Zadania Mechanika Gruntów

7/21/2019 Zadania Mechanika Gruntw

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d 0, 002

0, 002< d 0, 063

0, 063< d 2

2< d 63

63< d 200

200< d

630

630< d

d

d

p

dp

p%

d10 d30 d50 d60

Cu=U=d60d10

Cu

Cu = 1


6/114

Cu

Cu15

Cc= d230

d10 d60

Cc

1 Cc 3 Cu 4 Cu 6


7/114

mm V

ms Vs mw Vw ma Va

Vp=Vw+Va

V =Vs+Vp = Vs+Vw+Va

s=ms

Vs 2, 65 g

cm3= 2650

kg

m3

w =mw

Vw = 1, 00 g

cm3 = 1000kg

m3

a =ma

Va 0, 0012 g

cm3 = 1, 2

kg

m3

n=VpV

=Vw+Va

V

(0; 1)

e=

VpVs

=Vw+Va

Vs

(0;

)


8/114

n= e

1 +e oraz e=

n

1 n

n

e

wn =mwms

>100%

wsr wsr

wsr =e ws

wsr

Sr =VwVp

0; 1


9/114

n

e

wsr

n e wsr


10/114

Sr

0, 00

0, 01 0, 40 0, 41 0, 80

0, 81 1, 00

=ms+mw

V =

Vs s+Vw wV

= (1 n) s+Sr n w

d

s

Sr = 0

d =ms

V = (1 n) s

sr

Sr = 1

sr =ms+Vp

w

V = (1 n) s+n

= g

g

g= 10m

s2

gcm3

kNm3

s d sr

Gs= s

w= s

w

= (1 n) (s w)

=

g = (1 n) (s w)


11/114

s =ms

Vs

s = d (1 +e) = d1 n =

= (1 +e)

1 +w =

(1 n) (1 +w)

=mm

V

= d (1 +w) =1 +w1 +e

s== (1 n) (1 +w) s

w=

mw

ms

w= dd

=

(1

n)

s 1 =

=n Sr w

d

d=ms

V

d=

1 +w = (1 n) s = s

1 +e =

= s ww+wsr s

n=Vp

V

n= e

1 +e=s ds

=

= 1

s (1 +w)

e=Vp

Vs

e= n

1 n =s dd

=

=s (1 +w)

1

wsr=w Vpms

wsr=n wd

=n (1 +w) w

=

=e ws

=(1 +w) s

s

w

Sr =Vw

Vp= w

wsr

Sr =w se w =

dn w =

w dn w =

= w

n (1 +w) w

sr =ms+Vp w

V sr = (1 n)s+nw = d+nw

=ms Vs w

V = (1 n) (s w) =sr w

w wsr n e Sr


12/114

< d

sr < s

ID = Vmax VVmax Vmin

= emax eemax emin

= d d,mind,max d,min

Vmax

emax

d,min Vmin

emin d,max

ID

s ds

dmax

ds dmax


13/114

wL

wp

ws

Ip =wL wp

IL=wn wpwL wp =

wn wpIp

wn

Ic=wL wnwL wp =

wL wnIp

= 1 IL


14/114

1 dm3

wn =mm

ms 100% =1, 845kg 1, 674 kg

1, 674kg 100% 10, 2%

=mm

V =

1845g

1000cm3 1, 85 g

cm3

d= 100% +wn

100% = 1, 85100% + 10, 2%

100% = 1, 68 gcm3

mm = 210g

V= 100cm3

+105

wn e

wn=mwms

100% = 0, 18 210210 0, 18 210 100% = 21, 9%

=mm

V =

210

100= 2, 1

g

cm3

s=ms

Vs

w = 1

gcm3

Vs=V Vw


15/114

w =mw

Vw Vw =

mw

w

=0, 18 210

1

= 37, 8 g

cm

3

s=210 0, 18 210

100 37, 8 = 2, 76 g

cm3

d =ms

V =

210 0, 18 210100

= 1, 72 g

cm3

e=s d

d=

2, 76 1, 721, 72

= 0, 605

n=s d

s=

2, 76 1, 722, 76

= 0, 377

= (1

n)

(s

w) = (1

0, 377)

(2, 76

1, 0) = 1, 1 g

cm3

s= 2, 65 gcm3

gcm3

ID

d,min=

ms

V =

750

500= 1, 5

g

cm3


16/114

emaxs d,min

d,min=

2, 65 1, 51, 5

= 0, 766

d,max= ms

V V = 750

500 (0, 15 500) = 1, 76 g

cm3

emin=s d,max

d,max=

2, 65 1, 761, 76

= 0, 506

d= 100% 100% +w

= 100% 1, 83100% + 16%

= 1, 58 g

cm3

e = s d

d=

2, 65 1, 581, 58

= 0, 677

ID = emax eemax emin =

0, 766 0, 6770, 766 0, 506= 0, 342

ID = 0, 15 0, 35

ws = 7%

wp= 15% wL= 35% wn= 12%

Ip = wL wp= 35% 15% = 20%

I

= wn wpwL wp =

12% 15%20%

= 0, 15< 0

w < w = 12%p


17/114

L

p

I

= wn wpwL wp =

9, 5 9, 824 9, 8 = 0, 02

p

d g

cm3

s

d = 100% 100% +wn

=d (100% +wn)

