research article new numerical solution of von...

Research ArticleNew Numerical Solution of von Karman Equation ofLengthwise Rolling

Rudolf Pernis1 and Tibor Kvackaj2

1Alexander Dubcek University of Trencın Pri Parku 19 911 06 Trencın-Zablatie Slovakia2Technical University of Kosice Faculty of Metallurgy 042 00 Kosice Slovakia

Correspondence should be addressed to Tibor Kvackaj tiborkvackajtukesk

Received 10 June 2015 Accepted 16 August 2015

Academic Editor SutasnThipprakmas

Copyright copy 2015 R Pernis and T Kvackaj This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karmanequation was solved by many authors The solutions reported by authors are used simplifications for solution of Karman equationThe simplifications are based on two cases for approximation of the circular arch (a) by polygonal curve and (b) by parabolaThe contribution of the present paper for solution of two-dimensional differential equation of rolling is based on description ofthe circular arch by equation of a circle The new term relative stress as nondimensional variable was defined The result fromderived mathematical models can be calculated following variables normal contact stress distribution front and back tensionsangle of neutral point coefficient of the arm of rolling force rolling force and rolling torque during rolling process Laboratory coldrolled experiment of CuZn30 brass material was performed Work hardening during brass processing was calculated Comparisonof theoretical values of normal contact stress with values of normal contact stress obtained from cold rolling experiment wasperformed The calculations were not concluded with roll flattening

1 Introduction

It is likely that the author Karman [1] in the year 1925 wasthe first to submit equation of power balance in the rollgap and derived the two-dimensional differential equationwhich described lengthwise rolling process The followingsimplifications were received

(i) Rolled material has rectangular cross section withinitial thickness ℎ

0and is deformed by cylindrical

rolls to final thickness ℎ1without any action of front

and back tensions(ii) Spread is negligible(iii) Friction coefficient between rolls and rolled material

is constant(iv) Rolls are rigid without any elastic deformation and

bending(v) Rolled material is homogenous and without any

elastic deformation

(vi) Tresca condition of plasticity 1205901minus 1205903= 120590119886

(vii) Flow stress of material in rolling gap is constant(viii) Rolling speed is constant

Geometrical description of roll gap is given in Figure 1and essential formulas are presented in the appendix Fromgraphical scheme of roll gap the result is that all forceshave to be at balance which was the base for derivationtwo-dimensional differential equation describing lengthwiserolling process The procedure of derivation of the two-dimensional differential equation can be found in the classicalliterature of rolling for example Avitzur [2] Hensel andSpittel [3] and Mielnik [4] More recent literatures are thepublications Hajduk and Konvicny [5] and Kollerova etal [6] The stress state in rolling gap describes differentialequation

119889120590119909

119889119909minus120590119899minus 120590119909

119910sdot119889119910

119889119909plusmn120591

119910= 0 (1)

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 843720 20 pageshttpdxdoiorg1011552015843720

2 Journal of Applied Mathematics

Y

120572ld R

B dx x

A0

X

2(y+dy

)

2y 1

h1h0

R

120593d120593

Figure 1 Determination of geometric relationships

where plus sign (+) is for forward slip zone minus sign (minus)is for backward slip zone 120590

119899is normal contact stress on rolls

120591 is the shear stress between rolls and rolling material and119909 119910 are the coordinates of the cylinder touching the rolledmaterial

The Tresca condition of plasticity is used for solution of(1)

minus120590119909minus (minus120590

119899) = 120590119899minus 120590119909= 120590119886 (2)

where 120590119910

≐ 120590119899is the maximal principal stress (vertical

direction) 120590119909is the minimal principal stress (horizontal

direction) 120590119886is the flow stress [7ndash9] and 120591 is the shear stress

that is proportional to normal contact stress 120590119899according to

formula

120591 = 119891 sdot 120590119899 (3)

where 119891 is the friction coefficient between rolls and rolledmaterial

Substituting (2) into (3) the following formula will beobtained with only one stress variable 119889120590

119899

119889120590119899

119889119909minus120590119886

119910sdot119889119910

119889119909plusmn119891

119910sdot 120590119899= 0 (4)

Flow stress and friction coefficient are as a constant in (4)If differential equation (4) is divided by flow stress 120590

119886it is

possible to obtain relative normal contact stress 120590119899

120590119899=120590119899

120590119886

(5)

where 120590 is a function

Substituting (5) into (4) the following form is obtained

119889120590119899

119889119909minus1

119910sdot119889119910

119889119909plusmn119891

119910sdot 120590119899= 0 (6)

The application equation (5) can be excluded from (4) mate-rial constant (flow stress 120590

119886) Thus (6) become independent

from the rolled material This differential equation is a lineardifferential equation of first order with the right side andhas a general transcription 120590

119899= 120590119899(119909 119910) The coordinates

119909 and 119910 are defining the geometry of the cylinder which isconnected to the rolling materialThe variable 119910 is a functionof the 119909 coordinate If equation of a circle is substitutinginto (6) so differential equation is obtained which has noanalytical solution till nowThe authorOrowan [10] presentedthe solution of differential equation (1) based on analytical-graphical principle He defined procedure for calculationrolling force rolling torque and electric input for hot andcold rolling processes In the literature [11] showed solutionof differential equation (1) by simulation on analog computerThe authors of [12] solved this case with the use of the circulararc but the result is purely numerical solution The author of[13] presented simplified analytical solution with acceptanceof material work hardening during cold rolling process Thedevelopment in computational software allowed to solve thistask by using finite element method as was given in theliterature [14]

2 Experimental Material and Methods

The investigated material was annealed brass CuZn30 withchemical composition corresponding to standardDIN 17660WNo 20265 The sample sizes before rolling were ℎ times 119887 times 119897

= 34 times 30 times 150mm The samples were laboratory rolledusing duo mill (the diameter of rolls 119863 = 210mm) withconstant circumferential speed of rolls Vroll = 066ms atambient temperatures The temperature of a sample surfaceimmediately after rolling was measured using a thermovisioncamera and was from interval 119879 isin ⟨20 40⟩ [∘C] The coldrolling reductions of thickness were made in an interval120576cold isin ⟨0 30⟩[] as one pass deformation Deformationforces during samples processing by cold rolling were mea-sured by tensometric elements with signal registration inSpider apparatus The static tensile test was performed inaccordance with STN EN 10 002-1 standard

The differential equation of lengthwise rolling processgiven by formula (1) has no analytical solution when contactarc is described with the equation of the circle For solutionof this problem a new combined analytical and numericalapproach is presented

3 Analysis and Results

31 Analysis of the Present Solution of Differential EquationDescribing Stress Condition in Rolling Gap The presentsolutions of differential equation of rolling were based onthe same simplifications because the form of equation is

Journal of Applied Mathematics 3

S1

R120572

120572n

B

N A

h0 h1

xn

ld

Figure 2 The contact arc according to Korolev

complicated More than twenty solutions of differential equa-tion are described in the literature The present solutions ofdifferential equation (1) can be divided into following groups

(i) Geometry of cylinder the circular arc of the cylin-der is being approximated by simplifying curves asstraight line polygonal line and parabola or in somecases using equation of a circle

(ii) Friction conditions between rolls and rolled material

(a) The shear stress between rolls and rolling mate-rial is constant 120591 = 119891 sdot 120590

119886

(b) The shear stress between rolls and rolling mate-rial is changing 120591 = 119891 sdot 120590

119899

(c) Friction coefficient is 119891 = 0

(iii) Mathematical methods of solution analytical analyt-ical-graphical analytical-numerical numerical andsemiempirical

(iv) Work hardening of rolled material most of the solu-tion is based on assumption that material during pro-cessing is without work hardening In a few isolatedcases solution with work hardening of rolled materialwas used

In the following part most frequently ways for calculation of120590 function which are fall to first group will be analyzed

311 The Solution according to Korolev The author Korolev[15] presented solution based on approximation of circulararch by parallel step function which is shown in Figure 2Thecircular arch (BA) is approximated by polygonal line Neutralpoint N divides space of rolling gap on parts backward slipzone (BN) and forward slip zone (NA) Position of neutralpoint is determined by coordinate 119909

119899

119909119899

119897119889

=ℎ1

ℎ0+ ℎ1

(7)

The distribution functions describing relative normal contactstress in rolling gap were divided separately for backward andforward slip zone

The distribution function describing relative normal con-tact stress 120590

119899119861in backward slip zone of rolling gap is as

follows

120590119899119861

= 1198901198980(1minus119909119897119889) (8)

where

1198980= 2119891 sdot

119897119889

ℎ0

(9)

The distribution function describing relative normal contactstress 120590

119899119865in forward slip zone of rolling gap is as follows

120590119899119865

= 1198901198981 sdot(119909119897119889) (10)

where

1198981= 2119891 sdot

119897119889

ℎ1

(11)

The author of [15] on the basis of the previous distributionfunction derived an equation for calculation 120590 function asfollows

120590Kor =1

119898Kor(119890119898Kor minus 1) (12)

where119898Kor is constant

119898Kor = 119891 sdot119897119889

ℎav (13)

312The Solution according to Tselikov The authors Tselikovet al [16] presented solution based on approximation ofcircular arch by straight line function which is shown inFigure 3 The circular arch (BA) is approximated by straightline Neutral point N divides space of rolling gap on partsbackward slip zone (BN) and forward slip zone (NA)Position of neutral point is determined by coordinate 119909

119899and

thickness in neutral section ℎ119899was derived as follows

ℎ119899

ℎ1

= [1

119898 + 1sdot (1 + radic1 + (1198982 minus 1) sdot (

ℎ0

ℎ1

)

119898

)]

1119898

(14)

where119898 is constant

119898 equiv 119898Tsel =2119891 sdot 119897119889

ℎ0minus ℎ1

(15)

The distribution functions describing relative normal contactstress 120590

119899119865in forward slip zone and 120590

119899119861in backward slip zone

of rolling gap are as follows

120590119899119861

=1

119898[(119898 minus 1) sdot (

ℎ0

2119910)

119898

+ 1]

120590119899119865

=1

119898[(119898 + 1) sdot (

2119910

ℎ1

)

119898

minus 1]

(16)


S1

R120572

120572n

B

N A

h0 h1

hn

xn

ld

Figure 3 The contact arc according to Tselikov

S1

R120572

120572n

B

N A

h0 h1

xn

ld

Figure 4 The contact arc approximated by a parabola

The authors of [16] on the basis of the previous distributionfunctions derived an equation for calculation 120590 function asfollows

120590Tsel =2 (1 minus 120576)

120576 (119898 minus 1)sdotℎ119899

ℎ1

sdot [(ℎ119899

ℎ1

)

119898

minus 1] (17)

From (17) the result is that for condition119898 = 1 120590 function isdiscontinuous

313 The Solution according to Bland and Ford The authorsBland and Ford [17] presented solution based on approxima-tion of circular arch by parabolic function which is shownin Figure 4 The circular arch (BA) is approximated by partof parabola Neutral point N divides space of rolling gapon parts backward slip zone (BN) and forward slip zone

(NA) The authors defined neutral angle 120572119899according to the

following formula

120572119899

= radicℎ1

119877

sdot tg[

[

1

2sdot arctg(radic 119877

ℎ1

sdot 120572) minus1

4119891sdot radic

ℎ1

119877sdot ln

ℎ0

ℎ1

]

]

(18)





120590119899119861

=ℎ1+ 1198771205932

ℎ0

sdot 119890119891(119867∙

0minus119867∙)

120590119899119865

=ℎ1+ 1198771205932

ℎ1

sdot 119890119891sdot119867∙

(19)

where

119867∙= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120593)

119867∙

0= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120572)

(20)


120590Bland-Ford = radic1 minus 120576

120576119898[(1 minus 120576) sdot 119890

2119898sdotarctg(radic120576(1minus120576))

sdot int

119905120572

119905119899

(1 + 11989821199052) sdot 119890minus2119898sdotarctg(119898119905)

sdot 119889119905

+ int

119905119899

0

(1 + 11989821199052) sdot 1198902119898sdotarctg(119898119905)

sdot 119889119905]

