Сопротивление материалов / Nesmeyanov - Soprotivleniye materialov. Nestandartniye zadachi i podkhodi k ikh resheniyu 2005

	P		P	EI	l	P
ES		l
	ES		l			.
l	l			2EI
						A B
	Ɋɢɫ. 40		Ɋɢɫ. 41		Ɋɢɫ. 42

44.ɋɬɟɪɠɟɧɶ ɋ ɨɤɚɡɚɥɫɹ ɞɥɢɧɧɟɟ ɱɟɪɬɟɠɧɨɝɨ ɪɚɡɦɟɪɚ ɧɚ ɜɟɥɢɱɢɧɭ '. ɇɚɣɞɢɬɟ ɩɨɜɨɪɨɬɵ ɫɬɟɪɠɧɟɣ Ⱥ ɢ ȼ ɩɨɫɥɟ ɫɛɨɪɤɢ.

45.ɉɪɢ ɧɚɝɪɟɜɟ ɜɵɲɟ Tɷ ɫɬɨɣɤɚ ɢɡɨɝɧɭɬɚ. Ʉɚɤ ɜɵɝɥɹɞɢɬ ɷɩɸɪɚ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ?

46.Ɋɚɦɚ ɪɚɜɧɨɦɟɪɧɨ ɧɚɝɪɟɬɚ. Ʉɚɤ ɫɦɟɫɬɢɬɶ ɨɩɨɪɵ, ɱɬɨɛɵ ɧɚɩɪɹɠɟɧɢɹ ɨɬɫɭɬɫɬɜɨɜɚɥɢ? ɍɤɚɠɢɬɟ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɜɚɪɢɚɧɬɵ ɬɚɤɢɯ ɫɦɟɳɟɧɢɣ ɨɩɨɪ.

	B
			l
b a c	C	l T	l
	A
Q			l
Ɋɢɫ. 43	Ɋɢɫ.44	Ɋɢɫ.45	Ɋɢɫ.46

47.Ɋɚɦɚ ɧɚɝɪɭɠɟɧɚ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɨɣ. Ɉɩɪɟɞɟɥɢɬɟ ɢɡɦɟɧɟɧɢɹ ɭɝɥɨɜ

D ɢ E.

48.ɋɬɟɪɠɟɧɶ Ⱥȼ ɨɯɥɚɞɢɥɢ ɧɚ T0. ɇɚɣɞɢɬɟ ɬɟɩɥɨɜɵɟ ɧɚɩɪɹɠɟɧɢɹ.

	q	4l
3l	4l A
2l	3l S	2S	S
		4l	B	4l
	q
Ɋɢɫ.47		Ɋɢɫ.48

49. ȼɵɱɢɫɥɢɬɟ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɞɟɮɨɪɦɚɰɢɢ.

P

l l

l S 2S S

l l

P

Ɋɢɫ.49

		Pɚɡɞɟɥ 3. ɇȿɄɈɌɈɊɕȿ Ɋȿɒȿɇɂə
		ɇɢɠɟ ɩɪɢɜɟɞɟɧ ɪɹɞ ɡɚɞɚɱ, ɤɨɬɨɪɵɟ ɬɪɭɞɧɨ ɨɬɧɟɫɬɢ ɤ ɨɞɧɨɣ ɤɚɤɨɣ-ɥɢɛɨ ɬɟɦɟ,
	ɧɨ ɪɟɲɟɧɢɟ ɤɨɬɨɪɵɯ, ɧɚ ɧɚɲ ɜɡɝɥɹɞ, ɦɨɠɟɬ ɞɚɬɶ ɩɨɥɟɡɧɭɸ ɢɧɮɨɪɦɚɰɢɸ ɨ ɧɟɤɨɬɨ-
	ɪɵɯ ɧɟɫɬɚɧɞɚɪɬɧɵɯ ɫɩɨɫɨɛɚɯ ɪɚɫɱɟɬɨɜ. ɑɚɫɬɶ ɩɪɢɜɟɞɟɧɧɵɯ ɡɚɞɚɱ ɜɡɹɬɚ ɢɡ ɥɢɬɟɪɚ-
	ɬɭɪɵ, ɧɨ ɡɞɟɫɶ ɨɧɢ ɪɟɲɟɧɵ ɦɟɬɨɞɚɦɢ, ɧɟɫɤɨɥɶɤɨ ɨɬɥɢɱɚɸɳɢɦɢɫɹ ɨɬ ɢɡɜɟɫɬɧɵɯ.
		Ɂɚɞɚɱɚ 1. Ƚɢɛɤɢɣ ɫɬɟɪɠɟɧɶ ɥɟɠɢɬ ɧɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɣ ɩɨɜɟɪɯɧɨɫɬɢ (pɢɫ.1).
	Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɦɢɧɢɦɚɥɶɧɨɟ ɨɬɧɨɲɟɧɢɟ b/a, ɩɪɢ ɤɨɬɨɪɨɦ ɫɬɟɪɠɟɧɶ ɤɚɫɚɟɬɫɹ ɩɨ-
	ɜɟɪɯɧɨɫɬɢ ɬɨɥɶɤɨ ɜ ɞɜɭɯ ɫɟɱɟɧɢɹɯ [7].
		Ɋɟɲɟɧɢɟ. ɉɪɢ ɛɨɥɶɲɨɦ ɡɧɚɱɟɧɢɢ b (pɢɫ.1) ɫɬɟɪɠɟɧɶ ɤɚɫɚɟɬɫɹ ɩɨɜɟɪɯɧɨɫɬɢ ɜ
	ɞɜɭɯ ɫɟɱɟɧɢɹɯ; ɩɪɢ ɷɬɨɦ ɪɚɡɦɟɪ b ɦɨɠɟɬ ɛɵɬɶ ɭɜɟɥɢɱɟɧ ɛɟɡ ɢɡɦɟɧɟɧɢɹ ɫɯɟɦɵ ɤɨɧ-
	ɬɚɤɬɚ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɟɱɟɧɢɟ A ɧɚ ɪɢɫ.1 ɩɨɜɟɪɧɭɬɨ ɧɚ ɧɟɤɨɬɨɪɵɣ ɭɝɨɥ Dz 0. ɇɚ
	ɪɢɫ.2 ɩɨɤɚɡɚɧ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɪɚɡɦɟɪ b ɦɟɧɶɲɟ ɢɫɤɨɦɨɝɨ bmin. ȼ ɷɬɨɦ ɩɨɥɨɠɟɧɢɢ
					q		ɤɚɫɚɧɢɟ	ɩɪɨɢɫɯɨɞɢɬ		ɜ
	A	B					ɛɟɫɱɢɫɥɟɧɧɨɦ ɱɢɫɥɟ ɫɟ-
							ɱɟɧɢɣ. ɇɚɩɨɦɧɢɦ: ɜɨɡ-
							ɦɨɠɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɫɨ-
		a	b	r	R	b	ɫɪɟɞɨɬɨɱɟɧɧɨɣ		ɪɟɚɤɰɢɢ
							R ɨɛɫɭɠɞɚɥɚɫɶ		ɪɚɧɟɟ	ɜ
		Ɋɢɫ.1			Ɋɢɫ.2		ɡɚɞɚɱɟ,	ɩɪɟɞɫɬɚɜɥɟɧɧɨɣ
							ɧɚ ɪɢɫ.4.4 (ɝɥɚɜɚ 4). Ɇɟ-
	ɠɞɭ ɞɜɭɦɹ ɩɨɤɚɡɚɧɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ, ɨɱɟɜɢɞɧɨ, ɟɫɬɶ ɬɚɤɨɟ, ɤɨɝɞɚ ɤɨɧɬɚɤɬ ɩɪɨɢɫ-
	ɯɨɞɢɬ ɬɨɥɶɤɨ ɜ ɞɜɭɯ ɬɨɱɤɚɯ, ɧɨ ɞɚɥɶɧɟɣɲɟɟ ɭɦɟɧɶɲɟɧɢɟ ɪɚɡɦɟɪɚ b ɩɪɢɜɨɞɢɬ ɤ ɫɢ-
	A	q	B	ɬɭɚɰɢɢ ɧɚ ɪɢɫ.2. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɢɫɤɨɦɨɦ
				ɡɧɚɱɟɧɢɢ bmin		ɫɩɪɚɜɟɞɥɢɜɚ ɫɯɟɦɚ ɧɚ ɪɢɫ.1, ɜ ɤɨ-
				ɬɨɪɨɣ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɫɟɱɟɧɢɹ A ɪɚɜɟɧ ɧɭɥɸ. Ɋɚɫ-
				ɱɟɬɧɚɹ ɫɯɟɦɚ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ pɢɫ.3. ɋ ɩɨɦɨɳɶɸ
		a	b	ɢɧɬɟɝɪɚɥɚ Ɇɨɪɚ ɧɚɯɨɞɢɦ ɭɝɨɥ D ɢ ɩɪɢɪɚɜɧɢɜɚɟɦ
				ɟɝɨ ɧɭɥɸ. Ɉɬɜɟɬ: bmin | 0.71a.
		Ɋɢɫ.3
					Ɂɚɞɚɱɚ 2. ɋɬɟɪɠɟɧɶ BD ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧ
	ɜ ɬɨɱɤɟ B ɢ ɞɨ ɩɪɢɥɨɠɟɧɢɹ ɧɚɝɪɭɡɤɢ ɫɜɨɛɨɞɧɨ ɥɟɠɢɬ ɧɚ ɛɚɥɤɟ AC (pɢɫ.4). Ɉɩɪɟɞɟ-
	ɥɢɬɶ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɫɟɱɟɧɢɹ B ɩɪɢ ɧɚɝɪɭɠɟɧɢɢ ɜɟɪɯɧɟɝɨ ɫɬɟɪɠɧɹ ɦɨɦɟɧɬɨɦ M.
	ɉɨɩɟɪɟɱɧɵɟ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɟɣ ɨɞɢɧɚɤɨɜɵ.
		Ɋɟɲɟɧɢɟ. ɇɚ ɪɢɫ.5ɚ ɩɨɤɚɡɚɧɚ ɤɨɧɫɬɪɭɤɰɢɹ ɜ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ. ȼ
	D		B	ɡɨɧɟ ɤɨɧɬɚɤɬɚ ɩɪɨɝɢɛɵ ɢ ɭɝɥɵ ɩɨɜɨɪɨɬɚ ɫɨɜɩɚɞɚ-
				ɸɬ, ɩɨɷɬɨɦɭ ɩɪɚɜɚɹ ɱɚɫɬɶ ɜɟɪɯɧɟɣ ɢ ɥɟɜɚɹ ɱɚɫɬɶ
	A	l	C	ɧɢɠɧɟɣ ɛɚɥɨɤ ɦɨɝɭɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɨɞɧɚ
				ɛɚɥɤɚ, ɡɚɳɟɦɥɟɧɧɚɹ ɜ ɥɟɜɨɦ ɫɟɱɟɧɢɢ ɢ ɲɚɪɧɢɪɧɨ
		Ɋɢɫ.4	M
				ɡɚɤɪɟɩɥɟɧɧɚɹ ɜ ɩɪɚɜɨɦ (pɢɫ.5ɛ). Ɂɚɞɚɱɚ ɨ ɩɨɜɨɪɨ-
	ɬɟ ɫɟɱɟɧɢɹ B ɬɚɤɨɣ ɛɚɥɤɢ ɪɟɲɚɟɬɫɹ ɫɬɚɧɞɚɪɬɧɵɦ ɩɭɬɟɦ – ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɢɧɬɟ-
	ɝɪɚɥɚ Ɇɨɪɚ. ɉɨɫɬɪɨɟɧɢɟ ɷɩɸɪɵ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɦɨɠɟɬ ɛɵɬɶ ɭɫɤɨɪɟɧɨ ɫɥɟ-
	ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.

