Problemas Resueltos de Resistencia de Materiales II - Método de Las Deformaciones Angulares

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HIJK^M@AOKLIQ  @LGYD@^IQ K QDKUI HIJDIROKL ...................................... 1  ....................................................................... 1 Iau`aoklis Julh`milt`dis hi Wodskl y M`liy  M`liy  ....................................................................... ............................................................................................................... ................................................................... ............................................ 1 .......................................................................... ............................................................... ........... 2

HIJK^M@AOKLIQ @LGYD@^IQ K QDKUI HIJDIROKL IL UK^XOAKQ  ......... 00 .................................................................... ......... 00 Iau`aoklis Julh`milt`dis hi Wodskl y M`liy. M`liy. ...........................................................

Ytodoz`lhk d` jñrmud` hi M`xwidd p`r` a`h` tr`mk.  ............................................................ 00 ................................................................................................................. ........................................................... ...................................................... 0>

Ho`gr`m` hi juirz`s akrt`ltis.  ............................................................................................... .............................................................................................. 09

Akmprkn`lhk dks ^isudt`hks akl Qkjtw`ri s`p>555 .................................................. 0: Ho`gr`m` hi juirz` akrt`lti ........................................................................................ ....................................................................................... 0: Ho`gr`m` hi mkmiltk jdiatkr ...................................................................................... 07 ...................................................................................................... ........................................................... ........................................... 02  ...................................................................................................................... ..................................................................................................................... >8

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HIJK^M@AOKLIQ @LGYD@^IQ K QDKUI HIJDIROKL

Iau`aoklis Julh`milt`dis Julh`milt`dis hi Wodskl y M`liy D` iau`aoñl soguoilti is p`r` siaaoklis aklst`ltis

∊ ∊   = µ  ∗∗ ∗∗∞∗   =    ∊ ∊    =  µ  ∗∗ ∗∗∞∗   =    D`s iau`aoklis `ltirokris lk pirmotil promir`milti a`daud`r dks gorks il dks `pkyks k il luhks hi pñrtoaks y hispd`z`moiltks hi dks luhks hi pñrtoaks k sostim`s hklhi ixostil rktud`s.

Honuc`r id Ho`gr`m` hi Mkmiltk Jdiatkr y Juirz` Akrt`lti hinoh`milti `akt`h` p`r` d` jogur` mkstr`h` so IO= ati 05 tl

05 tl 05tkl-m

0> tl

05 tl

8 tl/m >m

 @ 

A

H

N

0.9 m

0.9 m

1m

1m

0.9 m

:m

>m

Mkmiltks hi impktr`moiltk pirjiatk  U

W `

n N

 @ 

> ∗∗   = ∞ >

  = 

N

 @  D@N m

D@N

∗∗> >

> ∗   = ∞ 0>

 = 

∗> 0>

D` a`rg` il pultu`d vkd`hk dk aklvirtomks il ul mkmiltk ixtirlk p`r` id `pkyk @

1

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Xr`mk @N3 ∗∗   + ∗∗  + Xr`mk @N3

  = ∞   = 

∗∗ +

 ∞    ∗∗ +

Xr`mk NA3

 = ∞  = 



∗∗ + ∗∗ +

Xr`mk AH3

∗   = ∞    =  ∗ 



∗.∗+. +. ∗ ∗.+ .+∗.  = ∞ ∗.∗ ∞ = ∞00. ∞00.201 201 tl ∞ m    ∗. ∗.∗.+  =  ∗+.   = 0>. 0>.?1 ?122 tl ∞ m  Xr`mk NA3

∗∗  = ∞ +  =

Xr`mk AH3

∞8.255 VXl∞m]





  = ∞ ∗ ∞0>.5.555 VXl VXl ∞ m]  = ∞0>   =  -00.201  =   0>.?12  =   -8.255

∗∗  =  +  = 7.>55 VXl∞m]

  =  ∗ 0>.5.555 VXl VXl ∞ m]  = 0>

VXl-m] VXl-m] VXl-m]

 = 7.>55   =   -0>.555   =   0>.555

VXl-m] VXl-m] VXl-m]

Ytodoz`lhk d` jñrmud` hi k iau`aoñl hi Mkbr k Iau`aoklis Julh`milt`dis hi Wodskl y M`liy p`r` a`h` tr`mk.

