ABSTRACT

The treatment 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.

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