(21)

where 119898 is constant and 119905119899 119905120572are limits of integration 119905 =

120593119891

119898 equiv 119898Bland-Ford = 119891 sdot radic119877

ℎ1

(22)

119905119899=120572119899

119891 (23)

119905120572=

1

119898sdot radic

120576

1 minus 120576 (24)

314 The Solution according to Sims The author Sims [18]presented solution of differential equation without consid-eration of friction coefficient Roll radius is considered with


elastic flattening 119877 = 1198771015840 The ratio ℎ

119899ℎ1was derived from

the following formula

ℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (25)

where

120572119899= radic

ℎ1

119877sdot tg( 120587

16sdot radic

ℎ1

119877sdot ln (1 minus 120576)

+1

2arctg(radic 119877

ℎ1

sdot 120572))

(26)



119899119861in backward slip

zone of rolling gap were calculated according to the followingformulae

120590119899119861

= 1 +120587

4ln((1 minus 120576) (1 +

119877

ℎ1

1205932))

+ 2radic119877

ℎ1

(arctg(radic 119877

ℎ1

sdot 120572) minus arctg(radic 119877

ℎ1

sdot 120593))

120590119899119865

= 1 +120587

4ln(1 + 119877

ℎ1

1205932) + 2radic

119877

ℎ1

sdot arctg(radic 119877

ℎ1

sdot 120593)

(27)

The author of [18] on the basis of the previous distributionfunctions derived an equation for calculation 120590 function asfollows

120590Sim =120587

2radic1 minus 120576

120576arctg(radic 120576

1 minus 120576) minus

120587

4minus radic

1 minus 120576

120576

sdot radic119877

ℎ1

sdot lnℎ119899

ℎ1

+1


120576sdot radic

119877

ℎ1

sdot ln 1

1 minus 120576

(28)

315 The Modification Smiryagin Solution by Pernis Theauthors of [19] for solution of (1) presented modification ofTselikov solution using a parabolic approximation functionof the circular arc as is shown in Figure 4 The distributionfunctions describing relative normal contact stress 120590

119899119865in

forward slip zone and 120590119899119861

in backward slip zone of rollinggap were calculated according to the following formulae

120590119899119861

= [1 minus (1199060

119898minus

1

1198982)] 119890119898(1199060minus119906) + 2 (

119906

119898minus

1

1198982)

120590119899119865

= (1 +1

1198982) 119890119898119906

minus 2 (119906

119898+

1

1198982)

(29)

where119898 1199060are constants

119898 equiv 119898Smir-Per =2119891 sdot 119897119889

radicℎ1sdot (ℎ0minus ℎ1)

(30)

1199060= arctg(radic 120576

1 minus 120576) (31)

At this point the Tselikov solution of differential equation(1) was completed The function for calculation of neutralangle and average relative normal contact stress has not beendetermined Additional calculations were carried out by theauthor of [20] which are derived by additional equationsdescribing neutral angle and average relative normal contactstress Location of the neutral point was determined throughusing the value 119906

119899(dimensionless number) and is expressed

by an explicit equation

(051198982 minus 119898 sdot 1199060+ 1) sdot 119890

119898(1199060minus119906119899) minus (051198982 + 1) sdot 119890119898sdot119906119899

+ 2119898 sdot 119906119899= 0

(32)

The neutral point can be obtained from the following for-mula

sin120572119899= radic

1 minus 120576

120576sdot tg119906119899sdot sin120572 (33)


120590Smir-Per =2

119898 sdot 1199060

sdot [(1 +2

1198982) sdot (119890119898sdot119906119899 minus 1) +

1199062

0

2

minus 119906119899sdot (119906119899+

2

119898)]

(34)

32 New Approach for Solution of von Karman DifferentialEquation The following approach represents analytic andnumerical solution of (1) where for description of the circulararc equation of circle is applied as is shown in Figure 5 Equa-tion (6) can be classified as the first-order linear differentialequation with right side

119889120590119899

119889119909plusmn119891

119910120590119899=1

119910sdot119889119910

119889119909 (35)

Many authors [21ndash23] have dealt with the solution of this typeof differential equation Firstly (35) will be solved as equationwithout right side

119889120590119899

119889119909plusmn119891

119910120590119899= 0 (36)

which can be divided into the two following differentialequations

119889120590119899119861

119889119909+119891

119910120590119899119861

= 0 (37)

119889120590119899119865

119889119909minus119891

119910120590119899119865

= 0 (38)


S1

R120572

120572n

B

N A

h0 h1

xn

ld

Figure 5 The contact arc described by equation of circle

The following solution of (37) is valid for backward slip zone

int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)

The calculation of integral of the left side of the equation canbe carried out as follows

ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861

= 119888 sdot 119890minus119891sdotint(119889119909119910)

(40)

If the integral is the exponent and also is a known function1198651(119909) so it is possible to write

1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)

Equation (40) is representing general form of (37) and 119888 isfunction of variable 119909

120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)

Differentiating the equation is obtained from119889120590119899

119889119909=119889119888 (119909)

119889119909sdot 119890minus119891sdot1198651(119909) minus 119888 (119909) sdot

1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)

Equations (42) and (43) are substituting into (35)119889119888 (119909)


1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)

from which function 119888(119909) can be determined as follows119889119888 (119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)

where 119862 is an integration constant that must be determinedfrom the boundary conditions for each segment separately(backward and forward slip zone)

Substituting (45) into (42) will obtain the general form of(35) for backward slip zone

120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)

Determination of ratio 119889119910119910 and function 1198651(119909) can be done

using the equation of the circle

1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)

Separating the variable 119910 from (47) and its differentiation isobtained

119910 = 119877 +ℎ1

2plusmn radic1198772 minus 1199092

119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)

The next solution is apparent from the description

119889119910

119910=

119909radic1198772 minus 1199092

119877 + ℎ12 minus radic1198772 minus 1199092

119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909

119877 (119898 minus radic1 minus (119909119877)2)

(50)

where the constant values are included in one constant119898

119898 = 1 +ℎ1

2119877 (51)

The simplification of (50) can be obtained with transforma-tion fromCartesian coordinates to polar coordinates with useof the following substitution with respect to Figure 6

119909

119877= sin120593

respectively 119889119909 = 119877 cos120593 sdot 119889120593

(52)

Substituting (52) into (50) will obtain the form

119889119910

119910=

sin120593

radic1 minus sin2120593sdot

119877 cos120593 sdot 119889120593

119877(119898 minus radic1 minus sin2120593)

=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)

where a new function 1198652(120593) is expressed as follows

1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1

Figure 6 Definition of the point K[119909 119910]

The function 1198651(120593) can be expressed from (41) according to

the solution defined in work [24] as follows

1198651 (119909) = int

119889119909

119910= int

119889119909

1199100minus radic1198772 minus 1199092

= int119889119909119877

119898 minus radic1 minus (119909119877)2

(55)

Transformation of function1198651(119909) fromCartesian coordinates

to function 1198651(120593) in polar coordinates will be obtained

1198651(120593) = int

cos120593 sdot 119889120593

119898 minus radic1 minus sin2120593= int

cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)

using the following substitutions

tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)

which are inserted into (57) and the integral is obtained

1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)

By decomposition of the integral of (59) to partial fractionsthe following formula is obtained

1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)

Figure 7 Graphical visualization of function 1198651(120593)

and calculation of subintegrals in (61) enters to the form

1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1

radic119886arctg 119911

radic119886) (62)


1198681= minus2arctg119911 + 2119898

radic1198982 minus 1arctg(radic119898 + 1

119898 minus 1sdot 119911) (63)

and also substituting (58) into (63) integral 1198681will be in the

form

1198681=

2119898


119898 minus 1sdot tg

120593

2) minus 120593 (64)

Integral 1198681represents the analytical form of the function

1198651(120593)

1198651(120593) =

2119898

radic1198982 minus 1sdot arctg(radic119898 + 1


120593

2) minus 120593 (65)

Transformation Cartesian coordinate 119909 to polar coordinate 120593in (46) and substituting function 119865

2(120593) are obtained from the

form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)

Limits of integration in (66) are ⟨120593 120572⟩ Solution of (38)describing the forward slip process can be obtained by theanalogical way as reported for solving (37)The final solutionof (38) of relative normal contact stress in forward slip zonewill be as follows

120590119899119865

= 119890119891sdot1198651(120593) sdot (119862

119865+ int 119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (67)

Limits of integration are ⟨120593 120572⟩ From (66) and (67) the resultis that the relative normal contact stress is proportional to theproduct of two functions 120590

119899asymp 119890119891sdot1198651 sdot 119865

2

Graphical visualization of function 1198651(120593) depending on

angle coordinate and ratio 119877ℎ1is given in Figure 7 Func-

tions 1198651(120593) throughout the observed interval are monotoni-

cally increasing With increasing of ratio 119877ℎ1also functions


0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)


1198651(120593) are rising The function 119865

1(120593) during rolling process

represents wrapping angle of roll 120573 by rolled material (120573is not gripping angle) Graphical visualization of function1198652(120593) depending on angle coordinate is given in Figure 8

Local maximum of functions 1198652(120593) increases with increasing

ratio 119877ℎ1 Location point expressing the maximum value

of the function 1198652(120593) is determined from the condition

1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)

where 1205930is coordinate in maximum and 119898 is constant of

differential equation given by (51)Maximal value of function 119865

2(120593) can be determined from

(54) for coordinate 1205930as follows

1198652max (1205930) =

1

radic1198982 minus 1 (69)

The broken line in Figure 8 shows the positions of themaximal values The integration constants in (66) and (67)must be determined fromTresca condition of plasticity Stressstate according to Figure 1 can be described for point A(output of material from rolling gap) and for point B (inputof material to rolling gap) using the following formulae

0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)

Equations (70) are normalized according to flow stress

120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)

The determination of integration constants is performedwith assumption without work hardening of material duringrolling process without forward and backward stretchingand without roll flattening In the input and output plane ofrolling material the stress is 120590

119909= 0 The integration constant

119862119861for the backward slip zone (120593 = 120572) is found from (66) as

follows

119862119861= 119890119891sdot1198651(120572) (72)

where 1198651(120572) is a known function given by (65) The integra-

tion constant 119862119865for the forward slip zone (120593 = 0) is found

from (67) as follows

119862119865= 1 (73)

The total form of equation for the calculation of the relativenormal contact stress 120590

119899119861for the backward slip zone can be

written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898

2minus1)sdotarctg(radic(119898+1)(119898minus1)sdottg(1205932))minus120593]

sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)

and the equation for the calculation of the relative normalcontact stress 120590

119899119865for the forward slip zone can be written as

follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)

The analytical solution of (74) and (75) is problematic becausethe integrals are very complicated and has no primitivefunction Therefore it is necessary to solve integrals 119868

119865and

119868119861using numerical methods

119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

respectively 119868119865= int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)

In (6) the first integral is calculated analytically from (36)and second integral is calculated numerically from (45) Thiscombined method for solving differential equations can becalled as analytical-numerical method for solving differentialequation of the rolling process described in (1)

33 Calculation of Neutral Section The two auxiliary con-stants 119886

0and 1198861were used for reducing of calculated time of

function 1198651(120593)

1198860=

2119898

radic1198982 minus 1

1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30

Figure 9 The distribution of the relative normal contact stress inrolling gap

Equation (65) is considerably simplified after the introduc-tion of constants

1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)

The numerical integration was used for calculation of definiteintegrals described by (74) and (75)

Graphical visualization of the distribution of the relativenormal contact stress in rolling gap depends on relativecoordinate 119909119897

119889and relative deformation is given in Figure 9

From graphical dependence the result is that neutral pointN is significantly dependent on the relative deformationMaterial output from the rolling gap is represented by a pointA and the angle coordinate120593 = 0Material input to the rollinggap is represented by a point B and the angle coordinate120593 = 120572 The influence of friction coefficient on position ofneutral point and distribution of the relative normal contactstress in rolling gap is given in Figure 10 with constant valueof relative deformation (120576 = 20) If the friction coefficient isincreased the peak value of relative normal contact stress israising but weak changes were observed in shifting of neutralpoint depending on relative coordinate 119909119897