∊   = ∊ =    = µ  ∗∗ ∗∗∞∗    ∊ ∊    =  µ  ∗∗ ∗∗∞∗   =    =  Hinohk ` qui lk ixosti `silt`moiltk il dks `pkyks k lk ixosti rktud`s p`r` aklsohir`r d` v`ro`ndi Xr`mk @N3

Xr`mkNA

    =  µ  ∗∗     ∗∗        = µ  ∗∗    ∗∗   

  = µ  ∗∗    ∗∗      = µ  ∗∗    ∗∗  

  = ∞00.201 ∗   ∗∗ ∗ ∗      = 0>.?12 ∗   ∗∗ ∗ ∗    

 = ∞8.255  ∗   ∗∗    = 7.>55 ∗   ∗∗ ∗ ∗   

Xr`mkAH3

  ∗∗       = µ  ∗∗      ∗∗      = µ  ∗∗   

--------------------------------------------------------------------------------------------------------- = ∞0>.555 ∗   ∗ ∗ ∗    8

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 = 0>.555 ∗   ∗ ∗ ∗    Qum`tkro` hi iquodonrok istàtoak Luhk @ Luhk N Luhk A Luhk H

M@N+Mixt(+05)+Mixt(+vkd M@N+Mixt(+05)+Mi xt(+vkd`hk)=5 `hk)=5 MN@+MNA=5 MAN+MAH=5 MHA+Mixt(-vkd`hk)=5 MHA+Mixt(-vk d`hk)=5

Hi d` sum`tkro` hi iquodonrok istàtoak tilimks d` soguoilti ixpr isoñl il sostim` hi iau`aoklis. 5.:::7 * β@ +

5.111111 *βN +

5.5555 *βA + 5.5555 * βH = 

-01.02795 5.5555 * βH = -2.01795 5.1111 * βH =  8.25555 5.:::7 * βH =  -8.55555

5.1111 * β@ +

0.8::::7 *βN +

5.8555 *βA +

5.5555 * β@ +

5.855555 *βN +

0.8::7 *βA +

5.5555 * β@ +

5.555555 *βN +

5.1111 *βA +

Jkrm` m`troao`d p`r` id sostim` hi iau`aoklis 5.:::::7 5.111111 5.555555 5.555555 5.555 555 β@ = -01.02795 5.111111 0.8::::7 5.855555 5.555555 5.555 555 βN = -2.01795 5.555555 5.855555 0.8::::7 5.111111 5.111 111 βA = 8.25555 5.555555 5.555555 5.111111 5.:::::7 5.::: ::7 βH = -8.55555 Ytodoz`lhk d` soguoilti ixprisoñl hitirmol`mks dks gorks il a`h` luhk

 = ∞ ∗  β@ βN βA βH

= = = =

0.70>8021 -5.8>821:: 5.01570?5 -5.5:919?9

-5.8>821:: 5.28?:71> -5.>:0817? -5 .>:0817? 5.01570?5

5.01570?5 -5.5:919?9 -01.02795 -5.>:0817 ? 5.01570?5 -5.>:0817? -2.01795 5.28?:71> -5.8>821:: -5.8>821 :: 8.25555 -8.55555 -5.8>821:: -5.8>821: : 0.70>8021

Gorks jol`dis β@ = βN = βA=  βH = 

-02.>1:90?: -1.52?8:572 :.02017>98? -?.5?5:2:>7

A`daud`lhk dks mkmiltks jol`dis il a`h` ixtrimk hid idimiltk

  =  =  =  =  =  =

-00.201 + 0>.?12 + -8.255 + 7.>55 + -0>.555 + 0>.555 +

5.::7 * β@ + 5.111 * β@ + 5.555 * β@ + 5.555 * β@ + 5.555 * β@ + 5.555 * β@ +

5.111 *βN + 5.::7 *βN + 5.255 *βN + 5.855 *βN + 5.555 *βN + 5.555 *βN +

5.555 *βA + 5.555 *βA + 5.855 *βA + 5.255 *βA + 5.::7 *βA + 5.111 *βA +

5.555 * βH 5.555 * βH 5.555 * βH 5.555 * βH 5.111 * βH 5.::7 * βH

9

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^iimpd`z`lhk dks gorks il d` iau`aoñl priaihilti

   =   =   =   =   =   =

-00.201 + 0>.?12 + -8.255 + 7.>55 + -0>.555 + 0>.555 +

5.::7 * -02.>17 + 5.111 * -02.>17 + 5.555 * -02.>17 + 5.555 * -02.>17 + 5.555 * -02.>17 + 5.555 * -02.>17 +

5.111 *-1.52? + 5.::7 *-1.52? + 5.255 *-1.52? + 5.855 *-1.52? + 5.555 *-1.52? + 5.555 *-1.52? +

5.555 *:.020 + 5.555 *:.020 + 5.855 *:.020 + 5.255 *:.020 + 5.::7 *:.020 + 5.111 *:.020 +

5.555 * -?.5?0 5.555 * -?.5?0 5.555 * -?.5?0 5.555 * -?.5?0 5.111 * -?.5?0 5.::7 * -?.5?0

Mkmiltks jol`dis

  =  =  =

->9.555 VXl-m] 8.7?? VXl-m] -8.7?? VXl-m]

