a sylinder
wall.
The objective
ofpresent study, is to investigate the influenceof autofretage treatment onthe
radial, circumferential andtotal stresses using von._mises yieldcriteria. Num.simulation
carried outon ABAQUS software to investigate thestresses distribution and
calculate the autofretage radius. The results revealthat, the autofretage treatmentof
thick_wall sylinder lead to decrease the
hoob and max.von._mises stresses and relocate them from the inner surface of
the sylinder to somewhere along it’s
thickness. The reduction in max.stresses is strongly depending on autofretage
pressure, it wasvarying from ( 3.6% at Pautofretage = 105 M.Pa.
to 19.2% at Pautofretage =
130 M.Pa. ) Also, it
has been found, there is no influenceof autofretage stages number on each of max.von._mises
stressand autofretage radius.
Key words: autofretage, radial, hoob and
axial stresses, von._mises yield criteria, autofretage radius, optimum autofretage
pressure.
1.
INTRODUCTION
The wide applications of
pressurized sylinder in chemical,
nuclear, armaments, fluid transmitting plants,
power plants and military equipment, in addition to the increasing scarcity and
high cost of materials lead
the designers to
concentrate their attentions to the elastic – plastic approach which offers
more efficient use of materials 1, 2.The treatment of producing residual
stresses in the wall of thick_walled sylinder before it is put in to usage is called autofretage, which it means; asuitable large enough
pressure to cause yielding within thewall, is applied to the inner surface of
the sylinder and then removed.
So that a compressive residual stresses are generated to a certain radial depth
at the sylinder wall. Then, duringthe
subsequent application of an operating pressure, the residual stresses will
reduce the tensile stresses generated asa result of applying operating pressure
1,3.
The influenceof
residual stresses onload-carry capacity of thick_walled sylinders have been
investigate by Ayob and Albasheer 4, using each analytical andNum.techniques.
The results of the study reveal three scenarios in the design of thick_walled sylinders.
Ayob and Elbasheer 5, used von._mises and Tresca yieldcriteria to develop a
procedure in whichthe autofretage pressure determined analytically resulting in
a reduced stress concentration. Then they coM.Pa.red the analytical results
with F.E.A. results. They concluded that, the autofretage treatment increase
the max.allowable internal pressure but it cannot increase the max.internal
pressure to case whole thickness of the sylinder to yield. Noraziah et al. 6 presented an
analytical autofretage procedure topredict the required autofretage pressure of
different levels of allowable pressure andthey validate their results with F.E.A.
results. They found three cases of autofretage in design of pressurized thick_
walled sylinders.
Zhu and Yang 7, using
each yield criteria von._mises and Tresca, presented an analytical equation for
optimum radius of elastic-plastic junction in autofretage sylinder , alsothey
studied the influence of autofretage on distribution of stress and load bearing
capacity. They concluded, to achieve optimum radius ofelastic – plastic
junction, an autofretage pressure a bit larger than operating pressure should
be applied before a pressure vessel is put in to use. Hu and Puttagunta 8
investigate the residual stresses in thick_ walled sylinder induced by internal autofretage pressure, also
they found the optimum autofretage pressure andthe max.reduction percentage of
the von._mises stress under elastic-limit working pressure. Md. Amin et al. 9
determined the optimum elasto_plasticradius and optimum autofretage pressure using
von._mises yield criteria , then they have been coM.Pa.red with Zhu and Yang’s
model 8. Also they observed that the percentage of max.von._mises stress
reduction increases as value of radius ratio (K) and working pressure
increases. F. Trieb et al. 10 discussed practical application of autofretage
on components for waterjet cutting. They reported that the life time of high
pressure components is improved by increasing autofretage depth due to
reduction of tangential stress at inner diameter, on other hand too high
pressure on outside diameter should be avoided to prevent cracks generate. In
addition to determine the optimum autofretage pressure and the optimum radius
of elastic-plastic junction , Abu Rayhan Md. et al.11 evaluated the influenceof
autofretage treatment in strain hardened thick_ walled pressure vessels using
equivalent von._mises stress as yield criteria. They found, the number of autofretage
stages has no influenceon max.von._mises stress and pressure capacity. Also,
they concluded that, optimum autofretage pressure depends on the working
pressure and on the ratio of outer to inner radius.