100% =

1, 65 (100% + 15%)100%

= 1, 9 g

cm3

e=s d

d

s=e d+d = 0, 6 1, 65 + 1, 65 = 2, 64 gcm3

S

= wn s

100% e w = 15% 2, 64100% 0, 6 1= 0, 66

200g

wsr = 50%

ms = 1000g s = 2, 60

gcm3

d

gcm3

w

gcm3

wsr =e ws

100% e= wsr sw 100%=

50% 2, 61 100% = 1, 3


18/114

n= e

1 +e=

1, 3

1 + 1, 3= 0, 57

d =ms

V V = ms

d=

1000g

1, 12 gcm3

893cm

mw,0 = wsr ms 200g = 50% 1000g 200 g = 300g

=

ms+mw,0

V =

1000 g + 300 g

893 cm3 1, 46 g

cm3

ID = 0, 54

Vmax= 1, 23dm3

Vmin = 0, 90dm

3

ms = 1, 40kg

s= 2, 65 g

cm3

ID = Vmax VVmax Vmin

V =Vmax ID (Vmax Vmin) = 1, 23 0, 54 (1, 23 0, 9) = 1, 05dm

d =ms

V

= 1, 4

1, 05

= 1, 33 g

cm3

n=s d

s=

2, 65 1, 332, 65

= 0, 50

e=s d

d=

2, 65 1, 331, 33

= 0, 99


19/114

= 1, 68 g

cm3

sr

s= 2, 65

gcm3

sr = 2, 05

gcm3

= 1, 89 g

cm3 d =

1, 7 gcm3

wn

wsr sr

wn= 11, 2% wsr = 21, 2% sr = 2, 06 gcm3

d= 1, 6

gcm3

= 1, 0 gcm3

wn = 10% n s s

n= 0, 4

s= 2, 67

gcm3

= 1, 76 g

cm3 = 17, 6 kN

m3 s= 26, 7

kNm3

= 10 kN

m3

mm = 165g

V = 80cm3

17%

wn e

wn= 20, 5% e= 0, 55

= 1, 06 gcm3


20/114

v= k i= k Hl

v k

i =

H

l

H

l

ms

v

vr = v

n

H=Hv+Hu+Hz = v2r2 g +

u

w g +Hz


21/114

Hz Hz Hz

u

vr

w g

H=Hu+Hz = u

w g +Hz

v=

w

u

l +w g z

l

w

u

z

k

k= w gw

[m2]

k10 10

T

k10

k10 = 1, 3591 + 0, 0337 T+ 0, 00022 T2 kT

Q= v A= Vt

Q

m3

s

A [m2]

V [m3]

t


22/114

km

s

k

md

>103 >100

104 103 10 100

105

104

1 10

106 105 101 1

108 106 103 101

1010 108 105 103

1011 109 106 104

1012 1011 107 106


23/114

k C d 210 [ md]

d10 C

C= 1200,

1< U


24/114

k

ki

Ai

Ai

Ai

k Li(Liki

)

sr s w


25/114

s w

j

w i kN

m3

kNm3

kNm2

jV kNm3

jV

jV jV

j

wi

F


26/114

j

F

j wi

10i10

2

i0, 5

l

l1+ l2 l1

i

Hl

68+2

0, 6

i

Hl

57+2

0, 56

i

H

l

4

6+2

0, 5

h2, 0


27/114

j

F

j wi

10i102 =i0, 5

i

Hl1+l2

l1 H+l2

6l1+l2

0, 5=l1+l212=H+ l2+l212=l2 3

j =

F


28/114

i =Hl

l=l1+l2+l3

i= 54+4+1

=0, 56

j =w i= 10 0, 56 = 5, 6 kN

m3

F =

j = 105,6

= 1, 79

A

100m2

k

105m

s

Q= 2, 16 m3

h

k

104ms


29/114

k

108m

s

A

1, 0m2


30/114

z

z = z= g z

z =n

i=1

(i zi) =n

i=1

(i g zi)

i i

u

= +u

=


31/114

u= w

zw

w zw

z =sr z w z= (sr w) z= z

z =sr zw+ (z zw) w zw = (z zw) + zw


32/114

z = (sr z+w hw) w (z+hw) = z


33/114

R = 3

2 Q cos

R2

z = 3

2 Q cos3

R2


34/114

z =q n

q

n zB

LB

n = 12

arctgLB

zB

1 +LB

2+

zB

2 +LB

zB

1 +LB

2+

zB

2 11 + zB

2 + 1LB

2+

zB

2


35/114

z =q m

q

m zB

LB

m= 2

arctg

LB

2 zB

1 +LB

2+ 4 z

B

2 + 2 LB zB

1 +LB

2+ 4 z

B

2

1

1 + 4 zB

2 + 1LB

2+ 4 z

B

2


36/114

w 10

kNm3

z 0, 0 u

z

z 18 1, 5 27kP a u z 27kP a

z 18 1, 5 + 20 1, 5 57kP a u 10 1, 5 15kP a z 57 15 = 42kP a

z 21 5 + 57 162kP a u 10 6, 5 65kP a z 162 65 = 97kP a


37/114

z z 17 2, 0 + 21 2, 0 76kP a.

