Сопротивление материалов. Часть 1. Учебное пособие
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ɡɵɜɚɸɳɢɣ ɪɚɫɬɹɠɟɧɢɟ ɜɟɪɯɧɢɯ ɜɨɥɨɤɨɧ, ɚ ɜ ɫɟɱɟɧɢɢ ȼ – ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɫɢɥɚ Ɋ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɧɢɡ. ȼɟɥɢɱɢɧɵ ɫɢɥɵ Ɋ ɢ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ RA ɢ RB ɩɪɨɳɟ ɜɫɟɝɨ ɨɩɪɟɞɟɥɢɬɶ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɨɫɬɪɨɢɜ ɷɩɸɪɭ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ, ɢɫɩɨɥɶɡɭɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ
Qy |
dM z |
tg , ɝɞɟ ɭɝɨɥ ɧɚɤɥɨɧɚ ɥɢɧɢɢ ɷɩɸɪɵ Ɇz ɤ ɨɫɢ ɛɚɥɤɢ. |
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ɇɚ ɭɱɚɫɬɤɟ Ⱥȼ (Ɇz |
ɜɨɡɪɚɫɬɚɟɬ, |
Ɇ=12ɤɇɦ Ɋ=40ɤɇ |
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q =10ɤɇ/ɦ |
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Qy > 0). |
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ɚ) Ⱥ |
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Qɭ |
Mȼ ɆȺ |
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18 ɤɇ. |
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RȺ=18ɤɇ |
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Rɋ=42ɤɇ |
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ɇɚ ɭɱɚɫɬɤɟ ȼɋ (Ɇz |
ɭɛɵɜɚɟɬ, Qy < 0). |
ɛ) |
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20 24 22ɤɇ. |
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ɇɚ ɭɱɚɫɬɤɟ ɋD (Ɇz ɜɨɡɪɚɫɬɚɟɬ, |
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Qy > 0). |
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QD = 0, QC = qa = 10 2 = 20 ɤɇ. |
Ɋɢɫ.3.20 |
ɋɯɟɦɚ ɧɚɝɪɭɡɤɢ ɢ ɷɩɸɪɚ ɩɨɩɟɪɟɱɧɵɯ |
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ɫɢɥ ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.3.20,ɚ,ɛ. |
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ɉɨ ɜɟɥɢɱɢɧɚɦ ɫɤɚɱɤɨɜ ɧɚ ɷɩɸɪɟ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ ɧɚɯɨɞɢɦ |
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Ɋ = 22 + 18 = 40 ɤɇ; RC = 20 + 22 = 42 ɤɇ, |
RȺ = QȺ = 18 ɤɇ. |
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ȽɅȺȼȺ 4 |
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ɈɉɊȿȾȿɅȿɇɂȿ ɇȺɉɊəɀȿɇɂɃ ȼ ȻȺɅɄȺɏ ɉɊɂ ɂɁȽɂȻȿ |
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ɂ ɊȺɋɑȿɌɕ ɇȺ ɉɊɈɑɇɈɋɌɖ |
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4.1. Ɉɫɧɨɜɧɵɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɮɨɪɦɭɥɵ |
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ɉɪɢ ɩɪɹɦɨɦ ɢɡɝɢɛɟ ɛɚɥɤɢ ɜ ɩɥɨɫɤɨɫɬɢ |
Ɉɯɭ ɜ ɟɺ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɹɯ ɞɟɣ- |
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ɫɬɜɭɸɬ ɧɨɪɦɚɥɶɧɵɟ ɢ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ |
M z |
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(4.1) |
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Qy Sɨɬɫ(y) |
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Ωyx |
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J zb(y) |
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Fɨɬɫ |
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ȼ ɷɬɢɯ ɮɨɪɦɭɥɚɯ Mz |
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Qy – ɢɡɝɢɛɚɸɳɢɣ ɦɨ- |
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yx |
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ɦɟɧɬ ɢ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ; |
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– ɫɬɚɬɢɱɟɫɤɢɣ ɦɨ- |
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ɦɟɧɬ ɨɬɫɟɱɺɧɧɨɣ ɩɥɨɳɚɞɢ ɫɟɱɟɧɢɹ |
Fɨɬɫ |
ɨɬɧɨɫɢ- |
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Ɋɢɫ.4.1 |
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ɬɟɥɶɧɨ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ |
Ɉz; |
b(y) – ɲɢɪɢɧɚ ɫɟɱɟɧɢɹ |
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(ɪɢɫ.4.1). |
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ɂɡ ɮɨɪɦɭɥɵ (4.1) ɫɥɟɞɭɟɬ, ɱɬɨ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ |
ςɯ ɩɪɢɧɢɦɚɸɬ |
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ɧɚɢɛɨɥɶɲɟɟ ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɹ ɜ ɧɢɠɧɢɯ (y = hɧ) ɢ ɜ ɜɟɪɯɧɢɯ (y = – hɜ) ɜɨ- |
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ɥɨɤɧɚɯ, ɧɚɢɛɨɥɟɟ ɭɞɚɥɺɧɧɵɯ ɨɬ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ (ɪɢɫ.4.2,ɚ,ɛ). |
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ɚ) |
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Mz > 0 |
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ɛ) |
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Mz< 0 |
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Ɋɢɫ.4.1 |
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h/3 |
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y |
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ɉɪɢ ɷɬɨɦ |
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Ɋɢɫ.4.2 |
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Ɇz |
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Ɇz , |
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(4.3) |
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ɝɞɟ |
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(4.4) |
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ȼɟɥɢɱɢɧɵ Wɧ ɢ Wɜ |
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ɧɚɡɵɜɚɸɬɫɹ ɦɨɦɟɧɬɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɟɱɟɧɢɹ ɞɥɹ |
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ɧɢɠɧɢɯ ɢ ɜɟɪɯɧɢɯ ɜɨɥɨɤɨɧ. |
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ɇɚɢɛɨɥɶɲɢɟ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɧɚɩɪɹɠɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪ- |
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ɦɭɥɟ |
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Vɧɛ |
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Ɇz |
, |
(4.5) |
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Wɧɦ |
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ɝɞɟ Wɧɦ – ɦɟɧɶɲɢɣ ɢɡ ɦɨɦɟɧɬɨɜ ɫɨɩɪɨɬɢɜɥɟɧɢɹ Wɧ ɢ Wɜ .