119889 The influence

of roll radius on position of neutral point and distributionof the relative normal contact stress in rolling gap is givenin Figure 11 with constant value of relative deformation andfriction coefficient Significant influence of the roll radiusto the peak value of the relative normal contact stress wasobserved but weak effect was observed in shifting of neutralpoint depending on relative coordinate 119909119897

119889

Determination of the neutral point N described by coor-dinate 120572

119899is based on equality of the relative normal contact

stress in backward and forward slip zone as follows

120590119899119865(120572119899) = 120590119899119861(120572119899) (79)

Substituting (74) and (75) into (79) will obtain the formula

119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010


where function 1198651(120593) is given by (65) and function 119865

2(120593) is

given by (66)The coordinate of neutral angle120572119899is integration

limit The analytical determination of the angle of neutralpoint 120572

119899resulting from (80) is practically impossible and

therefore must be used numerical solution The numericalsolution is based on the step method where interval 120593 isin

⟨0 120572⟩ is scanned with the step Δ119909 The general solution isrepresented with the change of variables as follows 120593 rarr 119909

and 120590 rarr 119910 The conception of numerical solution is shownin Figure 12

The curves of the relative normal contact stress in thesurrounding of pointN are substituted by equation of straightlines in the form119910 = 119896119909+119902 If119909 = 0 then coordinates of pointswill be [119910

1198611 0] and [119910

1198651 0] The slope of the straight lines (119891

and 119887) in interval ⟨119909 119909 + Δ119909⟩ can be described as follows

119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)


Table 1 The convergence of calculation of neutral angle 120572119899

119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894

(rad) (rad) (rad) (mdash) (mdash) (mdash)1 005 0069 778 523 mdash 2491 510 075 2515 040 042 0023 529 9662 004 0069 753 994 0000 024 530 2492 286 403 2514 257 232 0021 970 8303 003 0069 524 688 0000 229 306 2499 555 775 2506 948 744 0007 392 9694 002 0069 483 702 0000 040 985 2500 857 391 2505 644 227 0004 786 8375 001 0069 413 549 0000 070 153 2503 086 943 2503 412 591 0000 325 6486 0005 0069 410 604 0000 002 946 2503 180 603 2503 318 922 0000 138 3207 0001 0069 408 614 0000 001 990 2503 243 873 2503 255 651 0000 011 7788 00005 0069 408 458 0000 000 156 2503 248 840 2503 250 683 0000 001 8439 00001 0069 408 429 0000 000 001 2503 249 743 2503 249 781 0000 000 038

0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2

Figure 12 The neutral point

Equations of the straight lines in interval ⟨119909 119909 + Δ119909⟩ can bedescribed as follows

119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)

By comparing the equations of the straight lines 119910119865= 119910119861are

obtained refined local coordinate of the neutral point Δ119909119899

Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)

The improved value of the angle of neutral point N (120572119899) will

be obtained from formula

119909119899= 119909 + Δ119909

119899 (84)

The calculation of the neutral angle according to thedescribed algorithm was performed for the following rollingconditions roll radius 119877 = 105mm thickness of materialbefore rolling ℎ

0= 7mm relative deformation 120576 =

30 and friction coefficient 119891 = 03 Result convergence of

0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010

Figure 13 The dependence of angle of neutral point on relativedeformation and ratio 119877ℎ

1

calculation of neutral angle is given in Table 1 The precisionof calculation is evaluated by difference of values (120590

119899119865119894minus

120590119899119861119894

) Graphical visualization of the angle of neutral pointin dependence on relative deformation and ratio 119877ℎ

1is

given in Figure 13 Work hardening of rolled material is notconsideredThe neutral angle increases with rising of relativedeformation and decreasing of the ratio 119877ℎ

1 Approximate

formula for ratio 120572119899120572 ≐ 05 was presented by the authors of

[16]The validity of formula 120572

119899120572 ≐ 05 is shown in Figure 14

where ratio 120572119899120572 depends on relative deformation and ratio

119877ℎ1 The local maximum is significantly dependent on the

ratio 119877ℎ1

Thematerial thickness in neutral point can be determinedby the neutral angle as follows

ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)

Dividing (85) by output thickness ℎ1the ratio ℎ

119899ℎ1can be

obtained in the formℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)

Graphical visualization ratio ℎ119899ℎ1in dependence on relative

deformation and ratio 119877ℎ1for friction coefficient 119891 = 01 is

shown in Figure 15 The ratio ℎ119899ℎ1increases with rising of

relative deformation and decreasing of the ratio 119877ℎ1


0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010

Figure 14 Dependence of ratio angle of neutral point and grippingangle on relative deformation and ratio 119877ℎ

1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)

Figure 15 Graphical visualization of ratio ℎ119899ℎ1

If the coefficient of friction 119891 = 0 so the status canbe named as the ideal rolling This condition can be usedfor simplified solution of differential equation (4) when it ispossible to continue in solving (80) as follows

int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)

The value of variable of neutral angle 120572119899is one of integration

limit of function 1198652(120593)-equation (54) The calculation of

integral using substitution and their differential will be asfollows

119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593

119898 minus cos120593sdot 119889120593 = int

119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)

Figure 16 Neutral angle dependence on deformation and ratio119877ℎ1

Integration in limits 0 le 120593 le 120572119899for forward slip zone obtains

the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990

= ln (119898 minus cos120572119899) minus ln (119898 minus cos 0)

= ln119898 minus cos120572

119899

119898 minus 1

(89)

and for backward slip zone in limits 120572119899le 120593 le 120572

119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899

= ln (119898 minus cos120572) minus ln (119898 minus cos120572119899)

= ln 119898 minus cos120572119898 minus cos120572

119899

(90)

The neutral angle of ideal rolling process can be calculatedfrom comparison of (89) and (90) as follows

cos120572119899= 119898 minus radic(119898 minus 1) sdot (119898 minus cos120572) (91)

Substituting (51) to (91) and using simplification cos120572 asymp 119877ℎ1

the next formula will be obtained

120572119899= arccos(1 + (1 minus

1

radic1 minus 120576)ℎ1

2119877) (92)

Graphical visualization equation (92) is given in Figure 16The neutral angle is increased with rising of relative defor-mation and decreasing of ratio 119877ℎ

1 From comparison of

dependences shown in Figure 13 (119891 = 01) Figure 16 (119891 = 0)is the result that neutral angle is slightly higher if frictioncoefficient119891 = 0 Equation (92) can be considered as simplestform for direct calculation of neutral angle with satisfyingresults

34 Calculation of Average Normal Contact Stress The dis-tribution of relative normal contact stress to rolls in plasticdeformation zone is shown in Figures 9 10 and 11 by the


N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572

Figure 17 The average relative normal contact stress 120590

curves ANB The rolling force can be calculated from theaverage relative normal contact stress 120590

119899av along grippingangle 120572 (surface of rectangular OEFGO) as is given inFigure 17

It is supposed the egality of surfaces OANBGO equiv

OEFGO If it is valid that 120590 = 120590119899av then the following can

be written

120590 =1

120572 minus 0sdot int

120572

0

120590119899(120593) sdot 119889120593 (93)

The calculation of 120590 must be done independently for back-ward and forward slip zone according to the followingformula

120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)

Substituting (74) and (75) to (94) is obtained in the followingform

120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)

The limits of integration for a forward slip zone are definedby interval 120593 isin ⟨0 120572

119899⟩ and calculation of 120590

119899119865is performed

according to (75) Identically it is valid for backward slip zonewhere120593 isin ⟨120572

119899 120572⟩ and calculation of120590

119899119861is performed accord-

ing to (74)The definite integrals in (95) can be performed bysome numerical mathematical method Graphical visualiza-tion of (1) is given in Figure 18 Function 120590 throughout theobserved interval is monotonically increasing The relativenormal contact stress is increased with rising of relative

0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)

Figure 18 The dependence of function 120590 on relative deformation 120576and ratio 119877ℎ

1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010

Figure 19The dependence of function 120590 on ratio 119897119889ℎav and relative

deformation

deformation and decreasing of ratio 119877ℎ1 Dependence of

relative normal contact stress on ratio 119897119889ℎav and relative

deformation for three groups of friction coefficient are givenin Figure 19 From dependence the result is that the strongesteffect on relative normal contact stress has friction coefficient

For numerical integration of (95) Simpsonrsquos quadraturerule can be used The author of [24] presented Weddleformula which is more effective than Simpsonrsquos quadraturerule for numerical calculation of general function 119891(119909) Theadvantage of this numerical method is faster convergenceof calculation The continuous function 119891(119909) is numericallyintegrated in interval 119909 isin ⟨119886 119887⟩ where 119887 gt 119886 which is dividedon final number of subinterval with step ℎ as follows

ℎ =119887 minus 119886

119899 (96)

where 119899 is the whole positive number

int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the


Table 2 The convergence of calculation of 120590 function according toWeddle formula

119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1

1 6 4116 857 469 mdash2 12 4116 875 359 minus0000 017 8893 18 4116 875 502 minus0000 000 1434 24 4116 875 512 minus0000 000 0105 30 4116 875 514 minus0000 000 0026 36 4116 875 514 0000 000 000

whole positive number (119894 = 1 2 3 119899) The calculation ofrelative normal contact stress in backward slip zone-equation(66) according to Weddle formula is given in Table 2 Theprecision of calculation is evaluated according to differenceof values 120590

119899119861119894minus 120590119899119861119894minus1

The satisfied precision of calculation120590119899119861119894

is obtained after 119899 = 18 where (120590119899119861119894

minus 120590119899119861119894minus1

) asymp 110minus6

Boundary conditions of rolling process were given asfollows relative deformation 120576 = 60 ratio 119877ℎ

1= 10

friction coefficient119891 = 02 and angle coordinate 120593 = 005 radFrom comparison of numerical calculations the result is thatnumerical integration according to Weddle formula is fourtimes faster than numerical integration realized according toSimpsonrsquos quadrature rule

The previous calculations were carried out providedwithout work hardening of material during rolling process

In the next part the rolling process with acceptanceof isotropic work hardening of material will be describedAlexander [25] mentions Swift described work hardeningcurves according to formula

119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)

where 119884119878 is the uniaxial yield stress from static tensile testdepending on true strain 120576 = ln(ℎ

0ℎ1) 1198841198780is the uniaxial

yield stress from static tensile test resulting from annealingstate of material and 119861 119899 is the material constants

The average value of flow stress can be calculated asfollows

120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)

where 1205761is the true strain before actual rolling and 120576

2is the

final true strain after rollingFor calculation of flow stress the author of [25] used the

formula inwhich effective stress and effective strain are given

120590119886=

2

radic3sdot 1198841198780sdot (1 +

2

radic3sdot 119861 sdot 120576)

119899

(100)

Substituting (100) into (99) the following is obtained

120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2

radic3sdot 1198840sdot (1 +

2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912

Figure 20 Uniaxial stress-strain curve for brass CuZn30

and after integration average value of flow stress has a form

120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)

Let us assume that material is processing by rolling fromannealed state for which 120576

1= 0 is valid and transform upper

limit of integral 1205762= 120576 (102) is simplified to form

120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)

The measurements values of yield stress depending on truestrain of laboratory cold rolling experiment are given inFigure 20 For approximation of measurement values ofstress-strain curve (98) was used Final regression equationof stress-strain curve for brass CuZn30 has a form

119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)

Graphical visualization of (104) is given in Figure 20Substituting (104) into (103) the following formula is

obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)

Measured and calculated data from experimental cold rollingschedule are given in Table 3 The calculation of deformationresistance was carried out according to formula

120590119863=119865119903

119878 (106)

where 119865119903is the rolling force and 119878 is the contact surface of

rolled material with rollThe average relative normal contact stress calculated from

measurement data is given as follows

120590meas =120590119863

120590119886av

(107)


Table 3 Measured and calculated data from experimental cold rolling schedule

Number 120576 120576 119877ℎ1

119878 119865119903

120590119863

120590119886av 120590meas 120590 120590meas minus 120590

() (mdash) (mdash) (mm2) (kN) (MPa) (MPa) (mdash) (mdash) (mdash)1 1039 01097 3477 15710 65 41376 27941 1431 1251 01802 1746 01918 3763 23018 110 47789 32540 1423 1371 00523 2269 02573 4054 24776 137 55296 35538 1512 1468 00444 2738 03199 4303 28069 165 58784 38055 1506 1560 minus00545 3006 03575 4468 31700 178 56151 39441 1390 1617 minus0226

