TINH BIEN DANG CUA DAT
3.3. Bai toan phang va bai toan khong gian
3.3.1. Phuong trinh vi phan co ket
Nhu da trinh bay d tren, phuong trinh vi phan cd ket mot chiCu (theo cot ap tac dung) dirge bieu dien dudi dang cong thuc (6-21):
5H _
trong do: C v K
a ()Y n
- he so co ket doi vdi bai toan 1 chieu.
Doi vdi bai toan phang va bai toan khong gian, theo V.A. Florin, phuong trinh vi phan cua ly thuyet cd ket tham d dang don gian nhat la:
a) D oi voi bai toan phang:
<3H 1 50
<3t 2yn <9t + C'v V-H. (6-35)
trong do: C'v = — — - he sd cd ket trong bai toan phang:
2 Y n a
£0 - he so ap lire hong cua dat d trang thai tinh;
0 - tong cac ung suat chinh tai diem dang xet, dudi tac dung cua tai trong ngoai;
V - H =— —+ -— toan tu Laplaxo'doi vdi bai toan piling.
dz~ dy~
b) Doi voi bai toan kliong gian
— = — — + C ;V 2H, (6-36)
at 3yn at
K (l + 2£n)
trong d5: Cy = ---- b0/ - he so cd ket trong bai toan khong gian;
3Y„a
a-H a-H a -H , , . u ,
V “ H = — —-t---—4--- — - toan tu Laplaxo toan phan.
ax- dy~ dz~
Khi tmh cac lie so cd ket trong cac cong thuc (6-21), (6-35) va 6-36), cac dai luong dac truig: K, a0, va Yn duoc lay theo gia tri trung binh. Viec xet den tinh bien thien cua cac he so dac trung cua dat nen lun la van dd cuc ky phuc tap. Khi tim duoc gia tri cot ap 'di thdi gian t da cho bang viec giai phuong trinh vi phan co ket tuong ung, gia tri do lun i, tai diem dang xet, ta xac dinh theo cong thuc:
St = S 0O- (l- m ,)y na0^ HAz (6-37)
1 trong di: S^- do lun cuoi cung on dinh;
hn - do sau co hieu qua cua nen ep (do sau hit lun).
3.3.1. Truong hop tai trong phan bo deu hinh chit nhat
Co let cua dat no nirac dirai tac dung cua tai trong phan bo deu cuc bo trong bai toan khong ;ian cua nUa khong gian dong nhat, theo G.Gibxon va G.Makneimi, do nen chat U c cuadiem goc c thuoc dien tfch chir nhat chiu tai la:
r 1 e rr— r ert * 1 * k^dt ,fVt 2vt 2>/t
U c = ^ --- , (6-38)
00 1 1 A.
f erf r erf ^ dt , yfl 2Vt 2>/t C" t
trong d>: T =—V- - chi so thdi gian;
L2 t - thdi gian;
L - chi6u dai mat chfr nhat chiu tai;
Cy - he so co ket;
X - ty so giua chieu rong tren chi£u dai cua mat chir nhat chiu tai.
0,001 0,01 0,1 1,0 T1 [_ 10
Hinh 6-21. Cac dudng cong de xac dinh U0ddt no nude dudi diem goc c ciia mat chi? nhat chiu tdi.
De don gian tinh toan, ta dung nhung dirong cong cho phep xac dinh do co ket U 0 theo gia tri T doi vai mot so gia tri cua X (tir 0,2 —> 1,0), nhu tren hinh 6-21. Gia tri do lun toan bo cua dat (Sc) duai goc c mat chu nhat chiu tai, co the dung cong thuc cua mia khong gian hien dang tuyen tinh nhu sau:
s =cocp b (l- ^ ) (6 39)
E„
trong do: coc - lie so hinh dang (chir nhat coc = 0,5co0 theo l/b) doi vdi diem goc c;
P - tai trong don vi tren mat chfr nhat chiu tai;
b - chichi rong cua mat chir nhat chiu tai.