u w

h

10

2, 0

20kP a

z

z 76 20 56kP a

z z 17 2, 0 + 21 2, 0 + 21 4, 0 160kP a.

h

3, 0 + 8, 0

11, 0m

u w

h

10

11, 0

110kP a

z

z 160 110 50kP a

z

8 m

z

z 0

z z u z u

u w h 160kP a

h

160w

16010 16m


38/114

h1 h 8, 0 16, 0 8, 0 8, 0m

50 kP a

0

3, 0m 8, 0m

z

z u 0

z u

z 21 (4, 0 z) + 21 4, 0 168 21 z

u w h 10 11 110kP a

168 21 z 110 z 2, 76m


39/114

Izd 32 Qcos

3R2

R

3, 52 + 32 4, 6m tg 3,53 1, 667 49, 4

Izd 32 2500cos

3(49,4)4,62

15, 6kP a

IIzd q mz

3 m

B

3 m

L

4, 5 m

zB

1

LB

1, 5

m

IIzd 200 0, 42 84kP a

z 46kP a


40/114

z

q

100kP a

B

2, 0m

z

3 B

20 kNm3

z

z zd z

2, 73m

z zs zd

z

5m

5m


41/114

q

4, 0m 100kP a

z zs zd z 6m

6m

z


42/114

f= tg+c.

f


43/114

c

| | f

| | f

| | f

f tg+c

0, 5 (1 3) 0, 5 (1+3) sin+c cos


44/114

f= tg+c

c

u

0


45/114

n 100kP a

f

60 kP a

1 3

f n tg+c

c

0

60

100 tg = 31

31

c

0

1 3 n 100kP a f 60kP a

p

n+x

x

tg

xf

x60 =x36kP a p 100 + 36

136kP a

q

fcos

60cos31 70kP a

p=0,5(1+3)=136

q=0,5(13)=70

1 206kP a,

3 66kPa.


46/114

c

30 kP a

3 100 kP a 1

250 kP a

n

f

0, 5 (1 3) = 0, 5 (1+3) sin+c cos

75 175 sin+ 30 cos

sin2+ cos2

1

75=175sin+30cos

cos=

1sin2

sin

cos

x1 sin 0, 263= 15, 2x2 sin 0, 569= 34, 68

15, 2

cos

f

q

q

0, 5

(1

3) 75kP a

f q cos 75 cos15, 2

f 72, 4kP a

f n tg+c

72, 4 n tg15, 2+ 30

n 156, 1kP a


47/114

3 70 kP a 1 200 kP a 1 3

f 100kP a

3 70kP a 1 200kP a

p=0,5(1+3)=0,5(200+70)=135 kP aq=0,5(13)=0,5(20070)=65 kP a

sin

qp

65135 0, 4815 = 28, 8

1 3 f 100kP a

cos

fq

=q

fcos28,8

1000,876

114, 1kP a

p

qsin28,8

114,10,482

236, 8kP a

p=0,5(1+3)=236,8 kP aq=0,5(13)=114,1 kP a 1 350, 9 kP a

3 122, 7kP a

15

c

24, 52kP a

3

1 147, 1kP a

0, 5 (1 3) = 0, 5 (1+3) sin+c cos

1 3 (1+3) sin+ 2 c cos

1 1 sin 2 c cos 3+3 sin

1 (1+ 2 c ctg) sin 3 (1 +sin)

c

ctg


48/114

3 1(1+2cctg)sin

1+sin

3 147,1(147,1+224,52ctg15)sin15

1+sin15 48, 94kP a

3a= 80kP a 1a= 220kP a ua= 25kP a.

3b = 150kP a 1b = 360kP a ub = 40kP a.

u cu c

sin= qcctg+p

q = 1

3

2

p = 1+3

2


49/114

u cu

q

p

qa= 22080

2 = 70kP a

pa =

220+802

= 150kP a

qb = 360150

2 = 105kP a

pb=

360+1502

= 255kP a

u cu

sinu = qa

cuctgu+pa = 70

cuctgu+150sinu =

qbcuctgu+pb =

105cuctgu+255

x= cu ctgu

sinu =

70x+150

sinu = 105

x+255

70x+150

= 105x+255

x= 59, 52kP a

x

sinu= 70

59,52+150= 0, 3341 u = 19, 5o

x= cuctgu = 59, 52kP a cu = xctgu =xtgu= 59, 52tg19, 5o cu=

21, 1kP a

cu

i=i u u

3a= 80 25 = 55kP a 1a = 220 25 = 195kP a 3b = 150 40 = 110kP a 1b= 360 40 = 320kP a

q

p

qa= 195552 = 70kP a pa = 195+552 = 125kP a qb = 3201102 = 105kP a pb= 320+1102 = 215kP a

q=q

p =p u

sin = 70cctg+125 = 70x+125sin = qb

cctg+215 = 105x+215


50/114

70x+125

= 105x+215

x= 54, 56kP a

sin= 7054,56+125

= 0, 3898 = 22, 9o

c =x tg = 54, 56 tg22, 9o c = 23, 1kP a

c

u cu

u u

cu

n 100kP a

f 55kP a

3

f 100kP a

3 122, 7kP a

1 250kP a

15o

c

20kP a

3

3 116, 5kP a

tg

1 800kP a 3 200kP a

tg 0, 75

3

15o

c

20kP a

1

30

o

3

100kP a

3 145, 9kP a

1 300kP a

15o

c

0kP a

3

3 176, 64kP a


51/114

M0 =

h h0, [MP a]

hh0

M=

h h0, [MP a]

