Сопротивление материалов. Часть 1. Учебное пособие
.pdfɆɢɧɢɫɬɟɪɫɬɜɨ ɨɛɪɚɡɨɜɚɧɢɹ ɢ ɧɚɭɤɢ Ɋɨɫɫɢɣɫɤɨɣ Ɏɟɞɟɪɚɰɢɢ
ɇȺɐɂɈɇȺɅɖɇɕɃ ɂɋɋɅȿȾɈȼȺɌȿɅɖɋɄɂɃ ɆɈɋɄɈȼɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɋɌɊɈɂɌȿɅɖɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ
ɋɈɉɊɈɌɂȼɅȿɇɂȿ ɆȺɌȿɊɂȺɅɈȼ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ
ȼ ɬɪɟɯ ɱɚɫɬɹɯ
ɟ ɢɡɞɚɧɢɟ
ɑɚɫɬɶ1
Ɇɨɫɤɜɚ
2018
ɍȾɄ ȻȻɄ
ɋ
Ⱥɜɬɨɪɵ:
ɇ Ɇ Ⱥɬɚɪɨɜ Ƚɋ ȼɚɪɞɚɧɹɧ Ⱥ Ⱥ Ƚɨɪɲɤɨɜ Ⱥ ɇ Ʌɟɨɧɬɶɟɜ
Ɋɟɰɟɧɡɟɧɬɵ:
ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ ɩɪɨɮɟɫɫɨɪ ɋ ɇ Ʉɪɢɜɨɲɚɩɤɨ ɡɚɜɟɞɭɸɳɢɣ ɤɚɮɟɞɪɨɣ ©ɉɪɨɱɧɨɫɬɶ ɦɚɬɟɪɢɚɥɨɜ ɢ ɤɨɧɫɬɪɭɤɰɢɣª Ɋɨɫɫɢɣɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ ɞɪɭɠɛɵ ɧɚɪɨɞɨɜ ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ ɱɥɟɧ ɤɨɪɪɟɫɩɨɧɞɟɧɬ ɊȺȺɋɇ ɩɪɨɮɟɫɫɨɪ ɇ ɇ ɒɚɩɨɲɧɢɤɨɜ ɊɍɌ ɆɂɂɌ
ɋ ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɜ ɱɚɫɬɹɯ >ɇ Ɇ Ⱥɬɚɪɨɜ ɢ ɞɪ @ Ɇ ɜɨ ɨɛɪɚɡɨɜɚɧɢɹ ɢ ɧɚɭɤɢ Ɋɨɫ Ɏɟɞɟɪɚɰɢɢ ɇɚɰ ɢɫɫɥɟɞɨɜɚɬ Ɇɨɫɤ ɝɨɫ ɫɬɪɨɢɬ ɭɧ ɬ ɟ ɢɡɞ Ɇɨɫɤɜɚ ɂɡɞɚɬɟɥɶ ɫɬɜɨ Ɇɂɋɂ±ɆȽɋɍ
,6%1 ɑɚɫɬɶ ɫ ,6%1
Ɋɚɫɫɦɨɬɪɟɧɵ ɫɥɟɞɭɸɳɢɟ ɬɟɦɵ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɨɩɟ ɪɟɱɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɟɣ ɨɩɪɟɞɟɥɟɧɢɟ ɭɫɢɥɢɣ ɧɚɩɪɹɠɟɧɢɣ ɢ ɞɟɮɨɪɦɚ ɰɢɣ ɜ ɫɬɟɪɠɧɹɯ ɪɚɛɨɬɚɸɳɢɯ ɧɚ ɪɚɫɬɹɠɟɧɢɟ ɢ ɫɠɚɬɢɟ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɩɪɢ ɢɡɝɢɛɟ ɫɬɟɪɠɧɟɣ ɨɩɪɟɞɟɥɟɧɢɟ ɧɚɩɪɹɠɟɧɢɣ ɜ ɛɚɥɤɚɯ ɩɪɢ ɢɡɝɢɛɟ ɢ ɪɚɫ ɱɟɬɵ ɧɚ ɩɪɨɱɧɨɫɬɶ ɉɪɢɜɟɞɟɧɵ ɨɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɷɬɢɯ ɪɚɡɞɟɥɨɜ ɩɨɞɪɨɛ ɧɨ ɪɚɫɫɦɨɬɪɟɧɵ ɩɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