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Ɋɢɫ.4.3
Ⱦɥɹ ɫɟɱɟɧɢɣ, ɫɢɦɦɟɬɪɢɱɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ (ɪɢɫ.4.3), hɧ = hɜ = h/2 ɢ Wɧ = Wɜ = W = Jz/0,5h . ɉɪɢ ɷɬɨɦ ɧɚɢɛɨɥɶɲɢɟ ɧɚɩɪɹɠɟɧɢɹ ɨɩɪɟ-
ɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
Vɧɛ |
Ɇz |
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(4.6) |
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ȼ ɛɚɥɤɚɯ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ |
Wyx ɢɡɦɟɧɹɸɬ- |
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ɫɹ ɩɨ ɜɵɫɨɬɟ ɫɟɱɟɧɢɹ ɩɨ ɡɚɤɨɧɭ ɤɜɚɞɪɚɬɧɨɣ ɩɚɪɚɛɨɥɵ. ɇɚɢɛɨɥɶɲɢɟ ɡɧɚɱɟɧɢɹ ɨɧɢ ɢɦɟɸɬ ɜ ɬɨɱɤɚɯ ɧɚ ɭɪɨɜɧɟ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ (ɪɢɫ.4.4,ɚ).
Ɂɧɚɱɟɧɢɹ Wɧɛ = Wmax ɢ |
W1 ɧɚ ɭɪɨɜɧɟ ɫɨɩɪɹɠɟɧɢɹ ɩɨɥɤɢ ɢ ɫɬɟɧɤɢ ɞɜɭɬɚɜɪɚ |
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(ɪɢɫ.4.4,ɛ) ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ |
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W |
max |
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Qy S1 2 |
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Qy Sɩ |
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(4.7) |
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ɝɞɟ S1 2 – ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɩɨɥɨɜɢɧɵ ɫɟɱɟɧɢɹ ɞɜɭɬɚɜɪɚ; |
Sɩ – ɫɬɚɬɢɱɟɫɤɢɣ |
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ɦɨɦɟɧɬ ɩɨɥɤɢ. ɉɪɢ ɷɬɨɦ |
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Ɋɢɫ.4.4
43
ɉɪɢ ɢɡɝɢɛɟ ɛɚɥɤɢ ɜɟɥɢɱɢɧɵ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ς1 ɢ ς2 ɢ ɭɝɥɵ ɧɚɤɥɨɧɚ
ɧɨɪɦɚɥɟɣ ɤ ɝɥɚɜɧɵɦ ɩɥɨɳɚɞɤɚɦ |
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ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ |
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ς |
ςx |
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ς2x |
Ω2 |
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tg |
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ɉɪɢ ɪɚɫɱɺɬɟ ɢɡɝɢɛɚɟɦɵɯ ɷɥɟɦɟɧɬɨɜ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ ɧɚ ɩɪɨɱɧɨɫɬɶ ɩɪɢɦɟɧɹɟɬɫɹ ɦɟɬɨɞ ɪɚɫɱɺɬɚ ɩɨ ɩɪɟɞɟɥɶɧɵɦ ɫɨɫɬɨɹɧɢɹɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɭɫɥɨɜɢɟ ɩɪɨɱɧɨɫɬɢ ɩɨ ɧɨɪɦɚɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ |
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ςɧɛ ϑɫR , |
(4.11) |
ɝɞɟ R – ɪɚɫɱɺɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɚ ɛɚɥɤɢ, ɚ ϑɫ – ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɥɨɜɢɣ ɪɚɛɨɬɵ.
