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Сопротивление материалов. Часть 1. Учебное пособие

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Ɂɚɞɚɱɚ 2.4
Ʌɚɬɭɧɧɵɣ ɫɬɟɪɠɟɧɶ ɫɬɭɩɟɧɱɚɬɨ ɩɨɫɬɨɹɧɧɨɝɨ ɫɟɱɟɧɢɹ ɧɚɯɨɞɢɬɫɹ ɩɨɞ ɞɟɣɫɬ-
ɜɢɟɦ ɫɢɥɵ Ɋ = 30 ɤɇ. ɇɢɠɧɢɣ ɭɱɚɫɬɨɤ ɢɫɩɵɬɵɜɚɟɬ ɪɚɜɧɨɦɟɪɧɵɣ ɧɚɝɪɟɜ ɧɚ ɜɟ-
ɥɢɱɢɧɭ Ɍ = 20qɋ		ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɚɱɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ. Ɇɟɠɞɭ ɧɢɠɧɢɦ
ɬɨɪɰɨɦ ɫɬɟɪɠɧɹ ɢ ɠɟɫɬɤɢɦ ɨɫɧɨɜɚɧɢɟɦ ɢɦɟɟɬɫɹ ɦɚɥɵɣ ɡɚɡɨɪ							G = 0,012 ɫɦ
(ɪɢɫ.2.10,ɚ). ɉɨɫɬɪɨɢɦ ɷɩɸɪɵ N, V ɢ u.
ȼ ɪɚɫɱɟɬɚɯ ɩɪɢɦɟɦ ȿ = 1 105 Ɇɉɚ = 1 104 ɤɇ/ɫɦ2 ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɥɢɧɟɣɧɨ-
ɝɨ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɪɚɫɲɢɪɟɧɢɹ D = 1,65 10–5 1/ɝɪɚɞ.
a)	R1		ɛ)	N (ɤɇ)	ɜ)	ς (Ɇɉɚ) ɝ) u (ɫɦ)
	R1				18	9
	Ɉ				18	9
	Ɉ
=40ɫɦ		F1=20ɫɦ2
=40ɫɦ	1			+		+
1
l
		30ɤɇ
						0,0036
=40ɫɦ		F2=10ɫɦ2
	2	T=20o	–		–	+
		T=20o	–		–	+
2
l
		Γ
	Rɯ2		12		12		0,012
	Rɯ2
					Ɋɢɫ.2.10
Ɉɩɪɟɞɟɥɢɦ ɜɟɥɢɱɢɧɭ ɜɨɡɦɨɠɧɨɝɨ ɭɞɥɢɧɟɧɢɹ ɫɬɟɪɠɧɹ ɨɬ ɞɟɣɫɬɜɢɹ ɫɢɥɵ 5
ɢ ɨɬ ɧɚɝɪɟɜɚ ɧɢɠɧɟɝɨ ɭɱɚɫɬɤɚ.

'lɊ		ȇl1	30 40		0,006 ɫɦ;
		EF1		1 104 20
'l	Dl T			1,65 10 5 40 20 0,0132 ɫɦ;
T	2
'lɊ 'lT			0,006 0,0132 0,0192 ɫɦ.
ɉɨɫɤɨɥɶɤɭ ɫɭɦɦɚ 'lɊ				ɢ 'lɌ ɛɨɥɶɲɟ ɡɚɡɨɪɚ G = 0,012 ɫɦ, ɩɪɢ ɧɚɝɪɭɠɟɧɢɢ

ɫɬɟɪɠɧɹ ɷɬɨɬ ɡɚɡɨɪ ɡɚɦɤɧɟɬɫɹ, ɢ ɫɬɟɪɠɟɧɶ ɛɭɞɟɬ ɪɚɛɨɬɚɬɶ ɤɚɤ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩ-

ɪɟɞɟɥɢɦɵɣ ɫ ɞɜɭɦɹ ɨɩɨɪɧɵɦɢ ɪɟɚɤɰɢɹɦɢ R1 ɢ R2 (ɪɢɫ.2.10,ɚ).
ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ.
6ɏ = 0,	– R1 + 30 – R2 = 0 , R1 + R2 = 30 ɤɇ .

Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɭɫɥɨɜɢɟ ɞɟɮɨɪɦɚɰɢɢ ɫɬɟɪɠɧɹ l = Γ. Ɉɬɛɪɨɫɢɦ ɦɵɫɥɟɧɧɨ ɧɢɠɧɟɟ ɡɚɤɪɟɩɥɟɧɢɟ ɢ ɜɜɟɞɟɦ ɧɟɢɡɜɟɫɬɧɭɸ ɫɢɥɭ ɏ = R2. ȼɨɡɦɨɠɧɨɟ ɭɤɨɪɨɱɟɧɢɟ ɫɬɟɪɠɧɹ ɨɬ ɞɟɣɫɬɜɢɹ ɷɬɨɣ ɫɢɥɵ ɪɚɜɧɨ:

lɏ

ɏl1

ɏl2

ɏ 40

6 10 4 ɏ .

1 104 20

1 104 10

EF1 EF2

ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɟ ɞɟɮɨɪɦɚɰɢɢ ɫɬɟɪɠɧɹ, ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɢɧɰɢɩɚ ɧɟɡɚɜɢ-

ɫɢɦɨɫɬɢ ɞɟɣɫɬɜɢɹ ɫɢɥ ɩɨɥɭɱɢɦ

0,0192 6 10 4 ɏ

0,012 ɫɦ;

12 ɤɇ,

30 R2

30 12

18ɤɇ.

Ɉɩɪɟɞɟɥɢɦ ɩɪɨɞɨɥɶɧɵɟ ɫɢɥɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɩɪɟɞɟɥɚɯ ɭɱɚɫɬ-

ɤɨɜ ɫɬɟɪɠɧɹ ɢ ɜɟɥɢɱɢɧɵ ɭɞɥɢɧɟɧɢɣ ɭɱɚɫɬɤɨɜ.

