Механика грунтов. Пьянков С.А - 45 стр.

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89
Ɋɢɫ. 6.7. ɋɜɹɡɶ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ
E
ɫ ɜɟɥɢɱɢɧɨɣ ɢ ɧɚɩɪɚɜɥɟɧɢɟɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ
ɫɦɟɳɟɧɢɹ
u
ɫɬɟɧɤɢ: 1 – ɩɪɢɡɦɚ ɨɛɪɭɲɟɧɢɹ; 2 – ɩɪɢɡɦɚ ɜɵɩɢɪɚɧɢɹ; ɚ – ɞɟɣɫɬɜɢɟ ɚɤɬɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ;
ɛ – ɞɚɜɥɟɧɢɟ ɩɨɤɨɹ; ɜ – ɞɟɣɫɬɜɢɟ ɩɚɫɫɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ; ɝ – ɝɪɚɮɢɤ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ
ɮɨɪɦɢɪɨɜɚɧɢɟ ɩɪɢɡɦ ɨɛɪɭɲɟɧɢɹ ɢ ɜɵɩɢɪɚɧɢɹ
ȿɫɥɢ ɩɨɞɩɨɪɧɚɹ ɫɬɟɧɤɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ ɧɟ ɫɦɟɳɚɟɬɫɹ ɢ ɧɟ ɢɡɝɢɛɚɟɬɫɹ
(ɧɚɩɪɢɦɟɪ, ɩɥɢɬɧɵɣ ɮɭɧɞɚɦɟɧɬ ɤɨɪɨɛɱɚɬɨɝɨ ɫɟɱɟɧɢɹ), ɬɨ ɞɚɜɥɟɧɢɟ ɪɟɚɥɢɡɭɟɬɫɹ ɜ ɭɫɥɨɜɢɹɯ
ɨɬɫɭɬɫɬɜɢɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɫɦɟɳɟɧɢɹ ɩɪɢ 0 u (ɪɢɫ. 6.7, ɛ); ɟɝɨ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɞɚɜɥɟɧɢɟɦ
ɩɨɤɨɹ
0
E . ɉɪɢ ɷɬɨɦ ɞɨɩɭɫɤɚɟɬɫɹ ɨɩɪɟɞɟɥɹɬɶ ɨɪɞɢɧɚɬɭ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ ɧɚ
ɜɟɪɬɢɤɚɥɶɧɭɸ ɝɪɚɧɶ ɫɬɟɧɤɢ
x
V
, ɢɫɩɨɥɶɡɭɹ ɩɨɧɹɬɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɛɨɤɨɜɨɝɨ ɞɚɜɥɟɧɢɹ
[
. Ɍɨ-
ɝɞɚ, ɭɱɢɬɵɜɚɹ ɜɟɪɬɢɤɚɥɶɧɨɟ ɞɚɜɥɟɧɢɟ ɨɬ ɫɨɛɫɬɜɟɧɧɨɝɨ ɜɟɫɚ, ɝɪɭɧɬ
z
V
ɧɚ ɝɥɭɛɢɧɟ z ɨɬ ɩɨɜɟɪɯ-
ɧɨɫɬɢ ɡɚɫɵɩɤɢ, ɛɭɞɟɦ ɢɦɟɬɶ
zz
zx
J
Q
Q
[J[VV
1
, (6.10)
ɝɞɟ
J
– ɭɞɟɥɶɧɵɣ ɜɟɫ ɝɪɭɧɬɚ; v – ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ ɝɪɭɧɬɚ.