Ⱦɥɹ ɯɪɭɩɤɢɯ ɦɚɬɟɪɢɚɥɨɜ ɪɚɫɱɺɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ Rɪ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɶɲɟ, ɱɟɦ ɩɪɢ ɫɠɚɬɢɢ Rɫ . ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɨɥɠɧɵ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɹ ɩɪɨɱɧɨɫɬɢ ɩɨ ɧɚɢɛɨɥɶɲɢɦ ɪɚɫɬɹɝɢɜɚɸɳɢɦ ɢ ɧɚɢɛɨɥɶɲɢɦ ɫɠɢɦɚɸɳɢɦ ɧɚɩɪɹɠɟɧɢɹɦ
ςɧɛ δ ϑ |
R , |
ςɧɛ δ ϑ |
R . |
(4.12) |
ɪ |
ɫ ɪ |
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ȼ ɫɥɭɱɚɟ ɪɚɫɱɺɬɚ ɧɚ ɩɪɨɱɧɨɫɬɶ ɩɨ ɦɟɬɨɞɭ ɞɨɩɭɫɤɚɟɦɵɯ ɧɚɩɪɹɠɟɧɢɣ ɜ ɮɨɪ-
ɦɭɥɚɯ (4.11) ɢ (4.12) ɜɟɥɢɱɢɧɵ ɜ ɩɪɚɜɵɯ ɱɚɫɬɹɯ ɧɟɨɛɯɨɞɢɦɨ ɡɚɦɟɧɢɬɶ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ [ς], [ςɪ] ɢ [ςɫ].
Ⱦɥɹ ɛɚɥɨɤ ɢɡ ɦɚɬɟɪɢɚɥɚ, ɨɞɢɧɚɤɨɜɨ ɫɨɩɪɨɬɢɜɥɹɸɳɟɝɨɫɹ ɪɚɫɬɹɠɟɧɢɸ ɢ ɫɠɚɬɢɸ, ɫ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ, ɫɢɦɦɟɬɪɢɱɧɵɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ, ɭɫɥɨɜɢɟ ɩɪɨɱɧɨɫɬɢ (4.11) ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
ςɧɛ = |
Ɇɪɚɫɱ |
ϑɫR . |
(4.13) |
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Ⱦɥɹ ɫɟɱɟɧɢɣ, ɧɟɫɢɦɦɟɬɪɢɱɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ (ɬɢɩɚ ɬɚɜɪɚ ɢ ɞɪ.)
ςɧɛ = |
Ɇɪɚɫɱ |
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ȼ ɮɨɪɦɭɥɚɯ (4.13) ɢ (4.14) Ɇɪɚɫɱ = ϑf Ɇɧɛ – ɧɚɢɛɨɥɶɲɢɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ ɜ ɨɩɚɫɧɨɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɨɬ ɞɟɣɫɬɜɢɹ ɪɚɫɱɺɬɧɵɯ ɧɚɝɪɭɡɨɤ, ϑf – ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɞɺɠɧɨɫɬɢ ɩɨ ɧɚɝɪɭɡɤɟ.
ɋ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɭɫɥɨɜɢɣ ɩɪɨɱɧɨɫɬɢ (4.11) – (4.14) ɪɟɲɚɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɡɚɞɚɱɢ: ɩɪɨɜɟɪɤɚ ɩɪɨɱɧɨɫɬɢ ɛɚɥɤɢ; ɩɨɞɛɨɪ ɫɟɱɟɧɢɹ; ɨɩɪɟɞɟɥɟɧɢɟ ɧɟɫɭɳɟɣ ɫɩɨɫɨɛɧɨɫɬɢ (ɝɪɭɡɨɩɨɞɴɺɦɧɨɫɬɢ).
Ɋɟɲɟɧɢɟ ɩɟɪɜɨɣ ɡɚɞɚɱɢ ɫɜɨɞɢɬɫɹ ɤ ɩɪɨɜɟɪɤɟ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɣ ɩɪɨɱɧɨɫɬɢ ɩɪɢ ɢɡɜɟɫɬɧɵɯ ɧɚɝɪɭɡɤɚɯ, ɮɨɪɦɟ ɢ ɪɚɡɦɟɪɚɯ ɫɟɱɟɧɢɹ ɢ ɫɜɨɣɫɬɜɚɯ ɦɚɬɟɪɢɚɥɚ.
Ɋɟɲɟɧɢɟ ɜɬɨɪɨɣ ɡɚɞɚɱɢ ɫɜɨɞɢɬɫɹ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɪɚɡɦɟɪɨɜ ɫɟɱɟɧɢɹ ɡɚɞɚɧɧɨɣ ɮɨɪɦɵ ɩɪɢ ɢɡɜɟɫɬɧɵɯ ɧɚɝɪɭɡɤɚɯ ɢ ɫɜɨɣɫɬɜɚɯ ɦɚɬɟɪɢɚɥɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫ-
ɩɨɥɶɡɭɸɬɫɹ ɪɚɫɱɺɬɧɵɟ ɮɨɪɦɭɥɵ, ɩɨɥɭɱɚɟɦɵɟ ɢɡ ɭɫɥɨɜɢɣ ɩɪɨɱɧɨɫɬɢ (4.13) ɢɥɢ
(4.14)
44
W τ |
Mɪɚɫɱ |
ɢɥɢ W τ |
Mɪɚɫɱ |
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(4.15) |
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ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɝɪɭɡɨɩɨɞɴɺɦɧɨɫɬɢ ɛɚɥɤɢ ɜɧɚɱɚɥɟ ɢɡ ɭɫɥɨɜɢɹ ɩɪɨɱɧɨɫɬɢ (4.13) ɢɥɢ (4.14) ɧɚɯɨɞɢɬɫɹ ɜɟɥɢɱɢɧɚ ɪɚɫɱɺɬɧɨɝɨ ɡɧɚɱɟɧɢɹ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ
Mɪɚɫɱ δ ϑɫRW ɢɥɢ Mɪɚɫɱ δ ϑɫRWɧɦ , |
(4.16) |
ɡɚɬɟɦ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɧɚɱɟɧɢɸ Ɇɪɚɫɱ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɟɥɢɱɢɧɵ ɧɚɝɪɭɡɨɤ.