ɍɱɚɫɬɨɤ 1.

18 ɤɇ ,

0,9 ɤɇ/ɫɦ2

9 Ɇɉɚ ;

18 40

0,0036 ɫɦ.

1 104 20

ɍɱɚɫɬɨɤ 2.

ς 12

12 ɤɇ ,

1,2 ɤɇ/ɫɦ2

12 Ɇɉɚ ;

12 40

0,0048 0,0132

0,0084 ɫɦ.

1 104 10

ɉɪɨɜɟɪɢɦ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ ɞɟɮɨɪɦɚɰɢɢ ɫɬɟɪɠɧɹ.

l1 l2

0,0036 0,0084

0,012 ɫɦ

Γ.

Ɂɚɞɚɱɚ ɪɟɲɟɧɚ ɩɪɚɜɢɥɶɧɨ. ɗɩɸɪɵ

N ɢ

ς ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.2.10,ɛ,ɜ. Ɉɩ-

ɪɟɞɟɥɹɟɦ ɨɫɟɜɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɹ.

ɯ= 0 , u = u0 = 0 ;

ɯ= 40 ɫɦ , u1 = u0 + l1 = 0,0036 ɫɦ ;

ɯ= 80 ɫɦ , u2 = Γ = 0,012 ɫɦ .

ɗɩɸɪɚ u ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ.2.10,ɝ. Ɉɫɟɜɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ. ȼɫɟ ɩɨɩɟɪɟɱɧɵɟ ɫɟɱɟɧɢɹ ɩɟɪɟɦɟɳɚɸɬɫɹ ɜɧɢɡ.

Ɂɚɞɚɱɚ 2.5

ɋɬɟɪɠɧɟɜɚɹ ɫɢɫɬɟɦɚ ɫɨɫɬɨɢɬ ɢɡ ɠɟɫɬɤɨɣ ɛɚɥɤɢ Ⱥȼ, ɲɚɪɧɢɪɧɨ ɨɩɺɪɬɨɣ ɜ ɬɨɱɤɟ Ⱥ ɢ ɩɨɞɞɟɪɠɢɜɚɟɦɨɣ ɫɬɟɪɠɧɟɦ ɋȼ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ ɞɢɚɦɟɬɪɨɦ d = 22 ɦɦ (ɪɢɫ.2.11,ɚ). Ɉɩɪɟɞɟɥɢɦ ɞɨɩɭɫɬɢɦɭɸ ɜɟɥɢɱɢɧɭ ɫɢɥɵ 5 ɢɡ ɭɫɥɨɜɢɹ ɩɪɨɱɧɨɫɬɢ ɫɬɟɪɠɧɹ ɋȼ ɩɨ ɦɟɬɨɞɭ ɩɪɟɞɟɥɶɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɜɟɥɢɱɢɧɭ ɭɞɥɢɧɟɧɢɹ ɫɬɟɪɠ-

ɧɹ ɢ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɛɚɥɤɢ Ⱥȼ. ȼ ɪɚɫɱɟɬɚɯ ɩɪɢɦɟɦ R = 210 Ɇɉɚ = 21 ɤɇ/ɫɦ2,

ϑf = 1,4, ϑɫ = 0,9, ȿ = 2,1 105 Ɇɉɚ = 2,1 104 ɤɇ/ɫɦ2.

ɉɪɢ ɧɚɝɪɭɠɟɧɢɢ ɫɢɫɬɟɦɵ ɜɨɡɧɢɤɚɸɬ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ

RȺ

ɢ ɇȺ

ɧɚ ɨɩɨɪɟ

Ⱥ ɢ ɩɪɨɞɨɥɶɧɚɹ ɪɚɫɬɹɝɢɜɚɸɳɚɹ ɫɢɥɚ N ɜ ɫɬɟɪɠɧɟ

ɋȼ. ɗɬɢ ɜɟɥɢɱɢɧɵ ɦɨɠɧɨ

ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ 6ɏ = 0,

6Y = 0,

6Ɇ = 0.

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ N ɢɫɩɨɥɶɡɭɟɦ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ ɦɨɦɟɧɬɨɜ

6ɆȺ = 0 , Nr 2ȇ 0, N 2ȇ

ȇ 1,49ȇ,

1,34

1,5

ɝɞɟ

3sin D

3 0,447

1,34 ɦ – ɩɥɟɱɨ ɫɢɥɵ N, tgD

0,5,

sin D

0,447.

ɛ)

d =22ɦɦ

C •

1,5ɦ

•

2ɦ

P 1ɦ

•

• B1

Ɋɢɫ.2.11

ɂɫɩɨɥɶɡɭɟɦ ɭɫɥɨɜɢɟ ɩɪɨɱɧɨɫɬɢ ɫɬɟɪɠɧɹ

1,49ȇ

d Jc R 21 0,9 ,

3,8

ɝɞɟ

3,14 2,22

– ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ.

3,8 ɫɦ

Ɋɚɫɱɟɬɧɨɟ ɢ ɧɨɪɦɚɬɢɜɧɨɟ ɡɧɚɱɟɧɢɹ ɫɢɥɵ 5 ɪɚɜɧɵ:

ȇ d 21 0,9 3,8 48,2 ɤɇ;

1,49

ȇp

48,2

34,4 ɤɇ.

J f

1,4

ɉɪɢɧɢɦɚɟɦ ɫ ɨɤɪɭɝɥɟɧɢɟɦ 5ɧ = 34 ɤɇ

ɢ ɨɩɪɟɞɟɥɹɟɦ ɜɟɥɢɱɢɧɭ ɭɞɥɢɧɟɧɢɹ

ɫɬɟɪɠɧɹ ɨɬ ɞɟɣɫɬɜɢɹ ɷɬɨɣ ɫɢɥɵ. ɋɯɟɦɚ ɞɟɮɨɪɦɚɰɢɢ ɫɢɫɬɟɦɵ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ.2.11,ɛ.