M=

h h1, [MP a]


52/114

E0= M0, [MP a]

(1+)(12)1

E= M, [MP a]

= E0M0 =

E

M

a= e

[kP a1]

mv = e

(1 +e0) , [kP a1]

Es= E01 2 , [MP a]

kz = E0

B

(1

2), [

kN

m2/mb]

B

LB


53/114

h0 19, 699

mm

h 19, 699 19, 461

0, 238 mm

100 50 kP a

M0

h h0

500,238 19, 6994138 kP a

hh0

0,23819,699 1, 21%

s 2, 72

gcm3

2, 07 g

cm3

w 22%


54/114

d

100

100+wn

100

2,07

100+22

1, 7

g

cm3

e0 sd

d

2,721,71,7 0, 6

h h0h1 e e0e1 h

e

h0 1 +e0 h

h0

hh0

e1+e0

e

hh0

(1 +e0)

e

19,46119,18919,461

(1 + 0, 6) 0, 022

200 100 kP a

a

e

0,022100 2, 2 104 kP a1


55/114

M0 45000 kP a 0, 2

kz

kz E0

B(12)

E0 M0 (1+)(12)1 (1+0,2)(120,2)

10,2 0, 9E0 0, 9 45000 40500kP a

B

4m

LB

LB

5

2, 1

kz 40500

2,14(10,22) 5022 kNm2

/mb


56/114

qu = QuB L

qu

D


57/114

c

qu = c Nc+ D Nq

Nq =etg tg2(45o +

2) Nc= ctg

(Nq 1)

cu u 0

qu= 5, 14 cu+ D

Nq = 1 Nc= + 2 = 5, 14

c

qu=c Nc+ D Nq+ 0, 5 B N

qu = 1, 3 c Nc+ D Nq+ 0, 4 B N


58/114

qu = 1, 2 c Nc+ D Nq+ 0, 3 B N

Nq = e(

32)tg

2 cos2(45o + 2)

Nc= ctg (Nq 1) N= 0, 5 tg ( kp

cos2 1)

(32

),

kp

kp= (8 2 4 + 3, 8) tg2(60o +

2)

(8 2 4 + 3, 8),

cu u 0

qu c

cu

Nc= (3

2 + 1) = 5, 71 Nq = 1 N= 0

sc sq s ic iq i

dc dq d

c

qu = c Nc sc ic dc+ D Nq sq iq dq+ 0, 5 B N s i d


59/114

Nc,Nq N (Nq 1) tg(1, 4 )

sc = 1 + 0, 2 BL tg2(45o +

2) sq =s= 1 + 0, 1 B

L tg2(45o +

2)

ic= iq = (1 +

90o )2

i= (1

)2

dc= 1 + 0, 2 DB tg(45o +

2) dq =d= 1 + 0, 1 D

B tg(45o +

2)

cu u 0

qu c cu

Nc,Nq N 0sq s 1 sc

u 0 dq d 1 dc

u 0i 1 ic iq

L = L 2 eLorazB =B 2 eB

eL eB


60/114

B L

c

qu = c Nc sc ic dc bc gc+D Nq sq iq dq bq gq +0, 5 B Nsidbg

bc bq b gc gq g

1

Nq Nc N 1, 5 (Nq 1) tg

sc= 1 + Nq BNc L ic, sq = 1 + BL iq sin, s= 1 0, 4 BL i 0, 6.

sc= 1 +Nq LNc B ic, sq = 1 +

L

B iq sin, s= 1 0, 4 L

B i 0, 6.


61/114

ic=iq 1 iqNq 1 , iq = (1

0, 5 HB(L)V +A cb ctg )

1, i= (1 0, 7 HB(L)

V +A cb ctg )2 .

H

B

L

HB HL i

A

B L cb

215 225

30o 45o 1

2 5

i=

1 (0, 7

o/450o) HB(L)V +A cb ctg

2.

DB 1 D

L 1

dc = 1 + 0, 4 DB

, dq = 1 + 2 tg (1 sin)2 DB

, d= 1.

dc = 1 + 0, 4 DL

, dq = 1 + 2 tg (1 sin)2 DL

, d= 1.

DB

>1

DL

>1

dc= 1 + 0, 4 arctg(DB

), dq = 1 + 2 tg (1 sin)2 arctg(DB

), d= 1.

dc= 1 + 0, 4 arctg(DL

), dq = 1 + 2 tg (1 sin)2 arctg(DL

), d= 1.


62/114

gc = 1 o

147o, gq =g= (1 0, 5 tg)5.

bc = 1 o

147o, bq =e

2tg, b=e2,7tg.

cu u 0

qu= 5, 14 cu (1 +sc+dc ic bc gc) + D.

sc= 0, 2 BL ic lub sc= 0, 2 L

B ic.

ic

H

B H

L

ic = 0, 5 0, 5

1 HB(L)A cb .

dc= 0, 4 DB

lub dc= 0, 4 DL

.

gc= o

147o.

bc= o

147o.