 =   05.?5?   =   -05.?5?   = 2.555

VXl-m] VXl-m] VXl-m]

Akl dks mkmiltks jol`dis pkhimks a`daud`r dks isjuirzks il a`h` ixtrimk hi a`h` idimiltk. ID XOUK HI A@^G@Q W

Moc 

c 

o 

Mco 

Doc 

A`rg` Hostronuoh`  Poc

Pco

 

U

` 

Moc 

o 

n 

c 

Mco 

Doc 

A`rg` Uultu`d  Poc 

Pco 

Akrt`lti hinohk ` d`s a`rg`s

  =  ∗  6    = ∞ ∗ 

∗  = ∞ ∗ 6   =    

  =  ∞  ∗ (   ) ……………………    = ∞  ∞   ∗ (   ) …………………… ^iimpd`z`lhk il d`s hks iau`aoklis.

----------------------------------------------------------------------------------------------------------

0>∗0.9 ∞ 0 ∗ ∞>9.5558.7??   =   ∞  ∗     =  05∗8.9  : : : ∞>9.5558.7??  

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0>∗8.9 ∞ 0 ∗ ∞>9.5558.7??  = ∞ ∞  ∗     = ∞ 05∗0.9 ∞ : : : ∞>9.5558.7??  0 ∗ ∞8.7??05.?5?  =  ∞  ∗    =  05∗> ∞ ∞8.7??05.?5? 9 9   0 ∗ ∞8.7??05.?5?  = ∞ ∞  ∗    = ∞ 05∗> ∞ 9 9 ∞8.7??05.?5?   =  ∞  ∗    =  8 >∗ : ∞ 0: ∗ ∞05.?5?2.555 ∞05.?5?2.555   = ∞ ∞  ∗    = ∞ 8 >∗ : ∞ 0: ∗ ∞05.?5?2.555 ∞05.?5?2.555 

----------------------------------------------------------------------------------------------------------

Akrt`ltis jol`dis

  =   =   =  

01.2:7 VXl] -2.011 VXl] >.772 VXl]

 =    =   =  

-7.>>> VXl] 0>.829 VXl] -00.909 VXl]

Ho`gr`m` hi dks mkmiltks jdiatkris y juirz`s akrt`ltis hi d` vog`. Ho`gr`m` hi Juirz`s Akrt`ltis 01.2:7tl

0>.829tl

01.2:7tl >.772tl

>.772tl

1.2:7 tkl

 @

H

A

N

2.011tl 05.55tl

2.555tl

2.011tl

05.55tl

7.>>>tl

7.>>>tl

-00.909tl

Ho`gr`m` hi mkmiltks jdiatkris ->9.555

-05.?5?

-09.55 -8.7??

-2.555

-8.7??

-8.>55  @

+7.850

H

A

N

+1.919

+2.979

7

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OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[ [[[[[[[[[ [

05

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[ [[[[[[[[[ [

05

HIJK^M@AOKLIQ @LGYD@^IQ K QDKUI HIJDIROKL IL UK^XOAKQ Iau`aoklis Julh`milt`dis Julh`milt`dis hi Wodskl y M`liy. D` iau`aoñl soguoilti is p`r` siaaoklis aklst`ltis

∊ ∊   = µ  ∗∗ ∗∗∞∗   =    ∊ ∊    =  µ  ∗∗ ∗∗∞∗   =    U

W

Moc  

c  

o 

Mco 

Doc  

Ytodoz`lhk d` jñrmud` hi M`xwidd p`r` a`h` tr`mk.





µ

 







HIJK^M@AOKLIQ @LGYD@^IQ K QDKUI HIJDIROKL IL UK^XOAKQ Iau`aoklis Julh`milt`dis Julh`milt`dis hi Wodskl y M`liy. D` iau`aoñl soguoilti is p`r` siaaoklis aklst`ltis

∊   = ∊   = µ  ∗∗ ∗∗∞∗    ∊ ∊    =  µ  ∗∗ ∗∗∞∗   =    U

W

Moc  

o 

c  

Mco 

Doc  

Ytodoz`lhk d` jñrmud` hi M`xwidd p`r` a`h` tr`mk.

 = µ   ∗  ∗    ∞    =  µ   ∗  ∗    ∞   =  ∗  ∗   =  ∗  ∗   =  ∗  ∗  D`s iau`aoklis `ltirokris lk pirmotil promir`milti a`daud`r dks gorks il dks `pkyks k il luhks hi pñrtoaks y hispd`z`moiltks hi dks luhks hi pñrtoaks k sostim`s hklhi ixostil rktud`s.