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɷɩɸɪɚ ɞɚɜɥɟɧɢɹ ɧɚ ɫɬɟɧɤɭ ɩɪɢ ɨɞɧɨɪɨɞɧɨɦ ɝɪɭɧɬɟ ɡɚɫɵɩɤɢ ɛɭɞɟɬ ɢɦɟɬɶ
ɜɢɞ ɬɪɟɭɝɨɥɶɧɢɤɚ, ɢ ɩɪɢ ɜɵɫɨɬɟ ɫɬɟɧɤɢ h ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɷɩɸɪɵ ɞɚɜɥɟɧɢɹ ɩɨɤɨɹ ɨɩɪɟɞɟ-
ɥɢɬɫɹ ɤɚɤ
2
0
)1(2
hE
Q
QJ
. (6.11)
ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ ɜɨɡɦɨɠɧɨ ɬɚɤɠɟ ɫɦɟɳɟɧɢɟ ɫɬɟɧɤɢ ɜ ɫɬɨɪɨɧɭ ɨɬ ɡɚɫɵɩ-
ɤɢ, ɩɪɢɧɹɬɨɟ, ɤɚɤ ɩɪɚɜɢɥɨ, ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ (ɪɢɫ. 6.7, ɚ). Ʉɨɝɞɚ ɷɬɨ ɫɦɟɳɟɧɢɟ ɞɨɫɬɢɝɚɟɬ ɧɟɤɨ-
ɬɨɪɨɣ ɜɟɥɢɱɢɧɵ
a
u , ɜ ɝɪɭɧɬɟ ɡɚɫɵɩɤɢ ɮɨɪɦɢɪɭɟɬɫɹ ɨɛɥɚɫɬɶ ɨɛɪɭɲɟɧɢɹ ɝɪɭɧɬɚ, ɝɪɚɧɢɰɚ ɤɨ-
ɬɨɪɨɣ ɧɚɡɵɜɚɟɬɫɹ ɩɨɜɟɪɯɧɨɫɬɶɸ ɫɤɨɥɶɠɟɧɢɹ, ɚ ɫɚɦɚ ɨɛɥɚɫɬɶ – ɩɪɢɡɦɨɣ ɨɛɪɭɲɟɧɢɹ. Ⱦɚɜɥɟɧɢɟ,
ɩɟɪɟɞɚɜɚɟɦɨɟ ɩɪɢɡɦɨɣ ɨɛɪɭɲɟɧɢɹ ɧɚ ɝɪɚɧɶ ɫɬɟɧɤɢ, ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɚɤɬɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ, ɚɟɝɨ
ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɨɛɨɡɧɚɱɚɸɬ
a
E .
ɇɚɤɨɧɟɰ, ɟɫɥɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɚɤɢɯ-ɥɢɛɨ ɫɢɥ ɩɨɞɩɨɪɧɚɹ ɫɬɟɧɤɚ ɫɦɟɳɚɟɬɫɹ ɜ ɫɬɨɪɨɧɭ
ɝɪɭɧɬɚ, ɜ ɡɚɫɵɩɤɟ ɬɚɤɠɟ ɨɛɪɚɡɭɸɬɫɹ ɩɨɜɟɪɯɧɨɫɬɢ ɫɤɨɥɶɠɟɧɢɹ, ɢ ɩɪɢ ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢɧɟ ɩɟ-
ɪɟɦɟɳɟɧɢɹ
ɩ
u ɮɨɪɦɢɪɭɟɬɫɹ ɩɪɢɡɦɚ ɜɵɩɢɪɚɧɢɹ ɝɪɭɧɬɚ. ɉɪɢ ɷɬɨɦ ɪɟɚɤɰɢɹ ɝɪɭɧɬɚ ɞɨɫɬɢɝɚɟɬ
ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɚɫɫɢɜɧɨɦɭ ɞɚɜɥɟɧɢɸ (ɨɬɩɨɪɭ) ɝɪɭɧɬɚ, ɪɟɡɭɥɶɬɢ-
ɪɭɸɳɭɸ ɤɨɬɨɪɨɝɨ ɨɛɨɡɧɚɱɚɸɬ
ɩ
E (ɪɢɫ. 6.7, ɜ).
90
ɗɤɫɩɟɪɢɦɟɧɬɵ ɩɨɤɚɡɵɜɚɸɬ (ɪɢɫ. 6.7, ɝ), ɱɬɨ ɩɨɥɧɨɟ ɮɨɪɦɢɪɨɜɚɧɢɟ ɩɪɢɡɦɵ ɨɛɪɭɲɟɧɢɹ ɢ
ɪɚɡɜɢɬɢɟ ɚɤɬɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɩɪɢ ɨɱɟɧɶ ɧɟɛɨɥɶɲɢɯ ɩɟɪɟɦɟɳɟɧɢɹɯ ɫɬɟɧɤɢ, ɫɨ-
ɫɬɚɜɥɹɸɳɢɯ ɬɵɫɹɱɧɵɟ ɞɨɥɢ ɟɟ ɜɵɫɨɬɵ. ɇɚɩɪɨɬɢɜ, ɨɛɪɚɡɨɜɚɧɢɟ ɩɪɢɡɦɵ ɜɵɩɢɪɚɧɢɹ ɢ ɪɚɡɜɢ-
ɬɢɟ ɩɚɫɫɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɩɪɢ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ ɩɟɪɟɦɟɳɟɧɢɣ ɫɬɟɧɤɢ.