B

L


63/114

N qu

c

N= 2 (Nq 1) tg.

sc= 1 +Nq BNc L , sq = 1 +

B

L tg, s= 1 0, 4 B

L 0, 6.

sc= 1 +N

q L

Nc B , sq = 1 +L

B tg, s= 1 0, 4 L

B 0, 6.

iq = (1 HB(L)

V +A cb ctg )mB(L) , i= (1

HB(L)V +A cb ctg

mB(L)

.

mB =2 + B

L

1 + BL

, mL=2 + L

B

1 + LB

.

gc=iq 1 iq5, 14 tg , gq =g= (1 tg)2.


64/114

bc= 1

2 o5, 14 tg

, bq =b= (1

o

tg)2.

cu u 0

N= 2 sin.

sc= 0, 2 BL

, lub sc = 0, 2 LB

,

ic= (1 mB(L) HB(L)

A cb ctg )mB(L),

mB mL

gc=

5, 14,

bc=

5, 14,

d


65/114

cu u 0

qu = cu Nc+ D.

Nc= 5, 14 (1 + 0, 2 BL

) (1 + 0, 24 DB 0, 031 D

2

B2).

Nc= 5, 14 (1 + 0, 2 LB

) (1 + 0, 24 DL 0, 031 D

2

L2).

B

L

Nr m QfNB

Nr [kN] m

QfNB

QfNL

QfNB = BL

1 + 0, 3 BL

Nc c(r)u ic+

1 + 1, 5 B

L

ND (r)D Dmin iD+

1 0, 25 B

L

NB (r)B B iB

QfNB

[kN]

L 1mb


66/114

B B 2 eB L L 2 eL

B

L

B L

B

L

eB eL B L

ND =etg tg2

4+

2

Nc= (ND 1) ctg

NB = 0, 75 (ND 1) tg

Dmin c(

r)u

[kP a]

(r)D

kNm3 (r)B

B

kNm3

ic,iD, iB


67/114

c

10

28o

cu 105 u 0 19

kNm3

.

Nq etg tg2(45o +

2)

etg28

o tg2(45o + 28o2

) 14, 72

Nc ctg (Nq 1) ctg28o (14, 72 1) 25, 8

qu c Nc+ D Nq 10 25, 8 + 19 1, 0 14, 72 538kP a

qu 5, 14 cu+ D 5, 14 105 + 19 1, 0 559kP a

c

10

28o

cu 105 u 0 19kNm3

.

kp

kp (8 2 4 + 3, 8) tg2(60o + 2)

8 (28o

180o)2 4 (28o

180o) + 3, 8) tg2(60o + 28o

2 )

45, 68

Nq e(

32)tg

2cos2(45o+2)

e( 3228

o180o

)tg28o

2cos2(45o+28o2 ) 17, 81

Nc ctg (Nq 1) ctg28o (17, 81 1) 31, 61

N 0, 5 tg (kp

cos21) 0, 5 tg28o ( 45,68

cos228o1)

15, 31


68/114

qu c Nc+ D Nq+ 0, 5 B N

10 31, 61 + 19 1, 0 17, 81 + 0, 5 1, 5 19 15, 31 873 kP a

Nc (32 + 1)

5, 71 Nq 1 N

qu cu Nc + D Nq+ 0, 5 B N 105 5, 71+19 1 1 + 0, 5 1, 5 19 0 619kP a

c

10

28o

cu 105 u 0 19

kNm3

.

Nq etg tg2(45o +

2)

etg28

o tg2(45o + 28o2

) 14, 72

Nc ctg (Nq 1) ctg28o (14, 72 1) 25, 8

N (Nq 1) tg(1, 4 ) (14, 72 1) tg(1, 4 28o) 11, 19

ic iq i 1

sq s 1 + 0, 1 BL tg2(45o +

2)

1 + 0, 1 1,5

1,5 tg2(45o + 28o

2 )

1, 28

sc 1 + 0, 2 BL tg2(45o +

2)

1 + 0, 2 1,5

1,5 tg2(45o + 28o

2 )

1, 55

dc 1 + 0, 2 DB tg(45o +

2)

1 + 0, 2 1,01,5 tg(45o + 28o

2 )

1, 22

dq d 1 + 0, 1D

Btg(45o +

2

) 1 + 0, 1

1,0

1,5tg(45o + 28

o

2

) 1, 11


69/114

qu c Nc sc ic dc+ D Nq sq iq dq+ 0, 5 B N s i d

10 25, 8 1, 55 1, 0 1, 22+19 1, 0 14, 72 1, 28 1, 0 1, 11+0, 5 1, 5 19 11, 19 1, 28 1, 11

1112 kP a

Nc 5, 14 Nq 1 N 0

sq s 1

sc 1 + 0, 2 BL tg2(45o + u2 ) 1 + 0, 2 1,51,5 tg2(45o + 02) 1, 2

dq d 1

dc 1 + 0, 2 DB tg(45o + u2 ) 1 + 0, 2 1,01,5 tg(45o + 02) 1, 13

ic iq i 1

qu cu Nc sc ic dc+ D Nq sq iq dq+ 0, 5 B N s i d 105 5, 14 1, 2 1, 0 1, 13+19 1, 0 1, 0 1, 0 1, 0 1, 0 + 0, 5 1, 5 19 0 1, 0 1, 0 1, 0

751kP a

c

10

28o

cu 105 u 0 19

kNm3

.