120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30


Average value of flow stress is calculated according to (105)Theoretical value of 120590 function is calculated according to(95) with friction coefficient 119891 = 02 Numeric comparisonof measured and theoretically calculated values of 120590 functionis given in Table 3 and graphical comparison is shown inFigure 21

The differences between measured and calculated valuesof the average relative normal contact stress are from intervalΔ isin ⟨3 14⟩[] These deviations are resulting from thecalculation which does not take into account the flatteningrolls

35 Calculation of Front and Back Tensions Back tension isactuating in direction of119909-axis andmarked by label120590

1199090 Front

tension is actuating in direction of 119909-axis andmarked by label1205901199091 According to Figure 1 point B is valid 119909 = 119897

119889and tension

stress 120590119909

= 1205901199090 The tension stresses 120590

1199090and 120590

1199091can be

described by using flow stress as follows

1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585

1are coefficient for back and front tensions

0 le 1205850lt 1 (for without the back tension 120585

0= 0 and for

maximal back tension have to be valid 1205850lt 1) and 0 le 120585

1lt 1

(for without the front tension 1205851= 0 and for maximal front

tension have to be valid 1205851lt 1)

Using the Tresca condition of plasticity formulae of plas-ticity of material for backward and forward slip zones will beobtained as follows

120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)

Editing of (109) is received

120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)

The integration constants 119862119861and 119862

119865which are describing

front and back tensions can be determined from limitconditions given by (111) and (113) The integration constant119862119865for forward slip zone and for 120593 = 0 resulting from (67)

will be as follows

119862119865= 1 minus 120585

1 (114)

The integration constant 119862119861for backward slip zone and for

120593 = 120572 resulting from (66) will be as follows

119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)

The formula for calculation of relative normal contact stressat forward slip zone with consideration of front tension willbe as follows

120590119899119865

= 119890119891sdot1198651(120593)

sdot ((1 minus 1205851) + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)

And for calculation of relative normal contact stress atbackward slip zone with consideration of back tension willbe as follows

120590119899119861

= 119890minus119891sdot1198651(120593)

sdot ((1 minus 1205850) sdot 119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr

Figure 22 The dependence of function 120590 on ratio 119909119897119889and back

tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr

Figure 23 The dependence of function 120590 on ratio 119909119897119889and front

tension

The function 1198651(120593) was described by (65) and function 119865

2(120593)

by (64) The distribution of the relative normal contact stressin rolling gap depends on relative coordinate 119909119897

119889and back

tension is given in Figure 22 From graphical dependence theresult is that neutral point N is significantly dependent onthe back tension If coefficient of front tension is 120585

1= 0

then front tension stress will be 1205901199091

= 0 The back tensionstress described by flow stress will be given by coefficient 120585

0

The value of back tension is characterizing by point B Forexample for point B

4being 12058504 = 04 it means that back

tension will be as follows 1205901199090

= 04120590119886

If the back tension increases then average relative normalcontact stress is decreased The front and back tensions arenot active in neutral pointN

0 The neutral point is shifting to

exit side of rolling gap with increase of back tensionThe distribution of the relative normal contact stress in

rolling gap depends on relative coordinate 119909119897119889and front

tension is given in Figure 23 From graphical dependence theresult is that neutral point N is significantly dependent onthe front tension If coefficient of back tension is 120585

0= 0

then back tension stress will be 1205901199090

= 0 The front tensionstress described by flow stress will be given by coefficient 120585

1

The value of front tension is characterized by point A Forexample for point A



= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6

Figure 24 The dependence of function 120590 on ratio 119909119897119889back and

front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100

Figure 25 The dependence of function 120590 on friction coefficient

The common action of back and front tension on thedistribution of the relative normal contact stress in rolling gapdepends on relative coordinate 119909119897

119889that is given in Figure 24

The indexing of pointN119909119910

is as follows 119909 is the tension stresson enter side of material to rolling gap 119910 is the tension stresson exit side of material from rolling gapThis means thatN

00

is representing intersection point of curves without action offront and back tension 120585

0= 1205851= 0 forN

66it is for maximal

values of front and back tension 1205850= 1205851= 06

36 Calculation of Friction Coefficient The simple deter-mination of friction coefficient can be used with graphicaldependence given in Figure 25 for constant ratio 119877ℎ

1=

100 More precisely determination of friction coefficientdepends on deformation resistance of rolled material Thedeformation resistance of rolled material 120590

119863is characterized

by three-axial stress state and depends on friction coefficientbetween rolls and rolled material In the case of actiononly one-axial stress state is function of friction eliminatedand stress is defined as flow stress 120590

119886 If ratio 119877ℎ

1and

deformation 120576 are constant then the relative normal contactstress in rolling gap depends only on friction coefficientMathematical expression describing relationship between 120590

119863

and 120590119886can be given as follows

120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303

Figure 26 The dependence of function Δ120590 on friction coefficient

From (118) the following can be determined

1205900=120590119863

120590119886

(119)

Substituting (119) into (95) and subtraction of (119) from (95)is obtained

Δ120590 = 120590 (119891) minus 1205900 (120)

Graphical dependence of Δ120590 is given in Figure 26 Intersec-tion point of curve describing Δ120590 function with zero valueof 119909-axis is the determining value of friction coefficient 119891

2

The value of friction coefficient 1198912valid for given conditions

of rolling process For numerical calculation of frictioncoefficient Regula Falsi method was used Maximal value offriction coefficient results from

(i) Tresca condition of plasticity 1198911= 05

(ii) HMH condition of plasticity 1198911= 0577

The first approximation for calculation of friction coefficient1198912using Regula Falsi method will be as follows

1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)

The calculation of friction coefficient 1198912using Regula Falsi

method for rolling conditions given in Table 3 line number4 (1205900= 1506 ratio 119877ℎ

1= 4303 120576 = 2738) is shown

in Table 4 Calculation precision on three decimal positions(1198912= 0182) was obtained after 119899 = 7 steps

37 Calculation of Coefficient of the Arm of Rolling Force Therolling torque of rolling process depends on the coefficientof the arm of rolling force 120595 and the rolling force 119865

119903 The

coefficient of the arm of rolling force is calculated accordingto the following formula

120595 =119886

119897119889

(122)

where 119886 is the arm of rolling force and 119897119889is the length of

contact arc

Table 4The calculation of friction coefficient 1198912using Regula Falsi

method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)

(mdash) (mdash) (mdash) (mdash) (mdash) (mdash)0 0 05 minus044 1968 1433 027 0117 858 minus0174 4441 0117 858 05 minus0174 444 1433 027 0159 329 minus0064 8072 0159 329 05 minus0064 807 1433 027 0174 069 minus0023 1843 0174 069 05 minus0023 184 1433 027 0179 258 minus0008 1764 0179 258 05 minus0008 176 1433 027 0181 077 minus0002 8695 0181 077 05 minus0002 869 1433 027 0181 715 minus0001 0056 0181 715 05 minus0001 005 1433 027 0181 938 minus0000 3527 0181 938 05 minus0000 352 1433 027 0182 016 minus0000 1238 0182 016 05 minus0000 123 1433 027 0182 043 minus0000 043

The development of the relative normal contact stress incoordinates 120590

119899minus 120593 is shown in Figure 17 from which the

result is that surface is delimited by pointsOANBGO on theinterval 120593 isin ⟨0 120572⟩

The resultant of the force is determined from the anglegravity centre120593 = 120572

119886inwhich the force actsThe angle gravity

centre120572119886of the stress surfaceOANBGO is defined as the ratio

of static moment119872119878and the stress surface 119878

120590

120572119886=119872119878

119878120590

(123)

The stress surface 119878120590can be determined as follows

119878120590= 120572 sdot 120590 (124)

where 120572 is the gripping angle and 120590119899av is the average relative

normal contact stressThe static moment119872

119878then can be determined as follows

119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)

Result from Figure 27 can be described forward slip zoneby curve with points AN and angle coordinate 120593 isin ⟨0 120572

119899⟩

backward slip zone by curve with points NB and anglecoordinate 120593 isin ⟨120572

119899 120572⟩ The solution of integral equation

(125) is performed by numerical integration separately forbackward and forward slip zone as follows

119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)

where 120590119899119865(120593) is described by (75) and 120590

119899119861(120593) is described by

(74)The calculation of integrals is carried out by Weddle

formulae [24] Length of arm of force action is calculatedusing the following formula

119886 = 119877 sdot sin120572119886 (127)

The numerical calculation and visualization of the coefficientof arm of rolling force 120595 depends on the relative deformation


120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO

Figure 27 The determination of angle gravity centre 120572119886as a point

of rolling force 119865119903

0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)

Figure 28The dependence of the coefficient of arm of rolling force120595 on relative deformation

and ratio 119877ℎ1is shown in Figure 28 From graphical depen-

dence the result is that the coefficient of the arm of rollingforce is decreased with increase of relative deformation andration 119877ℎ

1

The coefficient of arm of rolling force can be determinedalso from the following formula [3]

120595 =120572119886

120572 (128)

The authors of [3] introduce calculation of 120595 the followingapproximated formula 120595 = 120572

119886120572 ≐ 05 From the graphical

dependence in Figure 28 the result is that value can beobtained only if deformation 120576 = 0

38 Calculation of Rolling Force and Rolling Torque Thecalculation of rolling force can be performed by the followingformula

119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1

Figure 29 Rolling gap parameters

and rolling torque of two rolls can be performed by thefollowing formula

119872119903= 2119865119903sdot 119886 (130)

The arm of rolling force resulting from (122) can be expressedas follows

119886 = 120595 sdot 119897119889 (131)

Substituting (129) and (131) into (130) formula for calculationof rolling torque is obtained

119872119903= 2119877 sdot 120590

119886sdot 120590 sdot 120595 sdot Δℎ sdot 119887av (132)

The value of coefficient of arm of rolling force 120595 is given inFigure 28

4 Discussion

The solution of difference equation (1) realized by the authorsof [15ndash18] was based on determining only one constant119898 which consists of product of friction coefficient andgeometrical characteristics of rolling gap

The constant according to Korolev [15] 119898Kor describedby (13) is depending on product of friction coefficient andratio 119897

119889ℎav The constant according to Tselikov et al [16]

119898Tsel described by (15) is depending on product of frictioncoefficient and ratio 2119897

119889(ℎ0minus ℎ1) The constant according to

Bland-Ford [17] 119898Bla-For described by (22) is depending onproduct of friction coefficient and ratioradic119877ℎ

1 The constant

according to Smiryagin-Pernisov [19] 119898Smir-Per described by(30) is depending on product of friction coefficient and ratio2119897119889radicℎ1(ℎ0minus ℎ1)

In the new solution of difference equation (1) presented bythis paper from calculation of 119898 constant is separating fric-tion coefficient and is described by (51) which is dependingonly on geometry of rolling gap given by ratio 119877ℎ

1

Each of constants is described by (13) (15) and (22)Equation (30) contains deformation in some form butthe deformation does not directly include (74) and (75)The interrelationship characteristics of rolling gap can beschematically shown in Figure 29


Table 5 The calculations 120590 function according to (95) and 120590Kor function by a Korolev

ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590

(mm) (mm) (mdash) (mm) (mdash) (mdash) (mdash) (mdash) (mdash) (mdash) (mdash)20 15259 02371 105 68813 4000 01 0400 12296 12742 0965050 41935 01613 105 25038 2000 02 0400 12296 12586 0977070 62550 01064 105 16786 13334 03 0400 12296 12482 09851

Table 6 The calculations 120590 function according to (95) and 120590Tsel function by Tselikov

ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Tsel 120590Tsel 120590 120590Tsel120590


Table 7 The calculations 120590 function according to (95) and 120590Bla-For function by Bland-Ford

ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Bla-For 120590Bla-For 120590 120590Bla-For120590