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

U`r` id pñrtoak mkstr`hk honuc`r dks ho`gr`m`s hi juirz`s akrt`ltis y mkmiltks jdiatkris d`s siaaoklis y prkpoih`his jísoa`s si h`l il d` jogur`. 5.25 tl/m

 @ 

9m

8m 15*85 am

N

:m 15*85 am

A

15*85 am

H

j'a=>05 Fg/am> 15*95 am

:m

I

 ZOG@ AKDYML@Q

DKLG.

OLI^AO@Q

^OG. ^ID.

5.15 5.15

D @N 8m 09

5.85 5.95 MOL=

5.550:55 O 5.5510>9 O 5.55> O

DNA 9m 0>

DAH :m 05

0.55 O 0.?9 O

DAI :m 05

O @N 0.555 O

ONA 0.555 O

OAH 0.555 O

09.555 O

0>.555 O

05.555 O

 =   =    

MAM=:5

OAI 0.?91 O 0?.910 O

W

N

 @  D@N m

0>

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

> ∗   = ∞ 0>

 = 

∗> 0>

Mkmiltks h impktr`moiltk pirjiatk, utodoz`lhk d` jogur` `ltirokr p`r` a`rg` hostronuoh` ulojkrmi.  µ = -0.5:::7 VXl-m] µ = ->.8 VXl-m] µ =  0.5::::7 VXl-m] µ =  >.8 VXl-m] VXl-m] µ =  -0.::::7 VXl-m] µ =  5 µ µ  = 0.:::::7 VXl-m]  = 5 VXl-m] Ytodoz`lhk d` iau`aoñl hi M`xwidd, p`r` a`h` idimiltk virtoa`d y bkrozklt`d.

 = µ   ∗ ∗ ∗6   =    =  µ    ∗ ∗ ∗6   =  Il isti a`sk lk ixosti hispd`z`moiltk bkrozklt`d y virtoa`d hi dks idimiltks, pkr dk t`ltk skdk si us`r` d` iau`aoñl `ltirokr.

   =   =    =    =    =   =    =    =

-0.5:::7+

15.555 * β@ +

09.55 * βN+

5.55 * βA+

5.55 * βH+

5.55 * βI

0.5::::7+

09.555 * β@+

15.55 * βN+

5.55 * βA+

5.55 * βH+

5.55 * βI

-0.::::7+

5.555 * β@+

>8.55 * βN+

0>.55 * βA+

5.55 * βH+

5.55 * βI

0.:::::7+

5.555 * β@+

0>.55 * βN+

>8.55 * βA+

5.55 * βH+

5.55 * βI

->.85555+

5.555 * β@+

5.55 * βN+

>5.55 * βA+

05.5 * βH+

5.55 * βI

>.85555+

5.555 * β@+

5.55 * βN+

05.55 * βA+

>5.55 * βH+

5.55 * βI

5.55555+

5.555 * β@+

5.55 * βN+

1?.5: * βA+

5.55 * βH+

5.55 * βI

5.55555+

5.555 * β@+

5.55 * βN+

0?.91 * βA

5.55 * βH+

5.55 * βI

Iquodonrok istàtoak. 15.555 * β@  +  + 09.555 * β@ + 5.555 * β@ + 5.555 * β@ +

09.55 * βN+ 98.55 * βN+ 0>.55 * βN+ 5.55 * βN +

5.55 * βA+ 0>.55 * βA+ 21.5: * βA+ 05.55 * βA+

5.55 * βH= 5.55 * βH= 05.55 * βH= >5.55 * βH=

0.5:::7 5.:5555 5.71111 ->.85555

Ddiv`mks ` ul mkhk hi sostim`s iau`aoklis p`r` d` silaodd` skduaoñl.

15.5555 09.5555 5.5555 5.5555

09.5555 98.5555 0>.5555 5.5555

5.5555 5.5555 0>.5555 5.5555 21.5:>9 05.5555 05.5555 05.5555 >5.5555

Ytodoz`lhk d` iau`aoñl

β@ βN βA βH

0.5:::7 = 5.:5555 5.71111 ->.85555

 = ∞ ∗  01

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

β@ βN βA βH

5.512?1077 -5.5000?7 5.5507>0>> -5.5552:5: 0.5:::7 -5.5000?:? 5.5>>1?17 -5.55188>81 5.5507>0>> 5.:5555 = 5.5507>0>> -5.55188> 5.50111?81 -5.55:::?7 5.71111 -5.5552:5: 5.5507>0> -5.55:::?70 5.5911182: ->.85555 Dks gorks il a`h` luhk hi d` istruatur`s `pkrtoa`h`. β@ =  βN =  βA =  βH =  βI = 

5.51208 -5.5590: 5.5>99: -5.01>72 5.55555 ^iimpd`z`lhk dks gorks il d` iau`aoklis `ltirokrmilti.