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɭɫɥɨɜɢɣ ɩɪɨɱɧɨɫɬɢ ɩɨ ɧɚɢɛɨɥɶɲɢɦ ɤɚɫɚɬɟɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦ
Ωɧɛ δ ϑɫRs , |
(4.17) |
ɝɞɟ Rs – ɪɚɫɱɺɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɚ ɩɪɢ ɫɞɜɢɝɟ.
Ɋɚɛɨɬɚ ɛɚɥɤɢ ɡɚ ɩɪɟɞɟɥɨɦ ɭɩɪɭɝɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ ɞɢɚɝɪɚɦɦɵ ς = f(Η). ɉɪɢɦɟɦ, ɱɬɨ ɞɥɹ ɦɚɬɟɪɢɚɥɚ ɛɚɥɤɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɢɚɝɪɚɦɦɭ ɉɪɚɧɞɬɥɹ (ɪɢɫ.4.5,ɚ). ɋɨɝɥɚɫɧɨ ɷɬɨɣ ɞɢɚɝɪɚɦɦɟ ɩɪɢ ɧɚɩɪɹɠɟɧɢɹɯ, ɦɟɧɶɲɢɯ ɩɪɟɞɟɥɚ ɬɟɤɭɱɟɫɬɢ (ςɯ < ςɬ), ɫɩɪɚɜɟɞɥɢɜ ɡɚɤɨɧ Ƚɭɤɚ, ɚ ɩɪɢ ςɯ = ςɬ ɞɟɮɨɪɦɚɰɢɢ ɧɟɨɝɪɚɧɢɱɟɧɧɨ ɜɨɡɪɚɫɬɚɸɬ.
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Ɋɢɫ.4.5 |
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Ɋɩɪɟɞ
Ɇɩɪɟɞ
ȼɫɥɭɱɚɟ ɢɡɝɢɛɚ ɛɚɥɤɢ ɩɪɢ ɩɨɫɬɟɩɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɧɚɝɪɭɡɤɢ ɧɚɩɪɹɠɟɧɢɹ
ɜɤɪɚɣɧɢɯ ɜɨɥɨɤɧɚɯ ɨɩɚɫɧɨɝɨ ɫɟɱɟɧɢɹ ɞɨɫɬɢɝɚɸɬ ɩɪɟɞɟɥɚ ɬɟɤɭɱɟɫɬɢ ςɬ ɢ ɜɨɡɧɢɤɚɸɬ ɩɥɚɫɬɢɱɟɫɤɢɟ ɞɟɮɨɪɦɚɰɢɢ. ɉɪɢ ɞɚɥɶɧɟɣɲɟɦ ɭɜɟɥɢɱɟɧɢɢ ɧɚɝɪɭɡɤɢ ɩɥɚɫɬɢɱɟɫɤɢɟ ɞɟɮɨɪɦɚɰɢɢ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɜɝɥɭɛɶ ɫɟɱɟɧɢɹ ɛɚɥɤɢ (ɪɢɫ.4.5,ɛ). ɉɪɟɞɟɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ (Ɇz = Ɇɩɪɟɞ) ɧɚɫɬɭɩɚɟɬ, ɤɨɝɞɚ ɧɚɩɪɹɠɟɧɢɹ ɜɨ ɜɫɟɯ ɜɨ-
ɥɨɤɧɚɯ ɜ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɱɚɫɬɹɯ ɛɚɥɤɢ ɞɨɫɬɢɝɚɸɬ ɡɧɚɱɟɧɢɹ ςɯ = ςɬ. ȼ ɨɩɚɫɧɨɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɜɨɡɧɢɤɚɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɩɥɚɫɬɢɱɟɫɤɢɣ ɲɚɪɧɢɪ (ɪɢɫ.4.5,ɜ).