Table 8 The calculations 120590 function according to (95) and 120590Sim function by Sims

ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590


From Figure 29 and the Appendix the result is thepossibility of mutual replacement of one parameter by theother but friction coefficient does not depend on any ofthese parameters The new solution of differential equation(1) based on calculation of the relative normal contactstress 120590 depending on two functions can be compared withcalculation of 120590 functions according to the authors of [15ndash18] The examples of calculations of 120590 functions accordingto the authors of [15ndash18] with comparison of calculation 120590

function according to (95) are given in Tables 5 6 7 and 8Graphical comparison of calculations of differential equation(1) resulting from various approaches defined by the authorsof [15ndash17] and calculation given by (95) is shown in Figure 30

The length of rolling gap is defined by relative coordinate119909119897119889 Entry side to rolling gap is marked by point B and exit

side from rolling gap by point A The maximal values of therelative normal contact stress 120590 were obtained by using (95)

5 Conclusion

The solutions of von Karman two-dimensional differentialequation of lengthwise rolling process were based on cer-tain simplifications which are describing the geometry ofsurface of circular cylinder Circular arch of roll gap was so

00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415

Pernis-KvackajKorolev

TselikovBland-Ford

02


far approximated by simplified curves (polygonal curve orparabola) because his accurate description using the equationcircle did not create an analytical solution of the differential


equation (1) Use of any approximated formulas leads toinaccuracy of calculations relative to normal contact stressalong the length of the contact arc Attempts have been madeto describe the contact arc of a circle by the equation ofthe circle but the exact solution of (1) has been replaced byapproximate solution using numerical simulation in softwareDeform

The proposed solution presents a new approach to thedescription of the circular arch of rolling gap by equation of acircle For this approach which is describing the circular archof rolling gap by equation of a circle the classical analyticalsolution of deferential equation cannot be used Thereforenew solution of deferential equation based on interactiontwo methods was suggested and realizedThe authors namedthis solution with term analytic numerical solution Also theappropriate solutionwas the transformation of the coordinatesystem from rectangular to polar coordinates The distribu-tion of the relative normal contact stress to rolls is dependenton two independent functions 120590

119899asymp 1198651sdot1198652The first function

has exponential form 1198651

= 119890119865(119891120593) and is describing the

influence of the friction between roll and rollingmaterialThesecond function has trigonometric form 119865

2= 1198652(119898 120593) and

is describing geometry of rolling gap and is independent onfriction coefficient

The results from the numerical calculations are carriedout geometric visualizations for neutral section normal con-tact stress distribution front and back tensions coefficient ofthe arm of rolling force rolling force and rolling torque independ on rolling conditions

Result from laboratory cold rolled experiment of CuZn30material was derived equation describing work hardening ofobserved material This equation was used for calculationof the relative normal contact stress 120590 with consideration ofwork hardening during material processing Comparison oftheoretical values of 120590 function calculated according to (95)with values of 120590 function calculated based on experimentalresults from cold rolled was performed The solution of (1)was not concluded with roll flattening

Appendix

Absolute deformation is as follows

Δℎ = ℎ0minus ℎ1 (A1)

Relative deformation is as follows

120576 =ℎ0minus ℎ1

ℎ0

(A2)

True strain is as follows

120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)

Ratio ℎ1ℎ0rarr ℎ

1ℎ0= 1 minus 120576

Average thickness is as follows

ℎav =ℎ0+ ℎ1

2 (A5)

Gripping angle is as follows

cos120572 = 1 minusΔℎ

2119877 (A6)

respectively

cos120572 = 1 minus120576

1 minus 120576sdotℎ1

2119877 (A7)

Length of contact arc is as follows

119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))

Ratio 119877ℎ1rarr 119877ℎ

1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2

respectively 119877ℎ1= 1205762(1 minus 120576) sdot 1(1 minus cos120572)


120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature

119886 Arm force [m]119887av Average width [m]119862119861 Integration constant (Backward slip zone)

[minus]

119862119865 Integration constant (Forward slip zone)

[minus]

119889119909 119889119910 Coordinate differentials 119909 and 119910 [minus]

119891 Friction coefficient [minus]119865119903 Rolling force [N]

ℎav Average thickness [m]ℎ119899 Thickness in neutral section [m]

ℎ0 ℎ1 Thickness before and after deformation

[m]119897119889 Length of contact arc [m]

119898 Constant of differential equation [minus]119872119903 Rolling torque [Nm]

119877 Radius of rollers [m]119909 119910 Rectangular coordinates [m]119884119878 Uniaxial yield stress [MPa]120572 Gripping angle [rad]


120572119886 Angle of arm coefficient [rad]

120572119899 Neutral angle [rad]

Δℎ Absolute reduction [m]120576 Relative reduction [minus]120576 True strain [minus]120593 Angle coordinate [rad]120590 Sigma function (average relative normal

contact stress) [minus]120590119886 Flow stress [MPa]

120590119886av Average flow stress [MPa]120590119863 Deformation resistant [MPa]

120590119899 Normal contact stress [MPa]

120590119899 Relative normal contact stress [minus]

120590119899119861 Relative normal contact stress (Backward

slip zone) [minus]120590119899119865 Relative normal contact stress (Forward

slip zone) [minus]1205901199090 Relative back tension [minus]

1205901199091 Relative front tension [minus]

120590119909 120590119910 Principal normal stress (120590

3 1205901) [MPa]

120591 Shear stress [MPa]1205850 Coefficient back tension [minus]

1205851 Coefficient front tension [minus]

120595 Arm force coefficient [minus]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was financially supported by the VEGA 1032514project and the project ldquoTechnological Preparation of Elec-trotechnical Steels with High Permeability for Electro-drives with Higher Efficiencyrdquo which is supported by theOperational Program ldquoResearch and Developmentrdquo ITMS26220220037 financed through European Regional Develop-ment Fund

References

[1] T Karman ldquoBeitrag zurTheorie desWalzvorgangesrdquo Zeitschriftfur Angewandte Mathematik und Mechanik vol 5 no 2 pp139ndash141 1925 (German)

[2] B AvitzurMetal Forming Processes andAnalysisMcGraw-HillNew York NY USA 1968

[3] A Hensel and T Spittel The Force and Energy Consumption inMetal Forming Processes Deutscher Verlag fur Grundstoffind-ustrie Leipzig Germany 1978 (German)

[4] EMMielnikMetalworking Science and Engineering McGraw-Hill New York NY USA 1991

[5] M Hajduk and J Konvicny Power Requirements for Hot RolledSteel SNTL Prague Czech Republic 1983 (Czech)

[6] M Kollerova M Zıdek B Pocta and V Dedek The RollingALFA Bratislava Slovakia 1991 (Slovak)

[7] T Kubina J Boruta and I Schindler ldquoNew possibilities ofevaluation of the degree of softening from torsion plastometrictestrdquo Acta Metallurgica Slovaca vol 13 no 4 pp 610ndash617 2007

[8] R Pernis J Kasala and J Boruta ldquoHigh temperature plasticdeformation of CuZn30 brassmdashcalculation of the activationenergyrdquoKovove MaterialymdashMetallic Materials vol 48 no 1 pp41ndash46 2010

[9] T Kvackaj I Pokorny and M Vlado ldquoSimulacie procesovtvarnenia pomocou krutovej plastometrierdquo Acta MetallurgicaSlovaca vol 6 no 3 pp 242ndash248 2000

[10] E Orowan ldquoThe calculation of roll pressure in hot and cold flatrollingrdquo Proceedings of the Institution of Mechanical Engineersvol 150 no 1 pp 140ndash167 1943

[11] HITACHI Analog Analysis of Rolling Theory by Analog Com-puter Karmanrsquos Differential Equation Hybrid Computer Tech-nical Information Series No 8 Hitaschi 1968

[12] O Harrer and I Pokorny ldquoThe course of stresses and velocitiesin billet rollingrdquo Acta Metallurgica Slovaca vol 9 no 1 pp 34ndash40 2003

[13] S I Gubkin Plastic Deformation of Metals MetallurgizdatMoscow Russia 1960 (Russian)

[14] R Kocisko J Bacso J Bidulska A Kovacova P Bella and MMolnarova ldquoInfluence of deformation degree on local deforma-tion heterogeneity during thin sheet rollingrdquo Chemicke Listyvol 105 no 17 pp s822ndashs823 2011

[15] A A Korolev The Machines and Equipments of Rolling of Fer-rous and Nonferrous Metallurgy Metallurgiya Moscow Russia1976 (Russian)

[16] A I Tselikov G S Nikitin and S E Rokotyan The Theory ofLengthwise Rolling MIR Publishers Moscow Russia 1981

[17] D R Bland and H Ford ldquoThe calculation of roll force andtorque in cold strip rolling with tensionrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash1631948

[18] R B Sims ldquoCalculation of roll force and torque in hot rollingmillsrdquoProceedings of the Institution ofMechanical Engineers vol168 no 6 pp 191ndash200 1954

[19] A P Smirjagin N Z Dnestrovskij A D Landichov et alProcessing Manual for Non-Ferrous Metals and Alloys Metallur-gizdat 1961 (Russian)

[20] R Pernis ldquoThe new hypothesis of normal stress distribution forcontact arc in rollingrdquo Acta Metallurgica Slovaca vol 17 no 2pp 68ndash76 2011

[21] L Matejıcka ldquoOn an open problem posed in the paperlsquoinequalities of power-exponential functionsrsquordquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 3article 75 5 pages 2008

[22] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press London UK 7th edition 2007

[23] L Matejıcka ldquoOn approximate solutions of degenerate inte-grodifferential parabolic problemsrdquo Mathematica Slovaca vol45 no 1 pp 91ndash103 1995

[24] AAngotComplement deMathematiques a lrsquousage des Ingenieursde lrsquoElectrotechnique et des Telecommunications MASSONParis France 1957

[25] J M Alexander ldquoOn the theory of rollingrdquo Proceedings of theRoyal Society of London A vol 326 pp 535ndash563 1972

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Y

120572ld R

B dx x

A0

X

2(y+dy

)

2y 1

h1h0

R

120593d120593

Figure 1 Determination of geometric relationships

where plus sign (+) is for forward slip zone minus sign (minus)is for backward slip zone 120590

119899is normal contact stress on rolls

120591 is the shear stress between rolls and rolling material and119909 119910 are the coordinates of the cylinder touching the rolledmaterial

The Tresca condition of plasticity is used for solution of(1)

minus120590119909minus (minus120590

119899) = 120590119899minus 120590119909= 120590119886 (2)

where 120590119910

≐ 120590119899is the maximal principal stress (vertical

direction) 120590119909is the minimal principal stress (horizontal

direction) 120590119886is the flow stress [7ndash9] and 120591 is the shear stress

that is proportional to normal contact stress 120590119899according to

formula

120591 = 119891 sdot 120590119899 (3)

where 119891 is the friction coefficient between rolls and rolledmaterial

Substituting (2) into (3) the following formula will beobtained with only one stress variable 119889120590

119899

119889120590119899

119889119909minus120590119886

119910sdot119889119910

119889119909plusmn119891

119910sdot 120590119899= 0 (4)

Flow stress and friction coefficient are as a constant in (4)If differential equation (4) is divided by flow stress 120590

119886it is

possible to obtain relative normal contact stress 120590119899

120590119899=120590119899

120590119886

(5)

where 120590 is a function

Substituting (5) into (4) the following form is obtained

119889120590119899

119889119909minus1

119910sdot119889119910

119889119909plusmn119891

119910sdot 120590119899= 0 (6)

The application equation (5) can be excluded from (4) mate-rial constant (flow stress 120590

119886) Thus (6) become independent

from the rolled material This differential equation is a lineardifferential equation of first order with the right side andhas a general transcription 120590