ɉɨɥɟɡɧɨ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɪɚɫɱɟɬɧɨ ɝɪɚɮɢɱɟɫɤɢɯ ɪɚɛɨɬ ɢ ɩɪɢ ɩɨɞɝɨɬɨɜɤɟ ɤ ɪɚɡɥɢɱɧɵɦ ɜɢɞɚɦ ɤɨɧɬɪɨɥɹ ɡɧɚɧɢɣ ɡɚɳɢɬɚ ɪɚɫɱɟɬɧɨ ɝɪɚɮɢɱɟɫɤɢɯ ɪɚɛɨɬ ɤɨɦɩɶɸɬɟɪɧɨɟ ɬɟɫɬɢɪɨɜɚɧɢɟ ɡɚɱɟɬɵ ɢ ɷɤɡɚɦɟɧɵ
Ⱦɥɹ ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɨɞɝɨɬɨɜɤɢ ɋɬɪɨɢɬɟɥɶɫɬɜɨ
ɍȾɄ
ȻȻɄ
,6%1 ɱ ɇɚɰɢɨɧɚɥɶɧɵɣ ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ
,6%1 Ɇɨɫɤɨɜɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ
ɫɬɪɨɢɬɟɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ
ɉɊȿȾɂɋɅɈȼɂȿ
ɉɨɫɨɛɢɟ ɫɨɫɬɨɢɬ ɢɡ 3-ɯ ɱɚɫɬɟɣ ɢ 13 ɝɥɚɜ ɩɨ ɬɟɦɚɦ ɪɚɫɱɟɬɧɨ-ɝɪɚɮɢɱɟɫɤɢɯ ɪɚɛɨɬ. Ʉɚɠɞɚɹ ɝɥɚɜɚ ɫɨɞɟɪɠɢɬ ɤɪɚɬɤɨɟ ɢɡɥɨɠɟɧɢɟ ɬɟɨɪɢɢ, ɝɞɟ ɩɪɢɜɟɞɟɧɵ ɨɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɢ ɭɪɚɜɧɟɧɢɹ ɢ ɩɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ, ɚɧɚɥɨɝɢɱɧɵɯ ɡɚɞɚɱɚɦ
ɜɪɚɫɱɟɬɧɨ-ɝɪɚɮɢɱɟɫɤɢɯ ɪɚɛɨɬɚɯ.
ȼɩɟɪɜɨɣ ɱɚɫɬɢ ɩɨɫɨɛɢɹ ɩɪɢɜɟɞɟɧɵ ɝɥɚɜɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɱɟɛɧɨɦɭ ɦɚɬɟɪɢɚɥɭ 1-ɝɨ ɫɟɦɟɫɬɪɚ ɢɡɭɱɟɧɢɹ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ – ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɟɣ, ɰɟɧɬɪɚɥɶɧɨɟ ɪɚɫɬɹɠɟɧɢɟ ɢ ɫɠɚɬɢɟ ɩɪɹɦɵɯ ɫɬɟɪɠɧɟɣ, ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɜ ɛɚɥɤɚɯ ɢ ɩɥɨɫɤɢɯ ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦɚɯ ɩɪɢ ɢɡɝɢɛɟ, ɧɚɩɪɹɠɟɧɢɹ ɜ ɛɚɥɤɚɯ ɩɪɢ ɢɡɝɢɛɟ ɢ ɪɚɫɱɟɬɵ ɧɚ ɩɪɨɱɧɨɫɬɶ.
ȼɤɨɧɰɟ ɤɚɠɞɨɣ ɱɚɫɬɢ ɩɨɫɨɛɢɹ ɩɪɢɜɟɞɟɧ ɫɨɪɬɚɦɟɧɬ ɫɬɚɥɶɧɵɯ ɩɪɨɤɚɬɧɵɯ ɫɬɟɪɠɧɟɣ – ɭɝɨɥɤɨɜ, ɞɜɭɬɚɜɪɨɜ ɢ ɲɜɟɥɥɟɪɨɜ.