		M		Ɍɚɤ ɤɚɤ ɜ ɩɪɨɥɟɬɟ ɛɚɥɤɢ ɧɚɝɪɭɡɤɚ
			a)	ɨɬɫɭɬɫɬɜɭɟɬ, ɬɨ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɹɜ-
				ɥɹɟɬɫɹ ɥɢɧɟɣɧɨɣ ɮɭɧɤɰɢɟɣ z (pɢɫ.5ɜ).
			ɛ)	ȿɞɢɧɢɱɧɚɹ ɷɩɸɪɚ ɦɟɬɨɞɚ ɫɢɥ ɢɦɟɟɬ ɜɢɞ,
		M		ɩɨɤɚɡɚɧɧɵɣ ɧɚ ɪɢɫ.5ɝ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
				ɤɨɧɬɪɨɥɟɦ	ɩɪɚɜɢɥɶɧɨɫɬɢ	ɩɨɫɬɪɨɟɧɢɹ
	"ɗɆ"	MB =M		ɷɩɸɪ ɜ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɵɯ ɫɢɫ-
M			ɜ)	ɬɟɦɚɯ, ɩɪɨɢɡɜɨɞɢɦɵɦ ɩɨ ɩɪɚɜɢɥɭ ȼɟɪɟ-
	"ɗɆ"	1		ɳɚɝɢɧɚ, ɧɭɠɧɨ ɭɦɧɨɠɢɬɶ ɩɥɨɳɚɞɶ ɗM1,
A
	1		ɝ)	ɧɚ ɨɪɞɢɧɚɬɭ ɗM ɩɨɞ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ
l	"ɗɆɜɫ”			ɗM1. Ɋɟɡɭɥɶɬɚɬ ɞɨɥɠɟɧ ɪɚɜɧɹɬɶɫɹ ɧɭɥɸ
1			ɞ)	(ɫɦɵɫɥ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ). ɗɬɨ
		1		ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ ɭɩɨɦɹ-
	Ɋɢɫ.5			ɧɭɬɚɹ ɨɪɞɢɧɚɬɚ ɪɚɜɧɚ ɧɭɥɸ. Ɉɬɫɸɞɚ ɫɥɟ-
				ɞɭɟɬ, ɱɬɨ ɗɆ ɩɟɪɟɫɟɤɚɟɬ ɨɫɶ ɛɚɥɤɢ ɜ ɫɟ-
ɱɟɧɢɢ, ɧɚɯɨɞɹɳɟɦɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ l/3 ɨɬ ɥɟɜɨɣ ɨɩɨɪɵ. Ɍɨɝɞɚ MA= –0.5MB=0.5M.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɝɥɚ ɩɨɜɨɪɨɬɚ ɩɟɪɟɦɧɨɠɚɟɦ ɷɩɸɪɵ ɗɆ ɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ
ɗM ɜɫ (pɢɫ.5ɞ). ɇɚɯɨɞɢɦ TB = Ml/(4EI).