Nq Nc

Nq etg tg2(45o +

2)

etg28

o tg2(45o + 28o2

)

14, 72

Nc ctg (Nq 1) ctg28o (14, 72 1) 25, 8

N 1, 5 (Nq 1) tg 1, 5 (14, 72 1) tg28o 10, 94


70/114

ic iq i 1

sc 1 +N

qB

NcL ic 1 + 14,721,525,81,5 1, 0 1, 57sq 1 +

BL iq sin 1 + 1,51,5 1, 0 sin28o 1, 47

s 1 0, 4 BL i 1 0, 4 1,51,5 1, 0 0, 6

DB

1,01,5

0, 67


71/114

B

L

m

0, 8

m

0, 9

(r) =(n) m= 20o 0, 9 = 18o

c(r) =c(n)

m= 20

0, 9 = 18kP a

(r)D =(r)B =

(n) m= 20 0, 9 = 18 kNm3

L=

L= 1, 0m

B =B 2 eB

Dmin= 2, 0m.

Nc= 13, 11 ND = 5, 26 NB = 1, 04

ic=iD =iB = 1, 0

Nr m QfNB QfNB 625 kNm

QfNB = (B 2 0, 1)1, 0{[1 + 0, 3 (B 2 0, 1)] 13, 11 18 1, 0 + [1 + 1, 5 (B 2 0, 1)] 5, 26

4, 68 B3 + 376, 37 B2 + 275, 34 B 695, 1 0

B 1, 05m


72/114

D

L

m 0, 9

m 0, 9

q

m 0, 9

eB

Nr mQfNB

m

0, 9

m 0, 9


73/114


74/114

Es

= 0, 1 + 0, 3 rel

rel TX25o45o25o

TX

Es

Es 2, 5

qc

Es 3, 5 qc


75/114

qc

Escu 1000 1500

Es [kN/m

2]

25000 35000

30000 40000

40000 45000

20000 25000

25000

35000

35000 40000

8000 12000

10000 12000

12000 15000


76/114

Es [kN/m

2]

3000 50000

6000 20000

3000 6000

6000 50000

4000 8000

2000 5000

L B

B

se=q B 1 2

Es I

q

B

Es

I

I1 I0 Isr Ir

I1= 1

L

B ln

1 +

( LB

)2 + 1

LB

+ln

L

B+

(

L

B)2 + 1


77/114

I0 = 2 I1

Isr= 1, 696 I1

Ir = 1, 57 I1

H

I1=I2+1 2

1 I3

I2= 1

L

B ln

1 +

( LB

)2 + 1

( LB

)2 + (HB

)2

LB (1 +

( LB

)2 + (HB

)2 + 1

+ln

LB +

( LB

)2 + 1

1 + ( HB

)2

LB

+

( LB

)2 + ( HB

)2 + 1

I3=HB

2 ctg LB

HB

( LB

)2 + ( HB

)2 + 1

I0 = 2 I1, Isr = 1, 696 I1, Ir = 1, 57 I1


78/114

h 0, 5 B

s=

si

zmax

zmaxd = 0, 3 zmax

i

si = s

i +s

i

s

i zsi

hi

Mi

i

zsi i

t 1

Mi hi i

s

i zdi hiM0i

i

zdi i

M0i


79/114

4 8

Es 20000kP a 0, 35

I1 184 ln

1+( 84)2+1

84

+ln

84

+

(84

)2 + 1

0, 766

se q

B

12

Es

I1 200

4

10,35220000

0, 766

0, 0269m

I0 2 I1 2 0, 766 1, 532

se q B 12Es I0 200 4 10,352

20000 1, 532

0, 0537

44

I1 144 ln

1+( 44 )

2+144

+ln

44

+

(44

)2 + 1

0, 56

se 2 (q B 12Es I1) 2 (200 4 10,352

20000 0, 56) 0, 0393

8 2

I1 182 ln

1+( 82)2+1

82

+ln

82

+

(82

)2 + 1

0, 98

se 2 (q B 12Es I1) 2 (200 2 10,352

20000 0, 98) 0, 0344


80/114

z

3 B

s

0

h 0, 5 B

z

3 B

zd q

z

1, 0

z

0, 0

1, 0

z

4, 5

0, 0

z

1, 0 0, 78

zd q 200 0, 78 156kP a

s

I zdihiM0i

1562,010000

31, 2mm

z

3, 25

4, 5 m

z

0, 0

1, 0

z

4, 5

0, 0

z

3, 25

0, 28

zd q 200 0, 28 56kP a


81/114

s

II zdi hiM0i

562,520000

7mm

s

s

I+s

II 31, 2 + 7 38, 2mm

z

1, 0

z 0, 0

1, 0

z

9, 0 0, 0

z 1, 0

0, 89

zd

q

200 0, 89

178kP a

s

I zdihiM0i

1782,010000

35, 6mm

z

4, 0

z

0, 0

1, 0

z

9, 0 0, 0

z 4, 0 0, 56

zd q 200 0, 56 112kP a

s

II zdi hiM0i

1124,020000

22, 4mm

s sI+s

II 35, 6 + 22, 4 58mm

B L q


82/114

M01 : M02 : M03

6 B


83/114

cv = k

w mv=

k M0w

cm2

s lub

m2

d

cv

k

w

mv mv hh0

10

1M0

h

h0

M0

Uv =f(Tv)