Ʉɚɤ ɩɪɚɜɢɥɨ (ɫɦ. ɪɢɫ. 6.6), ɧɚ ɩɨɞɩɨɪɧɭɸ ɫɬɟɧɤɭ ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɛɭɞɭɬ ɞɟɣɫɬɜɨɜɚɬɶ
ɚɤɬɢɜɧɨɟ ɞɚɜɥɟɧɢɟ ɫɨ ɫɬɨɪɨɧɵ ɡɚɫɵɩɤɢ (ɫɩɪɚɜɚ ɨɬ ɫɬɟɧɤɢ), ɫɬɪɟɦɹɳɟɟɫɹ ɫɞɜɢɧɭɬɶ ɢɥɢ ɨɩɪɨ-
ɤɢɧɭɬɶ ɫɬɟɧɤɭ, ɢ ɩɚɫɫɢɜɧɨɟ ɞɚɜɥɟɧɢɟ (ɫɥɟɜɚ ɨɬ ɫɬɟɧɤɢ), ɩɪɟɩɹɬɫɬɜɭɸɳɟɟ ɩɨɬɟɪɟ ɟɟ ɭɫɬɨɣɱɢ-
ɜɨɫɬɢ. ɉɨɷɬɨɦɭ ɪɚɫɱɟɬ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ ɹɜɥɹɟɬɫɹ ɧɟɨɬɴɟɦɥɟɦɨɣ ɱɚɫɬɶɸ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɨɝ-
ɪɚɠɞɚɸɳɢɯ ɤɨɧɫɬɪɭɤɰɢɣ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɚɤɬɢɜɧɨɝɨ ɢ ɩɚɫɫɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ ɝɪɭɧɬɚ ɧɚ ɫɨɨɪɭɠɟɧɢɟ ɨɛɵɱɧɨ ɩɪɢ-
ɧɢɦɚɟɬɫɹ ɦɨɞɟɥɶ ɬɟɨɪɢɢ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɪɟɚɥɢɡɭɟɦɚɹ ɜ ɪɚɦɤɚɯ ɫɬɪɨɝɢɯ ɢɥɢ ɩɪɢ-
ɛɥɢɠɟɧɧɵɯ ɪɟɲɟɧɢɣ. ɉɪɢ ɷɬɨɦ ɜɨɡɧɢɤɚɸɬ ɡɧɚɱɢɬɟɥɶɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɡɚɬɪɭɞɧɟɧɢɹ, ɫɜɹ-
ɡɚɧɧɵɟ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɨɱɟɪɬɚɧɢɹ ɥɢɧɢɣ ɫɤɨɥɶɠɟɧɢɹ ɜ ɦɚɫɫɢɜɟ ɝɪɭɧɬɚ, ɩɨɷɬɨɦɭ ɜ ɩɪɚɤɬɢɱɟ-
ɫɤɢɯ ɪɚɫɱɟɬɚɯ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɞɨɩɭɳɟɧɢɟ, ɜɜɟɞɟɧɧɨɟ ɟɳɟ ɒ. Ʉɭɥɨɧɨɦ, ɨ ɩɪɹɦɨɥɢɧɟɣɧɨɦ
ɨɱɟɪɬɚɧɢɢ ɥɢɧɢɣ ɫɤɨɥɶɠɟɧɢɹ. Ⱦɥɹ ɚɤɬɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ ɦɟɬɨɞɵ ɪɚɫɱɟɬɨɜ, ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɷɬɨɦ
ɞɨɩɭɳɟɧɢɢ, ɞɚɸɬ ɪɟɡɭɥɶɬɚɬɵ, ɛɥɢɡɤɢɟ ɤ ɫɬɪɨɝɢɦ ɪɟɲɟɧɢɹɦ. ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɚɫɫɢɜɧɨɝɨ
ɞɚɜɥɟɧɢɹ ɩɨɥɭɱɚɸɬ ɡɚɜɵɲɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɩɪɢɱɟɦ ɩɨɝɪɟɲɧɨɫɬɶ ɜɨɡɪɚɫɬɚɟɬ ɫ ɭɜɟɥɢɱɟɧɢɟɦ
ɭɝɥɚ ɜɧɭɬɪɟɧɧɟɝɨ ɬɪɟɧɢɹ ɝɪɭɧɬɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ,
ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɷɬɨɦ ɞɨɩɭɳɟɧɢɢ.