Ɂɚɞɚɱɚ 3. ɇɚɣɬɢ ɜɧɭɬɪɟɧɧɢɟ ɫɢɥɨɜɵɟ ɮɚɤɬɨɪɵ ɜ ɪɚɦɟ ɩɨɫɬɨɹɧɧɨɝɨ ɩɨɩɟɪɟɱ-
ɧɨɝɨ ɫɟɱɟɧɢɹ, ɧɚɝɪɭɠɟɧɧɨɣ ɫɢɥɨɣ P (ɪɢɫ.6).
Ɋɟɲɟɧɢɟ. Ɂɚɞɚɱɚ ɞɟɜɹɬɶ ɪɚɡ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɚ, ɟɫɥɢ ɪɟɲɚɬɶ ɟɟ ɬɪɚɞɢ-
			C		ɰɢɨɧɧɵɦ ɨɛɪɚɡɨɦ. Ɉɞɧɚɤɨ,
a	a	P			ɩɪɟɧɟɛɪɟɝɚɹ		ɩɟɪɟɦɟɳɟɧɢɹ-
P					ɦɢ ɨɬ ɪɚɫɬɹɠɟɧɢɹ (ɫɠɚɬɢɹ)
					ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɟɪɟɦɟɳɟ-
2a					ɧɢɹɦɢ ɨɬ ɢɡɝɢɛɚ, ɩɨɥɭɱɢɦ,
a	a	B	B	A	ɱɬɨ ɷɥɟɦɟɧɬɵ ɪɚɦɵ ɪɚɛɨɬɚ-
	A				ɸɬ ɬɨɥɶɤɨ ɧɚ ɪɚɫɬɹɠɟɧɢɟ-
Ɋɢɫ.6		Ɋɢɫ.7			ɫɠɚɬɢɟ (ɩɨɞɨɛɧɨ ɮɟɪɦɟ).
ɪɢɱɧɨɫɬɶ ɡɚɞɚɱɢ (pɢɫ.7).					ɍɱɬɟɦ	ɬɚɤɠɟ	ɤɨɫɨɫɢɦɦɟɬ-
Ɂɚɩɢɫɚɜ ɫɭɦɦɭ ɩɪɨɟɤɰɢɣ ɜɫɟɯ ɫɢɥ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɨɫɶ, ɧɚɣɞɟɦ

¦ x = 2AcosD – P = 0; A = P/(2cosD).

ɂɡ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ ɫɭɦɦɵ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ C ɩɨɥɭɱɢɦ 2Ba – 2Aa sin D = 0, ɨɬɤɭɞɚ B = P tg D/2. ɍɫɢɥɢɹ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɭɱɚɫɬɤɟ, ɤɚɤ ɫɥɟɞɭɟɬ ɢɡ ɤɨɫɨɣ ɫɢɦɦɟɬɪɢɢ, ɪɚɜɧɵ: – P/2 (ɥɟɜɚɹ ɱɚɫɬɶ) ɢ P/2 (ɩɪɚɜɚɹ).

Ɂɚɞɚɱɚ 4. Ɏɟɪɦɚ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɪɠɧɟɣ (ɪɢɫ.8). ɇɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ.

Ɋɟɲɟɧɢɟ. ȼ ɡɚɞɚɱɟ ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɧɟɢɡɜɟɫɬɧɵɯ ɢ ɫɬɟɩɟɧɶ ɫɬɚɬɢɱɟɫɤɨɣ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɢ ɬɚɤɠɟ ɛɟɫɤɨɧɟɱɧɚ; ɫɬɚɧɞɚɪɬɧɵɟ ɦɟɬɨɞɵ ɡɞɟɫɶ ɧɟ ɝɨɞɹɬɫɹ.

Ɋɢɫ.10

Ɋɢɫ.9

		ɇɨ ɫ ɷɬɢɦ ɠɟ ɫɜɹɡɚɧɚ ɨɫɨɛɟɧɧɨɫɬɶ, ɩɨɦɨɝɚɸɳɚɹ ɟɟ ɪɟɲɢɬɶ. Ɋɚ-
		ɡɨɛɶɟɦ ɤɨɧɫɬɪɭɤɰɢɸ ɧɚ ɞɜɟ ɱɚɫɬɢ: ɷɥɟɦɟɧɬ ɢɡ ɬɪɟɯ ɫɬɟɪɠɧɟɣ ɢ
		ɨɫɬɚɥɶɧɨɟ (pɢɫ.9, 10). Ɍɨɱɤɚ A ɫɦɟɳɚɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ P
		ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɠɟɫɬɤɨɫɬɶɸ ɮɟɪɦɵ c ('A=P/c), ɧɨ ɠɟɫɬɤɨɫɬɶ
	60ɨ 60ɨ	ɮɟɪɦɵ ɧɚ ɪɢɫ.10 ɬɚ ɠɟ, ɱɬɨ ɢ ɢɫɯɨɞɧɨɣ ɮɟɪɦɵ (ɬɨ ɠɟ ɛɟɫɱɢɫ-
		ɥɟɧɧɨɟ ɦɧɨɠɟɫɬɜɨ ɫɬɟɪɠɧɟɣ), ɬ.ɟ. 'B=P1/c. Ɇɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ
	Ⱥ	ɬɪɟɯɫɬɟɪɠɧɟɜɚɹ ɮɟɪɦɚ (ɪɢɫ.9) ɜ ɬɨɱɤɟ B ɢɦɟɟɬ ɭɩɪɭɝɭɸ ɨɩɨɪɭ
	Ɋ	(pɢɫ.11) ɫ ɠɟɫɬɤɨɫɬɶɸ c. Ɉɧɚ ɦɨɠɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɩɚɪɚɥ-
	Ɋɢɫ.8	ɥɟɥɶɧɨɟ ɫɨɟɞɢɧɟɧɢɟ ɞɜɭɯ ɤɨɧɫɬɪɭɤɰɢɣ (pɢɫ.12); ɩɪɢɥɨɠɢɜ ɜɦɟ-
		ɫɬɨ F, Fc ɟɞɢɧɢɱɧɵɟ ɫɢɥɵ, ɧɚɣɞɟɦ ɩɨɞɚɬɥɢɜɨɫɬɢ Oɥ, Oɩ ɷɬɢɯ

ɤɨɧɫɬɪɭɤɰɢɣ ɤɚɤ ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɟɤ A, Ac ɜ ɥɟɜɨɣ ɢ ɜ ɩɪɚɜɨɣ ɤɨɧɫɬɪɭɤɰɢɹɯ. ɇɨɪɦɚɥɶɧɵɟ ɫɢɥɵ ɜ ɨɛɨɢɯ ɫɬɟɪɠɧɹɯ ɥɟɜɨɣ ɮɟɪɦɵ ɪɚɜɧɵ ɟɞɢɧɢɰɟ;

Oɥ='A=	P
=l/(ES)+l/(ES)=2l/(ES)	1
	B
(ɞɥɹ ɥɟɜɨɣ ɮɟɪɦɵ) ɢ
Oɩ='Ac =1/c+l/(ES)	A	B
(ɫɭɦɦɢɪɭɟɦ ɜɵɬɹɠɤɭ	P	P1

ɩɪɭɠɢɧɵ ɢ ɫɬɟɪɠɧɹ).

Ɉɛɳɚɹ ɠɟɫɬɤɨɫɬɶ ɪɚɜɧɚ ɫɭɦɦɟ ɠɟɫɬɤɨɫɬɟɣ ɷɬɢɯ ɞɜɭɯ ɮɟɪɦ, ɬ.ɟ.

c B

A

P

Ɋɢɫ.11

c= 1/Oɥ+1/Oɩ= ES/(2l)+1/(1/c+ l/(ES)),

ɨɬɤɭɞɚ ɧɟɬɪɭɞɧɨ ɧɚɣɬɢ c (ɪɟɲɢɜ ɤɜɚɞɪɚɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢ ɨɬɛɪɨɫɢɜ ɨɬɪɢɰɚɬɟɥɶɧɵɣ ɤɨɪɟɧɶ) c= ES/l. Ɍɨ ɠɟ ɡɧɚɱɟɧɢɟ ɠɟɫɬɤɨɫɬɢ ɦɨɠɧɨ ɛɵɥɨ ɩɨɥɭɱɢɬɶ, ɪɚɫɤɪɵɜ ɦɟɬɨɞɨɦ ɫɢɥ ɫɬɚɬɢɱɟɫɤɭɸ ɧɟɨɩɪɟɞɟɥɢɦɨɫɬɶ ɜ ɡɚɞɚɱɟ ɧɚ ɪɢɫ 11.