   = -0.5::::7   = 0.5::::7   = -0.:::::7   = 0.:::::7   = ->.8   = >.8  = 5.55  = 5.55

+0.0880519 +5.97>5907 +5.55 +5.55 +5.55 +5.55 5.55 5.55

Vr`h] Vr`h] Vr`h] Vr`h] Vr`h] hi M`xwidd hitirmol`hks

-5.57781:2 -5.098271: -5.0>1?5 -5.5:0?8?9 5.55 +5.55 5.55 5.55

+5.55 +5.55 +5.55 +5.55 +5.55 +5.55 +5.15:7>0 +5.55 +5.55 +5.:0188> +5.55 +5.55 +5.900>50 -0.1>725 +5.55 +5.>99:50 ->.:99:5 +5.55 5.??2885 5.5555555 5.55 5.8??>>5 5.5555555 5.55

Dks mkmiltks jol`dis il a`h` ixtrimk hi dks idimiltks

  =  =  =  =

5.55555 0.82128 -0.82128 >.>020:

VXl-m] VXl-m] VXl-m] VXl-m]

 =  =  =  =

-1.>0::5 5.55555 5.??288 5.8??>>

VXl-m] VXl-m] VXl-m] VXl-m]

U`r` hitirmol`r d`s akrt`ltis jol`dis il a`h` ixtrimk hid idimiltk n`st`r` utodoz`r skdk d`s iau`aoklis hi d` istàtoa`. Dks akrt`ltis jol`dis il a`h` ixtrimk hi dks idimiltks. Id prkaihomoiltk hid aàdaudk hi d`s akrt`ltis si prkaihi `d ogu`d qui il d` vog` hi d prkndim` `ltirokr.

  =  =  =  =

0.>>?51? -0.?75?: 0.291017 ->.08:2:

VXkl] VXkl] VXkl] VXkl]

 =  =  =  =

>.?1:0 -0.2:1? -5.>8?:0 -5.>8?:0

VXkl] VXkl] VXkl] VXkl] 08

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Ho`gr`m` hi juirz`s akrt`ltis. >.?1: tkl 0.>>? tkl

0.291 tkl

x=>.10: m x=0.91: m

 @ 

x=1.:75 m

A

N

>.087 tkl

0.2:8 tkl

H

0.?70 tkl

I

5.>95 tkl

Ho`gr`m` hi Mkmiltks Jdiatkris. -1.>07 t-m

-0.828 t-m

->.>02 t-m

5.55 t-m

5.55 t-m

 @ 

N 5.?88 t-m

m-

A t

5.::> t-m

2 ? ?.

>.070 t-m

5

H

mt ? ?

I

8. 5-

09

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Akmprkn`lhk dks ^isudt`hks akl Qkjtw`ri s`p>555

Ho`gr`m` hi juirz` akrt`lti

0:

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Ho`gr`m` hi mkmiltk jdiatkr

07

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Ytodoz`lhk id mïtkhk hi d`s Hijkrm`aoklis @lgud`ris honuc`r id Ho`gr`m` hi Mkmiltk Jdiatkr y juirz` akrt`lti hinoh`milti `akt`hk  > tl/m : tl-m A

N

> tl/m

: tl-m H

8m

8m >m

I

>m

> tl

> tl

:m >m

>m

1 tl/m

1 tl/m J

G

05 m

2m

>m

2m

>m

H

B

Mkmiltks hi impktr`moiltk pirjiatk U W

`

n N

 @ 

N

 @  D@N m

D@N

>

>

>

∗∗   = ∞ ∗   =  ∗   = ∞ ∗∗   =       >  > 0> 0>

>

W

D/>

D/>

N

 @  D

  = ∞

∗∗

 ∗∗    =  

Dks mkmiltks hi impktr`moiltk pirjiatk si hitirmol`l utodoz`lhk dks gràjoaks priaihiltis. >