'l	Nl	1,49 34 335,4		0,213ɫɦ,
	EF		2,1 104 3,8

ɝɞɟ l 1,52 32 3,354 ɦ – ɞɥɢɧɚ ɫɬɟɪɠɧɹ. Ɉɩɪɟɞɟɥɢɦ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɠɟɫɬɤɨɣ ɛɚɥɤɢ Ⱥȼ.

BB	'l	0,213	0,477 ɫɦ ;
1	sin D	0,447

tgE

BB1

0,477

0,00159,

E = 0,09ɨ.

300

ɍɝɨɥ ɩɨɜɨɪɨɬɚ ɛɚɥɤɢ ɨɱɟɧɶ ɦɚɥ.

Ɂɚɞɚɱɚ 2.6

ɋɬɟɪɠɧɟɜɚɹ ɫɢɫɬɟɦɚ (ɪɢɫ.2.12,ɚ) ɫɨɫɬɨɢɬ ɢɡ ɠɟɫɬɤɨɣ ɛɚɥɤɢ

Ⱥȼ, ɢɦɟɸɳɟɣ

ɲɚɪɧɢɪɧɨ ɧɟɩɨɞɜɢɠɧɭɸ ɨɩɨɪɭ Ⱥ, ɢ ɞɜɭɯ ɫɬɟɪɠɧɟɣ

ɋD ɢ ED,

ɩɨɞɞɟɪɠɢɜɚɸ-

ɳɢɯ ɛɚɥɤɭ. Ʉ ɛɚɥɤɟ ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ

ɧɨɪɦɚɬɢɜɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɣ ɪɚɜɧɨ

400 ɤɇ. Ɉɩɪɟɞɟɥɢɦ ɭɫɢɥɢɹ ɜ ɫɬɟɪɠɧɹɯ ɢ ɩɨɞɛɟɪɟɦ ɢɯ ɫɟɱɟɧɢɹ ɜ ɜɢɞɟ ɞɜɭɯ

ɫɬɚɥɶɧɵɯ ɪɚɜɧɨɛɨɤɢɯ ɭɝɨɥɤɨɜ. Ɉɩɪɟɞɟɥɢɦ ɜɟɥɢɱɢɧɭ ɪɚɡɪɭɲɚɸɳɟɣ ɫɢɥɵ. ȼ

ɪɚɫɱɟɬɚɯ ɩɪɢɦɟɦ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɞɟɠɧɨɫɬɢ ɩɨ ɧɚɝɪɭɡɤɟ

Jf = 1,2 , ɪɚɫɱɺɬɧɨɟ ɫɨ-

ɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɚ

R = 210 Ɇɉɚ = 21 ɤɇ/ɫɦ2 , ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɥɨɜɢɣ ɪɚɛɨ-

ɬɵ Jɫ = 0,9, ɩɪɟɞɟɥ ɬɟɤɭɱɟɫɬɢ Vɬ = 240 Ɇɉɚ = 24 ɤɇ/ɫɦ2

ɢ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ

ɩɥɨɳɚɞɹɦɢ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɟɣ F2/F1 = 1,2.

ɛ)

• E

• C

2ɦ

45ɨ

A •

•B

•45ɨ l2

2ɦ

1ɦ P=400ɤɇ

•

• B1

Ɋɢɫ.2.12

ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɧɚɝɪɭɡɤɢ ɧɚ ɨɩɨɪɟ

ɜɨɡɧɢɤɚɸɬ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ

RȺ

ɇȺ, ɚ ɜ ɫɬɟɪɠɧɹɯ – ɪɚɫɬɹɝɢɜɚɸɳɢɟ ɭɫɢɥɢɹ

ɢ N2. ɉɨɫɤɨɥɶɤɭ ɞɥɹ ɢɯ ɨɩɪɟɞɟ-

ɥɟɧɢɹ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɨɥɶɤɨ ɬɪɢ ɧɟɡɚɜɢɫɢɦɵɯ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ

6ɏ

= 0,

6Y = 0,

6Ɇ = 0, ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɨɣ.

ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɫɢɥɢɣ N1

ɢ N2.

6ɆȺ = 0 ;

N1r1 N2r2 ȇrP 0 ,

ɝɞɟ

2sin 45$

2 0,707

1,414 ɦ,

2 ɦ ɢ rP

3ɦ – ɩɥɟɱɢ ɫɢɥ N1, N2

5 ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ⱥ.

Ɋɚɫɱɟɬɧɨɟ ɡɧɚɱɟɧɢɟ ɫɢɥɵ Ɋ ɪɚɜɧɨ

ȇp

ȇɧJf

400 1,2

480 ɤɇ.

ɉɨɥɭɱɢɥɢ ɨɞɧɨ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɜɭɯ ɧɟɢɡɜɟɫɬɧɵɯ N1 ɢ N2:

1,44N1 + 2N2 = 480 3 = 1440 ɤɇɦ.

Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɪɚɫɫɦɨɬɪɢɦ ɫɯɟɦɭ ɞɟɮɨɪɦɚɰɢɢ ɫɢɫɬɟɦɵ.

ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ 5 ɠɟɫɬɤɚɹ ɛɚɥɤɚ Ⱥȼ ɩɨɜɟɪɧɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ⱥ ɧɚ ɦɚɥɵɣ ɭɝɨɥ E. ɋɬɟɪɠɧɢ ɩɪɢ ɷɬɨɦ ɩɨɥɭɱɚɬ ɭɞɥɢɧɟɧɢɹ 'l1 ɢ 'l2 (ɪɢɫ.2.12,ɛ).