Ɂɧɚɹ ɠɟɫɬɤɨɫɬɶ ɮɟɪɦɵ, ɧɚɣɞɟɦ ɫɦɟɳɟɧɢɟ ɬɨɱɤɢ A ('A=P/c=Pl/(ES)) ɢ ɫɢɥɭ

		Fc ɧɚ pɢɫ.12 (ɨɧɚ ɠɟ – P1 ɧɚ pɢɫ.10) ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɨ-
A	c	ɞɚɬɥɢɜɨɫɬɶɸ Oɩ=1/ɫ+l/(ES)=2l/(ES):
	A'	P1=Fc='Ac ɫɥ='Ac /Oɥ=Pl ES/(ES 2l)=P/2.
	F	Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɚɫɬɶ ɤɨɧɫɬɪɭɤɰɢɢ, ɩɨɤɚɡɚɧɧɚɹ ɧɚ
	F'
	Ɋɢɫ.12	ɪɢɫ.10, ɧɚɝɪɭɠɟɧɚ ɜɞɜɨɟ ɦɟɧɶɲɟ, ɱɟɦ ɧɚ pɢɫ.9 ɢ ɦɚɤɫɢ-

ɦɚɥɶɧɵɟ ɭɫɢɥɢɹ – ɜ ɬɪɟɯ ɧɢɠɧɢɯ ɫɬɟɪɠɧɹɯ – ɪɚɜɧɵ ɩɨ P/2. Ɇɚɤɫɢɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɪɚɜɧɨ P/(2S).

Ɂɚɞɚɱɚ 5. Ʉɨɧɫɨɥɶɧɵɣ ɫɬɟɪɠɟɧɶ (ɪɢɫ.13) ɜɵ-			1
ɩɨɥɧɟɧ ɢɡ ɞɜɭɯ ɦɟɬɚɥɥɨɜ; ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ ɜ ɜɟɪɯ-		h	h/2
ɧɟɣ ɩɨɥɨɜɢɧɟ ɜɞɜɨɟ ɦɟɧɶɲɟ, ɱɟɦ ɜ ɧɢɠɧɟɣ. ɇɚɣɬɢ
ɧɚɩɪɹɠɟɧɢɹ.	b	l	2
Ɋɟɲɟɧɢɟ. ɋɨɨɛɪɚɠɟɧɢɹ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɝɢɩɨɬɟ-			Ɋ
		Ɋɢɫ. 13
ɡɟ ɩɥɨɫɤɢɯ ɫɟɱɟɧɢɣ ɜ ɫɬɟɪɠɧɹɯ, ɨɫɬɚɸɬɫɹ ɜ ɫɢɥɟ ɢ
ɞɥɹ ɛɢɦɟɬɚɥɥɢɱɟɫɤɨɣ ɛɚɥɤɢ. Ɂɧɚɱɢɬ, ɞɟɮɨɪɦɚɰɢɢ ɥɢɧɟɣɧɨ ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɜɵɫɨɬɟ,
ɧɨ ɧɟɣɬɪɚɥɶɧɚɹ ɥɢɧɢɹ ɦɨɠɟɬ ɭɠɟ ɧɟ ɧɚɯɨɞɢɬɶɫɹ ɧɚ ɩɨɥɨɜɢɧɟ ɜɵɫɨɬɵ (ɫɢɦɦɟɬɪɢɹ

ɫɬɟɪɠɧɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɫɪɟɞɧɟɝɨ ɫɟɱɟɧɢɹ ɡɞɟɫɶ ɨɬɫɭɬɫɬɜɭɟɬ). Ɉɛɨ-

ɡɧɚɱɢɜ ' – ɫɦɟɳɟɧɢɟ ɧɟɣɬɪɚɥɶɧɨɣ ɥɢɧɢɢ ɫ ɫɟɪɟɞɢɧɵ ɜɧɢɡ, ɡɚɩɢɲɟɦ

H=F(y+'),

(1)

ɝɞɟ F, ɤɚɤ ɢ ɪɚɧɶɲɟ, ɤɪɢɜɢɡɧɚ ɛɚɥɤɢ, ɚ ɜɟɥɢɱɢɧɚ y ɨɬɫɱɢɬɵɜɚɟɬɫɹ ɨɬ ɫɪɟɞɢɧɵ

ɜɜɟɪɯ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ,

V=ȿF(y+'),

ȿ=ȿ0 ɩɪɢ y>0;

E=2E0 ɩɪɢ y<0.

(2)

ɇɨɪɦɚɥɶɧɚɹ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ ' :

h/ 2

0

³ ȿ0F(y+')dy+

³ 2ȿ0F(y+')dy=0,

0

h/ 2

ɨɬɤɭɞɚ '=h/12. Ɂɧɚɹ ', ɧɟɬɪɭɞɧɨ ɧɚɣɬɢ ɫɜɹɡɶ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɫ ɤɪɢɜɢɡɧɨɣ

h/2

0

M=³ VydS=F ³ ȿ0y(y+')dS+F ³

2ȿ0y(y+')dS=11/8E0IF ,

(3)

ɢɥɢ

S

0

h/ 2

F=8M/(11ȿ0I), I{ bh3/12.

(4)

Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɵ ɢɫɤɚɥɢ ɦɨɦɟɧɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɪɟɞɧɟɣ, ɚ ɧɟ ɧɟɣɬɪɚɥɶɧɨɣ ɥɢ-

ɧɢɢ (ɜɟɥɢɱɢɧɚ y ɨɬɫɱɢɬɵɜɚɟɬɫɹ ɨɬ ɫɩɚɹ). ɗɬɨ ɧɟ ɜɟɞɟɬ ɤ ɨɲɢɛɤɟ, ɩɨɫɤɨɥɶɤɭ ɪɚɜɧɨ-

ɞɟɣɫɬɜɭɸɳɚɹ (ɧɨɪɦɚɥɶɧɚɹ) ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ ɢ ɦɨɦɟɧɬ ɧɚɩɪɹɠɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɥɸɛɨɣ ɨɫɢ ɨɞɢɧɚɤɨɜ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɣɞɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɤɪɢɜɢɡɧɵ ɨɬ

7

7

h/12

ɜɟɥɢɱɢɧɵ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ. Ɂɧɚɹ ɟɟ, ɢɡ ɜɵɪɚɠɟ-

h/2

ɧɢɹ (2) ɧɚɯɨɞɢɦ ɧɚɩɪɹɠɟɧɢɹ. ɂɯ ɷɩɸɪɚ ɩɨɤɚɡɚɧɚ ɧɚ

ɪɢɫ.14,

ɡɞɟɫɶ

ɞɥɹ

ɤɪɚɬɤɨɫɬɢ

ɨɩɭɳɟɧ

ɦɧɨɠɢɬɟɥɶ

h/2

2

-

E=h/12 (8M/(11I))=4M/(33W), W{ bh2/6. ɉɨɫɬɪɨɢɜ

5

10

Ɋɢɫ.14

ɷɩɸɪɭ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ (ɨɛɵɱɧɵɦ ɨɛɪɚɡɨɦ), ɨɬ-

ɫɸɞɚ ɧɚɯɨɞɢɦ ɜɫɟ ɧɚɩɪɹɠɟɧɢɹ.