II. Limits of pressureand Distribution
of stress in non – autofretaged sylinder
2.1. Limits of pressureof non – autofretage
sylinder
According to Von._Mises yield criteria,
Each of the internal pressure requires to yield the inner surface of the sylinder
( i.e. partial autofretage ), PYi
, and that to yield the whole wall of the sylinder ( i.e. completely autofretage ), PYo
, can be calculated from equations ( 1& 2 )4, 7
PYi
=
……………………. ( 1 )
PYo
=
……………………. ( 2 )
2.2. Distribution of stress
of non – autofretage sylinder
The
radial stress ?r, circumferential ( hoop
) stress ?? and axial stress ?z,
distributions in non _autofretage sylinder subjected to an operating pressure, Pi,
are given by Lame’s formulations which is available in 3, 4, 5, 6, 7 . As
shown in Fig. ( 1 ), it is obvious that the
tensile hoob, ??,
compressive radial , ?r,
and max. Von._Mises stresses have
their max. values at the inner surface of the sylinder . The hoop stress has
always positive value which represents
as tensile stress while the stress in the radial direction is always
compressive. Also the hoop tensile stress’s value is greater than radial
compressive stress’s value.
Fig.
1: Distribution of stress on non-autofretage thick-walled sylinder subjected to operating pressure.
Fig. 2: Geometry of inspectedmodel.
III. Finite Element
Analysis and Materials of Num.Simulation
Models
Fig. ( 2 )
illustrates the geometry of inspectedsylinder that is made up of carbon steel with young’s
modulus of ( 203 GPa ), Poisson’s ratio of ( 0.33 ) and yield stress of ( 325 M.Pa.
) 12 . It subjected to internal pressure ( Pi ). The material is assumed
homogeneous and isotropic. To compute the required results, Num.simulation is
carried out on ABAQUS ver.6.9 13. The inspected cases are consider as 2D –
planar problem with quadratic element have been used ( CPS8R–8– nodes )
IV.
Validation of Num.Simulation
In the
present study, the validation of software has been done by coM.Pa.ring the
analytical calculation results which obtained by solutions of equations are
available in literatures 3, 4, 5, 6 7, with results of Num.solution using
ABAQUS ver.6.9.
From Fig.
( 3 ) , it is obvious that, the theor. and Num.calculations of circumferential, radial and max. Von._Mises
stresses for different internal pressure are very closed and overlap each
other. It means, a good agreement is found between the results, and the static
analysis shows that, the percentage of errors between the result of analytical and Num.solution are les than
0.5%. This low percentage of errors affirm, there are no significsnt
differences between the theor. results and those obtained by simulation.
Consequently, FE modeling using ABAQUS software can be used to study the influenceof
autofretage treatment on the distribution of stress and location of autofretage
radius ( Ra ) of thick_walled sylinder subjected to operating pressure.
a
b
Fig. 3 : Validation of Num.solution results with theor.
results at different operating pressure; a – operating pressure = 80 M.Pa., b
– operating pressure = 100 M.Pa..
V. Results and Discussions
5.1. Min.. Autofretage Pressure
By calaculating the min.. pressure
that needed to yield the inner surface of the tested sylinder ( PYi ) from equation (1) , it was
found equal to ( 104.243 M.Pa. ). That is mean, the influenceof autofretage pressure
will start at (104.243 M.Pa.), then the plastic deformation spreads through the
sylinder thickness. Fig. (4) shows that,
the simulation solution of influenceof autofretage pressure on max. Von._Mises stress
for different operating pressure, it is obvious that , there is no influenceof autofretage
pressure on max. Von._Mises stress generating in the sylinder due to the operating pressure as long as it is
less than ( 104 M.Pa. ) for each value of operating pressure.Then , when it is
exceed ( Pautofretage ? 104 M.Pa.
) the maximunm Von._Mises stress decreases depending on the autofretage pressure,
the bigger value of autofretage pressure, the lower of max. Von._Mises stress.
In addition to that , it has been
observed from Table 1 that, the max. Von._Mises stress decreases with
increasing the autofretage pressure even Pautofretage reache value
of about ( 130 M.Pa. ) then starts increasing, which it means, this value of autofretage
pressure represents the optimum autofretage pressure 5,6. This results agree
with result was found by 1, 9, 11.
Fig.
4 : Simulation solution results of autofretage pressures’ influenceon Max. von._mises
stress at different operating pressure.
Tab. 1 : F.E.A. results of influence of Autofretage Pressure
on Max. Von._Mises Stress
No.
Operating Pressure, M.Pa.
Autofretage Pressure, M.Pa.
Max. von._mises Stress, M.Pa.
1.
90
120
247.00
2.
90
125
241.40
3.
90
130
238.8
4.
90
131
240.20
5.
90
132
241.40
6.
100
120
273.10
7.
100
125
265.20
8.
100
130
260.00
9.