ɉɨɫɨɛɢɟ ɧɚɩɢɫɚɧɨ ɚɜɬɨɪɫɤɢɦ ɤɨɥɥɟɤɬɢɜɨɦ ɤɚɮɟɞɪɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɆȽɋɍ. Ȼɨɥɶɲɭɸ ɩɨɦɨɳɶ ɩɪɢ ɧɚɩɢɫɚɧɢɢ ɢ ɩɨɞɝɨɬɨɜɤɟ ɤ ɢɡɞɚɧɢɸ ɭɱɟɛɧɨɝɨ ɩɨɫɨɛɢɹ ɨɤɚɡɚɥɢ ɚɜɬɨɪɚɦ ɤɨɥɥɟɝɢ ɩɨ ɤɚɮɟɞɪɟ – ɩɪɨɮɟɫɫɨɪ Ɉ.ȼ. Ɇɤɪɬɵɱɟɜ, ɞɨɰɟɧɬɵ Ⱥ.ə. Ⱥɫɬɚɯɨɜɚ, Ⱥ.ȼ. ɂɥɶɹɲɟɧɤɨ ɢ Ⱥ.Ƚ. ɉɚɭɲɤɢɧ.
ȼɩɨɫɨɛɢɢ ɢɫɩɨɥɶɡɨɜɚɧɚ ɫɢɫɬɟɦɚ ɟɞɢɧɢɰ ɋɂ, ɚ ɬɚɤɠɟ ɬɪɚɞɢɰɢɨɧɧɵɟ ɞɥɹ ɤɭɪɫɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɨɛɨɡɧɚɱɟɧɢɹ: ɫɢɥɚ – P, ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ – F. ɋɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɨɫɧɨɜɧɵɦɢ ɦɟɯɚɧɢɱɟɫɤɢɦɢ ɜɟɥɢɱɢɧɚɦɢ ɜ ɟɞɢɧɢɰɚɯ ɋɂ ɢ ɜ ɬɟɯɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɩɪɢɜɟɞɟɧɵ ɜ ɫɥɟɞɭɸɳɟɣ ɬɚɛɥɢɰɟ:
ɇɚɢɦɟɧɨɜɚɧɢɟ |
ȿ ɞ ɢ ɧ ɢ ɰ ɚ |
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ɋɨɨɬɧɨɲɟɧɢɟ |
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ɜɟɥɢɱɢɧɵ |
ɇɚɢɦɟɧɨɜɚɧɢɟ |
Ɉɛɨɡɧɚɱɟɧɢɟ |
ɟɞɢɧɢɰ |
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ɋɢɥɚ, ɧɚɝɪɭɡɤɚ, ɜɟɫ |
ɇɶɸɬɨɧ |
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ɇ |
1ɇ | 0,1 ɤɝɫ |
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1ɤɇ | 0,1ɬɫ |
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Ʌɢɧɟɣɧɚɹ ɧɚɝɪɭɡɤɚ |
ɇɶɸɬɨɧ ɧɚ ɦɟɬɪ |
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ɇ/ɦ |
1ɇ/ɦ | 0,1ɤɝɫ/ɦ |
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1ɤɇ/ɦ | 0,1ɬɫ/ɦ |
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Ɇɨɦɟɧɬ ɫɢɥɵ, ɦɨɦɟɧɬ |
ɇɶɸɬɨɧ-ɦɟɬɪ |
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ɇɦ |
1ɇɦ | 0,1ɤɝɫ ɦ |
ɩɚɪɵ ɫɢɥ |
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1ɤɇɦ | 0,1ɬɫ ɦ |
ɇɚɩɪɹɠɟɧɢɟ, ɞɚɜɥɟɧɢɟ |
ɉɚɫɤɚɥɶ |
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ɉɚ |
1ɉɚ | 0,1ɤɝɫ/ɦ2 |
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1Ɇɉɚ | 10ɤɝɫ/ɫɦ2 |
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ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɧɚɩɪɹɠɟɧɢɣ ɜ ɤɚɱɟɫɬɜɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɢɫɩɨɥɶɡɭɟɬɫɹ ɬɚɤɠɟ ɤɇ/ɫɦ2 (1 ɤɇ/ɫɦ2 = 10 Ɇɉɚ).