Ⱦɨɛɚɜɢɦ ɤ ɫɤɚɡɚɧɧɨɦɭ, ɱɬɨ ɟɫɥɢ, ɧɚɨɛɨɪɨɬ, ɜ ɬɨɱɤɟ y= – h/12 ɩɪɢɥɨɠɢɬɶ ɪɚɫ-

1

ɬɹɝɢɜɚɸɳɭɸ ɫɢɥɭ (ɪɢɫ.15), ɬɨ ɫɬɟɪ-

h/2

ɠɟɧɶ ɧɟ ɛɭɞɟɬ ɢɡɝɢɛɚɬɶɫɹ, ɨɧ ɛɭɞɟɬ

P

2

ɜɵɬɹɝɢɜɚɬɶɫɹ ɪɚɜɧɨɦɟɪɧɨ ɩɨ ɜɵɫɨ-

b

l

2

ɬɟ: H=H0, V=ȿH0 (ɷɩɸɪɵ H ɢ V ɩɨɤɚ-

0

ɡɚɧɵ ɧɚ ɪɢɫ.16). ɇɟɬɪɭɞɧɨ ɭɫɬɚɧɨ-

ɜɢɬɶ, ɱɬɨ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɩɨɤɚ-

Ɋɢɫ. 15

Ɋɢɫ.16

ɡɚɧɧɵɯ

ɧɚɩɪɹɠɟɧɢɣ

ɩɪɢɥɨɠɟɧɚ

ɢɦɟɧɧɨ ɜ ɬɨɱɤɟ y= – h/12. Ɉɧɚ ɪɚɜɧɚ 3/2E0H0S=N (ɨɛɨɡɧɚɱɟɧɨ S=bh). Ɉɬɫɸɞɚ ɧɚ-

ɯɨɞɢɦ ɦɧɨɠɢɬɟɥɶ J ɧɚ ɪɢɫ. 16 (J = H0E0 =2N/(3S)) ɢ ɫɜɹɡɶ ɦɟɠɞɭ ɫɢɥɨɣ ɢ ɞɟɮɨɪ-

ɦɚɰɢɟɣ: H0=N/(3E0S/2). ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɜɟɫɬɢ ɪɚɫɱɟɬɵ ɧɚ ɩɪɨɱɧɨɫɬɶ ɢ ɠɟɫɬɤɨɫɬɶ. ȼ

ɱɚɫɬɧɨɫɬɢ, ɦɚɤɫɢɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɪɚɜɧɨ 2J = 4N/(3S) – ɜɵɲɟ ɫɪɟɞɧɟɝɨ N/S.

ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɜɧɟɲɧɹɹ ɧɚɝɪɭɡɤɚ ɦɨɠɟɬ ɨɩɪɟɞɟɥɹɬɶ ɜ ɤɚɠɞɨɦ ɫɟɱɟɧɢɢ ɧɨɪ-

ɦɚɥɶɧɭɸ ɫɢɥɭ ɢ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ. ɇɚɩɪɹɠɟɧɢɹ ɩɪɢ ɷɬɨɦ ɫɤɥɚɞɵɜɚɸɬɫɹ ɢɡ ɩɨ-

ɤɚɡɚɧɧɵɯ

ɞɜɭɯ

ɷɩɸɪ

(ɪɢɫ.

14,

16)

ɫ

ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ

ɦɧɨɠɢɬɟɥɹɦɢ

E=4M/(33W) ɢ J = 2N/(3S).

Ɂɚɞɚɱɚ 6. Ȼɚɥɤɚ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ buh ɧɚɝɪɟɜɚɟɬɫɹ ɩɨ ɡɚ-

	ɤɨɧɭ, ɩɨɤɚɡɚɧɧɨɦɭ ɧɚ ɪɢɫ.17. Ɉɩɪɟɞɟɥɢɬɶ ɦɚɤɫɢɦɚɥɶ-		m
	ɧɨɟ ɧɚɩɪɹɠɟɧɢɟ.	h
	Ɋɟɲɟɧɢɟ. Ɂɚɞɚɱɭ ɭɞɨɛɧɨ ɪɟɲɢɬɶ ɦɟɬɨɞɨɦ ɮɢɤ-	l	ȼ
	ɬɢɜɧɵɯ ɧɚɝɪɭɡɨɤ (ɫɦ. ɩ.5.9 ɩɟɪɜɨɝɨ ɪɚɡɞɟɥɚ). Ⱦɥɹ ɷɬɨɝɨ A b
	ɫɥɟɞɭɟɬ ɧɚɝɪɭɡɢɬɶ ɫɜɨɛɨɞɧɵɣ ɬɨɪɟɰ ɮɢɤɬɢɜɧɨɣ ɧɚ-	Ɋɢɫ.17

	ɝɪɭɡɤɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ q =EDTb (ɧɚ ɟɞɢɧɢɰɭ ɜɵɫɨɬɵ h, ɚ ɧɟ ɩɥɨɳɚɞɢ) – ɪɢɫ.18ɚ –
		ɢ ɧɚɣɬɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɮɢɤɬɢɜɧɵɟ ɧɨɪɦɚɥɶɧɵɟ
		ɧɚɩɪɹɠɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɹɯ. ɇɚ ɪɢɫ.18ɛ
	ɚ)	q* ɩɨɤɚɡɚɧɚ ɷɤɜɢɜɚɥɟɧɬɧɚɹ ɡɚɞɚɱɚ, ɝɪɭɡɨɜɚɹ ɢ ɟɞɢ-
		ɧɢɱɧɚɹ ɷɩɸɪɵ. ɉɨɫɥɟ ɩɟɪɟɦɧɨɠɟɧɢɹ ɩɨɫɥɟɞɧɢɯ

"ɗ"

q*h2/12

Mɪ

M1 l

ɛ)

Ɋɢɫ.18

q*h2/24 q*h/2

q*h2/12

Ɋɢɫ.19

q*

x1*

ɗN *

ɗM*

ɧɚɯɨɞɢɦ X1 ɢɡ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ (X1 =q h2/(8l)) ɢ ɫɬɪɨɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɷɩɸɪɵ ɮɢɤɬɢɜɧɵɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ (ɪɢɫ. 19).

Ɂɚɦɟɬɢɦ, ɱɬɨ ɫɨɫɬɚɜɥɹɬɶ ɢ ɪɟɲɚɬɶ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ, ɟɫɥɢ ɟɳɟ ɪɚɡ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢɟɦ, ɩɪɢɦɟɧɟɧɧɵɣ ɜ ɡɚɞɚɱɟ 2 (ɪɢɫ.5): ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɫɥɟɜɚ ɞɨɥɠɧɨ ɛɵɬɶ ɜ ɞɜɚ ɪɚɡɚ, ɦɟɧɶɲɟ, ɱɟɦ ɫɩɪɚɜɚ, ɱɬɨɛɵ ɩɨɞ ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɟɞɢɧɢɱɧɨɣ ɷɩɸɪɵ ɛɵɥ ɧɨɥɶ ɧɚ ɫɭɦɦɚɪɧɨɣ ɷɩɸɪɟ.

ɂɡ ɩɨɥɭɱɚɸɳɢɯɫɹ ɮɢɤɬɢɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ V*=N*/S+M*(z)y 12/(bh2) ɫɥɟɞɭɟɬ ɜɵɱɟɫɬɶ ɜɟɥɢɱɢɧɭ EDT=ED(y+h/2)Tm/h. ɇɚɩɪɢɦɟɪ, ɜ ɫɟɱɟɧɢɢ Ⱥ ɩɨɥɭɱɢɦ ɷɩɸɪɭ ɧɚɩɪɹɠɟɧɢɣ, ɩɨɤɚɡɚɧɧɭɸ ɧɚ ɪɢɫ.20. Ɂɞɟɫɶ G = EDTm.