>

>

>

1∗:   =∞?.55   =  ∗ =  1∗:   =?.55   = ∞ ∗ = ∞  0> 0> 0> 0>

02

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

> 00∗>∗2> 00∗∗  = ∞ 0?> = 0?>   =∞7.11

> 00∗>∗2> 9∗∗  =  0?> = 0?>   =1.11

> > ∗∗ >∗>∗>  = ∞ > = ∞ >>> = ∞0.5

> ∗∗  =  > =

> > ∗∗ >∗>∗>  = ∞ > = ∞ >>> = ∞0.5

> ∗∗  =  > =

> > ∗ 1∗:  = ∞ 0> = ∞ 0>   =∞?.55

 = 

> 00∗>∗2> 00∗∗  = ∞ 0?> = 0?>   =∞1.11



>∗>∗>> = 0.5 >>>



>∗>∗>> = 0.5 >>>

∗> =  1∗:>  =?.55 0> 0>

> 00∗>∗2> 9∗∗  =  0?> = 0?>   =7.11

  =    =  =    =   =     =   =

-?.55 Vtl-m]  = 5.55 Vtl-m] ?.55 Vtl-m]  = -0.55 Vtl-m] -7.11 Vtl-m]   = 0.55 Vtl-m] 1.11 Vtl-m]  = -1.11 Vtl-m] -0.55 Vtl-m]  = 7.11 Vtl-m] 0.55 Vtl-m]  = -?.55 Vtl-m]  = ?.55 Vtl-m] 5.55 Vtl-m] Hitirmol`rimks d`s rogohiais rid`tov`s hi a`h` tr`mk hi `auirhk ` d` kroilt`aoñl qui bimks tkm`hk hi dks idimiltks virtoa`dis y bkrozklt`dis. U`r` hitirmol`r d`s rogohiais rid`tov`s si bozk id mosmk prkaihomoiltk hi dks icimpdks `ltirokris.

F@N = 85 ∗ O FAH  = 0> ∗ O

6 6

FNA = 09 ∗ O FHG = :5 ∗ O FIB = 85 ∗ O

6 6

FAJ  = :5 ∗ O FHI = 09 ∗ O

Ytodoz`lhk d` jñrmud` hi M`xwidd p`r` a`h` tr`mk. B`aoïlhkdks mosmks prkaihomoiltks il dks prkndim`s `ltirokris tilimks d`s iau`aoklis hi dks mkmiltks.

   =    =   =   =   =

85.5 *βN 25.5 *βN 15.5 *βN 09.5 *βN 5.5 *βN

5.5 *βA 5.5 *βA 09.5 *βA 15.5 *βA :5.5 *βA

5.5 * βH 5.5 * βH 5.5 * βH 5.5 * βH 5.5 * βH

5.5 * βI 5.5 * βI 5.5 * βI 5.5 * βI 5.5 * βI

5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 0>5.5 * βJ

5.5 *βG  5.5 *βG  5.5 *βG  5.5 *βG  5.5 *βG 

-?.55 ?.55 -7.11 1.11 -0.55 0?

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

  =   =   =   =   =   =   =   =   =

5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN 5.5 *βN

0>5.5 *βA 5.5 * βH >8.5 *βA 0>.5 * βH 0>.5 *βA >8.5 * βH 5.5 *βA 0>5.5 * βH 5.5 *βA :5.5 * βH 5.5 *βA 15.5 * βH 5.5 *βA 09.5 * βH 5.5 *βA 5.5 * βH 5.5 *βA 5.5 * βH

5.5 * βI 5.5 * βI 5.5 * βI 5.5 * βI 5.5 * βI 09.5 * βI 15.5 * βI 25.5 * βI 85.5 * βI

:5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ 5.5 * βJ

5.5 *βG  5.5 *βG  5.5 *βG  :5.5 *βG  0>5.5 *βG  5.5 *βG  5.5 *βG  5.5 *βG  5.5 *βG 

0.55 5.55 5.55 -0.55 0.55 -1.11 7.11 -?.55 ?.55

Qum`tkro` hi iquodonrok istàtoak il dks soguoiltis luhks. Luhk N MN@+MNA+Mixt(+vkd`hozk)=5 MN@+MNA+Mixt(+vk d`hozk)=5 Luhk A MAN+MAH+MAJ+Mixt(MAN+MAH+MAJ+Mixt(-:)=5 :)=5 Luhk H MHA+MHI+MHG+Mixt(+:)=5 Luhk I MIH+MIB+Mixt(-vkd`hk)=5 MIH+MIB+Mixt(-v kd`hk)=5 Luhk J MJA=5 Luhk G MGH=5 Hi d`s sum`tkro` hi iquodonrok istàtoak b`aimks ul sostim` hi iau`aoklis. 005.5 *βN+

09.5 * βA+

5.5 * βH+

5.5 *βI+

5.5 * βJ+

09.5 *βN+

078.5 * βA+

0>.5 * βH+

5.5 *βI+

:5.5 * βJ+

5.5 *βN+

0>.5 * βA+

078.5 * βH+

09.5 *βI+

5.5 * βJ+

5.5 *βN+

5.5 * βA+

09.5 * βH+

005.5 *βI+

5.5 * βJ+

5.5 *βN+

:5.5 * βA+

5.5 * βH+

5.5 *βI+

0>5.5 * βJ+

5.5 *βN+

5.5 * βA+

:5.5 * βH+

5.5 *βI+

5.5 * βJ+

5.5 * βG= 

-9.::7 5.5 * βG=  0.::7 :5.5 * βG=  -0.::7 9.::7 5.5 * βG=  0.555 5.5 * βG=  0>5.5 * βG=  -0.555

Qostim` hi m`troz

005.55 09.55 5.55 5.55 5.55 5.55

09.55 078.55 0>.55 5.55 :5.55 5.55

5.55 5.55 5.55 5.55 0>.55 5.55 :5.55 5.55 078.55 09.55 5.55 :5.55 09.55 005.55 5.55 5.55 5.55 5.55 0>5.55 5.55 :5.55 5.55 5.55 0>5.5