3
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ȽɅȺȼȺ 1 |
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ȽȿɈɆȿɌɊɂɑȿɋɄɂȿ ɏȺɊȺɄɌȿɊɂɋɌɂɄɂ ɉɈɉȿɊȿɑɇɕɏ ɋȿɑȿɇɂɃ |
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ɋɌȿɊɀɇȿɃ |
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1.1. Ɉɫɧɨɜɧɵɟ ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɮɨɪɦɭɥɵ |
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Ɉɫɧɨɜɧɵɦɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɣ |
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ɫɬɟɪɠɧɟɣ (ɪɢɫ.1.1), ɢɫɩɨɥɶɡɭɟɦɵɦɢ ɩɪɢ ɪɚɫɱɟɬɟ ɫɬɟɪɠɧɟɣ ɧɚ ɩɪɨɱɧɨɫɬɶ ɢ ɠɟɫɬ- |
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ɤɨɫɬɶ, ɹɜɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ. |
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ɉɥɨɳɚɞɶ ɫɟɱɟɧɢɹ F. |
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y |
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x |
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ɋɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɩɥɨɳɚɞɢ ɫɟ- |
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ɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ Ɉx ɢ Oy |
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dF |
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Sx |
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³³ ydF , |
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Sy |
³³xdF . |
(1.1) |
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C |
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y |
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F |
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F |
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Ɉɫɟɜɵɟ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ |
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C |
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y |
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J x |
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³³ y2dF , |
J y |
≥≥ x2dF . |
(1.2) |
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F |
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F |
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Ɉ |
xC |
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x |
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ɐɟɧɬɪɨɛɟɠɧɵɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ |
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J xy |
³³xydF . |
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(1.3) |
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Ɋɢɫ.1.1 |
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F |
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ɉɨɥɹɪɧɵɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ |
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J p |
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³³r2dF |
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Jx J y . |
(1.4) |
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F |
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ɋɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɢɦɟɸɬ ɪɚɡɦɟɪɧɨɫɬɶ ɞɥɢɧɵ ɜ ɬɪɟɬɶɟɣ ɫɬɟɩɟɧɢ (ɫɦ3), ɚ |
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ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ – ɟɞɢɧɢɰɵ ɞɥɢɧɵ ɜ ɱɟɬɜɟɪɬɨɣ ɫɬɟɩɟɧɢ (ɫɦ4). ɋɬɚɬɢɱɟɫɤɢɟ |
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ɦɨɦɟɧɬɵ ɢ ɰɟɧɬɪɨɛɟɠɧɵɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ, ɨɬɪɢ- |
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ɰɚɬɟɥɶɧɵɦɢ ɢ ɪɚɜɧɵɦɢ ɧɭɥɸ. Ɉɫɟɜɵɟ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɜɫɟɝɞɚ ɹɜɥɹɸɬɫɹ ɩɨɥɨ- |
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ɠɢɬɟɥɶɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ. |
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Ʉɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ |
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Ɉɫɢ, ɩɪɨɯɨɞɹɳɢɟ ɱɟɪɟɡ ɰɟɧɬɪ ɬɹɠɟɫɬɢ |
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ɫɟɱɟɧɢɹ, ɧɚɡɵɜɚɸɬɫɹ ɰɟɧɬɪɚɥɶɧɵɦɢ ɨɫɹɦɢ. |
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ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶ- |
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ɧɭɥɸ. |
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ɑɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɹɜɥɹ- |
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ɸɬɫɹ ɨɫɢ ɫɢɦɦɟɬɪɢɢ ɫɟɱɟɧɢɹ. |
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ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ |
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ɫɟɱɟɧɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ |
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ɦɨɦɟɧɬɚɦɢ ɢɧɟɪɰɢɢ ɩɪɢ ɩɚɪɚɥɥɟɥɶɧɨɦ ɩɟ- |
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Ɋɢɫ.1.2 |
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ɪɟɧɨɫɟ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ.1.2): |
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ɝɞɟ ɚ ɢ b – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ Ɉ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉ1ɯ1y1. |
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Ⱦɜɟ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɟ ɨɫɢ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɵɯ ɰɟɧɬɪɨɛɟɠɧɵɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɪɚɜɟɧ ɧɭɥɸ, ɧɚɡɵɜɚɸɬɫɹ ɝɥɚɜɧɵɦɢ ɨɫɹɦɢ ɢɧɟɪɰɢɢ. Ɉɫɟɜɵɟ
ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɨɫɟɣ ɢɦɟɸɬ ɷɤɫɬɪɟɦɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ Jmax = J1 ɢ Jmin = J2 . Ɉɧɢ ɧɚɡɵɜɚɸɬɫɹ ɝɥɚɜɧɵɦɢ ɦɨɦɟɧɬɚɦɢ ɢɧɟɪɰɢɢ.