2		3	Ɂɚɞɚɱɚ 7. Ɍɟɪɦɨɪɟɥɟ, ɪɚɫɱɟɬɧɚɹ ɫɯɟɦɚ ɤɨ-
	4	4	ɬɨɪɨɝɨ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ.21, ɧɚɝɪɟɜɚɟɬɫɹ ɧɚ 'T.
			Ɉɩɪɟɞɟɥɢɬɶ	ɦɚɤɫɢɦɚɥɶɧɵɣ	ɩɪɨɝɢɛ
			(D1=2D2=2D0, E1=E2=E).
		=	Ɋɟɲɟɧɢɟ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɹ
		3	ɢɫɩɨɥɶɡɭɟɬɫɹ ɢɧɬɟɝɪɚɥ Ɇɨɪɚ: ɞɟɮɨɪɦɚɰɢɢ ɢ
			ɫɦɟɳɟɧɢɹ ɛɟɪɭɬɫɹ ɢɡ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ, ɚ ɫɢɥɵ ɢ
		4	ɧɚɩɪɹɠɟɧɢɹ	ɢɡ

Ɋɢɫ.20	ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ		1
	(ɪɢɫ.22ɛ). Ʉɚɤ ɨɬ-		h/2	h
ɦɟɱɚɥɨɫɶ ɜ ɩ. 5.8 ɪɚɡɞɟɥɚ 1, ɟɫɥɢ ɢɡɜɟɫɬɧɨ ɬɨɱɧɨɟ
ɪɟɲɟɧɢɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɢ, ɬɨ ɧɟ ɧɭɠɧɨ ɛɟɫ-		b	l 2	ȼ
ɩɨɤɨɢɬɶɫɹ ɨ ɫɨɜɦɟɫɬɧɨɫɬɢ ɞɟɮɨɪɦɚɰɢɣ (ɜ ɧɚɲɟɦ			Ɋɢɫ. 21
ɫɥɭɱɚɟ, ɬɟɩɥɨɜɵɯ) ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ, ɫɱɢɬɚɹ, ɱɬɨ

ɫɦɟɳɟɧɢɟ ɧɚɯɨɞɢɬɫɹ ɤɚɤ ɢɧɬɟɝɪɚɥ

	X=³³ VɜɫHɌdzdS.
	l S
Ɂɞɟɫɶ Vɜɫ=Ɇɜɫ/I y, HɌ=2D0'T ɩɪɢ y>0 ɢ D0'T				ɩɪɢ y<0.
ɂɧɬɟɝɪɢɪɭɹ ɩɨ ɫɟɱɟɧɢɸ ɧɚ ɭɱɚɫɬɤɟ ɛɚɥɤɢ ɞɥɢɧɨɣ dz, ɩɨɥɭɱɢɦ
		h / 2		h / 2
	dX{ dz	³ Ɇɜɫ/I yHɌdS= Ɇɜɫ/I dzb ³			HɌydy.
		h / 2		h / 2
ɉɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ ɫɨɞɟɪɠɢɬ ɞɜɚ ɫɥɚɝɚɟɦɵɯ:
	h/ 2
	³ 2D0'Tydy=2D0'T(h2/2) 1/2=D0'Th2/4;
	0
	0
	³ D0'Tydy= – D0'T(– h/2)2 1/2= – D0'Th2/8.
	h/ 2
ȼ ɢɬɨɝɟ
	dX= Ɇɜɫ/(bh3) 12bD0'Th2/8 dz=3/2ɆɜɫD0'T/h dz.
ɉɨɫɤɨɥɶɤɭ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ, ɫɢɥɚ ɜɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɟ ɩɪɢɤɥɚɞɵɜɚɟɬɫɹ
ɩɨɫɪɟɞɢɧɟ,				Ɋ*				Ɋ*
	³ Ɇɜɫdz=l/4 l 1/2=l2/8,			1				1
								Ɋ*
	l			Ɋ2*
ɩɨɥɭɱɚɟɦ ɡɧɚɱɟɧɢɟ ɩɪɨɝɢɛɚ ɜ ɫɟɪɟɞɢɧɟ ɩɪɨ-						ɚ)	l	2
ɥɟɬɚ		2	(5)	ɗM *		M *
	Xɫ=³ dX=3/(16h) D0'Tl .					1
	l
ȿɫɥɢ ɷɬɭ ɡɚɞɚɱɭ ɪɟɲɚɬɶ ɦɟɬɨɞɨɦ ɮɢɤ-						ɋ
ɬɢɜɧɵɯ ɧɚɝɪɭɡɨɤ, ɬɨ ɪɟɲɟɧɢɟ ɨɤɚɡɵɜɚɟɬɫɹ
				ɗM ɜɫ
ɩɪɨɳɟ. ȼɦɟɫɬɨ ɧɚɝɪɟɜɚ ɫɥɟɞɭɟɬ ɩɪɢɥɨɠɢɬɶ ɧɚ
ɬɨɪɰɚɯ	ɮɢɤɬɢɜɧɵɟ	ɪɚɫɬɹɝɢɜɚɸɳɢɟ	ɫɢɥɵ				l/4
P1*=2D0'TE0S/2, P2*=D0'TE0S/2			(S=bh,			ɛ)
ɪɢɫ.22ɚ). Ɇɨɞɭɥɢ ɭɩɪɭɝɨɫɬɢ ɩɨɥɨɫ ɨɞɢɧɚɤɨ-						Ɋɢɫ.22
ɜɵ, ɩɨɷɬɨɦɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɢɯ ɤɚɤ ɨɞɧɭ ɛɚɥɤɭ ɫɟɱɟɧɢɟɦ buh (ɬɨ ɟɫɬɶ I=bh3/12).
ȼɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɟ (ɪɢɫ.22ɛ) ɜɨɡɧɢɤɚɟɬ ɬɨɥɶɤɨ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ, ɩɨ-
ɷɬɨɦɭ ɮɢɤɬɢɜɧɚɹ ɧɨɪɦɚɥɶɧɚɹ ɫɢɥɚ ɧɚɫ ɧɟ ɢɧɬɟɪɟɫɭɟɬ; ɮɢɤɬɢɜɧɵɣ ɢɡɝɢɛɚɸɳɢɣ
ɦɨɦɟɧɬ M*= (P1* – P2*) h/4. ɉɟɪɟɦɧɨɠɢɜ ɷɬɢ ɷɩɸɪɵ, ɩɨɥɭɱɢɦ
	Xc=2(P1* – P2*)h/4 l/2 l/8/(EI).
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɨɬɜɟɬ (5).
ɇɚɩɨɦɧɢɦ, ɱɬɨ, ɧɟɫɦɨɬɪɹ ɧɚ ɮɢɤɬɢɜɧɨɫɬɶ ɧɚɝɪɭɡɤɢ (ɢ ɧɚɩɪɹɠɟɧɢɣ), ɫɦɟɳɟ-
ɧɢɹ ɢ ɞɟɮɨɪɦɚɰɢɢ ɜ ɷɬɨɦ ɦɟɬɨɞɟ ɨɬɧɸɞɶ ɧɟ ɮɢɤɬɢɜɧɵ.

Ɂɚɞɚɱɚ 8. ɇɚɣɬɢ ɜ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɟ ɦɚɤɫɢɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ.

Ɋɟɲɟɧɢɟ. Ɏɢɤɬɢɜɧɵɟ ɧɚɩɪɹɠɟɧɢɹ V*=M*/I y+N*/S (N*=P1*+P2*) ɜ ɷɬɨɣ ɡɚɞɚɱɟ ɩɨɫɬɨɹɧɧɵ ɩɨ ɞɥɢɧɟ, ɡɧɚɱɢɬ, ɜɫɟ ɫɟɱɟɧɢɹ ɪɚɜɧɨɨɩɚɫɧɵ. ɂɫɬɢɧɧɵɟ ɦɟɧɶɲɟ ɮɢɤɬɢɜɧɵɯ ɧɚ ED'T (ɫɦ. ɪɚɡɞɟɥ 1, ɩ. 5.8). ȼ ɢɬɨɝɟ (ɪɢɫ.23, ɝɞɟ G=D0E'T) ɤ ɧɚɩɪɹɠɟɧɢɹɦ ɨɬ ɮɢɤɬɢɜɧɨɝɨ ɢɡɝɢɛɚ M*/Ix y, ɢɡɦɟɧɹɸɳɢɦɫɹ ɩɨ ɜɵɫɨɬɟ ɥɢɧɟɣɧɨ, ɞɨɛɚɜɥɹɸɬɫɹ ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɨɬ ɞɜɭɯ ɞɪɭɝɢɯ ɫɥɚɝɚɟɦɵɯ (ɨɧɢ, ɤɚɤ ɢ

ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɨɬɜɟɱɚɸɬ ɧɭɥɟɜɨɣ ɧɨɪɦɚɥɶɧɨɣ ɫɢɥɟ). ɇɚɯɨɞɢɦ: ɦɚɤɫɢɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɪɚɜɧɨ D0E'T/2 – ɧɚ ɫɬɵɤɟ ɦɟɬɚɥɥɨɜ.