Tv

Tv = cvh2 t= k

w mv t

h2

t

h

h


84/114

Uv =Sk =stsc

=t

=

u

.

st tsc t t

u

t

te=

h2

h2e

t

h

te he

Cc= e0 e1

log1 log0=

e0 e1log

10

0

1

e

C = e

log t

e

e0 e1) log t


85/114

Cr

pc

Cc

si sc

ss

sc=e0 e1

1 +e0 h0=

Cc h01 +e0

log 1

0

h sc

eh

1+e0h0

sc=mv (1 0) h0=1 0

M0 h0

ss=C h

1 +e log t

e

h


86/114

e0

Cc mv

e

h

eh

1+e0h0

e1,044

1+0,84119

e

0, 1011

Cc e0e1

log

10

0,1011

log

428107

0, 1679

mv e

(1+e0) 0,1011

(428107)(1+0,841) 1, 71 104m2

kN

e

Cc e0e1

log

10

0,8410,71log

1107

1 645, 1kP a

Tv


87/114

90kP a

sr w hw2 10 92 45kP a

sc

srhw

M0

459

15000

0, 027m

2, 7cm

Tv 2

Uu2 0, 996

st Uu2 sc 0, 996 2, 7 2, 69 cm

t

Tv h2wM0k

2921015000107 1080000s 12, 5d

q2

q1

60kP a

U

88, 7%

h0e 20

U

88, 7%

q1

60kP a


88/114

q1 t1

q2 t2

14 t1

tte

h2

h2e

h

he

q1

t1 teh2

he

2220,022

20000

q1

sIc q1hM0

6023000 0, 04m

t1

U 0, 887

st1sIc

st1 0, 887 0, 04 0, 035m

U

0, 887

Tv

Tv 0, 8

cv

cv Tvh2t1

0,82220000

0, 00016 m

2

godz 3, 84 103m2

d


89/114

q2 t2 14 t1 5000godz.

st2 st1

0, 035m

Tv cvh2t2 1,610422 5000 0, 2

U

0, 504

q2

U 0, 504

st2sIIc

sIIc 0,0350,504 0, 069m

q2

h

sIIc

M0 h h0 q2sIIc h0q2

MosIIch0

30000,0692

103, 5kP a

q2 103, 5kP a t2 5000godz.

st2 0, 035m

18kNm3

Hn q2

103,518

5, 75m

q2

t1 q1 50kP a

U1 st1sc1

0, 9

t2 st2 q2 75kP a

st1 st1 st2

k


90/114

t1 99, 4dni t2 33, 2dni

q

q

he 20mm Ue 0, 9

q

U

0, 211

G q 150kP a

he0 20mm

q

150kP a

19mm

U

88, 7%


91/114

s

54, 24mm


92/114

e0= ( h+q) K0

E0=

1

2 H2 +q H

K0


93/114

q

H

K0 =

1

K0=1 2 3 (1 sin)

1 2 3

1 2 3

1 2 3

Ka=tg2

450

2


94/114

ea= ( h+q) Ka

Ea=

1

2 H2 +q H

Ka

ea = ( h+q) Ka 2 c

Ka

Ea =

1

2 H2 +q H

Ka 2 c H

Ka+

2 c2

Hc= 2 c Ka


95/114

Kp =tg2

450 +

2

ep= ( h+q) Kp

Ep =

1

2 H2 +q H

Kp

ep = ( h+q) Kp+ 2 c

Kp


96/114

Ep = 12

H2 +q

H Kp+ 2 c H Kp

eIa(2) 0

eIa(3) I hI KIa 2 cI

KIa

eII

a(3)

I

hI

KII

a 2 cII

KIIa


97/114

eIIa(4)

I hI +II hII KIIa 2 cII KIIa

e

I

p(1)

2 cI

KIp

eIp(3) I hI KIp + 2 cI

KIp

eIIp(3) I hI KIIp + 2 cII

KIIp

eIIp(4)

I hI +II hII KIIp + 2 cII KIIp


98/114

240

c

10kP a

18 kN

m3

Ka

tg2

450

2

tg2

450

240

2

0, 422

ea h Ka 2 c

Ka

h

4mea 18 4 0, 422 2 10

0, 422

30, 4 13

17, 4kP a

Hc 2c

Ka

210180,422 1, 71m

Ea 12 H2 Ka 2 c H

Ka+

2c2

12 18 42 0, 422 2 10 4 0, 422 + 2102

18

19, 9kN

m

ra 13 (4 1, 71) 0, 76m


99/114

Kp tg

2(450 + 2

) tg2(450 + 24

2)

