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					%
		σ3		. % !
			σ2	" , -
σ1			σ1	$ & # , $
				σ1>σ 2>σ 3 ( . 3.9).
				+ -
σ2							-

		σ3		, -

	,. 3.9			, & :
						σ ýêâ	= σ 1 ≤ [σ ].	(3.24)

	$ $,
! &
					σ3	≤ [σ− ],		(3.25)

	[σ− ] – " .
	( " ",

! " ! " "# "# -

# &! # # . 0 " 3

1830 .

!& ! εmax , . . &

εmax

[ε], .

:
	εmax	≤ [ε].	(3.26)

% ε = σ / Ε , !:
	[ε] = [σ].		(3.27)
		Ε

% !& !& ( -

)						. (3.16)
:
εmax =	1	[σ1 − µ(σ2 + σ3 )]; εmin	=	1	[σ3 − µ(σ1 + σ2 )].		(3.28)

	Ε			Ε
% " (3.26) (3.27) (3.28),						& -
:
σ ",II = σ 1 − µ(σ 2 + σ 3 )≤ [σ ];							(3.29)

, $ " -

, # $ -

" [σ − ]:

σ ",II = σ 3 − µ(σ 1 + σ 2 )≤ [σ − ].

(3.30)

&! # # -

(!, , " " ). '" - -

(1680) - (1860).

-. -

!& ! " . % !

, !& !

τmax "&$ "# [τ ]

, . .
τmax ≤ [τ]	(3.31)

! ! " ! " -

τ = σ / 2 (3.2), ! [τ] = [σ]/ 2 ,

* τmax ) (3.11)

τmax =	σ1 − σ3	.

2
%
σ .III = σ1 − σ3 ≤ [σ].			(3.32)

4 & ! "

"# . $

, " ! σ2 . 0 ! )-

/ (1776) 0 (1869).

! -

), ) ". + !

) :

U =	1 + µ	[σ1	2 + σ2	2 + σ3	2 − (σ1σ2 + σ2σ3 + σ3σ1 )].	(3.33)

	3Ε

% )-

! :

						1 + µ
					[U ]=			[σ]2 .	(3.34)
					[U ]=		3Ε	[σ]2 .	(3.34)
%		" U ≤ [U ], :

	σ1		2 + σ2		2 + σ3	2 − σ1σ2 − σ2σ3 − σ3σ1 ≤ [σ],			(3.35)

			1
σ ",IV =			1	(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ≤ [σ].					(3.36)

			2
			2

& ! ",

(3.32), "# , - # !, !.

< " . (1904) - -

(1913) "$ . – - .

- ,

σ1	σ3 , σ2	-
. ( " " "# "#		# -
, & ! σ2	"& 12 – 15 %. '" -
$ :
σ ", = σ1 − [σ+ ]σ3 ≤ [σ].		(3.37)
	[σ− ]

% "# # [σ+ ] [σ− ] )

) (3.32), ! "# ! "# -

. 0 - " ! , -

$# $ $, ! -

# # .

) . . 3.9 -

σ1 = 20 -%, σ3 = - 45 -%, σ2 = - 10 -%.

% ! !,

[σ+ ] = 35 -%, [σ− ] = 120 -%.

% $ $ $ , -

"# -

. "# # " -

" "& .

1 .

σmax = 20 -%. σmin = 45 -%. [σ+ ] = 35 -%, [σ− ]=120 -%.

, σmax ≤ [σ+ ] σmin ≤ [σ− ] "$ !& -

+ σ1 − µ(σ2 + σ3 ) ≤ [σ+ ].

20 – 0,25 (-10 - 45) = 20 + 13,8 = 33,8 -% < 35 -%.

% σ3 − µ(σ2 + σ1 ) ≤ [σ− ]. -45– 0,25 (-10 + 20) = 47,5 -%< 120 -%.

+ ".
6 .
		σ1 - σ3 ≤ [σ+ ],			20 – (-45) = 65 > 35 -%.
+ ",
		[( 65 – 35 )/ 35]# 100 % = 86 %.
IV .
	1
	1		(σ1 − σ2 )2 +	(σ2	− σ3 )2 + (σ3 − σ1 )2 ≤ [σ+ ].


	2
	2
1
1		(20 + 10)2 + (− 10 + 45)2 + (− 45 − 20)2 = 57,1 -% > 35 -%.


2
2

+ ".

- .

σ1 − [σ+ ]σ3 ≤ [σ+ ].

[σ− ]

20 – ( 30 # ( - 45)) / 120 = 33,1 -% < 35 -% –

", . . ! - -.

2 , , -

, "

. ! -

σ1 :σ 2 :σ 3 . 2 , -

. ! " ,

" -

, . !

( ),

, . !

, -

4./-" 1 7 8,",7-"-7 7 8 1 9

4.1.) ) ) ) *

! -

# -

" -

– -

1, -

" ! ", -

", ! " . . ( " -

	",	, ,
		"
,. 4.1	.

! "-

! “ ” , “&”, -

– . % -

# " #-

:	z .	=	Sy	,	y .	=	Sz	,	(4.1)
							A
			A

y . z . – # , z; Sy = zdA

Sz = ydA – ! y z

( . 4.1). ( ! , # # , Syo Szo

" $.