100
131
260.80
10.
100
132
261.00
5.2. Influenceof Autofretage treatment
on stress distribution
Fig.s ( 5, 6 & 7 ) demonstrates the influenceof
autofretage treatment on distribution of stress of thicked–walled sylinder subjected to operating pressure of ( 100 M.Pa.
). It is obvious, the autofretage treatment leads to decrease the value of max.
Von._Mises stress and relocated the compressive circumferential & max. Von._Mises
stresses from the inner surface of the sylinder to somewhere through it’s thickness. This new
location of max. Von._Mises stress called Autofretage radius, Ra
. It does not depend on operating pressure while it is strongly affected by autofretage
pressure as shown in Table 2, which shows the values of autofretage radius, Ra
, with different values
of autofretage pressure. Also, it is found , the reduction in max. Von._Mises stresses
varying from ( 3.6 % at Pautofretage =105 M.Pa. ) to ( 19.2% at Pautofretage
=130 M.Pa. ). It is vital to see that , there is no significant influenceof
autofretage treatment on radial stress as that seen on the circumferential
stress.
Fig. 5 :Influenceof Autofretage Pr. on hoob & Radial stresses at
operating Pressure = 100 M.Pa..
Fig. 6 : Influenceof Autofretage Pr. on max. Von._Mises stress at operating Pressure = 100 M.Pa..
Table 2 : F.E.A. results of influenceof Autofretage Pressure
on Max. Von._Mises Stress
No.
Operating Pressure, M.Pa.
Autofretage Pressure, M.Pa.
Max. Von._Mises Stress, M.Pa.
Autofretage Radius, mm
Reduction in Max. Von._Mises stress %
1.
90
without
290.00
100
—
2.
90
105
278.975
101.99836
3.8 %
3.
90
110
264.108
103.99686
8.9 %
4.
90
120
246.88
111.9915
14.8 %
5.
90
130
238.792
125.9761
17.65 %
6.
100
without
321.836
100
—
7.
100
105
310.00
101.99836
3.6 %
8.
100
110
294.020
103.99686
8.6 %
9.
100
120
273.116
111.9915
15.2 %
10.
100
130
259.992
125.9761
19.2 %
a
b
c
d
Fig. 7 : F.E.A.of influenceof autofretage
Pressure on max. Von._Mises stress and location of autofretage radius at
operating Pressure = 100 M.Pa. ; a- without autofrettage, b- Pautofretage = 110 M.Pa., c –
Pautofretage = 120 M.Pa., d –
Pautofretage = 130 M.Pa..
5.3. Influenceof Autofretage stages
on max. Von._Mises stress
To investigate the influenceof autofretage
stages on max. Von._Mises stress,
the inspectedsylinder was subjected to (
100 M.Pa. ) as operating pressure and autofretage pressures of ( 110, 120 and 130 M.Pa. ) are done by
two steps, at first step,the autofretage pressure has been applied in one stage, while at
second step it was done by three loading stages ( see Table 3 ). As can be noticed clearly in Table 3
and Fig. ( 7 ), the Num.results confirm there is no influenceof autofretage stages
on the max. Von._Mises stress generated in the sylinder due to operating pressure. This results are
very close to the with results have been
found by 3.
Tabe 3 : F.E.A. results of influenceof Autofretage stages
on Max. Von._Mises Stress
No. of case
Autofretage pressure, M.Pa.
First
stage
Unloading
step
M.Pa.
Autofretage
pressure, M.Pa.
second stage
Unloading step
M.Pa.
Loading of Operating Pressure, M.Pa.
Max. Von._Mises Stress,
M.Pa.
Case I
110
0
–
–
100
294.020
Case II
120
0
–
–
100
273.116
Case III
130
0
–
–
100
259.992
Case IV
105
0
110
0
100
294.033
Case V
105
0
120
0
100
273.05
Case VI
105
0
130
0
100
260.254
Fig. 7 : Num.solution results of influenceof autofretage
stagse on Max. von_mises
stresses and autofretage radius at operating Pressure = 100 M.Pa.
VI. Conclusion
The results of present
investigation can be summarized as :-
1. The autofretage treatment
on thick_walled sylinder leads to
decrease the circumferential and max. Von._Mises stresses and relocate them from the inner surface of the sylinder
to somewhere along it’s thickness, which
called as, autofretage radius, Ra .
2. The autofretage radius, Ra ,
is strongly affected by autofretage pressure
while it does not depend on the operating pressure..
3. There is no influenceof autoffrettage
stages on max. Von._Mises stress developed in the sylinder subjected to an operating pressure.
References