ȿɫɥɢ ɝɥɚɜɧɵɟ ɨɫɢ ɩɪɨɯɨɞɹɬ ɱɟɪɟɡ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ, ɬɨ ɨɧɢ ɧɚɡɵɜɚɸɬɫɹ ɝɥɚɜɧɵɦɢ ɰɟɧɬɪɚɥɶɧɵɦɢ ɨɫɹɦɢ ɫɟɱɟɧɢɹ.
ȼɟɥɢɱɢɧɵ ɝɥɚɜɧɵɯ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ J1 ɢ J2 ɢ ɭɝɥɵ ɧɚɤɥɨɧɚ ɝɥɚɜɧɵɯ ɨɫɟɣ D1 ɢ D2 ɤ ɨɫɢ Ɉɯ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ
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Ɉɫɶ ɫɢɦɦɟɬɪɢɢ ɫɟɱɟɧɢɹ ɢ ɥɸɛɚɹ ɨɫɶ, ɟɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ, ɫɨɫɬɚɜɥɹɸɬ ɩɚɪɭ ɝɥɚɜɧɵɯ ɨɫɟɣ. Ⱦɥɹ ɫɟɱɟɧɢɣ, ɢɦɟɸɳɢɯ ɛɨɥɟɟ ɞɜɭɯ ɨɫɟɣ ɫɢɦɦɟɬɪɢɢ, ɚ ɬɚɤɠɟ ɩɪɢ ɪɚɜɟɧɫɬɜɟ ɝɥɚɜɧɵɯ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ J1 = J2 ɜɫɟ ɰɟɧɬɪɚɥɶɧɵɟ ɨɫɢ ɹɜɥɹɸɬɫɹ ɝɥɚɜɧɵɦɢ.
ɇɢɠɟ ɩɪɢɜɟɞɟɧɵ ɫɩɪɚɜɨɱɧɵɟ ɞɚɧɧɵɟ ɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɩɪɨɫɬɵɯ ɫɟɱɟɧɢɣ.