βN  βA βH  βI

-9.::7 0.::7 -0.::7 = 9.::7 0.555 β J  -0.555 β G 

Ytodoz`lhk d` soguoilti ixprisoñl hitirmol`mks dks gorks il a`h` luhk

 = ∞ ∗  βN = βA =  βH = βI = βJ = βG = 

5.55?>>1 -5.555?:2 5.55552> -5.555500 5.555828 -5.555580 -5.555?:2 5.5575?9 -5.555:55 5.55552> -5.551982 5.555155 5.55552> -5.555:55 5.5575?9 -5.555?:2 5.555155 -5.551982 -5.555500 5.55552> -5.555?:2 5.55?>>1 -5.555580 5.555828 5.555828 -5.551982 5.555155 -5.555580 5.505057 -5.555095 -5.555580 5.555155 -5.551982 5.555828 -5.555095 5.505057

-9.::7 0.::7 -0.::7 9.::7 0.555 -0.555 >5

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Gorks jol`dis il a`h` luhk. βN βA βH βI βJ βG

= = = = = =

-5.59199 5.508?> -5.508?> 5.59199 5.55527 -5.55527

^`h ^`h ^`h ^`h ^`h ^`h

Xiloilhk dks gorks hitirmol`hks k jol`dis, pkhimks a`daud`r dks mkmiltks il a`h` ixtrimk hidks idimiltks virtoa`dis y bkrozklt`dis hid sostim` `pkrtoa`hk.

   =    =   =   =  =  =   =   =   =   =   =   =   =   =

85.5 *-5.59199+

5.5 *5.508?>+

5.5 * -5.508?>+

5.5 * 5.59199+

5.5 * 5.55527+

5.5 *-5.55527+

-?.55

25.5 *-5.59199+

5.5 *5.508?>+

5.5 * -5.508?>+

5.5 * 5.59199+

5.5 *5.55527+

5.5 *-5.55527+

?.55

15.5 *-5.59199+

09.5 *5.508?>+

5.5 * -5.508?>+

5.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

-7.11

09.5 *-5.59199+

15.5 *5.508?>+

5.5 * -5.508?>+

5.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

1.11

5.5 *-5.59199+

:5.5 *5.508?>+

5.5 *-5.508?>+

5.5 *5.59199+

0>5. *5.55527+

5.5 *-5.55527+

-0.55

5.5 *-5.59199+

0>5.5 *5.508?>+

5.5 *-5.508?>+

5.5 *5.59199+

:5.5 *5.55527+

5.5 *-5.55527+

0.55

5.5 *-5.59199+

>8.5 *5.508?>+

0>.5 *-5.508?>+

5.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

5.55

5.5 *-5.59199+

0>.5 *5.508?>+

>8.5 *-5.508?>+

5.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

5.55

5.5 *-5.59199+

5.5 *5.508?>+

0>5.5 *-5.508?>+

5.5 *5.59199+

5.5 *5.55527+

:5.5 *-5.55527+

-0.55

5.5 *-5.59199+

5.5 *5.508?>+

:5.5 *-5.508?>+

5.5 *5.59199+

5.5 *5.55527+

0>5.5 *-5.55527+

0.55

5.5 *-5.59199+

5.5 *5.508?>+

15.5 *-5.508?>+

09.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

-1.11

5.5 *-5.59199+

5.5 *5.508?>+

09.5 *-5.508?>+

15.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

7.11

5.5 *-5.59199+

5.5 *5.508?>+

5.5 *-5.508?>+

25.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

-?.55

5.5 *-5.59199+

5.5 *5.508?>+

5.5 *-5.508?>+

85.5 *5.59199+

5.5 *5.55527+

5.5 *-5.55527+

?.55

Mkmiltks jol`dis

  =     =    =    =    =     =    =  

-00.08>5 8.70:5 -2.70:5 >.?772 5.5555 >.2810 5.07?0

Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m]

 =    =    =    =    =    =   = 

-5.07?0 ->.2810 5.5555 ->.?772 2.70:5 -8.70:5 00.08>5

Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m] Vtl-m]

Xiloilhk dks mkmiltks jol`dis p`s`mks ` a`daud`r dks isjuirzks il a`h` ixtrimk hi dks idimiltks virtoa`dis y bkrozklt`dis hid sostim` `pkrtoa`hk.