Ɂɚɞɚɱɚ 9. ɉɭɫɬɶ ɜ ɬɟɪɦɨɪɟɥɟ (ɪɢɫ.21) ɦɨɞɭɥɢ

ɭɩɪɭɝɨɫɬɢ ɞɜɭɯ ɦɟɬɚɥɥɨɜ ɪɚɡɥɢɱɚɸɬɫɹ: E2=2E1= =2E0. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ Vmax ɢ ɩɪɨɝɢɛ ɜ ɫɟɪɟɞɢɧɟ Xc.

Ɋɟɲɟɧɢɟ. ȼ ɷɬɨɣ ɡɚɞɚɱɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɟɞɵɞɭɳɢɯ ɡɚɞɚɱ 5, 7, 8. Ɏɢɤɬɢɜɧɵɟ ɧɚɝɪɭɡɤɢ ɩɪɢ ɪɚɜɧɨɦɟɪɧɨɦ ɧɚɝɪɟɜɟ ɫɜɨɞɹɬɫɹ ɤ ɫɢɥɚɦ ɩɨ ɬɨɪɰɚɦ (ɪɢɫ.22ɚ), ɤɨɬɨɪɵɟ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɡɚɞɚɱɢ 7, ɨɤɚɡɵɜɚɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦɢ:

3 /4

/2

/4

P

*=E

D

'TS/2=E D

'TS=E

D

'TS/2=P

*.

1

1

1

0 0

2

2

ɫɢɥɚ N*

2

+

Ɂɧɚɱɢɬ,

ɮɢɤɬɢɜɧɚɹ

ɧɨɪɦɚɥɶɧɚɹ

ɪɚɜɧɚ

P1*+P2*=2E0D0'TS. ɇɨ ɩɪɢ ɡɚɞɚɧɧɨɦ ɪɚɡɥɢɱɢɢ

-

+

-

/2

ɦɨɞɭɥɟɣ ɭɩɪɭɝɨɫɬɢ (ɬɚɤɨɦ ɠɟ, ɤɚɤ ɢ ɜ ɡɚɞɚɱɟ 5)

=

+

ɮɢɤɬɢɜɧɵɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɧɟ ɪɚɜɟɧ ɧɭɥɸ,

+

/2

ɟɝɨ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ

-

3 /4

/4

ɜ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɧɚ ɜɟɥɢɱɢɧɭ h/12 ɧɢɠɟ

(M ) (N ) TE

ɫɪɟɞɢɧɵ (ɫɦ. ɪɢɫ.14). Ɉɬɫɸɞɚ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ,

*

*

h/12=E0D0'TSh/6.

Ɋɢɫ.23

Ɇ =N

ȼɵɱɢɫɥɹɹ ɩɟɪɟɦɟɳɟɧɢɟ Xc ɫ ɩɨɦɨɳɶɸ ɢɧɬɟ-

ɝɪɚɥɚ Ɇɨɪɚ, ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɨɞɧɢɦ ɢɡ ɫɨɦɧɨɠɢɬɟɥɟɣ ɜ ɢɧɬɟɝɪɚɥɟ ɹɜɥɹ-

ɟɬɫɹ ɞɟɮɨɪɦɚɰɢɹ ɨɫɟɜɨɣ ɥɢɧɢɢ (ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɡɚɞɚɱɟ

– ɪɢɫ.22ɛ – ɢɦɟɟɬɫɹ ɥɢɲɶ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ, ɬɚɤɨɣ ɞɟɮɨɪɦɚɰɢɟɣ ɹɜɥɹɟɬɫɹ ɤɪɢ-

ɜɢɡɧɚ). ɇɚɩɪɢɦɟɪ, ɤɪɢɜɢɡɧɚ ɜ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɟ (ɮɢɤɬɢɜɧɵɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ

Ɇ*) ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɥɹ ɛɢɦɟɬɚɥɥɢɱɟɫɤɨɣ ɛɚɥɤɢ ɫ ɡɚɞɚɧɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɦɨɞɭ-

ɥɟɣ ɭɩɪɭɝɨɫɬɢ ɜɵɪɚɠɟɧɢɟɦ (4), ɝɞɟ ɢɫɤɨɦɵɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɫɥɟɞɭɟɬ ɡɚɦɟ-

ɧɢɬɶ ɧɚ ɮɢɤɬɢɜɧɵɣ. Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ ɢɧɬɟɝɪɚɥ Ɇɨɪɚ, ɧɚɣɞɟɦ:

	Xc=³ FɆɜɫdz=l2/8 Ɇ*8 /(11E0I)=l2E0D0'TSh/(11E0I 6)=2/(11h)D0'Tl2.
	l
	ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɧɚɩɪɹɠɟɧɢɣ ɭɞɨɛɧɨ ɧɚɱɚɬɶ ɫ ɮɢɤɬɢɜɧɵɯ,
	ɨɬɨɛɪɚɠɚɟɦɵɯ ɷɩɸɪɚɦɢ ɧɚ ɪɢɫ. 14 ɢ 16 ɫ ɦɧɨɠɢɬɟɥɹɦɢ 4Ɇ*/(33W)
	ɢ 2N*/(3S) (ɢɡ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 5). ɂɫɬɢɧɧɵɟ ɧɚɯɨɞɹɬɫɹ ɩɨɫɥɟ ɜɵɱɢ-							5	-
	ɬɚɧɢɹ ɨɬɫɸɞɚ ɜɟɥɢɱɢɧɵ ED'T, ɜ ɧɚɲɟɣ ɡɚɞɚɱɟ ɩɨɫɬɨɹɧɧɨɣ ɩɨ ɜɵ-							+
									3
	ɫɨɬɟ (2E0D0'T). ȼ ɢɬɨɝɟ ɩɨɥɭɱɢɦ ɷɩɸɪɭ, ɩɨɤɚɡɚɧɧɭɸ ɧɚ ɪɢɫ. 24,								3
	ɝɞɟ G=2/11E0D0'T. Ɋɚɡɧɵɟ ɦɟɬɚɥɥɵ ɢɦɟɸɬ ɨɛɵɱɧɨ ɢ ɪɚɡɥɢɱɧɵɟ
	ɞɨɩɭɫɤɚɟɦɵɟ ɧɚɩɪɹɠɟɧɢɹ, ɩɨɷɬɨɦɭ ɫɥɟɞɭɟɬ ɧɚɯɨɞɢɬɶ ɦɚɤɫɢɦɚɥɶɧɨɟ							Ɋɢɫ.24
	ɧɚɩɪɹɠɟɧɢɟ	ɜ	ɤɚɠɞɨɣ	ɢɡ	ɩɨɥɨɜɢɧ	ɫɬɟɪɠɧɹ;	ɧɚɣɞɟɦ
	Vmax=10/11E0D0'T – ɞɥɹ ɧɢɠɧɟɣ ɩɨɥɨɜɢɧɵ ɢ 6/11E0D0'T – ɞɥɹ ɜɟɪɯɧɟɣ.

Ɂɚɞɚɱɚ 10. ɉɪɢ ɢɡɝɨɬɨɜɥɟɧɢɢ ɤɨɧɫɬɪɭɤɰɢɢ (ɪɢɫ.25) ɪɚɦɭ ɭɞɚɥɨɫɶ ɫɞɟɥɚɬɶ ɬɨɱɧɨ, ɧɨ ɨɩɨɪɵ 1, 2 ɢ 3 ɨɤɚɡɚɥɢɫɶ ɫɦɟɳɟɧɧɵɦɢ ɧɚ ɜɟɥɢɱɢɧɵ '1, '2 ɢ '3. Ʉɚɤ ɫɦɟɫɬɢɬɶ ɩɨɫɥɟɞɧɸɸ ɨɩɨɪɭ (4), ɱɬɨɛɵ ɫɨɛɪɚɬɶ ɤɨɧɫɬɪɭɤɰɢɸ ɛɟɡ ɭɫɢɥɢɣ?