2, 37

ep h Kp+ 2 c

Kp 18 4 2, 37 + 2 10

2, 37

170, 6 + 30, 8

201, 4kP a

Ep

12 H2 Kp+ 2 c H

Kp

12 18 42 2, 3 7 + 2 10 4 2, 37 341, 3+123, 2

464, 5kN

m

Ep1 Ep2

Ep1 123, 2kNmrp1 H2 2m

Ep2 341, 3kNmrp2 H3 1, 33m

Ep Ep1+Ep2

Ep

rp

Ep(i)rp(i)

Ep(i)

123,22+341,3

1,33

123,2+341,3

1, 51m


100/114

KIa tg2

450

2

tg2

450 3002

0, 333

eIa(1) q Ka 10 0, 333 3, 33kP a

eIa(2) (1 h1+q) KIa (17 2 + 10) 0, 333 14, 65kP a

EIa eIa(1)

+eIa(2)

2 h1 3,33+14,65

2 2 17, 98kNm

EIa

rI

13 h12e

Ia(1)+eIa(2)

eIa(1)

+eIa(2)

13 223,33+14,653,33+14,65 0, 79m

KIIa tg2

450 2

tg2

450 150

2

0, 589

eIIa(2) (1 h1+q) KIIa 2 c2 KIIa (17 2 + 10) 0, 589 2 20

0, 589

25, 92 30, 7 4, 78kP a

eIIa(3) (1 h1+2 h2+q) KIIa 2 c2

KIIa

(17 2 + 20 3 + 10) 0, 589 2 20 0, 589 61, 26 30, 7

30, 56kP a

4, 78 x30, 56 3 x

x 0, 41m

EIIa 12 eIIa(3) h 12 30, 56 2, 59 39, 58kNm

EIIa

rII

13 h

13 2, 59

0, 86m

Ea E

Ia+ E

IIa 17, 98 + 39, 58 57, 56

kNm

Ea

r

E(i)a r(i)E(i)a

17,983,79+39,580,8617,98+39,58

1, 78m


101/114

h

Ka tg2

450 2

tg2

450 2802

0, 361

Kp tg2

450 + 2

tg2

450 2802

2, 77

h ep ea

h Kp ( h+q) Ka

h

qKaKpKa

1200,361162,77160,361

=1, 12m


102/114

Ka tg2

450

2 tg2

450 280

2 0, 361

Kp tg2

450 + 2

tg2

450 2802

2, 77

Ep Ea

1

2 H2

Kp 12 H2 +q H Ka12 16 H2 2, 77

12 16 H2 + 120 H 0, 361

19, 272 H2 + 43, 32 H

0

b2 4 a c 1953, 71

H

b

2a 2, 27m

Ea

a

0


103/114

q

ea2(G)

ea2(Pd) Ea.

c

0

Ea.

fa Ea fp Ep

fa 1, 2 fp 0, 86


104/114

Fmin

Fmin Fdop = 1, 1 2, 0

W

W

N

S

S

T

T

T =N tg [kN/m]


105/114

W = V [kN/m]

V

V =l d 1, 0

S=W sin[kN/m]

N=W cos[kN/m]

S T

W sin W cos tg

tg tg

F=T

S =

W cos tgW sin =

tg

tg

Fdop

1.3 1, 5

F =

+w tg

tg

Pj Vj V


106/114

F = T

S+Pj

A

H

A

4, 5 H O

A

1 2


107/114

O

O1


108/114


109/114

F =Mutrz

Mobr

Fdop

1, 1 1, 3

Si Wi sini Wi Vi Ni Wi cosi Ti Ni tgi Wi cosi tgi

Ti Ni

tgi+li

ci Wi

cosi

tgi+li

ci

i

Vi i li ii

i

F =Ti

R

Si R

F =

ni=1(Wi cosi tgi+li ci)n

i=1(Wi sini)


110/114

27o

F 1, 5

F

tgtg

tg

tgF

tg27o

1,5 0, 3397

arctg (0, 3397) 18, 8o

30o

F

1, 1

1 :m

F

tg

tg

tg

tgF

tg30o

1,1 0, 5249

arctg (0, 5249)

27o

tg

1m

0, 5249

1 : 1, 9

lgd 1, 00, 51, 0

350

sr 21

kNm3

16o


111/114

sr w 21 9, 81 11, 19kNm3

S W sin

V sin

1, 0 0, 5 11, 19 sin16o

1, 54kN

m

Pj =V j j =w i i= Hl ,l= 1m H=sin16oPj = 1, 0 0, 5 10 sin16o = 1, 34 kNm

T N tg

W cos tg

V cos tg

1, 0 0, 5 11, 19 cos16o tg35o

3, 78kN

m

F

+w

tgtg

11,1911,19+9,81

tg35otg16o

1, 3

F

tgtg

tg35o

tg16o 2, 44


112/114

19, 5kN

m3,

25o

c

10kP a

b

1, 8m

h

4, 04m

V5 b

h

1, 0

1, 8

4, 04

1, 0

7, 3m3

W5 V5 7, 3 19, 5 142, 35kNm

cos5 bl5l5 1,8cos17o 1, 88m

F

(W5cos5tg+l5c5)(W5sin5)

142,35cos17otg25o+1,8810142,35sin17o

82,2841,62

1, 97

S

T


113/114

350

sr 20

kNm3

F

1, 2

1 :m


114/114

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Zadania Mechanika Gruntów