% "# “ ” “ ” -

! $ #:

# ! y z, ,

,. 4.2

,. 4.3

J y = z2dA	J z = y2 dA ;								(4.2)
A				A
A
#
J P = ρ 2dA ,									(4.3)
A
, J P = (y2 + z2 )dA = J z + J y ;									(4.4)
A
"
J zy = yz dA .									(4.5)
A
, -
, , -
! ( . 4.2).
	h 2					bh3
J z = y2dA =	y2bdy =					bh3	.		(4.6)
J z = y2dA =	y2bdy =						.		(4.6)
A	− h 2			12
1
	J y =	hb	3		.				(4.7)

		12
		12
( Jp	R
( . 4.3):
	R							πR	4
J p = ρ 2dA = ρ 2 2π ρ dρ =								πR		. (4.8)
J p = ρ 2dA = ρ 2 2π ρ dρ =										. (4.8)
A	0			2

2 !

dA = 2πρdρ .

(4.4), ,

J z = J y =	πR4	.	(4.9)
	4

4.2. # ) ) ( ) ) ) * . ) -

+ !

% ! , "# #

!, ( . 4.4):

J z1

y12 dA = (à + y0 )2 dA = a2 dA + 2ay0dA +

y2 dA.

(4.10)

0 y0, z0 – ! ", ,

y0 dA = 0 . " ( 4.10)

& :

J z1

= J z0 + a2 A,

(4.11)

J y

= J y

+ b2 A.

(4.12)

= "

Jz y

= Jz

+ abA.

(4.13)

% α -

" y1 z1 "# #

! ( . 4.5)

,. 4.4

z1 = z cosα + ysinα ,

y1 = y cosα − z sinα .

(4.14)

, " -

" !

z1 y1 "# Jz Jy

α :

J z

= J z cos2 α + J y sin2 α − J zy sin 2α ,

= J z sin 2 α + J y cos2 α + J zy sin 2α.

(4.15)

= " :

,. 4.5

Jz y

= z1y1dA =

Jz − J y

sin 2α + Jzy os2α

(4.16)

(4.15) :

J z1 + J y1

= J z

+ J y ,

(4.17)

. . "# .

4.3. / * ' ) ) *

% ! α0, J z1

J y1 ! ! " , # ! -

$ $ α J z1 J y1			(4.15) ! $.
%:
tg2α0	= −	2J zy		(4.18)
		J z	− J y

! , "# α0 , -

, , (4.17), – -

. 0 "$ , " " !

#, " ) " (4.15) α=α0 , "$ -

0 " (4.18) α0 " (4.16), " J z1 y1 ! $. -

!, ! "# "

% ) " (4.15) α=α0 " cos2α sin2α -)$ , ) " "#

+ J

− J

+ J 2zy .

(4.19)

max

min

. " , # , "$ "

! " . , ) " " -

$ .

%. ! #

, , , -

# . 4 -

: Sz Sy– -

z y; Jz, Jy, J5 Jzy - , # -

# . ! -

" . - ,

# , #. - , # "-

( " # # -), , – #-

#. 4 "

5.% /. 7"$1

5.1.1 ! '

" -

! "# . $ ! ! " ,

! " " $, "$ " ( . 5.1). %-

$ ,

- "

)

	τ =	Q	,	(5.1)

		A
	Q – -
	– ( -
	$ ) ;
,. 5.1	1 – ! .

% " ) (3.15) -

! "

+ σ

− σ

σ 1

+ τ

0 σα = 0 , σ β = 0 ,		σ 1 = +τ ; σ 3 = −τ ; σ 2 = 0 .
" "# ) (3.14)
tg2ψ 0 =	− 2τ		= −∞ , ψ 0 = −45

	σα − σ β

"# " . 5.1, .

) . 5.1, . -

!$, !

δ " $" , & γ = δ –

! " .

! ! -

):
τ = G γ ,	(5.2)

" . , G – ! – -, # $ )

. G = 8 104 M%. - ! E,

)) ) µ

!: G = E / 2(1 + µ) .

5.2. .

( 4 ' ' '

/ – ,

$ ! – ( . 5.2, ). ( "

" "# . # -

" ! " # –

# , b .

!$ 1-1 . ( $

!. & &$

– 1 ( . 5.2, ). -

" $.

,. 5.2

Σ = 0. 21 + 1 = 0 . 1 = - 21.

" , -

" & # # 2j , "#

. % -

" ! $ , # -

$ . %

# II III & , ! ! "

2 = −21 + 22 ,

3 = −21 + 22 − 23 .

% ) ( $ ) #

( . 5.2, ). ', $ , " . ( , $

( . 5.3). , dx ( . 5.3 ). /-

& , " !-

	" , . .
	ρτ dA = M ,							(5.3)
	&
	τ –	! ,
	$
	dA -
	ρ.
	< " " !						#
	τ -
	1,	)$
						dx		( .
	5.4). %, -
	) > , -
	– ", dx
	.
	". ( $-
	%. γ, -
	. .1.
,. 5.3
	tgγ =		CC1	, γ =	CC1		.	(5.4)

			BC
					dx

ρ = (

dϕ ,

CC1 = ρdϕ .

(5.4) (5.5)

ρdϕ

γ =

τ = Gρ

dϕ

% (5.8) (5.3)

= ρρG dϕ dA = G dϕ

ρ 2dA

,. 5.4

(5.5)

(5.6)

(5.7)

(5.8)

. (5.9)

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