Ɉɩɪɟɞɟɥɟɧɢɟ ɚɤɬɢɜɧɨɝɨ ɞɚɜɥɟɧɢɹ ɧɚ ɜɟɪɬɢɤɚɥɶɧɭɸ ɝɥɚɞɤɭɸ ɫɬɟɧɤɭ ɩɪɢ ɝɨɪɢɡɨɧ-
ɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɫɵɩɤɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɟɣɲɢɣ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɡɚɫɵɩɤɚ ɩɪɟɞɫɬɚɜɥɟɧɚ ɢɞɟɚɥɶɧɨ ɫɵɩɭɱɢɦ ɝɪɭɧ-
ɬɨɦ (ɪɢɫ. 6.8). ɉɨɫɤɨɥɶɤɭ ɩɪɢɧɹɬɨ, ɱɬɨ ɫɬɟɧɤɚ ɢɦɟɟɬ ɚɛɫɨɥɸɬɧɨ ɝɥɚɞɤɭɸ ɝɪɚɧɶ, ɬ. ɟ. ɬɪɟɧɢɟ
ɝɪɭɧɬɚ ɨ ɫɬɟɧɤɭ ɨɬɫɭɬɫɬɜɭɟɬ (0
zx
W
), ɜɟɪɬɢɤɚɥɶɧɵɟ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɩɥɨɳɚɞɤɢ ɹɜɥɹɸɬɫɹ
ɝɥɚɜɧɵɦɢ. Ɍɨɝɞɚ ɦɚɤɫɢɦɚɥɶɧɨɟ ɝɥɚɜɧɨɟ ɧɚɩɪɹɠɟɧɢɟ, ɞɟɣɫɬɜɭɸɳɟɟ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɥɨ-
ɳɚɞɤɭ ɜ ɬɨɱɤɟ ɤɨɧɬɚɤɬɚ ɝɪɭɧɬɚ ɫɨ ɫɬɟɧɤɨɣ ɧɚ ɝɥɭɛɢɧɟ z ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɫɵɩɤɢ, ɛɭɞɟɬ ɪɚɜɧɨ
z
JV
1
, (6.12)
ɝɞɟ
J
– ɭɞɟɥɶɧɵɣ ɜɟɫ ɝɪɭɧɬɚ.
ȼ ɩɪɟɞɟɥɚɯ ɩɪɢɡɦɵ ɨɛɪɭɲɟɧɢɹ ɈȺȼ ɝɪɭɧɬ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟ-
ɫɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɢɧɢɦɚɥɶɧɨɟ ɝɥɚɜɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɜ ɷɬɨɣ ɬɨɱɤɟ
3
V
, ɪɚɜɧɨɟ ɚɤɬɢɜɧɨɦɭ
ɞɚɜɥɟɧɢɸ
a
V
, ɛɭɞɟɬ ɫɜɹɡɚɧɨ ɫ ɦɚɤɫɢɦɚɥɶɧɵɦ ɝɥɚɜɧɵɦ ɧɚɩɪɹɠɟɧɢɟɦ ɭɫɥɨɜɢɟɦ ɩɪɟɞɟɥɶɧɨɝɨ
ɪɚɜɧɨɜɟɫɢɹ.
ɍɫɥɨɜɢɟ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɞɥɹ
ɫɵɩɭɱɢɯ ɝɪɭɧɬɨɜ
31
31
sin
VV
VV
M
ɦɨɠɧɨ ɜɵɪɚ-
ɡɢɬɶ ɜ ɜɢɞɟ
)
2
45(
sin1
sin1
2
113
M
V
M
M
VV
$
tg . (6.13)
Ɍɨɝɞɚ ɫ ɭɱɟɬɨɦ ɮɨɪɦɭɥɵ (6.12) ɚɤɬɢɜɧɨɟ
ɞɚɜɥɟɧɢɟ ɝɪɭɧɬɚ ɧɚ ɫɬɟɧɤɭ ɜ ɬɨɱɤɟ, ɪɚɫɩɨɥɨ-
ɠɟɧɧɨɣ ɧɚ ɝɥɭɛɢɧɟ z ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɫɵɩɤɢ,
ɛɭɞɟɬ ɪɚɜɧɨ
)
2
45(
2
3
M
JVV
$
tgz
a
. (6.14)
Ɋɢɫ. 6.8. ɋɯɟɦɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɚɤɬɢɜɧɨɝɨ
ɞɚɜɥɟɧɢɹ ɫɵɩɭɱɟɝɨ ɝɪɭɧɬɚ
ɧɚ ɝɥɚɞɤɭɸ ɫɬɟɧɤɭ

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