ɂɡ ɫɯɟɦɵ ɞɟɮɨɪɦɚɰɢɢ ɧɚɯɨɞɢɦ:

	ɨ					'l1	ɨ
'l	DD sin 45 ,	'l	2	DD	,		sin 45 0,707 .
'l	DD sin 45 ,	'l		DD	,		sin 45 0,707 .
1	1			1		'l2
						'l2

ɉɨɥɭɱɟɧɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɹɜɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ ɞɟɮɨɪɦɚɰɢɢ ɫɢɫɬɟɦɵ. ȼɵɪɚɡɢɦ 'l1 ɢ 'l2 ɱɟɪɟɡ ɭɫɢɥɢɹ ɜ ɫɬɟɪɠɧɹɯ.

				'l	N1l1			,							'l		2	N2l2			,
				1		EF1											2	EF2
						EF1												EF2
ɝɞɟ l	22 22			2,828 ɦ,					l	2		2 ɦ – ɞɥɢɧɵ ɫɬɟɪɠɧɟɣ.
1										2
ɂɫɩɨɥɶɡɭɟɦ ɭɫɥɨɜɢɟ ɞɟɮɨɪɦɚɰɢɢ ɫɢɫɬɟɦɵ.
		'l1		N1l1		EF2			N1				l1			F2			N1		2,828	1,2 0,707,
																					2,0	1,2 0,707,
		'l	2	EF N			l	2	N		2		l	2			F N			2	2,0
			2	1	2			2			2			2			1			2
								N1			0,417N2 .

ɉɨɞɫɬɚɜɥɹɟɦ ɩɨɥɭɱɟɧɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɜ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɢ ɨɩɪɟɞɟɥɹɟɦ ɪɚɫɱɟɬɧɵɟ ɡɧɚɱɟɧɢɹ ɭɫɢɥɢɣ ɜ ɫɬɟɪɠɧɹɯ.

1,414 0,417N2 + 2N2 = 1440 , N2 = 556 ɤɇ , N1 = 0,417 556 = 232 ɤɇ .

Ɉɩɪɟɞɟɥɹɟɦ ɬɪɟɛɭɟɦɵɟ ɩɥɨɳɚɞɢ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɟɣ ɢɡ ɭɫɥɨɜɢɣ ɩɪɨɱɧɨɫɬɢ.

F t	N1		232	12,3ɫɦ2 ,	F	t	N2		556	29,4 ɫɦ2 .
1	Jc R 21 0,9				2		Jc R 21 0,9

ɉɪɨɜɟɪɹɟɦ ɜɵɩɨɥɧɟɧɢɟ ɩɪɢɧɹɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ F1 ɢ F2.

F2	29,4	2,39 !1,2 .
F	12,3	2,39 !1,2 .
1

ɉɪɢ ɩɨɞɛɨɪɟ ɫɟɱɟɧɢɹ ɩɟɪɜɨɝɨ ɫɬɟɪɠɧɹ ɟɝɨ ɩɥɨɳɚɞɶ ɧɚɞɨ ɭɜɟɥɢɱɢɬɶ ɢ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ

F		F2	29,4	24,5 ɫɦ2 .
	1,2		1,2
1

ɉɨɫɤɨɥɶɤɭ ɤɚɠɞɵɣ ɫɬɟɪɠɟɧɶ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɨɞɢɧɚɤɨɜɵɯ ɪɚɜɧɨɛɨɤɢɯ ɭɝɨɥɤɨɜ (ɪɢɫ.2.13), ɪɚɡɞɟɥɢɦ ɬɪɟɛɭɟɦɵɟ ɩɥɨɳɚɞɢ ɫɟɱɟɧɢɣ ɩɨɩɨɥɚɦ ɢ ɩɨ ɫɨɪɬɚɦɟɧɬɭ ɩɪɢɦɟɦ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɟɣ.

ɉɪɨɜɟɪɢɦ ɩɪɨɱɧɨɫɬɶ ɫɬɟɪɠɧɟɣ ɢ ɨɩɪɟɞɟɥɢɦ ɜɟɥɢ-
ɱɢɧɵ ɢɯ ɭɞɥɢɧɟɧɢɣ.	Ɋɢɫ.2.13
ɱɢɧɵ ɢɯ ɭɞɥɢɧɟɧɢɣ.
ɉɟɪɜɵɣ ɫɬɟɪɠɟɧɶ	ŋŊ90u90u7
F	2 12,3 24,6ɫɦ2 ;
1

232

9,43ɤɇ/ɫɦ2 Jc R

18,9 ɤɇ/ɫɦ2 ;

24,6

'l1

N1l1

232 282,8

0,106 ɫɦ .

EF1

1,2 2,1 104 24,6

ȼɬɨɪɨɣ ɫɬɟɪɠɟɧɶ

ŋŊ110u110u7

2 15,2 30,4 ɫɦ2 ;

556

18,3ɤɇ/ɫɦ2 Jc R

18,9 ɤɇ/ɫɦ2 ;

30,4

'l2

N2l2

556 200

0,145 ɫɦ .

EF2

1,2 2,1 104 30,4

ɉɪɨɱɧɨɫɬɶ ɫɬɟɪɠɧɟɣ ɨɛɟɫɩɟɱɟɧɚ. ɋɨɝɥɚɫɧɨ ɋɇɢɉ ɨɩɪɟɞɟɥɟɧɢɟ ɭɞɥɢɧɟɧɢɣ ɩɪɨɢɡɜɟɞɟɧɨ ɨɬ ɞɟɣɫɬɜɢɹ ɧɨɪɦɚɬɢɜɧɨɣ ɧɚɝɪɭɡɤɢ. ȼ ɫɢɥɭ ɷɬɨɝɨ ɪɚɫɱɟɬɧɵɟ ɡɧɚɱɟ-

ɧɢɹ ɭɫɢɥɢɣ ɪɚɡɞɟɥɟɧɵ ɧɚ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɞɟɠɧɨɫɬɢ ɩɨ ɧɚɝɪɭɡɤɟ Jf. Ɇɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ ɫɬɚɥɢ ɩɪɢɧɹɬ ɪɚɜɧɵɦ ȿ = 2,1 105 = 2,1 104 ɤɇ/ɫɦ2.