119899= 120590119899(119909 119910) The coordinates

119909 and 119910 are defining the geometry of the cylinder which isconnected to the rolling materialThe variable 119910 is a functionof the 119909 coordinate If equation of a circle is substitutinginto (6) so differential equation is obtained which has noanalytical solution till nowThe authorOrowan [10] presentedthe solution of differential equation (1) based on analytical-graphical principle He defined procedure for calculationrolling force rolling torque and electric input for hot andcold rolling processes In the literature [11] showed solutionof differential equation (1) by simulation on analog computerThe authors of [12] solved this case with the use of the circulararc but the result is purely numerical solution The author of[13] presented simplified analytical solution with acceptanceof material work hardening during cold rolling process Thedevelopment in computational software allowed to solve thistask by using finite element method as was given in theliterature [14]

2 Experimental Material and Methods

The investigated material was annealed brass CuZn30 withchemical composition corresponding to standardDIN 17660WNo 20265 The sample sizes before rolling were ℎ times 119887 times 119897

= 34 times 30 times 150mm The samples were laboratory rolledusing duo mill (the diameter of rolls 119863 = 210mm) withconstant circumferential speed of rolls Vroll = 066ms atambient temperatures The temperature of a sample surfaceimmediately after rolling was measured using a thermovisioncamera and was from interval 119879 isin ⟨20 40⟩ [∘C] The coldrolling reductions of thickness were made in an interval120576cold isin ⟨0 30⟩[] as one pass deformation Deformationforces during samples processing by cold rolling were mea-sured by tensometric elements with signal registration inSpider apparatus The static tensile test was performed inaccordance with STN EN 10 002-1 standard

The differential equation of lengthwise rolling processgiven by formula (1) has no analytical solution when contactarc is described with the equation of the circle For solutionof this problem a new combined analytical and numericalapproach is presented

3 Analysis and Results

31 Analysis of the Present Solution of Differential EquationDescribing Stress Condition in Rolling Gap The presentsolutions of differential equation of rolling were based onthe same simplifications because the form of equation is


S1

R120572

120572n

B

N A

h0 h1

xn

ld






119886


119899






119899

119909119899

119897119889

=ℎ1

ℎ0+ ℎ1

(7)




follows

120590119899119861

= 1198901198980(1minus119909119897119889) (8)

where

1198980= 2119891 sdot

119897119889

ℎ0

(9)



120590119899119865

= 1198901198981 sdot(119909119897119889) (10)

where

1198981= 2119891 sdot

119897119889

ℎ1

(11)


120590Kor =1

119898Kor(119890119898Kor minus 1) (12)


119898Kor = 119891 sdot119897119889

ℎav (13)


119899and


ℎ119899

ℎ1

= [1


ℎ0

ℎ1

)

119898

)]

1119898

(14)


119898 equiv 119898Tsel =2119891 sdot 119897119889

ℎ0minus ℎ1

(15)





120590119899119861

=1

119898[(119898 minus 1) sdot (

ℎ0

2119910)

119898

+ 1]

120590119899119865

=1

119898[(119898 + 1) sdot (

2119910

ℎ1

)

119898

minus 1]

(16)


S1

R120572

120572n

B

N A

h0 h1

hn

xn

ld


S1

R120572

120572n

B

N A

h0 h1

xn

ld



120590Tsel =2 (1 minus 120576)

120576 (119898 minus 1)sdotℎ119899

ℎ1

sdot [(ℎ119899

ℎ1

)

119898

minus 1] (17)




following formula

120572119899

= radicℎ1

119877

sdot tg[

[

1


ℎ1

sdot 120572) minus1

4119891sdot radic

ℎ1

119877sdot ln

ℎ0

ℎ1

]

]

(18)





120590119899119861

=ℎ1+ 1198771205932

ℎ0

sdot 119890119891(119867∙

0minus119867∙)

120590119899119865

=ℎ1+ 1198771205932

ℎ1

sdot 119890119891sdot119867∙

(19)

where

119867∙= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120593)

119867∙

0= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120572)

(20)



120576119898[(1 minus 120576) sdot 119890


sdot int

119905120572

119905119899


sdot 119889119905

+ int

119905119899

0

(1 + 11989821199052) sdot 1198902119898sdotarctg(119898119905)

sdot 119889119905]

(21)


120593119891


ℎ1

(22)

119905119899=120572119899

119891 (23)

119905120572=

1

119898sdot radic

120576

1 minus 120576 (24)






ℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (25)

where

120572119899= radic

ℎ1

119877sdot tg( 120587

16sdot radic

ℎ1

119877sdot ln (1 minus 120576)

+1

2arctg(radic 119877

ℎ1

sdot 120572))

(26)





120590119899119861

= 1 +120587

4ln((1 minus 120576) (1 +

119877

ℎ1

1205932))

+ 2radic119877

ℎ1

(arctg(radic 119877

ℎ1


ℎ1

sdot 120593))

120590119899119865

= 1 +120587

4ln(1 + 119877

ℎ1

1205932) + 2radic

119877

ℎ1


ℎ1

sdot 120593)

(27)


120590Sim =120587




120587

4minus radic

1 minus 120576

120576

sdot radic119877

ℎ1

sdot lnℎ119899

ℎ1

+1


120576sdot radic

119877

ℎ1

sdot ln 1

1 minus 120576

(28)


119899119865in



120590119899119861

= [1 minus (1199060

119898minus

1

1198982)] 119890119898(1199060minus119906) + 2 (

119906

119898minus

1

1198982)

120590119899119865

= (1 +1

1198982) 119890119898119906

minus 2 (119906

119898+

1

1198982)

(29)




(30)

1199060= arctg(radic 120576

1 minus 120576) (31)




(051198982 minus 119898 sdot 1199060+ 1) sdot 119890


+ 2119898 sdot 119906119899= 0

(32)


sin120572119899= radic

1 minus 120576

120576sdot tg119906119899sdot sin120572 (33)


120590Smir-Per =2

119898 sdot 1199060

sdot [(1 +2

1198982) sdot (119890119898sdot119906119899 minus 1) +

1199062

0

2

minus 119906119899sdot (119906119899+

2

119898)]

(34)


119889120590119899

119889119909plusmn119891

119910120590119899=1

119910sdot119889119910

119889119909 (35)


119889120590119899

119889119909plusmn119891

119910120590119899= 0 (36)


119889120590119899119861

119889119909+119891

119910120590119899119861

= 0 (37)

119889120590119899119865

119889119909minus119891

119910120590119899119865

= 0 (38)


S1

R120572

120572n

B

N A

h0 h1

xn

ld



int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)


ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861


(40)


1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)


120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)


119889119909=119889119888 (119909)


1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)



1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)


119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)



120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)



1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)


119910 = 119877 +ℎ1


119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)


119889119910

119910=

119909radic1198772 minus 1199092


119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909


(50)


119898 = 1 +ℎ1

2119877 (51)


119909

119877= sin120593


(52)


119889119910

119910=

sin120593


119877 cos120593 sdot 119889120593


=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)


1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S1

R120572

120572n

B

N A

h0 h1

xn

ld






119886


119899






119899

119909119899

119897119889

=ℎ1

ℎ0+ ℎ1

(7)




follows

120590119899119861

= 1198901198980(1minus119909119897119889) (8)

where

1198980= 2119891 sdot

119897119889

ℎ0

(9)



120590119899119865

= 1198901198981 sdot(119909119897119889) (10)

where

1198981= 2119891 sdot

119897119889

ℎ1

(11)


120590Kor =1

119898Kor(119890119898Kor minus 1) (12)


119898Kor = 119891 sdot119897119889

ℎav (13)


119899and


ℎ119899

ℎ1

= [1


ℎ0

ℎ1

)

119898

)]

1119898

(14)


119898 equiv 119898Tsel =2119891 sdot 119897119889

ℎ0minus ℎ1

(15)





120590119899119861

=1

119898[(119898 minus 1) sdot (

ℎ0

2119910)

119898

+ 1]

120590119899119865

=1

119898[(119898 + 1) sdot (

2119910

ℎ1

)

119898

minus 1]

(16)


S1

R120572

120572n

B

N A

h0 h1

hn

xn

ld


S1

R120572

120572n

B

N A

h0 h1

xn

ld



120590Tsel =2 (1 minus 120576)

120576 (119898 minus 1)sdotℎ119899

ℎ1

sdot [(ℎ119899

ℎ1

)

119898

minus 1] (17)




following formula

120572119899

= radicℎ1

119877

sdot tg[

[

1


ℎ1

sdot 120572) minus1

4119891sdot radic

ℎ1

119877sdot ln

ℎ0

ℎ1

]

]

(18)





120590119899119861

=ℎ1+ 1198771205932

ℎ0

sdot 119890119891(119867∙

0minus119867∙)

120590119899119865

=ℎ1+ 1198771205932

ℎ1

sdot 119890119891sdot119867∙

(19)

where

119867∙= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120593)

119867∙

0= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120572)

(20)



120576119898[(1 minus 120576) sdot 119890


sdot int

119905120572

119905119899


sdot 119889119905

+ int

119905119899

0

(1 + 11989821199052) sdot 1198902119898sdotarctg(119898119905)

sdot 119889119905]

(21)


120593119891


ℎ1

(22)

119905119899=120572119899

119891 (23)

119905120572=

1

119898sdot radic

120576

1 minus 120576 (24)






ℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (25)

where

120572119899= radic

ℎ1

119877sdot tg( 120587

16sdot radic

ℎ1

119877sdot ln (1 minus 120576)

+1

2arctg(radic 119877

ℎ1

sdot 120572))

(26)





120590119899119861

= 1 +120587

4ln((1 minus 120576) (1 +

119877

ℎ1

1205932))

+ 2radic119877

ℎ1

(arctg(radic 119877

ℎ1


ℎ1

sdot 120593))

120590119899119865

= 1 +120587

4ln(1 + 119877

ℎ1

1205932) + 2radic

119877

ℎ1


ℎ1

sdot 120593)

(27)


120590Sim =120587




120587

4minus radic

1 minus 120576

120576

sdot radic119877

ℎ1

sdot lnℎ119899

ℎ1

+1


120576sdot radic

119877

ℎ1

sdot ln 1

1 minus 120576

(28)


119899119865in



120590119899119861

= [1 minus (1199060

119898minus

1

1198982)] 119890119898(1199060minus119906) + 2 (

119906

119898minus

1

1198982)

120590119899119865

= (1 +1

1198982) 119890119898119906

minus 2 (119906

119898+

1

1198982)

(29)




(30)

1199060= arctg(radic 120576

1 minus 120576) (31)




(051198982 minus 119898 sdot 1199060+ 1) sdot 119890


+ 2119898 sdot 119906119899= 0

(32)


sin120572119899= radic

1 minus 120576

120576sdot tg119906119899sdot sin120572 (33)


120590Smir-Per =2

119898 sdot 1199060

sdot [(1 +2

1198982) sdot (119890119898sdot119906119899 minus 1) +

1199062

0

2

minus 119906119899sdot (119906119899+

2

119898)]

(34)


119889120590119899

119889119909plusmn119891

119910120590119899=1

119910sdot119889119910

119889119909 (35)


119889120590119899

119889119909plusmn119891

119910120590119899= 0 (36)


119889120590119899119861

119889119909+119891

119910120590119899119861

= 0 (37)

119889120590119899119865

119889119909minus119891

119910120590119899119865

= 0 (38)


S1

R120572

120572n

B

N A

h0 h1

xn

ld



int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)


ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861


(40)


1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)


120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)


119889119909=119889119888 (119909)


1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)



1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)


119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)



120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)



1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)


119910 = 119877 +ℎ1


119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)


119889119910

119910=

119909radic1198772 minus 1199092


119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909


(50)


119898 = 1 +ℎ1

2119877 (51)


119909

119877= sin120593


(52)


119889119910

119910=

sin120593


119877 cos120593 sdot 119889120593


=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)


1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S1

R120572

120572n

B

N A

h0 h1

hn

xn

ld


S1

R120572

120572n

B

N A

h0 h1

xn

ld



120590Tsel =2 (1 minus 120576)

120576 (119898 minus 1)sdotℎ119899

ℎ1

sdot [(ℎ119899

ℎ1

)

119898

minus 1] (17)




following formula

120572119899

= radicℎ1

119877

sdot tg[

[

1


ℎ1

sdot 120572) minus1

4119891sdot radic

ℎ1

119877sdot ln

ℎ0

ℎ1

]

]

(18)