Biet gia tri do lun Sc va do cd ket Uc de dang tmh diroc do lun cua diem goc ung vdi thdi gian t bat ky:
S, = SCUC, (6-40)
Cung dung phirong phap diem goc va nhung dudng cong tren hinh 6-21, xac dinh dugc do lun doi vdi bat ky diem nao cua dat. Tat nhien co the tmh Sc don gian hon, neu dung cac he so ldp tuong dirong (phirong phap ldp tirong duong) cua dat.
3.3.3. Trudng hop bdi toan doi xung true ciia ly thuyet co ket
Day la bai toan co y nghla thirc tien to Ion khi thiet ke gieng cat, bac tham la cac gieng tieu niroc thing dung, lam tang qua trinh cd ket tham va sue chiu tai cua ndn dat
yeu no nirac. Trong truong hap nay (gieng cat), phuong trinh vi phan cua bai toan co ket khong gian dirge bieu thi dudi dang:
d?,
dt C d-P„ 1 dPK dzr r dr +c
'' dz2
(6-41)
trong do: Cr, Cz - cac lie so cd ket (hirdng tam va thang di'mg - doc true);
r - ban kfnh gieng (cat).
Giai phuong trinh (6-41) dua tren dinh ly phan tach dong khong gian hudng tam ra thanh dong chay phang va dong tuyen tfnh, bang cach lay tich phan bang sd nhan duoc nhung so lieu dd lap bieu do do cd ket:
Hirdng tam U r va doc true U r Theo do nhan duoc do cd ket tong quat (U, ) nhu sau:
U tq = 1 - (1 - U r)( 1 — U z), Neu xet tdi do bdn kien true nen cua dat duoc tieu nude Pkt va hien tuong gradien thuy luc ban dau (I0), thi bai toan se phuc tap hon nhieu. Ta chi xet trudng hop bien dang thang dung deu, rat thuan tien va da sir dung trong thuc te tfnh toan thiet ke. Khi bd mat khoi thoat nude duoc phu mot ldp tham nude (chieu day khong nho hon 2r), nhu tren hinh 6-22, thi ldp dem nay se san bang do lun. V i vay, gieng cat thoat nude thang dirng cd the xuat phat tir dieu kien bien dang deu, vdi gia thiet la tat ca bd mat phan thoat nude ciia khoi dat lun xuong nhu nhau. Luc nay, phuong trinh (6-41) duoc thay bang phuong trinh vi phan gan diing bac nhat, ldi giai khep km ciia nd la gia tri trung binh ciia do cd ket hudng tam U r ciia toan bo ldp thoat nude, nhu sau:
(4-42)
turn fljUBKU E E r a j Sm
'/////
J\ t
A
Hinh 6-22. Set do bo tri gieng eat thoat nude thang dung
k 2 Yn1!) [
P 3 P IR - R- J x (1 e - M ',) + £ k L
P (6-43)
trong do:
R - ban kinh anh hudng ciia gieng thoat nude (nira cu li giua cac true gieng);
P - ap suat ngoai phan bd deu (tai trong don vi).
M = K a 0Yn
R n — RR 3p
8
Neu I0 = 0, di6u nay phu hap vai mot sd loai dat, trong mot sd trudng hop thi phuong trinh (6-40) co dang don gian hon:
( l- e " M'') + h o a c U = 1 - - ^ c M' ' . (6-44) P
Gia tri do co ket U z (thcing dirng) theo bieu thuc (6-25):
trong do:
U z= 1 - - ^ e " ^ , (6-45)
7Z~
M' = ^ 4h~
Trong trudng hop dang xet, co the tmh do liin doi vdi thdi gian bat ky theo cong thirc:
St = SU tq = a0hPUtq. (6-46)
Lai giai cua plnrong trinh vi phan bai toan cd ket doi xirng true trong dieu kien d canh gieng thoat nude khong bi\ng do kin d trong gieng, nhung co xet den do ben kien true cua dat Pkl va I() ciing nhan diroc, song d dang phuc tap hon.