ɉɪɨɜɟɪɢɦ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ ɞɟɮɨɪɦɚɰɢɢ ɫɢɫɬɟɦɵ.

	'l1	0,106	0,73 z 0,707 .
	'l2	0,145	0,73 z 0,707 .
Ɋɚɫɯɨɠɞɟɧɢɟ ɫɨɫɬɚɜɥɹɟɬ:
'%		0,73 0,707		100% 3,3% .
		0,707

Ɉɩɪɟɞɟɥɢɦ ɜɟɥɢɱɢɧɭ ɪɚɡɪɭɲɚɸɳɟɣ (ɩɪɟɞɟɥɶɧɨɣ) ɫɢɥɵ. ɋɨɝɥɚɫɧɨ ɞɢɚɝɪɚɦɦɟ ɉɪɚɧɞɬɥɹ ɩɪɢɧɢɦɚɟɬɫɹ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɧɚɩɪɹɠɟɧɢɹ ɜ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ ɩɪɟɞɟɥɭ ɬɟɤɭɱɟɫɬɢ Vɬ, ɚ ɭɫɢɥɢɹ ɜ ɫɬɟɪɠɧɹɯ ɪɚɜɧɵ N1ɬ = VɬF1 ɢ N2ɬ = VɬF2 .

ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ.

6ɆȺ = 0 , VɬF1r1 VɬF2r2 ȇɩɪɟɞrɊ .

Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ

ȇɩɪɟɞ	Vɬ F1r1 F2r2	24 24,6 1,414 30,4 2	765 ɤɇ .
	rɊ	3

Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɩɚɫɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɨɪɦɚɬɢɜɧɨɦɭ ɡɧɚɱɟɧɢɸ ɫɢɥɵ ɪɚɜɟɧ

n	ȇɩɪɟɞ	765	1,91.
n	ȇ	400	1,91.
	ȇ	400
	ɧ

ȽɅȺȼȺ 3 ȼɇɍɌɊȿɇɇɂȿ ɍɋɂɅɂə ɉɊɂ ɉɊəɆɈɆ ɂɁȽɂȻȿ ȻȺɅɈɄ

ɂɉɅɈɋɄɂɏ ɋɌȿɊɀɇȿȼɕɏ ɋɂɋɌȿɆ

3.1.Ɉɫɧɨɜɧɵɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɮɨɪɦɭɥɵ

ɉɪɢ ɩɥɨɫɤɨɦ ɩɪɹɦɨɦ ɢɡɝɢɛɟ ɜ ɩɥɨɫɤɨɫɬɢ Ɉɯɭ ɜ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥ-

ɤɢ ɜɨɡɧɢɤɚɸɬ ɞɜɚ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɹ – ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ Qy ɢ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ Mz (ɪɢɫ.3.1).

				a)			a)	Mz > 0
				a)			Mz	Mz > 0
			z			Qy		Mz
			z			Qy		Mz
	ɨ							Ɋɚɫɬɹɧɭɬɵɟ
			Mz x	ɛ) Qy > 0			ɛ)	ɜɨɥɨɤɧɚ
			Mz x	ɛ) Qy > 0			ɛ)	ɜɨɥɨɤɧɚ
		Qy				Qy	Mz	Mz
		Qy


								Mz < 0
					Qy < 0			Mz < 0
					Qy < 0
Ɋɢɫ.3.1					Ɋɢɫ.3.2			Ɋɢɫ.3.3

ɉɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɨɣ, ɟɫɥɢ ɨɧɚ ɫɬɪɟɦɢɬɫɹ ɩɨɜɟɪɧɭɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɱɚɫɬɶ ɛɚɥɤɢ ɩɨ ɯɨɞɭ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ (ɪɢɫ.3.2). ɂɡɝɢɛɚɸɳɢɣ

ɦɨɦɟɧɬ ɫɱɢɬɚɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ, ɟɫɥɢ ɨɧ ɜɵɡɵɜɚɟɬ ɪɚɫɬɹɠɟɧɢɟ ɧɢɠɧɢɯ ɜɨɥɨɤɨɧ ɛɚɥɤɢ (ɪɢɫ.3.3).

ɉɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɜ ɥɸɛɨɦ ɫɟɱɟɧɢɢ	a)
ɛɚɥɤɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɩɪɨɟɤɰɢɣ
ɜɫɟɯ ɧɚɝɪɭɡɨɤ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɨɞɧɨɣ ɢɡ	ɨ


ɱɚɫɬɟɣ ɛɚɥɤɢ, ɧɚ ɧɨɪɦɚɥɶ ɤ ɟɺ ɨɫɢ.
ɂɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɜ ɥɸɛɨɦ ɫɟɱɟ-	Ɋ1		a
ɂɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɜ ɥɸɛɨɦ ɫɟɱɟ-	Ɋ1
ɧɢɢ ɛɚɥɤɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɦɨɦɟɧ-



ɬɨɜ ɜɫɟɯ ɧɚɝɪɭɡɨɤ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɨɞɧɨɣ



ɢɡ ɱɚɫɬɟɣ ɛɚɥɤɢ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɬɹ-	ɛ)	Ɋ2



ɠɟɫɬɢ ɞɚɧɧɨɝɨ ɫɟɱɟɧɢɹ.