120590119899119861

=ℎ1+ 1198771205932

ℎ0

sdot 119890119891(119867∙

0minus119867∙)

120590119899119865

=ℎ1+ 1198771205932

ℎ1

sdot 119890119891sdot119867∙

(19)

where

119867∙= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120593)

119867∙

0= 2radic

119877

ℎ1

arctg(radic 119877

ℎ1

sdot 120572)

(20)



120576119898[(1 minus 120576) sdot 119890


sdot int

119905120572

119905119899


sdot 119889119905

+ int

119905119899

0

(1 + 11989821199052) sdot 1198902119898sdotarctg(119898119905)

sdot 119889119905]

(21)


120593119891


ℎ1

(22)

119905119899=120572119899

119891 (23)

119905120572=

1

119898sdot radic

120576

1 minus 120576 (24)






ℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (25)

where

120572119899= radic

ℎ1

119877sdot tg( 120587

16sdot radic

ℎ1

119877sdot ln (1 minus 120576)

+1

2arctg(radic 119877

ℎ1

sdot 120572))

(26)





120590119899119861

= 1 +120587

4ln((1 minus 120576) (1 +

119877

ℎ1

1205932))

+ 2radic119877

ℎ1

(arctg(radic 119877

ℎ1


ℎ1

sdot 120593))

120590119899119865

= 1 +120587

4ln(1 + 119877

ℎ1

1205932) + 2radic

119877

ℎ1


ℎ1

sdot 120593)

(27)


120590Sim =120587




120587

4minus radic

1 minus 120576

120576

sdot radic119877

ℎ1

sdot lnℎ119899

ℎ1

+1


120576sdot radic

119877

ℎ1

sdot ln 1

1 minus 120576

(28)


119899119865in



120590119899119861

= [1 minus (1199060

119898minus

1

1198982)] 119890119898(1199060minus119906) + 2 (

119906

119898minus

1

1198982)

120590119899119865

= (1 +1

1198982) 119890119898119906

minus 2 (119906

119898+

1

1198982)

(29)




(30)

1199060= arctg(radic 120576

1 minus 120576) (31)




(051198982 minus 119898 sdot 1199060+ 1) sdot 119890


+ 2119898 sdot 119906119899= 0

(32)


sin120572119899= radic

1 minus 120576

120576sdot tg119906119899sdot sin120572 (33)


120590Smir-Per =2

119898 sdot 1199060

sdot [(1 +2

1198982) sdot (119890119898sdot119906119899 minus 1) +

1199062

0

2

minus 119906119899sdot (119906119899+

2

119898)]

(34)


119889120590119899

119889119909plusmn119891

119910120590119899=1

119910sdot119889119910

119889119909 (35)


119889120590119899

119889119909plusmn119891

119910120590119899= 0 (36)


119889120590119899119861

119889119909+119891

119910120590119899119861

= 0 (37)

119889120590119899119865

119889119909minus119891

119910120590119899119865

= 0 (38)


S1

R120572

120572n

B

N A

h0 h1

xn

ld



int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)


ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861


(40)


1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)


120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)


119889119909=119889119888 (119909)


1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)



1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)


119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)



120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)



1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)


119910 = 119877 +ℎ1


119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)


119889119910

119910=

119909radic1198772 minus 1199092


119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909


(50)


119898 = 1 +ℎ1

2119877 (51)


119909

119877= sin120593


(52)


119889119910

119910=

sin120593


119877 cos120593 sdot 119889120593


=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)


1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













ℎ119899

ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (25)

where

120572119899= radic

ℎ1

119877sdot tg( 120587

16sdot radic

ℎ1

119877sdot ln (1 minus 120576)

+1

2arctg(radic 119877

ℎ1

sdot 120572))

(26)





120590119899119861

= 1 +120587

4ln((1 minus 120576) (1 +

119877

ℎ1

1205932))

+ 2radic119877

ℎ1

(arctg(radic 119877

ℎ1


ℎ1

sdot 120593))

120590119899119865

= 1 +120587

4ln(1 + 119877

ℎ1

1205932) + 2radic

119877

ℎ1


ℎ1

sdot 120593)

(27)


120590Sim =120587




120587

4minus radic

1 minus 120576

120576

sdot radic119877

ℎ1

sdot lnℎ119899

ℎ1

+1


120576sdot radic

119877

ℎ1

sdot ln 1

1 minus 120576

(28)


119899119865in



120590119899119861

= [1 minus (1199060

119898minus

1

1198982)] 119890119898(1199060minus119906) + 2 (

119906

119898minus

1

1198982)

120590119899119865

= (1 +1

1198982) 119890119898119906

minus 2 (119906

119898+

1

1198982)

(29)




(30)

1199060= arctg(radic 120576

1 minus 120576) (31)




(051198982 minus 119898 sdot 1199060+ 1) sdot 119890


+ 2119898 sdot 119906119899= 0

(32)


sin120572119899= radic

1 minus 120576

120576sdot tg119906119899sdot sin120572 (33)


120590Smir-Per =2

119898 sdot 1199060

sdot [(1 +2

1198982) sdot (119890119898sdot119906119899 minus 1) +

1199062

0

2

minus 119906119899sdot (119906119899+

2

119898)]

(34)


119889120590119899

119889119909plusmn119891

119910120590119899=1

119910sdot119889119910

119889119909 (35)


119889120590119899

119889119909plusmn119891

119910120590119899= 0 (36)


119889120590119899119861

119889119909+119891

119910120590119899119861

= 0 (37)

119889120590119899119865

119889119909minus119891

119910120590119899119865

= 0 (38)


S1

R120572

120572n

B

N A

h0 h1

xn

ld



int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)


ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861


(40)


1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)


120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)


119889119909=119889119888 (119909)


1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)



1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)


119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)



120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)



1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)


119910 = 119877 +ℎ1


119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)


119889119910

119910=

119909radic1198772 minus 1199092


119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909


(50)


119898 = 1 +ℎ1

2119877 (51)


119909

119877= sin120593


(52)


119889119910

119910=

sin120593


119877 cos120593 sdot 119889120593


=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)


1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S1

R120572

120572n

B

N A

h0 h1

xn

ld



int119889120590119899119861

120590119899119861

= minus119891 sdot int119889119909

119910 (39)


ln (120590119899119861) = minus119891 sdot int

119889119909

119910 120590119899119861


(40)


1198651(119909) = int

119889119909

119910

respectively 1198891198651(119909)

119889119909=1

119910

(41)


120590119899= 119888 (119909) sdot 119890

minus119891sdot1198651(119909) (119888 = 119888 (119909)) (42)


119889119909=119889119888 (119909)


1198891198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) (43)



1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) + 119888 (119909)

sdot1198651(119909)

119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

(44)


119889119909sdot 119890minus119891sdot1198651(119909) =

1

119910sdot119889119910

119889119909

119888 (119909) = int 119890119891sdot1198651(119909) sdot

119889119910

119910+ 119862

(45)



120590119899119861

= 119890minus119891sdot1198651(119909) sdot (119862

119861+ int 119890119891sdot1198651(119909) sdot

119889119910

119910) (46)



1199092+ (119910 minus 119910

0)2= 1198772 (47)

1199100= 119877 +

ℎ1

2 (48)


119910 = 119877 +ℎ1


119889119910 =119909 sdot 119889119909

radic1198772 minus 1199092

(49)


119889119910

119910=

119909radic1198772 minus 1199092


119889119909

=119909119877

radic1 minus (119909119877)2

sdot119889119909


(50)


119898 = 1 +ℎ1

2119877 (51)


119909

119877= sin120593


(52)


119889119910

119910=

sin120593


119877 cos120593 sdot 119889120593


=sin120593

119898 minus cos120593sdot 119889120593 = 119865

2(120593) sdot 119889120593

(53)


1198652(120593) =

sin120593119898 minus cos120593

(54)


S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S

120593R

LK

B

h0

y xA

h1




1198651 (119909) = int

119889119909

119910= int

119889119909


= int119889119909119877


(55)



1198651(120593) = int

cos120593 sdot 119889120593


cos120593119898 minus cos120593

sdot 119889120593 (56)

where

1198681= int

cos120593119898 minus cos120593

sdot 119889120593 (57)


tg120593

2= 119911

cos120593 =1 minus 1199112

1 + 1199112

119889120593 =2119889119911

1 + 1199112

(58)


1198681=

2

119898 + 1int

1 minus 1199112

(1 + 1199112) sdot (119886 + 1199112)sdot 119889119911 (59)

where

119886 =119898 minus 1

119898 + 1 (60)


1198681=

2

119898 + 1(

2

119886 minus 1int

119889119911

1 + 1199112+1 + 119886

1 minus 119886int

119889119911

119886 + 1199112) (61)

0 005 010 015 020 025 030120593 (mdash)

0

10

20

30

40

F1

(mdash)

300

51020

50

100

150

200

Rh

1(mdash

)



1198681=

2

119898 + 1(

2

119886 minus 1arctg119911 + 1 + 119886

1 minus 119886sdot1


radic119886) (62)


1198681= minus2arctg119911 + 2119898


119898 minus 1sdot 119911) (63)


form

1198681=

2119898



120593

2) minus 120593 (64)


1198651(120593)

1198651(120593) =

2119898



120593

2) minus 120593 (65)



form

120590119899119861

= 119890minus119891sdot1198651(120593) sdot (119862

119861+ int 119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593) (66)


120590119899119865

= 119890119891sdot1198651(120593) sdot (119862


2(120593) sdot 119889120593) (67)


119899asymp 119890119891sdot1198651 sdot 119865

2






0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 010 015 020 025 030120593 (mdash)

0

5

10

15

20

25

F2

(mdash)

5

1020

50

100

150200

300

500Rh

1(mdash

)








1198891198652(120593)119889120593 = 0

cos1205930=

1

119898 (68)





1198652max (1205930) =

1



0 minus (minus120590119899119861) = 120590119899119861

= 120590119886

0 minus (minus120590119899119865) = 120590119899119865

= 120590119886

(70)


120590119899119861

120590119886

=120590119886

120590119886

997904rArr 120590119899119861

= 1 (120593 = 120572)

120590119899119865

120590119886

=120590119886

120590119886

997904rArr 120590119899119865

= 1 (120593 = 0)

(71)




follows

119862119861= 119890119891sdot1198651(120572) (72)




119862119865= 1 (73)



written as follows

120590119899119861

= 119890minus119891sdot[(2119898radic119898


sdot (119890119891sdot[(2119898radic119898


+ int

120572

120593

119890119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(74)



follows

120590119899119865

= 119890119891sdot[(2119898radic119898


sdot (1

+ int

120593

0

119890minus119891sdot[(2119898radic119898


sdotsin120593

119898 minus cos120593sdot 119889120593)

(75)


119865and


119868119861= int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593

(76)




function 1198651(120593)

1198860=

2119898


1198861= radic

119898 + 1

119898 minus 1

(77)


00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 1000

05

10

15

20

25

xld (mdash)

120590n

(mdash)

A B

120590nF

N1

N2

N3

1205761

1205762

1205763

120590nB

f = 03 R = 105mm h0 = 7mm1205761 = 10 1205762 = 20 1205763 = 30



1198651(120593) = 119886

0sdot arctg(119886

1sdot tg

120593

2) minus 120593 (78)







119889




120590119899119865(120572119899) = 120590119899119861(120572119899) (79)


119890119891sdot1198651(120572119899) sdot (1 + int

120572119899

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

= 119890minus119891sdot1198651(120572119899) sdot (119890

119891sdot1198651(120572) + int

120572

120572119899

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(80)

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

120590n

(mdash)

120590nFN1

N2

N3

f1

f2

f3

120590nB

120576 = 20 R = 105mm h0 = 7mmf1 = f2 = f3 =

A B

01 02 03


08

10

12

14

16

18

20

R3

R2

R1

f = 120576 = 30

R1 = 30mmR2 = 100mmR3 = 300mm

00 02 04 06 08 10xld (mdash)

120590nF

N1

N2

N3

120590nB

120590n

(mdash)

A B

010



2(120593) is








1198611 0] and [119910



119896119865=1199101198652minus 1199101198651

Δ119909

119896119861=1199101198612minus 1199101198611

Δ119909

(81)