ɉɪɟɞɟɥɶɧɵɣ ɦɨɦɟɧɬ ɜ ɩɥɚɫɬɢɱɟɫɤɨɦ ɲɚɪɧɢɪɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
Ɇɩɪɟɞ ςɬWɬ . |
(4.18) |
ȼɟɥɢɱɢɧɚ Wɬ – ɧɚɡɵɜɚɟɬɫɹ ɩɥɚɫɬɢɱɟɫɤɢɦ ɦɨɦɟɧɬɨɦ ɫɨɩɪɨɬɢɜɥɟɧɢɹ. Ⱦɥɹ ɫɢɦɦɟɬɪɢɱɧɵɯ ɫɟɱɟɧɢɣ ɨɧɚ ɪɚɜɧɚ
Wɬ 2S1 2 2³³ydF |
(4.19) |
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Ɉɩɪɟɞɟɥɢɜ ɩɪɟɞɟɥɶɧɵɣ ɦɨɦɟɧɬ, ɦɨɠɧɨ ɧɚɣɬɢ ɜɟɥɢɱɢɧɭ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɩɚɫɚ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɤ ɧɚɢɛɨɥɶɲɟɦɭ ɢɡɝɢɛɚɸɳɟɦɭ ɦɨɦɟɧɬɭ ɨɬ ɞɟɣɫɬɜɢɹ ɧɨɪɦɚɬɢɜɧɵɯ ɧɚɝɪɭɡɨɤ:
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Ɂɚɞɚɱɚ 4.1. |
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Ⱦɥɹ ɛɚɥɤɢ, ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ.4.6, |
ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ |
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ɫɟɱɟɧɢɟ ɛɚɥɤɢ ɜ ɜɢɞɟ ɫɬɚɥɶɧɨɝɨ ɩɪɨɤɚɬɧɨɝɨ ɞɜɭɬɚɜɪɚ. ȼ ɪɚɫɱɺɬɚɯ ɩɪɢɦɟɦ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɞɺɠɧɨɫɬɢ ɩɨ ɧɚɝɪɭɡɤɟ Jf = 1,2, ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɥɨɜɢɣ ɪɚɛɨɬɵ Jɫ = = 1,0, ɪɚɫɱɺɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ R = 210 Ɇɉɚ = 21 ɤɇ/ɫɦ2, ɪɚɫɱɺɬɧɨɟ ɫɨɩɪɨɬɢɜ-
ɥɟɧɢɟ ɩɪɢ ɫɞɜɢɝɟ Rs = 130 Ɇɉɚ = 13 ɤɇ/ɫɦ2, ɩɪɟɞɟɥ ɬɟɤɭɱɟɫɬɢ Vɬ = 230 Ɇɉɚ = 23 ɤɇ/ɫɦ2.
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Ɋɚɫɱɺɬ ɧɚɱɧɺɦ ɫ ɨɩɪɟɞɟɥɟɧɢɹ ɨɩɨɪɧɵɯ |
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ɪɢɫ.4.6. Ɉɩɚɫɧɵɦ ɫɟɱɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɫɟɱɟ- |
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ɧɢɟ ɧɚɞ ɨɩɨɪɨɣ ȼ, |
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ɛɨɥɶɲɟɝɨ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɢ ɧɚɢ- |
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ɛɨɥɶɲɟɣ ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɵ: |
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Ɋɢɫ.4.6 |
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Ɇɪɚɫɱ = ɆɧɛJf = 20·1,2 = 24 ɤɇɦ; |
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Qɪɚɫɱ = QɧɛJf = 40·1,2 = 48 ɤɇ |
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ɢ ɬɪɟɛɭɟɦɵɣ ɦɨɦɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
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ɉɨ ɫɨɪɬɚɦɟɧɬɭ ɩɪɢɧɢɦɚɟɦ ɞɜɭɬɚɜɪ I18 |
JcR |
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ɢ ɜɵɩɢɫɵɜɚɟɦ ɧɟɨɛɯɨɞɢɦɵɟ ɝɟɨ- |
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ɦɟɬɪɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɟɱɟɧɢɹ (ɪɢɫ.4.7). |
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h = 18 ɫɦ , |
b = 9 ɫɦ , |
d = 0,51 ɫɦ , |
t = 0,81 ɫɦ ; |
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Jz = 1290 ɫɦ4 , |
W = 143 ɫɦ3 , |
S1 2 |
81,4ɫɦ3. |
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Ⱦɥɹ ɨɩɚɫɧɨɝɨ ɫɟɱɟɧɢɹ, ɝɞɟ |
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ɢ Mz |
ɢɦɟɸɬ ɧɚɢɛɨɥɶɲɢɟ ɡɧɚɱɟ- |
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Vɯ |
ɢ Wɭɯ. |
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ɡɧɚɱɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɢ ɩɪɨɜɟɪɢɦ |
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ɩɪɨɱɧɨɫɬɶ ɛɚɥɤɢ ɩɨ ɧɨɪɦɚɥɶɧɵɦ |
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152 |
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ɢ ɤɚɫɚɬɟɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦ: |
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Vɧɛ |
Mɪɚɫɱ |
24 102 |
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Ɋɢɫ.4.7 |
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16,8ɤɇ/ɫɦ2 |
168Ɇɉɚ Jc R |
210Ɇɉɚ; |
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QɪɚɫɱS1 2 |
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48 81,4 |
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5,93ɤɇ/ɫɦ2 |
59,3Ɇɉɚ J |
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R |
130 Ɇɉɚ ; |
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Jzd |
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1290 0,51 |
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48 62,7 |
4,57 ɤɇ/ɫɦ2 |
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45,7 Ɇɉɚ . |
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1290 0,51 |
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ɗɩɸɪɵ ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ Vɯ ɢ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ Wɭɯ ɜ ɫɬɟɧɤɟ ɞɜɭɬɚɜɪɚ ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.4.7.