Ɋɚɫɫɦɚɬɪɢɜɚɹ, ɧɚɩɪɢɦɟɪ, ɪɚɜɧɨɜɟɫɢɟ	ɨ


ɥɟɜɨɣ ɱɚɫɬɢ ɛɚɥɤɢ (ɪɢɫ.3.4), ɩɨɥɭɱɢɦ
¦Y 0 , P1 P2 Qy 0 ,	Ɋ1
¦Y 0 , P1 P2 Qy 0 ,	Ɋ1
Qy P1 P2 .
¦MC 0 , P1x P2 x a M Mz	0 , Mz

Ɋ2

Ɋ3 m

Ɋ3

Ɋ4

ɜ)

Ɇ =

Ɋ3d m

Mz Mz

y m

Ɋɢɫ.3.4

P1x P2 x a M .

Ɇɟɠɞɭ ɢɡɝɢɛɚɸɳɢɦ ɦɨɦɟɧɬɨɦ Mz, ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɨɣ Qy ɢ ɪɚɫɩɪɟɞɟɥɺɧɧɨɣ ɧɚɝɪɭɡɤɨɣ q ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ:

dQy	q ,	dM	z	Q		,	d	2M	z	q .	(3.1)
dx		dx			y		dx2

ɗɬɢ ɡɚɜɢɫɢɦɨɫɬɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɷɩɸɪ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ.

	3.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
	Ɂɚɞɚɱɚ 3.1		Ⱦɥɹ ɤɨɧɫɨɥɶɧɨɣ ɛɚɥɤɢ, ɢɡɨɛɪɚɠɺɧɧɨɣ
	30ɤɇ/ɦ	10ɤɇɦ
			ɧɚ ɪɢɫ.3.5, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy ɢ Mz.
		B	ɉɪɢ	ɩɨɫɬɪɨɟɧɢɢ ɷɩɸɪ ɩɨɩɟɪɟɱɧɵɯ
ɆȺ	A	C	ɫɢɥ Qy	ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ Mz ɜ
	RA	20ɤɇ	ɤɨɧɫɨɥɶɧɵɯ ɛɚɥɤɚɯ ɨɩɪɟɞɟɥɟɧɢɟ ɨɩɨɪ-
	2 ɦ	2 ɦ	ɧɵɯ ɪɟɚɤɰɢɢ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ, ɨɞɧɚɤɨ, ɟɫ-
			ɥɢ ɷɬɢ ɪɟɚɤɰɢɢ ɢɡɜɟɫɬɧɵ, ɬɨ ɨɧɢ ɦɨɝɭɬ

		20	ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ ɩɪɨɜɟɪɤɢ ɩɪɚɜɢɥɶɧɨ-
			ɫɬɢ ɷɩɸɪ Qy ɢ Mz.
40		(ɤɇ)	ƒY 0 , RA 30 2 20 0 ,
				RA	40 ɤɇ;

30		10	ƒMA 0 ,MA 30 2 1 20 2 10 0 ,
				MA	30 ɤɇɦ.
		M
	3,33		ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy ɢ Mz ɜ ɯɚɪɚɤ-
		(ɤɇɦ)
			ɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ, ɧɚɱɢɧɚɹ ɫɨ ɫɜɨ-
	Ɋɢɫ.3.5		ɛɨɞɧɨɝɨ ɤɨɧɰɚ.

ɋɟɱɟɧɢɟ ɯ = 4 ɦ , Qy = 0 , Mz = – 10 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ). ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɩɪɚɜɚ) , Qy = 0 , Mz = – 10 ɤɇɦ .

ɋɟɱɟɧɢɟ ɯ = 2 ɦ (ɫɥɟɜɚ) , Qy = – 20 ɤɇ , Mz = – 10 ɤɇɦ .

ɋɟɱɟɧɢɟ ɯ = 0 , Qy = – 20 + 30Â2 = 40 ɤɇ , Mz = – 10 – 30Â2Â1 + 20Â2 =

= – 30 ɤɇɦ (ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).

ɇɚ ɭɱɚɫɬɤɟ ȼɋ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɫɨɝɥɚɫɧɨ ɜɬɨɪɨɣ ɢɡ ɮɨɪɦɭɥ (3.1) ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ. ɇɚ ɭɱɚɫɬɤɟ Ⱥȼ ɫ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɺɧɧɨɣ ɧɚɝɪɭɡɤɨɣ (q = const) ɫɨɝɥɚɫɧɨ ɩɟɪɜɨɣ ɢ ɬɪɟɬɶɟɣ ɢɡ ɮɨɪɦɭɥ (3.1) ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ – ɩɨ ɡɚɤɨɧɭ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ ɫ ɜɵɩɭɤɥɨɫɬɶɸ, ɨɛɪɚɳɺɧɧɨɣ ɜ ɫɬɨɪɨɧɭ ɞɟɣɫɬɜɢɹ ɧɚɝɪɭɡɤɢ.

ɗɩɸɪɚ Mz ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦ ɜ ɫɟɱɟɧɢɢ ɯ0, ɝɞɟ Qy = 0. ȼɟɥɢɱɢɧɭ ɯ0 ɦɨɠɧɨ
ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ Qy:
	ɯ0	40	,	ɯ	0	1,33ɦ, 2 ɯ 0,67 ɦ.

	2 ɯ0	20				0

ɉɪɢ ɷɬɨɦ
Ɇmax Ɇz 1,33		10 30		0,672		20 0,67 3,33ɤɇɦ.
				2

ɗɩɸɪɵ Qy ɢ Mz ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.5.