119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894Δ120593119894

120572119899119894

120572119899119894minus 120572119899119894minus1

120590119899119861119894

120590119899119865119894

120590119899119865119894

minus 120590119899119861119894


0

Y

x x + Δx

Δx

Δxn

yB2

b

N

fyB1

yF1

yF2



119910119865=1199101198652minus 1199101198651

Δ119909sdot 119909 + 119910

1198651

119910119861=1199101198612minus 1199101198611

Δ119909sdot 119909 + 119910

1198611

(82)



Δ119909119899=

1199101198611minus 1199101198651

1199101198652minus 1199101198612+ 1199101198611minus 1199101198651

sdot Δ119909 (83)



119909119899= 119909 + Δ119909

119899 (84)




0 10 20 30 40 50

120576 ()

000

005

010

015

020

120572n

(mdash)

f =5

10

20

50100200500

Rh1 (mdash)

000

286

573

859

114

120572n

(∘)

010


1


119899119865119894minus

120590119899119861119894


1is


1 Approximate






ratio 119877ℎ1


ℎ119899= ℎ1+ 2119877 (1 minus cos120572

119899) (85)


119899ℎ1can be


ℎ1

= 1 +2119877

ℎ1

(1 minus cos120572119899) (86)






0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 6050

120576 ()

5

10

20

50

100

200

500

Rh

1(mdash

)

f =

040

045

050

055

060

120572n120572

(mdash)

010


1

0 10 20 30 40 50

120576 ()

5 102050100200500

Rh

1(mdash

)

f = 010

100

105

110

115

120

125

hnh

1(mdash

)



int

120572119899

0

1198652(120593) sdot 119889120593 = int

120572

120572119899

1198652(120593) sdot 119889120593 (87)




119905 = 119898 minus cos120593

119889119905 = sin120593 sdot 119889120593

119868 = intsin120593


119889119905

119905= ln (119905)

= ln (119898 minus cos120593)

(88)

0 10 20 30 40 50

120576 ()

510

20

100200500

Rh

1(mdash

)

50

f = 0

000

005

010

015

020

120572n

(mdash)



the form

119868119865= int

120572119899

0

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]1205721198990


= ln119898 minus cos120572

119899

119898 minus 1

(89)


119868119861= int

120572

120572119899

sin120593119898 minus cos120593

sdot 119889120593 = [ln (119898 minus cos120593)]120572120572119899



119899

(90)





120572119899= arccos(1 + (1 minus

1


2119877) (92)






N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










N

120590nF(120593) 120590nB(120593)

F

GO

E

A B

120590n

av120590max

120590a=1

120590a=1

120572n

120572






be written

120590 =1


120572

0

120590119899(120593) sdot 119889120593 (93)


120590 =1

120572sdot [int

120572119899

0

120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120590119899119861(120593) sdot 119889120593] (94)


120590 =1

120572sdot [int

120572119899

0

(119890119891sdot1198651(120593)

sdot (1 + int

120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593

+ int

120572

120572119899

(119890minus119891sdot1198651(120593)

sdot (119890119891sdot1198651(120572) + int

120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)) sdot 119889120593]

(95)








0 10 20 30 40 50

120576 ()

1020

7550

150

100

200

300

Rh

1(mdash

)

f = 010

10

12

14

16

18

20

120590(mdash

)


1

00 05 10 15 20 25 30ldhav (mdash)

10

12

14

16

18

120590(mdash

)

f = 040

10203040

10203040

10203040

120576(

)

f = 025

f = 010


deformation





ℎ =119887 minus 119886

119899 (96)


int

119887

119886

119891 (119909) sdot 119889119909 =3

10ℎ (1199100+ 51199101+ 1199102+ 61199103+ 1199104+ 51199105

+ 21199106+ 51199107+ 1199108+ 61199109+ 11991010+ 511991011+ 211991012+ 511991013

+ 11991014+ sdot sdot sdot + 6119910

119899minus3+ 119910119899minus2

+ 5119910119899minus1

+ 119910119899)

(97)

where 1199100= 119891(119886) 119910

1= 119891(119886 + ℎ) 119910

2= 119891(119886 + 2ℎ) 119910

3=

119891(119886 + 3ℎ) 119910119894= 119891(119886 + 119894 sdot ℎ) and 119910

119899= 119891(119887) 119894 is the



119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 119899 120590119899119861119894

120590119899119861119894

minus 120590119899119861119894minus1



119899119861119894minus 120590119899119861119894minus1



minus 120590119899119861119894minus1

) asymp 110minus6


1= 10




119884119878 = 1198841198780sdot (1 + 119861 sdot 120576)

119899 (98)





120590119886av =

1

1205762minus 1205761

int

1205762

1205761

120590119886119889120576 (99)


2is the



120590119886=

2

radic3sdot 1198841198780sdot (1 +

2


119899

(100)


120590119886av =

1

1205762minus 1205761

int

1205762

1205761

2


2


119899

sdot 119889120576 (101)

00 02 04 06 08

120576 (mdash)

0

100

200

300

400

500

600

YS

(MPa

)

CuZn30

YS = 156(1 + 5008356 middot 120576)034971

R2 = 09912



120590119886av =

1198841198780

(119899 + 1) sdot (1205762 minus 1205761) sdot 119861

[(1 +2


119899+1

minus (1 +2


119899+1

]

(102)




120590119886av =

1198841198780

(119899 + 1) sdot 120576 sdot 119861[(1 +

2


119899+1

minus 1] (103)


119884119878 = 156 sdot (1 + 5008356 sdot 120576)034971 (104)


obtained

120590119886av =

230775120576

[(1 + 5783152 sdot 120576)134971 minus 1] (105)


120590119863=119865119903

119878 (106)




120590meas =120590119863

120590119886av

(107)



Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Number 120576 120576 119877ℎ1

119878 119865119903

120590119863



120576 (mdash)

120590m

eas120590

(mdash)

120590meas

120590

010 015 020 025 030 035 04010

12

14

16

18

20

h0 = 337mm = 066msR = 105mm

CuZn30





1199090 Front


119889and tension

stress 120590119909


1199090and 120590

1199091can be


1205901199090

= 1205850sdot 120590119901

1205901199091

= 1205851sdot 120590119901

(108)

where 1205850and 120585



0= 0 and for


1lt 1




120590119899119861minus (minus120590

1199090) = 120590119899119861+ 1205901199090

= 120590119901

120590119899119865

minus (minus1205901199091) = 120590119899119865

+ 1205901199091

= 120590119901

(109)


120590119899119861

120590119886

=120590119886

120590119886

minus1205850120590119886

120590119886

(110)

997904rArr 120590119899119861

= 1 minus 1205850 (111)

120590119899119865

120590119886

=120590119886

120590119886

minus1205851120590119886

120590119886

(112)

997904rArr 120590119899119865

= 1 minus 1205851 (113)




will be as follows

119862119865= 1 minus 120585

1 (114)



119862119861= (1 minus 120585

0) sdot 119890119891sdot1198651(120572) (115)


120590119899119865

= 119890119891sdot1198651(120593)


120593

0

119890minus119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(116)


120590119899119861

= 119890minus119891sdot1198651(120593)


120572

120593

119890119891sdot1198651(120593) sdot 119865

2(120593) sdot 119889120593)

(117)


00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25120590

(mdash)

1205851 = 0

120576 = 20 R = 105mmf = 03

AN6

N4

N2

N0120590nF 120590nB

B0

B2

B4

B6

B0B2B4B6

12058500 = 0

12058502 = 02

12058504 = 04

12058506 = 06

h0 = 7mm

rarr

rarr

rarr

rarr


tension

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

1205850 = 0

120576 = 20 R = 105mmf = 03h0 = 7mm

N6

N4

N2

N0

A0

A2

A4

A6

A0

A2

A4

A6

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

B

rarr

rarr

rarr

rarr


tension


2(120593)


119889and back


1= 0



0




= 04120590119886






0= 0



1




= 04120590119886

00 02 04 06 08 10xld (mdash)

00

05

10

15

20

25

120590(mdash

)

12058510 = 0

12058512 = 02

12058514 = 04

12058516 = 06

12058500 = 012058502 = 0212058504 = 0412058506 = 06

120590nF 120590nB

N60

N66

N06

N00

1205850 = 1205851

120576 = 20 R = 105mm f = 03h0 = 7mm

B0

B2

B4

B6

A0

A2

A4

A6


front tension

0 01 02 03 04 05f (mdash)

10

15

20

25

30

35

40

120590(mdash

)

5

10

15

20304050120576 ()

Rh1 = 100






00


0= 1205851= 0 forN

66it is for maximal



1=




119886 If ratio 119877ℎ

1and


119863


120590119863= 120590119886sdot 1205900 (118)


0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05f2

f (mdash)

minus10

minus05

00

05

10

15

Δ120590(f

)(mdash

)

1205900 = 1506

120576 = 2738

f0 = 0fmax

= 0577

Rh1 = 4303



1205900=120590119863

120590119886

(119)


Δ120590 = 120590 (119891) minus 1205900 (120)


2






1198912=1198910sdot Δ120590 (119891

1) minus 1198911sdot Δ120590 (119891

0)

Δ120590 (1198911) minus Δ120590 (119891

0)

(121)



1= 4303 120576 = 2738) is shown



119903 The


120595 =119886

119897119889

(122)


contact arc


method

1198991198910

1198911

Δ120590(1198910) Δ120590(119891

1) 119891

2Δ120590(1198912)









120590

120572119886=119872119878

119878120590

(123)


119878120590= 120572 sdot 120590 (124)




119872119878= int

120572

0

120593 sdot 120590119899(120593) sdot 119889120593 (125)


119899⟩




119872119878= int

120572119899

0

120593 sdot 120590119899119865(120593) sdot 119889120593 + int

120572

120572119899

120593 sdot 120590119899119861(120593) sdot 119889120593 (126)


119899119861(120593) is described by



119886 = 119877 sdot sin120572119886 (127)



120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590n

120590nF(120593)

A

N

120590nB(120593)

B

120593

120590a=1

120590a=1

Fr

120590ni(120593i)

Δ120593

120593i

120572n

120572a

120572

GO



0 10 20 30 40 50

120576 ()

047

048

049

050

051

Ψ(mdash

)

f = 010

500

200

100

50

20105

Rh

1(mdash

)




1


120595 =120572119886

120572 (128)





119865119903= 120590119886sdot 120590 sdot 119897119889sdot 119887av (129)

f

120572

120576

Rh1



119872119903= 2119865119903sdot 119886 (130)


119886 = 120595 sdot 119897119889 (131)


119872119903= 2119877 sdot 120590



4 Discussion







1 The constant



1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Kor 120590Kor 120590 120590Kor120590



ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1




ℎ0

ℎ1

120576 119877 119877ℎ1

119897119889ℎav 119891 119898Sim 120590Sim 120590 120590Sim120590






5 Conclusion


00 01 02 03 04 05 06 07 08 09 10xld (mdash)

120590(mdash

)

10

12

14

16

18

20

A B

f = R = 105mm120576 = 30h0 = 7mm

120572 = 01415


TselikovBland-Ford

02










2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















2= 1198652(119898 120593) and




Appendix




120576 =ℎ0minus ℎ1

ℎ0

(A2)


120576 = ln(ℎ0

ℎ1

) (A3)

respectively

120576 = minusln (1 minus 120576) (A4)


1ℎ0= 1 minus 120576


ℎav =ℎ0+ ℎ1

2 (A5)



2119877 (A6)

respectively

cos120572 = 1 minus120576


2119877 (A7)


119897119889= radic119877Δℎ minus

Δℎ2

4 (A8)

respectively

119897119889≐ radic119877Δℎ (A9)

Ratio 119897119889ℎav rarr 119897

119889ℎav = 2(radic119877Δℎ(ℎ

0+ ℎ1))


1= (2 minus 120576)

24120576(1 minus 120576) sdot (119897

119889ℎav)2



120576 =1

1 + ℎ12119877 sdot 1 (1 minus cos120572)

(A10)

Nomenclature


[minus]


[minus]





















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















3 1205901) [MPa]






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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