Механика грунтов. Пьянков С.А - 25 стр.

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49
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɢɩɨɬɟɡɨɣ
ɜ
),
ı
R
=
A
cosȕ/
R
2. Ʉɪɨɦɟ ɷɬɨɝɨ,
r
=
R
Ӧ sinȕ. ɉɨɞɫɬɚɜɥɹɹ ɜ
ɭɪɚɜɧɟɧɢɟ (3.1) ɜɵɪɚɠɟɧɢɹ ɞɥɹ
ı
R
ɢ
r
ɢ ɜɵɩɨɥɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɩɨɥɭɱɢɦ:
,0sin
cos
2
2
2
0
2
2
³
PdR
R
A
EE
E
S
S
(4.2)
.0sincos2
2
0
2
³
PdA
EEES
S
ȼɵɩɨɥɧɹɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ ɜ ɭɪɚɜɧɟɧɢɢ (4.2):
u
= cosȕ,
du
= í sinȕ Ӧ
d
ȕ. ɉɪɨɞɨɥɠɚɹ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ
Ⱥ
:
0
3
2
3
P
u
A
S
; (4.3)
;0
3
cos
2
2
3
PA
x
E
S
ȼɵɪɚɡɢɦ cosȕ ɱɟɪɟɡ ɨɪɞɢɧɚɬɭ
z
: cosȕ =
z
/
R
. ɋ ɭɱɟɬɨɦ ɷɬɨɝɨ, ɮɨɪɦɭɥɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɧɚɩɪɹɠɟɧɢɹ
ı
R
ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
32
2
3
2
cos3
R
zP
R
P
R
SS
E
V
. (4.4)
ɉɪɚɤɬɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɟ, ɧɚɤɥɨ-
ɧɟɧɧɨɣ ɤ ɩɥɨɳɚɞɤɟ, ɧɚ ɤɨɬɨɪɨɣ ɞɟɣɫɬɜɭɸɬ ɧɚɩɪɹɠɟɧɢɹ
ı
R
, ɩɨɞ ɭɝɥɨɦ ȕ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɢ-
ɩɨɬɟɡɨɣ
ɛ
) ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɟ
ıƍ
R
ɫɨɜɩɚɞɚɟɬ ɩɨ ɧɚ-
ɩɪɚɜɥɟɧɢɸ ɫ ɜɟɤɬɨɪɨɦ ɧɚɩɪɹɠɟɧɢɹ
ı
R
, ɚɟɝɨɦɨɞɭɥɶɪɚɜɟɧ
ıƍ
R
=
ı
R
Ӧ cosȕ. ɉɪɨɟɤɰɢɢ ɝɥɚɜɧɨɝɨ
ɜɟɤɬɨɪɚ ɧɚɩɪɹɠɟɧɢɣ
ıƍ
R
ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ ɹɜɥɹɸɬɫɹ ɤɨɦɩɨɧɟɧɬɚɦɢ ɬɟɧɡɨɪɚ ɧɚɩɪɹɠɟɧɢɣ
ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɟ. ɉɨɫɤɨɥɶɤɭ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɢɣ
ıƍ
R
ɫɨɜɩɚɞɚɟɬ ɩɨ ɧɚ-
ɩɪɚɜɥɟɧɢɸ ɫ ɪɚɞɢɭɫɨɦ ɜɟɤɬɨɪɨɦ
R
, ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɜɟɤɬɨɪɚ ɧɚɩɪɹɠɟɧɢɣ ɨɩɪɟɞɟɥɹ-
ɸɬɫɹ ɮɨɪɦɭɥɚɦɢ:

;/,cos Ryy
R
c
V

;/,cos Rxx
R
c
V

Rzz
R
/,cos
c
V
. (4.5)
ɋ ɭɱɟɬɨɦ ɩɨɥɭɱɟɧɧɵɯ ɜɵɲɟ ɡɚɜɢɫɢɦɨɫɬɟɣ, ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ ɧɚɩɪɹɠɟɧɢɣ ɧɚ
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɳɚɞɤɟ ɛɭɞɭɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɮɨɪɦɭɥɚɦɢ
;
2
3
5
2
R
zP
R
z
R
x
R
z
RRz
c
S
VVV
5
2
2
3
R
zxP
R
z
R
x
R
x
RRzx
c
S
VVW
; (4.6)
5
2
2
3
R
zyP
R
z
R
y
R
y
RRzy
c
S
VVW
.
50
Ɏɨɪɦɭɥɭ ɞɥɹ
ı
z
ɨɛɵɱɧɨ ɬɚɛɭɥɢɪɭɸɬ. Ⱦɥɹ ɷɬɨɝɨ ɜɵɩɨɥɧɹɸɬ ɫɥɟɞɭɸɳɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ:
;1
2
2
22
z
r
zzrR
(4.7)