Ɂɚɞɚɱɚ 3.2
Ⱦɥɹ ɤɨɧɫɨɥɶɧɨɣ ɛɚɥɤɢ,					ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ			12ɤɇ/ɦ	24 ɤɇɦ
ɪɢɫ.3.6, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy					ɢ	Mz.		12ɤɇ/ɦ	24 ɤɇɦ
ɪɢɫ.3.6, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy					ɢ	Mz.	ɜ ɯɚɪɚɤɬɟɪ-
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy					ɢ	Mz	ɜ ɯɚɪɚɤɬɟɪ-	A	B C
ɧɵɯ ɫɟɱɟɧɢɹɯ	ɛɚɥɤɢ,			ɧɚɱɢɧɚɹ		ɫɨ	ɫɜɨɛɨɞɧɨɝɨ	A	B C	ɆC
ɧɵɯ ɫɟɱɟɧɢɹɯ	ɛɚɥɤɢ,			ɧɚɱɢɧɚɹ		ɫɨ	ɫɜɨɛɨɞɧɨɝɨ		RC
ɤɨɧɰɚ.									RC
ɤɨɧɰɚ.								3 ɦ	1 ɦ
ɋɟɱɟɧɢɟ x = 0, Qy= 0, Mz = 0.								3 ɦ	1 ɦ
ɋɟɱɟɧɢɟ x = 0, Qy= 0, Mz = 0.
ɋɟɱɟɧɢɟ x = 3 ɦ (ɫɥɟɜɚ), Qy = 12 3 = 36 ɤɇ,										36
		Mz = 12 3 1,5 = 54 ɤɇɦ								36
		Mz = 12 3 1,5 = 54 ɤɇɦ
	(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).
ɋɟɱɟɧɢɟ x = 3ɦ (ɫɩɪɚɜɚ),					Qy= 36 ɤɇ,					66 (ɤɇ)
		Mz = 54 24 = 30 ɤɇɦ.							54	66 (ɤɇ)
		Mz = 54 24 = 30 ɤɇɦ.							54
ɋɟɱɟɧɢɟ x = 4ɦ,			Qy = – 36 ɤɇ,
		Mz = 12 3 2,5 24 = 66 ɤɇɦ								M
	(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).								30	M
	(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).									(ɤɇɦ)
ɇɚ ɭɱɚɫɬɤɟ		Ⱥȼ		ɫ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟ-						(ɤɇɦ)
ɥɟɧɧɨɣ ɧɚɝɪɭɡɤɨɣ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ (3.1) ɩɨ-								Ɋɢɫ.3.6
ɩɟɪɟɱɧɚɹ ɫɢɥɚ		Qy	ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ					Ɋɢɫ.3.6
ɩɟɪɟɱɧɚɹ ɫɢɥɚ		Qy	ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ

ɡɚɤɨɧɭ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ Ɇz ɩɨ ɡɚɤɨɧɭ

ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ ɫ ɜɵɩɭɤɥɨɫɬɶɸ, ɨɛɪɚɳɟɧɧɨɣ ɜ ɫɬɨɪɨɧɭ ɞɟɣɫɬɜɢɹ ɧɚɝɪɭɡɤɢ; ɩɪɢ ɷɬɨɦ ɜɨ ɜɫɟɯ ɫɟɱɟɧɢɹɯ ɨɧ ɜɵɡɵɜɚɟɬ ɪɚɫɬɹɠɟɧɢɟ ɜɟɪɯɧɢɯ ɜɨɥɨɤɨɧ. ɇɚ ɭɱɚɫɬɤɟ ȼɋ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ ɨɬɫɭɬɫɬɜɭɟɬ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ, ɩɪɢɱɟɦ ɜ ɫɟɱɟɧɢɢ ȼ ɢɦɟɟɬɫɹ ɫɤɚɱɨɤ, ɪɚɜɧɵɣ ɩɨ

ɜɟɥɢɱɢɧɟ ɩɪɢɥɨɠɟɧɧɨɦɭ ɦɨɦɟɧɬɭ 24 ɤɇɦ.

ɗɩɸɪɵ Qy ɢ Mz ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.6. ɂɡ ɷɬɢɯ ɷɩɸɪ ɫɥɟɞɭɟɬ, ɱɬɨ Rɋ =

= 36 ɤɇ, Ɇɋ = 66 ɤɇɦ.

Ɂɚɞɚɱɚ 3.3				18ɤɇ/ɦ		12ɤɇ
Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ, ɢɡɨɛɪɚɠɟɧ-				18ɤɇ/ɦ		12ɤɇ
Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɛɚɥɤɢ, ɢɡɨɛɪɚɠɟɧ-		A
ɧɨɣ ɧɚ ɪɢɫ.3.7, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy ɢ Mz.		A			C	B	x
ɧɨɣ ɧɚ ɪɢɫ.3.7, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ Qy ɢ Mz.					C	B	x
Ɋɚɫɱɟɬ ɲɚɪɧɢɪɧɨ ɨɩɺɪɬɨɣ ɛɚɥɤɢ ɧɟɨɛɯɨ-		RA					RB
ɞɢɦɨ ɧɚɱɢɧɚɬɶ ɫ ɨɩɪɟɞɟɥɟɧɢɹ ɨɩɨɪɧɵɯ ɪɟɚɤ-		RA	y	4 ɦ		2 ɦ	RB
ɰɢɣ.				x0	20	_	32
6ɆȺ = 0,	18 4 2 12 4 6Rȼ = 0,			x0	20	_	32
6ɆȺ = 0,	18 4 2 12 4 6Rȼ = 0,						Q
	Rȼ = 32 ɤɇ.		+				(ɤɇ)
6Ɇȼ = 0,	18 4 4 12 2 6RȺ = 0,	52					(ɤɇ)
6Ɇȼ = 0,	18 4 4 12 2 6RȺ = 0,	52
	RȺ = 52 ɤɇ.						M
6Y = 0 (ɩɪɨɜɟɪɤɚ),					+		M
18 4 12 32 52 = 84 84 = 0.					+	64	(ɤɇɦ)
						64
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy ɢ Mz ɜ ɯɚɪɚɤɬɟɪ-					75,37	ɢɡɥɨɦ
ɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.					Ɋɢɫ.3.7
					Ɋɢɫ.3.7
							29