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Ʉɪɭɝ (ɪɢɫ.1.5) |
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ɉɨɥɭɤɪɭɝ (ɪɢɫ.1.6) |
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Ƚɟɨɦɟɬɪɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɟɱɟɧɢɣ ɩɪɨɤɚɬɧɵɯ ɫɬɟɪɠɧɟɣ (ɞɜɭɬɚɜɪɚ, |
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ɲɜɟɥɥɟɪɚ, ɭɝɨɥɤɚ) ɩɪɢɜɟɞɟɧɵ ɜ ɫɨɪɬɚɦɟɧɬɟ. |
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1.2. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ |
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Ɂɚɞɚɱɚ 1.1 |
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Ɉɩɪɟɞɟɥɢɦ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɫɟ- |
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ɱɟɧɢɹ ɜ ɜɢɞɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ɫ ɤɪɭɝɨɜɵɦɢ ɜɵɪɟɡɚɦɢ (ɪɢɫ.1.7). |
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Ɋɚɡɦɟɪɵ ɫɟɱɟɧɢɣ ɧɚ ɪɢɫɭɧɤɚɯ 1.7 y 1.12 |
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ɝɥɚɜɧɵɦɢ ɰɟɧɬɪɚɥɶɧɵɦɢ ɨɫɹɦɢ ɜɫɟɝɨ |
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ɦɨɭɝɨɥɶɧɢɤɚ ɢ ɤɪɭɝɨɜɵɯ ɜɵɪɟɡɨɜ ɨɬɧɨ- |
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ɫɢɬɟɥɶɧɨ ɢɯ ɫɨɛɫɬɜɟɧɧɵɯ ɨɫɟɣ ɨɩɪɟɞɟ- |
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ɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (1.8) ɢ (1.10) |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɨɩɪɟ- |
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ɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (1.6). |
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40000 3 1017 |
36949 ɫɦ4 ; |
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3J 2 |
2F a2 |
360000 3 1017 2 113 182 |
283725ɫɦ4 . |
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Ɂɚɞɚɱɚ 1.2 |
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Ɉɩɪɟɞɟɥɢɦ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ |
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ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ, ɩɨɤɚɡɚɧɧɨɝɨ ɧɚ ɪɢɫ.1.8. |
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Ɋɚɡɨɛɶɟɦ ɫɟɱɟɧɢɟ ɧɚ ɬɪɢ ɩɪɨɫɬɵɟ ɮɢɝɭɪɵ: |
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ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɫ ɪɚɡɦɟɪɚɦɢ 12υ8 ɫɦ |
ɢ ɞɜɚ ɪɚɜ- |
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ɧɨɛɟɞɪɟɧɧɵɯ ɬɪɟɭɝɨɥɶɧɢɤɚ ɫ ɪɚɡɦɟɪɚɦɢ |
12υ6 ɫɦ. |
6 |
O1 |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɢ ɩɥɨɳɚɞɢ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ɢ |
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ɬɪɟɭɝɨɥɶɧɢɤɨɜ |
ɨɬɧɨɫɢɬɟɥɶɧɨ |
ɢɯ |
ɫɨɛɫɬɜɟɧɧɵɯ |
8 |
O |
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ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ |
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(1.8) ɢ (1.9). |
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12 83 |
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8 123 |
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512 ɫɦ |
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96 ɫɦ2 ; |
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Ɋɢɫ.1.8 |
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72 ɫɦ |
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48 |
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ɉɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɪɚɜɧɚ |
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F 96 2 36 168 ɫɦ2 .
Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ Ɉɯ, Ɉɭ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (1.6).
J |
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2 J |
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F b2 |
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512 2 72 36 62 |
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3248ɫɦ4 ; |
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2J y2 |
1152 2 216 1584 ɫɦ4 . |
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Ɂɚɞɚɱɚ 1.3
Ɉɩɪɟɞɟɥɢɦ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ
ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɚɥɶɧɨɝɨ ɫɬɟɪɠɧɹ, ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ ɱɟɬɵɪɟɯ ɪɚɜɧɨɛɨɤɢɯ ɭɝɨɥɤɨɜ L100υ100υ10 ɢ ɥɢɫɬɚ ɫɟɱɟɧɢɟɦ 300υ10 ɦɦ (ɪɢɫ.1.9).
ȼɵɩɢɲɟɦ ɢɡ ɫɨɪɬɚɦɟɧɬɚ ɩɥɨɳɚɞɶ ɢ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɭɝɨɥɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɨɛɫɬɜɟɧɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ Ɉ1ɯ1 ɢ Ɉ1ɭ1:
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179 |
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F1 = 19,2 |
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ɫɦ |
ɫɦ . |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ Ɉɯ ɢ Ɉɭ ɢ ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ ɥɢɫɬɚ ɪɚɜɧɵ
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2250 |
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2,5 ɫɦ |
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L100υ100υ10 |
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ɉɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɪɚɜɧɚ |
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F = 4 19,2 + 30 = 106,8 ɫɦ2. |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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300υ10ɦɦ |
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ɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (1.6). |
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4179 19,2 12,172 |
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14341ɫɦ4 ; |
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2, |
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4179 19,2 3,332 |
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Ɋɢɫ.1.9 |
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1570 ɫɦ4 . |
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Ɂɚɞɚɱɚ 1.4 |
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Ɉɩɪɟɞɟɥɢɦ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɢ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫ ɨɞɧɨɣ ɨɫɶɸ ɫɢɦɦɟɬɪɢɢ
(ɪɢɫ.1.10).