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɢ ɭɝɥɵ ɧɚɤɥɨɧɚ ɧɨɪɦɚɥɟɣ ɤ ɝɥɚɜɧɵɦ ɩɥɨɳɚɞɤɚɦ ɜ ɨɩɚɫɧɨɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɜ ɫɬɟɧɤɟ ɞɜɭɬɚɜɪɚ ɧɚ ɭɪɨɜɧɟ ɟɺ ɫɨ-
ɩɪɹɠɟɧɢɹ ɫ ɩɨɥɤɨɣ (ɭ = 9 – 0,81 = 8,19 ɫɦ) |
ɜ ɡɨɧɟ ɪɚɫɬɹɠɟɧɢɹ. |
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Ɇɪɚɫɱ |
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24 102 |
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15,2ɤɇ/ɫɦ |
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= 152 Ɇɉɚ , |
Wɭɯ = W1 = 45,7 Ɇɉɚ ; |
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1290 |
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V1,2 |
152 |
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45,7 |
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76 r88,7, |
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164,7 Ɇɉɚ , V2 |
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Wyɯ |
45,7 |
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0,277, |
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15,5 ; |
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164,7 |
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Wyɯ |
45,7 |
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74,5 . |
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12,7 |
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45,7Ɇɉɚ |
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ɇɚ |
ɪɢɫ.4.8 ɩɨɤɚɡɚɧɵ |
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ɧɨɪɦɚɥɶɧɵɟ, ɤɚɫɚɬɟɥɶɧɵɟ ɢ |
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152Ɇɉɚ |
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164,7Ɇɉɚ |
ɝɥɚɜɧɵɟ ɧɚɩɪɹɠɟɧɢɹ, ɞɟɣɫɬ- |
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ɜɭɸɳɢɟ ɧɚ ɝɪɚɧɹɯ ɛɟɫɤɨɧɟɱ- |
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45,7Ɇɉɚ |
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ɨ |
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ɧɨ ɦɚɥɵɯ ɷɥɟɦɟɧɬɨɜ, ɜɵɞɟ- |
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15,5 |
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ɥɟɧɧɵɯ ɢɡ ɫɬɟɧɤɢ ɞɜɭɬɚɜɪɚ ɧɚ |
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Ɋɢɫ.4.8 |
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ɭɪɨɜɧɟ ɫɨɩɪɹɠɟɧɢɹ ɫɩɨɥɤɨɣ. |
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47
ȼɵɱɢɫɥɹɟɦ ɡɧɚɱɟɧɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜ ɩɥɚɫɬɢɱɟɫɤɨɣ ɫɬɚɞɢɢ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ. ɋ ɭɱɺɬɨɦ ɮɨɪɦɭɥ (4.18) ɢ (4.19), ɢɦɟɟɦ
Ɇɩɪɟɞ = 2VɬS1 2 = 2·23·81,4 = 3744,4 ɤɇɫɦ = 37,4 ɤɇɦ .
ɉɥɚɫɬɢɱɟɫɤɢɣ ɲɚɪɧɢɪ ɨɛɪɚɡɭɟɬɫɹ ɜ ɫɟɱɟɧɢɢ ɧɚɞ ɨɩɨɪɨɣ ȼ, ɝɞɟ ɞɟɣɫɬɜɭɟɬ ɧɚɢɛɨɥɶɲɢɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ.
ȼɵɱɢɫɥɢɦ ɜɟɥɢɱɢɧɭ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɩɚɫɚ
nɆɩɪɟɞ 37,4 1,87.