	ɋɟɱɟɧɢɟ x = 0,			Qy = RȺ = 52 ɤɇ,			Ɇz = 0.
	ɋɟɱɟɧɢɟ x = 6 ɦ,			Qy = Rȼ = 32 ɤɇ, Ɇz = 0.
	ɋɟɱɟɧɢɟ ɯ = 4 ɦ (ɫɩɪɚɜɚ),				Qy	= 32 ɤɇ, Ɇz = 32 2 = 64 ɤɇɦ
	ɋɟɱɟɧɢɟ ɯ = 4 ɦ (ɫɥɟɜɚ),					(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
	ɋɟɱɟɧɢɟ ɯ = 4 ɦ (ɫɥɟɜɚ),				Qy = 32 12 = 20 ɤɇ,			Ɇz = 64 ɤɇɦ.
	ɇɚ ɭɱɚɫɬɤɟ		Ⱥɋ ɫ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɨɣ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ
ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ ɫɨ ɫɦɟɧɨɣ ɡɧɚɤɚ ɫ ɩɥɸɫɚ ɧɚ ɦɢɧɭɫ. ɂɡɝɢɛɚɸ-
ɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɡɚɤɨɧɭ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ ɢ ɢɦɟɟɬ ɷɤɫɬɪɟɦɚɥɶɧɨɟ
ɡɧɚɱɟɧɢɟ ɜ ɫɟɱɟɧɢɢ, ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ.
	ɇɚ ɭɱɚɫɬɤɟ		ɋȼ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ ɨɬɫɭɬɫɬɜɭɟɬ,					ɩɨɷɬɨɦɭ ɩɨɩɟɪɟɱɧɚɹ
ɫɢɥɚ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɚ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣ-
ɧɨɦɭ ɡɚɤɨɧɭ. ȼ ɫɟɱɟɧɢɢ ɋ, ɝɞɟ ɞɟɣɫɬɜɭɟɬ ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɫɢɥɚ 12 ɤɇ, ɧɚ ɷɩɸɪɟ
Qy	ɢɦɟɟɬɫɹ ɫɤɚɱɨɤ, ɪɚɜɧɵɣ ɩɨ ɜɟɥɢɱɢɧɟ ɩɪɢɥɨɠɟɧɧɨɣ ɫɢɥɟ, ɚ ɧɚ ɷɩɸɪɟ								Ɇz
ɢɦɟɟɬ ɦɟɫɬɨ ɢɡɥɨɦ.							Qy ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɭ ɫɟɱɟɧɢɹ
	ɂɡ ɩɨɞɨɛɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɧɚ ɷɩɸɪɟ						Qy ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɭ ɫɟɱɟɧɢɹ
ɯ0 , ɝɞɟ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ, ɢ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ ɨɩɪɟɞɟɥɹɟɦ
ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ.
			ɯ0	52	,	ɯ0 = 2,89 ɦ .
			4 ɯ0	20
		Ɇmax= Ɇz(2,89) = 52 2,89 18 2,89 2,89 = 75,37 ɤɇɦ
							2
					(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
	ɗɩɸɪɵ Qy		ɢ Ɇz ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ. 3.7.
	Ɂɚɞɚɱɚ 3.4.						Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɺɪɬɨɣ ɛɚɥɤɢ, ɢɡɨ-
12ɤɇɦ		18ɤɇɦ					Ⱦɥɹ ɲɚɪɧɢɪɧɨ ɨɩɺɪɬɨɣ ɛɚɥɤɢ, ɢɡɨ-
12ɤɇɦ		18ɤɇɦ				ɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.3.8, ɩɨɫɬɪɨɢɦ ɷɩɸ-
						ɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.3.8, ɩɨɫɬɪɨɢɦ ɷɩɸ-
A			C	B		ɪɵ Qy ɢ Mz.
	RA 3 ɦ		3 ɦ	RB			Ɉɩɪɟɞɟɥɹɟɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ.
	RA 3 ɦ		3 ɦ	RB		6ɆȺ = 0, 12 18 6Rȼ = 0,
5		_					Rȼ = 5 ɤɇ.
5		_			Q	6Ɇȼ = 0, 12 18 + 6RȺ = 0,
					Q		RȺ = 5 ɤɇ.
					(ɤɇ)		RȺ = 5 ɤɇ.
			3		(ɤɇ)	6Y = 0 (ɩɪɨɜɟɪɤɚ),		5 5 = 0.
						6Y = 0 (ɩɪɨɜɟɪɤɚ),		5 5 = 0.
					M		ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy ɢ Ɇz ɜ
	+				M		ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ Qy ɢ Ɇz ɜ
12	+		+		(ɤɇɦ)	ɯɚɪɚɤɬɟɪɧɵɯ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.
12		15			(ɤɇɦ)		ɋɟɱɟɧɢɟ ɯ = 0, Qy = – RA = – 5 ɤɇ,
		15					ɋɟɱɟɧɢɟ ɯ = 0, Qy = – RA = – 5 ɤɇ,
		Ɋɢɫ.3.8						Ɇz = 12 ɤɇɦ
							(ɪɚɫɬɹɧɭɬɵ ɧɢɠɧɢɟ ɜɨɥɨɤɧɚ).
	ɋɟɱɟɧɢɟ ɯ = 3 ɦ (ɫɥɟɜɚ),				Qy = 5 ɤɇ,		Ɇz = 12 5 3 = 3 ɤɇɦ
							(ɪɚɫɬɹɧɭɬɵ ɜɟɪɯɧɢɟ ɜɨɥɨɤɧɚ).

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