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Ɋɢɫ.1.10 |
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Ⱦɚɧɧɨɟ ɫɟɱɟɧɢɟ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɫ ɪɚɡɦɟɪɚɦɢ 24υ18 ɫɦ ɢ ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɜɵɪɟɡ ɫ ɪɚɡɦɟɪɚɦɢ 12υ12 ɫɦ.
ɉɥɨɳɚɞɶ ɫɟɱɟɧɢɹ ɪɚɜɧɚ
F = 24 18 – 12 12 = 288 ɫɦ2.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ, ɤɨɬɨɪɵɣ ɧɚɯɨɞɢɬɫɹ ɧɚ ɨɫɢ ɫɢɦɦɟɬɪɢɢ Ɉɭ, ɩɪɢɦɟɦ ɜ ɤɚɱɟɫɬɜɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɨɫɢ ɨɫɶ Ɉ1ɯ1, ɩɪɨɯɨɞɹɳɭɸ ɩɨ ɨɫɧɨɜɚɧɢɸ ɮɢɝɭɪ.
ɋɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ ɨɩɪɟɞɟɥɢɦ ɤɚɤ ɪɚɡɧɨɫɬɶ ɫɬɚɬɢɱɟɫɤɢɯ ɦɨɦɟɧɬɨɜ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ɢ ɤɜɚɞɪɚɬɚ
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F y F y |
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24 18 9 12 12 6 3024 ɫɦ3 . |
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y0 |
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3024 |
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8
Ɉɫɢ Ɉɯ ɢ Ɉɭ ɹɜɥɹɸɬɫɹ ɝɥɚɜɧɵɦɢ ɰɟɧɬɪɚɥɶɧɵɦɢ ɨɫɹɦɢ ɫɟɱɟɧɢɹ. |
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ɉɨ ɬɪɟɬɶɟɣ ɢɡ ɮɨɪɦɭɥ (1.8) ɨɩɪɟɞɟɥɢɦ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢ- |
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ɬɟɥɶɧɨ ɨɫɢ Ɉ1ɯ1: |
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24 183 |
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39744 |
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ɫɦ . |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɨɩɪɟ- |
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ɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (1.6). |
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39744 288 10,52 |
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Ɂɚɞɚɱɚ 1.5 |
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Ɉɩɪɟɞɟɥɢɦ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɢ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɚɥɶɧɨɣ ɛɚɥɤɢ, ɫɨɫɬɚɜɥɟɧɧɨɣ |
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ɢɡ ɞɜɭɯ ɞɜɭɬɚɜɪɨɜ I27 ɢ ɫɬɚɥɶɧɨɝɨ ɥɢɫɬɚ ɫɟɱɟɧɢɟɦ 400υ12 ɦɦ (ɪɢɫ.1.11). |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɢ ɩɥɨɳɚɞɢ ɫɟɱɟ- |
400υ12ɦɦ |
ɭ2 |
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ɧɢɣ ɞɜɭɬɚɜɪɚ ɢ ɥɢɫɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɨɛ- |
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ɫɬɜɟɧɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɫɨɨɬɜɟɬ- |
13,51,2 |
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ɫɬɜɟɧɧɨ ɪɚɜɧɵ: |
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8,83 |
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5010 ɫɦ , |
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F1 = 40,2 ɫɦ2 ; |
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ȱ27 |
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ɯ1 |
5,27 |
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40 1,23 |
5,76 ɫɦ4 |
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1,2 403 |
6400 ɫɦ |
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F2 = 40 1,2 = 48 ɫɦ2 . |
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Ɋɢɫ.1.11 |
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ɉɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɪɚɜɧɚ |
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F = 2 40,2 + 48 = 128,4 ɫɦ2.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɨɩɪɟɞɟɥɢɦ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Ɉ1ɯ1, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪɵ ɬɹɠɟɫɬɢ ɞɜɭɬɚɜɪɨɜ:
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1,2∙ |
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x1 |
F y |
2 |
48♦13,5 |
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÷ |
676,8 |
ɫɦ . |
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2 ≠ |