>0

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

  =   ∞    ∗     =  ∗ ∞  ∗ ∞00.08>58.70:5  = ∞ ∞    ∗     = ∞ ∗ ∞  ∗ ∞00.08>58.70:5  =  ∞   ∗    =   ∗ > ∗ 2 ∞  ∗ ∞2.70:5>.?772  = ∞ ∞   ∗    = ∞  ∗ > ∗ 2 ∞  ∗ ∞2.70:5>.?772 ∞2.70:5>.?772  =  ∞   ∗    = 5 5 ∞  ∗ 5.07?0∞5.07?0  = ∞ ∞   ∗    = ∞5 ∞5 ∞  ∗ 5.07?0∞5.07?0  =  ∞   ∗    =   ∗ > ∗ 2 ∞  ∗ ∞>.?7722.70:5 ∞>.?7722.70:5  = ∞ ∞   ∗    = ∞  ∗ > ∗ 2 ∞  ∗ ∞>.?7722.70:5  =  ∞   ∗    =  ∗ ∞  ∗ ∞8.70:500.08>5 ∞8.70:500.08>5  = ∞ ∞   ∗    = ∞ ∗ ∞  ∗ ∞8.70:500.08>5 ∞8.70:500.08>5  =  ∞  ∗    =  ∗ ∞  ∗ 5.5555>.2810 5.5555>.2810   = ∞ ∞  ∗    = ∞ ∗ ∞  ∗ 5.5555>.2810   =  ∞   ∗    =  ∗ ∞  ∗ ∞>.28105.5555  = ∞ ∞   ∗    = ∞ ∗ ∞  ∗ ∞>.28105.5555 ∞>.28105.5555   =    =   =   =   =   =    = 

05.57055 -7.?>? :.707 -0.>21 5.>2? -0.700 5.555

VXl] VXl] VXl] VXl] VXl] VXl] VXl]

  =  5.555   =  0.700   =  -5.>2?   =  0.>21   =  -:.707  =   7.?>?  =   -05.570

VXl] VXl] VXl] VXl] VXl] VXl] VXl]

>>

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[

Ho`gr`m` hi juirz`s akrt`ltis

:.707tl

8.55tl 0.>21tl

0.>21tl

-7.?>?tl

N

I A

H

l t l ?

l

8.55tl

k t

> t

0.>21tl 0.>21tl 0 0

0 . .7

0 7 0

l t t ? ? 2

2

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l >.

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l

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0

7

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l

t

?.

0 7

7 5 . 5. 5

5

H

0

B

0 -

Ho`gr`m` hi Mkmiltks Jdiatkris

-2.70:

-2.70: -8.55tl

->.?72

->.?72

-8.55

-5.07?

      :        0       7   .        8     -

N

>.091 >.9:8

A

      1        8       2 .  

H

+5.07?

      >     -

      >     -

      2        7        9   .        5 

      >       :        7   .        9       +

      >       8       0  .        0       0     -

J

 @ 

      1        8       2 .  

      5        5   .        5      -

      2        7        9   .        5 

I

>.091 >.9:8

      :        0       7   .        8     -

      >       :        7   .        9       + G

      >       8       0  .        0       0     -

      5        5        5   .        5 

B

>1

OLGILOI^Í@ AOZOD

Q @LH^K D @LHIK @.

 [[[[[[[[[[[[[[[[[[[[[[[[[  [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[ [[[[[



^isostilao` hi m`tiro`dis O y OO. (@. @rti`g`. L, U. Oniroak A, Gklz`dis, @. Migk A.)



 @làdosos istruatur`d (Xikro` y Urkndim`s ^isuidtks, Olg. O lg. No`ggok  @rnudó G.-YLO)



A`daudk hi Istruatur`s Bopiristàtoa`s (Zkdumil OO, Olg. No`ggok  @rnudó G.-YLO)



A`daudk hi Istruatur`s Bopiristàtoa`s (Zkdumil OOO, Olg. No`ggok  @rnudó G.-YLO)



 @pultis hi Ad`sis hi ^isostilao` hi M`tiro`dis O y OO  ‑  —Ylovirsoh`h L`aokl`d hi Bu`la`vidoa`” (>508 -O,

>508-OO) OLG.

A@N@DDI^K Q@LABIS, Km`r. 

 ‑—Ylovirsoh`h  @pultis hi ad`sis hi @làdosos Istruatur`d O - OO  ‑—Ylovirsoh`h L`aokl`d

hi

Bu`la`vidoa`

—>509 -O,

OLG. A@N@DDI^K

Q@LABIS, Km`r. 

 @pultis hi Ad`sis hi ^isostilao` hi M`tiro`dis O y OO  ‑  —Ylovirsoh`h L`aokl`d hi Bu`la`vidoa`”

OLG. NILHISÓ NKS@,

^iyhir I. 

 ‑—Ylovirsoh`h  @pultis hi ad`sis hi @làdosos Istruatur`d O  ‑  OO  ‑—Ylovirsoh`h L`aokl`d hi Bu`la`vidoa` —>509 -O,

OLG. NILHISÓ NKS@,

^iyhir I.

>8