Ɋɟɲɟɧɢɟ. Ɂɚɞɚɱɚ ɫɬɚɬɢɱɟɫɤɢ ɧɟɨɩɪɟɞɟɥɢɦɚ (ɤ=1) ɢ ɧɟɢɡɜɟɫɬɧɚɹ ɏ1 – ɟɞɢɧɫɬɜɟɧɧɚɹ ɧɚɝɪɭɡɤɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɪɚɦɭ – ɧɚɯɨɞɢɬɫɹ ɢɡ ɤɚɧɨɧɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ

	G11ɏ1=¦Ri1'i. ɉɨɞɚɬɥɢɜɨɫɬɶ G11 ɧɟ ɪɚɜɧɚ ɧɭɥɸ, ɡɧɚɱɢɬ, ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɏ1 ɛɵɥɚ
	2	3l		ɪɚɜɧɚ ɧɭɥɸ, ɧɭɠɧɨ, ɱɬɨɛɵ ɫɭɦɦɚ ¦Ri1'i ɛɵɥɚ ɧɭɥɟɜɨɣ. ȼɵ-
				ɛɪɚɜ ɨɫɧɨɜɧɭɸ ɫɢɫɬɟɦɭ, ɨɬɛɪɚɫɵɜɚɹ ɜɫɟ ɱɟɬɵɪɟ ɫɦɟɳɚɸɳɢɟ-
	3l		1	ɫɹ ɫɜɹɡɢ, ɢ ɡɚɩɢɫɚɜ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ
				¦ɏ=R3= 0, ¦Y= R1 – R2+ R4=0, ¦MA= R2 2l+ R1 l= 0,
		A		ɧɚɣɞɟɦ:
			3	R3=0, R1=2X1, R2= – X1, R4= – 3X1.
		2l		Ɂɚɞɚɜ X1=1, ɧɚɯɨɞɢɦ ɪɟɚɤɰɢɢ Ri1: 2, –1, 0, –3. Ɉɬɫɸɞɚ
		Ɋɢɫ.25		ɭɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɢɦɟɟɬ ɜɢɞ
				¦Ri1'i=2'1 – '2 – 3'4=0.
				Ɉɬɜɟɬ: '4=(2'1 – '2)/3.
	Ɂɚɞɚɱɚ 11. ɉɥɚɫɬɢɧɚ (ɪɢɫ.26) ɩɪɢɤɪɟɩɥɟɧɚ ɤ ɫɬɨɣɤɟ ɬɪɟɦɹ ɡɚɤɥɟɩɤɚɦɢ. Ɂɚ-
	ɤɥɟɩɤɚ ɫɪɟɡɚɟɬɫɹ ɩɪɢ ɫɢɥɟ Q=15 kH. ɇɚɝɪɭɡɤɚ P ɧɚ ɩɥɚɫɬɢɧɭ ɪɚɜɧɚ 10 kH. Ɉɩɪɟ-
	ɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɩɚɫɚ ɩɪɨɱɧɨɫɬɢ ɤɨɧɫɬɪɭɤɰɢɢ.
	Ɋɟɲɟɧɢɟ. Ɏɨɪɦɚ ɩɥɚɫɬɢɧɵ ɢ ɬɨɱɤɚ ɩɪɢɥɨɠɟɧɢɹ ɫɢ-					Ɋ
	ɥɵ P ɧɚ ɥɢɧɢɢ ɟɟ ɞɟɣɫɬɜɢɹ ɛɟɡɪɚɡɥɢɱɧɵ ɞɥɹ ɫɭɳɟɫɬɜɚ ɡɚ-
	ɞɚɱɢ,	ɩɨɷɬɨɦɭ ɤɨɧɫɬɪɭɤɰɢɹ ɦɨɠɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ				L=1,5l
	ɫɢɦɦɟɬɪɢɱɧɚɹ (ɪɢɫ.27), ɚ ɧɚɝɪɭɡɤɚ –				ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɚɹ.
						2l
	Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɩɚɫɚ ɧɭɠɧɨ ɧɚɣɬɢ ɩɪɟ-
	ɞɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɢɥɵ P, ɩɪɢ ɤɨɬɨɪɨɦ ɨɛɪɚɡɭɟɬɫɹ ɩɥɚ-
	ɫɬɢɱɟɫɤɢɣ ɦɟɯɚɧɢɡɦ: ɫɪɟɡ ɞɜɭɯ ɢɥɢ ɬɪɟɯ ɡɚɤɥɟɩɨɤ (ɫɪɟɡ					Ɋɢɫ. 26
	ɨɞɧɨɣ ɦɟɯɚɧɢɡɦɚ ɧɟ ɨɛɪɚɡɭɟɬ). Ɇɟɯɚɧɢɡɦ ɞɨɥɠɟɧ ɛɵɬɶ
	ɬɚɤɠɟ ɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵɦ: ɷɬɨ ɩɨɜɨɪɨɬ ɩɥɚɫɬɢɧɵ ɜɨɤɪɭɝ ɬɨɱɤɢ, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ
	ɫɢɦɦɟɬɪɢɢ.
	ȿɫɥɢ ɷɬɨɣ ɬɨɱɤɨɣ ɹɜɥɹɟɬɫɹ ɫɪɟɞɧɹɹ ɡɚɤɥɟɩɤɚ, ɬɨ ɫɪɟɡɚɸɬɫɹ ɞɜɟ, ɢ ɫɢɥɵ, ɞɟɣɫɬ-
	ɜɭɸɳɢɟ ɧɚ ɩɥɚɫɬɢɧɭ, ɩɨɤɚɡɚɧɵ ɧɚ pɢɫ.28. Ɋɚɜɧɨɜɟɫɢɟ (ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɨɬɧɨɫɢ-
	ɬɟɥɶɧɨ, ɧɚɩɪɢɦɟɪ, ɫɪɟɞɧɟɣ ɡɚɤɥɟɩɤɢ) ɬɪɟɛɭɟɬ, ɱɬɨɛɵ 2lQ ɛɵɥɨ ɪɚɜɧɨ 1.5lP0, ɨɬɤɭ-
	ɞɚ P0=4Q/3=20 kH. ɇɨ ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɪɟɞɧɹɹ ɡɚɤɥɟɩɤɚ ɧɟ ɪɚɡɪɭɲɚɹɫɶ, ɜɵ-
	ɞɟɪɠɢɜɚɟɬ ɫɢɥɭ 20 kH, ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ.
			Q	P0	ȼ ɩɨɥɧɨɦ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɢɧɟɦɚɬɢɱɟ-
					ɫɤɢɦ ɦɟɬɨɞɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɟɞɟɥɶɧɨɣ ɧɚ-
			Q		ɝɪɭɡɤɢ ɪɚɫɫɦɨɬɪɢɦ ɞɪɭɝɨɣ ɦɟɯɚɧɢɡɦ: ɫɪɟɡ
					ɜɫɟɯ ɬɪɟɯ ɡɚɤɥɟɩɨɤ ɢ ɩɨɜɨɪɨɬ ɩɥɚɫɬɢɧɵ ɜɨ-
		P		P0	ɤɪɭɝ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ A (ɪɢɫ.29). ɍɫɥɨɜɢɹ
					ɪɚɜɧɨɜɟɫɢɹ:

Ɋɢɫ. Ɋɢɫ. 28 1.5P0 l=2lQs; Q(1+2c)=P0.

Ɂɞɟɫɶ s – ɫɢɧɭɫ, ɚ c – ɤɨɫɢɧɭɫ ɭɝɥɚ D (ɩɨɤɚɡɚɧɧɨɝɨ ɧɚ ɪɢɫ.29ɚ). ɂɫɤɥɸɱɚɹ ɭɝɨɥ (s2+c2=1), ɧɚɣɞɟɦ