K
z
P
zrz
zP
Z
225
225
3
/12
3
S
V
, (4.8)
ȼ ɞɚɥɶɧɟɣɲɟɦ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ ɪɚɫɱɟɬ-
ɧɭɸ ɫɯɟɦɭ ɡɚɞɚɱɢ ɩɪɢɜɨɞɹɬ ɤ ɛɨɥɟɟ ɩɪɨɫɬɨɦɭ ɜɢɞɭ (ɪɢɫ.
4.6). ȼɟɪɬɢɤɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɪɚɫɱɟɬɧɨɣ ɬɨɱɤɟ
Ɇ
ɨɩ-
ɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ
2
Z
N
K
Z
V
.
Ʉɨɷɮɮɢɰɢɟɧɬ
Ʉ
, ɡɚɜɢɫɹɳɢɣ ɨɬ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚ-
ɪɚɦɟɬɪɚ
r/z
, ɩɪɢɜɨɞɢɬɫɹ ɜ ɫɩɪɚɜɨɱɧɵɯ ɞɚɧɧɵɯ.
Z
– ɝɥɭɛɢɧɚ ɬɨɱɤɢ;
r
– ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ ɞɨ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɫɢɥɵ;
Ɇ
– ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɬɨɱɤɚ;
N
– ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɚɹ ɫɢɥɚ.
Ɉɩɪɟɞɟɥɟɧɢɟ ɧɚɩɪɹɠɟɧɢɣ ɜ ɦɚɫɫɢɜɟ ɝɪɭɧɬɚ ɨɬ ɞɟɣɫɬɜɢɹ ɧɟɫɤɨɥɶɤɢɯ ɜɟɪɬɢɤɚɥɶ-
ɧɵɯ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɝɪɚɧɢɰɟ ɝɪɭɧɬɨɜɨɝɨ ɨɫɧɨɜɚɧɢɹ (ɩɪɢɧɰɢɩ
ɋɟɧ-ȼɟɧɚɧɚ – ɩɪɢɧɰɢɩ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɞɟɣɫɬɜɢɹ ɫɢɥ).
Ɋɢɫ. 4.7. ɋɯɟɦɚ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɧɚɩɪɹɠɟɧɢɣ ɜ ɦɚɫɫɢɜɟ ɝɪɭɧɬɚ ɨɬ ɞɟɣɫɬɜɢɹ
ɧɟɫɤɨɥɶɤɢɯ ɜɟɪɬɢɤɚɥɶɧɵɯ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯ ɫɢɥ
ȿɫɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɨɬɪɨɩɧɨɝɨ ɥɢɧɟɣɧɨ-ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɩɪɢɥɨ-
ɠɟɧɨ ɧɟɫɤɨɥɶɤɨ ɫɢɥ (
N
1
,
N
2
, ...,
N
n
), ɬɨ ɩɪɢ ɩɪɹɦɨɣ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɦɟɠɞɭ ɧɚɩɪɹɠɟɧɢɹ-
ɦɢ ɢ ɞɟɮɨɪɦɚɰɢɹɦɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɢ ɧɚɣɬɢ ɡɧɚɱɟɧɢɟ
ı
z
ɜɥɸ-
ɛɨɣ ɬɨɱɤɟ
Ɇ
ɩɪɨɫɬɵɦ ɫɭɦɦɢɪɨɜɚɧɢɟɦ:
22
22
2
11
)(
Z
NK
Z
NK
Z
NK
nn
MZ
V
, (4.9)
Ɋɢɫ. 4.6. ɋɯɟɦɚ ɤ ɨɩɪɟɞɟɥɟɧɢɸ
ɧɚɩɪɹɠɟɧɢɣ ɜ ɦɚɫɫɢɜɟ ɝɪɭɧɬɚ ɩɪɢ
ɞɟɣɫɬɜɢɢ ɟɞɢɧɢɱɧɨɣ
ɜɟɪɬɢɤɚɥɶɧɨɣ ɫɢɥɵ

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