Ɇɧɛ 20
Ɂɚɞɚɱɚ 4.2 |
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Ⱦɥɹ ɛɚɥɤɢ, ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ.4.9, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ |
Qy |
ɢ Mz |
, ɩɨɞɛɟɪɺɦ |
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ɫɟɱɟɧɢɟ ɛɚɥɤɢ ɜ ɜɢɞɟ ɫɬɚɥɶɧɨɝɨ ɩɪɨɤɚɬɧɨɝɨ ɞɜɭɬɚɜɪɚ. |
Jf = 1,2, |
Jɫ = 0,9, |
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ȼ ɪɚɫɱɺɬɚɯ ɩɪɢɦɟɦ |
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20ɤɇ |
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45ɤɇ |
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R = 210 Ɇɉɚ = 21 ɤɇ/ɫɦ2, |
Rs = 130 Ɇɉɚ = 13 |
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30ɤɇ/ɦ |
20ɤɇɦ |
ɤɇ/ɫɦ2, Vɬ = 230 Ɇɉɚ = 23 ɤɇ/ɫɦ2. |
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Ⱥ |
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ɯ |
Ɉɩɪɟɞɟɥɢɦ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɢ ɩɨɫɬɪɨɢɦ |
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ɷɩɸɪɵ Qy ɢ Ɇz: |
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1,5ɦ |
2ɦ |
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2ɦ |
1,5ɦ |
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35 |
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– 45·2 – 20 = 0, |
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¦Ɇȼ = 0, – RȺ·4 + 20·5,5 + 30·2·3 + |
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+ 45·2 – 20 = 0, |
RȺ = 90 ɤɇ ; |
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¦Y = 0 (ɩɪɨɜɟɪɤɚ), 20 + 30·2 +45 – 90 – |
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ɗɩɸɪɵ Qy ɢ Ɇz |
ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ.4.9. |
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20 |
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ɉɨɩɟɪɟɱɧɚɹ ɫɢɥɚ Qy ɢɦɟɟɬ ɧɚɢɛɨɥɶɲɟɟ ɡɧɚ- |
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ɱɟɧɢɟ ɜ ɫɟɱɟɧɢɢ ɧɚɞ ɨɩɨɪɨɣ Ⱥ (ɯ = 1,5 ɦ), ɚ |
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ȼɵɱɢɫɥɢɦ ɪɚɫɱɺɬɧɵɟ ɡɧɚɱɟɧɢɹ ɧɚɢɛɨɥɶ- |
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ɲɟɝɨ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɢ ɩɨɩɟɪɟɱɧɨɣ ɫɢ- |
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Ɋɢɫ.4.9 |
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ɥɵ ɞɥɹ ɨɩɚɫɧɨɝɨ ɫɟɱɟɧɢɹ (ɯ = 3,5 ɦ) |
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Ɇɪɚɫɱ = ɆɧɛJf = 50·1,2 = 60 ɤɇɦ ; Qɪɚɫɱ = QɧɛJf = 35·1,2 = 42 ɤɇ .
Ɍɪɟɛɭɟɦɵɣ ɦɨɦɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɪɚɜɟɧ
W t |
Mɪɚɫɱ |
60 |
102 |
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0,9 |
21 |
318 |
ɫɦ . |
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JɫR |
48
ɉɨ ɫɨɪɬɚɦɟɧɬɭ ɩɪɢɧɢɦɚɟɦ ɞɜɭɬɚɜɪ |
I27 ɢ ɜɵɩɢɫɵɜɚɟɦ ɧɟɨɛɯɨɞɢɦɵɟ ɝɟɨ- |
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ɦɟɬɪɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɟɱɟɧɢɹ (ɪɢɫ.4.10,ɚ). |
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h = 27 ɫɦ , |
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d = 0,6 ɫɦ , |
t = 0,98 ɫɦ ; |
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Jz = 5010 ɫɦ4 , |
W = 371 ɫɦ3 , |
S1 2 |
210ɫɦ3. |
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ȼɵɱɢɫɥɹɟɦ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɩɥɨɳɚɞɢ ɫɟɱɟɧɢɹ ɩɨɥɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɧɟɣɬɪɚɥɶɧɨɣ ɨɫɢ Ɉz |
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Sɩ |
§h |
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§ |
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0,98 |
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159,4 |
3 |
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2 |
¸ |
12,5 0,98¨ |
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¸ |
ɫɦ . |
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12,5ɫɦ |
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162 |
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44,6 |
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0,6ɫɦ |
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150 |
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22,3 |
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27ɫɦ |
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29,3 |
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58,6 |
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0,98ɫɦ |
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150 22,3 |
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44,6 |
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162 |
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Ɋɢɫ.4.10 |
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Ⱦɥɹ ɨɩɚɫɧɨɝɨ ɫɟɱɟɧɢɹ ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ |
Vɯ |
ɢ |
Wyx. Ɉɩɪɟɞɟɥɢɦ ɧɚɢɛɨɥɶɲɟɟ |
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ɡɧɚɱɟɧɢɟ ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɢ ɩɪɨɜɟɪɢɦ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ ɩɪɨɱɧɨɫɬɢ: |
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Vɧɛ |
Mɪɚɫɱ |
60 102 |
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16,2 |
2 |
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162Ɇɉɚ JcR |
0,9 210 189Ɇɉɚ ; |
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W |
371 |
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ɤɇ/ɫɦ |
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ɇɚɣɞɺɦ ɯɚɪɚɤɬɟɪɧɵɟ ɡɧɚɱɟɧɢɹ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ: |
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Wmax |
QɪɚɫɱS1 2 |
42 210 |
2,93ɤɇ/ɫɦ2 29,3Ɇɉɚ ; |
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Jzd |
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5010 0,6 |
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W |
QɪɚɫɱSɩ |
42 159,4 |
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2,23ɤɇ/ɫɦ2 22,3Ɇɉɚ . |
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1 |
Jzd |
5010 0,6 |
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ɗɩɸɪɵ ɧɚɩɪɹɠɟɧɢɣ |
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ɢ |
Wyx |
ɞɥɹ ɨɩɚɫɧɨɝɨ ɫɟɱɟɧɢɹ (ɯ = 3,5 ɦ) ɩɪɢɜɟɞɟɧɵ |
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ɧɚ ɪɢɫ.4.10,ɛ,ɜ. |
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ɜ ɫɟɱɟɧɢɢ ɯ = 1,5 ɦ, ɝɞɟ Qy ɢɦɟɟɬ ɧɚɢɛɨɥɶɲɟɟ |
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Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɷɩɸɪɵ Wyx |
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ɡɧɚɱɟɧɢɟ, ɢ ɩɪɨɜɟɪɤɢ ɭɫɥɨɜɢɹ ɩɪɨɱɧɨɫɬɢ ɩɨ ɤɚɫɚɬɟɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦ ɧɚɣɞɺɦ Wɧɛ ɢ W1. ɍɱɢɬɵɜɚɹ, ɱɬɨ ɞɥɹ ɷɬɨɝɨ ɫɟɱɟɧɢɹ
Qɪɚɫɱ = QɧɛJf = 70·1,2 = 84 ɤɇ ,
ɩɨɥɭɱɢɦ
49
Wɧɛ |
QɪɚɫɱS1 2 |
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84 210 |
5,86 ɤɇ/ɫɦ2 |
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Jzd |
5010 0,6 |
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58,6 Ɇɉɚ Jc Rs 0,9 130 117 Ɇɉɚ ; |
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W |
QɪɚɫɱSɩ |
84 159,4 4,46ɤɇ/ɫɦ2 44,6 Ɇɉɚ . |
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1 |
Jzd |
5010 0,6 |
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ɗɩɸɪɚ Wyx ɞɥɹ ɫɟɱɟɧɢɹ ɯ = 1,5 ɦ ɩɨɤɚɡɚɧɚ ɧɚ ɪɢɫ.4.10,ɝ.
ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɢ ɭɝɥɵ ɧɚɤɥɨɧɚ ɧɨɪɦɚɥɟɣ ɤ ɝɥɚɜ-
ɧɵɦ ɩɥɨɳɚɞɤɚɦ ɜ ɨɩɚɫɧɨɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ, ɜ ɫɬɟɧɤɟ, ɧɚ ɭɪɨɜɧɟ ɟɺ ɫɨɩɪɹɠɟɧɢɹ ɫ ɩɨɥɤɨɣ (ɭ = 13,5 – 0,98 = 12,52 ɫɦ) ɜ ɡɨɧɟ ɪɚɫɬɹɠɟɧɢɹ.
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ
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Ɇɪɚɫɱ |
y |
60 102 |
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2 |
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Vɯ |
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12,52 |
15,0 |
ɤɇ/ɫɦ |
= 150 Ɇɉɚ , Wɭɯ = W1 = – 22,3 Ɇɉɚ , |
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Jz |
5010 |
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ɩɨɥɭɱɢɦ |
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V1,2 |
150 |
r |
§150 |
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2 |
75 r 78,2; |
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2 |
¨ |
¸ |
22,3 |
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© 2 |
¹ |
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V1 153,2 Ɇɉɚ , |
V2 3,2Ɇɉɚ ; |
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Wyɯ |
22,3 |
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tgD |
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0,146, |
D |
8,3 ; |
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1 V |
153,2 |
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1 |
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Wyɯ |
22,3 |
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tgD2 |
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6,969 , |
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81,7 . |
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V2 |
3,2 |
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22,3Ɇɉɚ |
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3,2Ɇɉɚ |
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ɇɚ ɪɢɫ.4.11 |
ɩɨɤɚɡɚɧɵ |
ɧɨɪ- |
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ɦɚɥɶɧɵɟ, ɤɚɫɚɬɟɥɶɧɵɟ ɢ ɝɥɚɜɧɵɟ |
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150Ɇɉɚ |
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153,2Ɇɉɚ |
ɧɚɩɪɹɠɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɢɟ |
ɧɚ |
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ɝɪɚɧɹɯ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɷɥɟ- |
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8,3ɨ |
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22,3Ɇɉɚ |
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ɦɟɧɬɨɜ, ɜɵɞɟɥɟɧɧɵɯ ɢɡ ɫɬɟɧɤɢ |
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81,7ɨ |
ɞɜɭɬɚɜɪɚ ɧɚ ɭɪɨɜɧɟ ɫɨɩɪɹɠɟɧɢɹ ɫ |
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Ɋɢɫ.4.11 |
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ɩɨɥɤɨɣ. |
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ȼɵɱɢɫɥɹɟɦ |
ɡɧɚɱɟɧɢɟ |
ɩɪɟ- |
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ɞɟɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜ ɩɥɚɫɬɢɱɟɫɤɨɣ ɫɬɚɞɢɢ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ.
Ɇɩɪɟɞ = 2VɬS1 2 = 2·23·210 = 9660 ɤɇɫɦ = 96,6 ɤɇɦ .
ɉɥɚɫɬɢɱɟɫɤɢɣ ɲɚɪɧɢɪ ɨɛɪɚɡɭɟɬɫɹ ɜ ɨɩɚɫɧɨɦ ɫɟɱɟɧɢɢ (ɯ = 3,5 ɦ) , ɝɞɟ ɞɟɣɫɬɜɭɟɬ ɧɚɢɛɨɥɶɲɢɣ ɢɡɝɢɛɚɸɳɢɣ ɦɨɦɟɧɬ.
ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɩɚɫɚ ɪɚɜɧɚ
nɆɩɪɟɞ 96,6 1,93.
Ɇɧɛ 50
50