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ɉɨ ɜɬɨɪɨɣ ɢɡ ɮɨɪɦɭɥ (1.5) ɩɨɥɭɱɢɦ
y0 |
Sx1 |
676,8 |
5,27 ɫɦ. |
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F |
128,4 |
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9
Ɉɫɢ |
Ɉɯ ɢ Ɉɭ ɹɜɥɹɸɬɫɹ ɝɥɚɜɧɵɦɢ ɰɟɧɬɪɚɥɶɧɵɦɢ ɨɫɹɦɢ. Ɇɨɦɟɧɬɵ ɢɧɟɪ- |
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ɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɢɯ ɨɫɟɣ ɪɚɜɧɵ |
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Jx |
2 5010 40,2 5,272 (5,76 48 8,832 ) |
16000 ɫɦ4 ; |
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J y |
2 260 40,2 102 6400 |
14960 ɫɦ4 . |
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Ɂɚɞɚɱɚ 1.6 |
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Ⱦɥɹ ɫɬɟɪɠɧɹ ɧɟɫɢɦɦɟɬɪɢɱɧɨɝɨ ɫɟɱɟɧɢɹ, ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ ɞɜɭɬɚɜɪɚ |
I50 ɢ |
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ɧɟɪɚɜɧɨɛɨɤɨɝɨ ɭɝɨɥɤɚ |
L200υ125υ16 |
(ɪɢɫ.1.12,ɚ), ɨɩɪɟɞɟɥɢɦ ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ |
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ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ, ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɢ |
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ɩɨɥɨɠɟɧɢɟ ɷɬɢɯ ɨɫɟɣ. |
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a) |
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y2 |
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ɛ) |
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y1 |
y |
21,79 |
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ɦɚɫɲɬɚɛ |
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6,71 |
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103 2·103 |
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50 |
7,24 |
14,55 |
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2,99 |
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14,69 22,01 |
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Jxy=15300 |
2=700 |
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O J2= |
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1=200 |
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7,32 |
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Jy=19000 |
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Jx=56000 |
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J1=62000 |
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Ɋɢɫ.1.12 |
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Ɇɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɢ ɩɥɨɳɚɞɢ ɫɟɱɟɧɢɣ ɞɜɭɬɚɜɪɚ ɢ ɭɝɨɥɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɯ |
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ɫɨɛɫɬɜɟɧɧɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɨɫɟɣ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ: |
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Jx1 |
39727 ɫɦ4 , |
J y1 |
1043ɫɦ4 , |
Jx1y1 |
0, |
F1 = 100 ɫɦ2 ; |
Jx2 |
617 ɫɦ4 , |
J y2 |
2026 ɫɦ4 , |
Jx2 y2 |
644 ɫɦ4 , F2 = 49,8 ɫɦ2 . |
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ɉɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɪɚɜɧɚ
F = 100 + 49,8 = 149,8 ɫɦ2 .
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɜɵɛɟɪɟɦ ɜ ɤɚɱɟɫɬɜɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɯ ɨɫɟɣ ɨɫɢ ɞɜɭɬɚɜɪɚ Ɉ1ɯ1 ɢ Ɉ1ɭ1. ɉɨ ɮɨɪɦɭɥɚɦ (1.5) ɩɨɥɭɱɢɦ
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F x |
2 |
49,8 21,79 |
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x0 |
1 |
2 |
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7,24 ɫɦ; |
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149,8 |
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F y |
2 |
49,8 22,01 |
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7,32 ɫɦ. |
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149,8 |
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10