Download - Zadania Mechanika Gruntów
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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
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Cu
Cu15
Cc= d230
d10 d60
Cc
1 Cc 3 Cu 4 Cu 6
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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;
)
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n= e
1 +e oraz e=
n
1 n
n
e
wn =mwms
>100%
wsr wsr
wsr =e ws
wsr
Sr =VwVp
0; 1
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n
e
wsr
n e wsr
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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)
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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
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< 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
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wL
wp
ws
Ip =wL wp
IL=wn wpwL wp =
wn wpIp
wn
Ic=wL wnwL wp =
wL wnIp
= 1 IL
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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
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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
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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
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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
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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
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= 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
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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
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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
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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
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k C d 210 [ md]
d10 C
C= 1200,
1< U
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k
ki
Ai
Ai
Ai
k Li(Liki
)
sr s w
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s w
j
w i kN
m3
kNm3
kNm2
jV kNm3
jV
jV jV
j
wi
F
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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
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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
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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
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k
108m
s
A
1, 0m2
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z
z = z= g z
z =n
i=1
(i zi) =n
i=1
(i g zi)
i i
u
= +u
=
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u= w
zw
w zw
z =sr z w z= (sr w) z= z
z =sr zw+ (z zw) w zw = (z zw) + zw
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z = (sr z+w hw) w (z+hw) = z
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R = 3
2 Q cos
R2
z = 3
2 Q cos3
R2
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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
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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
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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
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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
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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
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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
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z
q
100kP a
B
2, 0m
z
3 B
20 kNm3
z
z zd z
2, 73m
z zs zd
z
5m
5m
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q
4, 0m 100kP a
z zs zd z 6m
6m
z
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f= tg+c.
f
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c
| | f
| | f
| | f
f tg+c
0, 5 (1 3) 0, 5 (1+3) sin+c cos
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f= tg+c
c
u
0
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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.
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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
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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
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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
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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
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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
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M0 =
h h0, [MP a]
hh0
M=
h h0, [MP a]
M=
h h1, [MP a]
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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
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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%
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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
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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
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qu = QuB L
qu
D
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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
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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
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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
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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.
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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.
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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D
L
m 0, 9
m 0, 9
q
m 0, 9
eB
Nr mQfNB
m
0, 9
m 0, 9
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Es
= 0, 1 + 0, 3 rel
rel TX25o45o25o
TX
Es
Es 2, 5
qc
Es 3, 5 qc
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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
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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
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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
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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
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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
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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
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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
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M01 : M02 : M03
6 B
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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
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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
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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
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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
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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
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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
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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
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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%
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s
54, 24mm
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e0= ( h+q) K0
E0=
1
2 H2 +q H
K0
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q
H
K0 =
1
K0=1 2 3 (1 sin)
1 2 3
1 2 3
1 2 3
Ka=tg2
450
2
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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
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Kp =tg2
450 +
2
ep= ( h+q) Kp
Ep =
1
2 H2 +q H
Kp
ep = ( h+q) Kp+ 2 c
Kp
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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
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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
-
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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
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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
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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
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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
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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
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q
ea2(G)
ea2(Pd) Ea.
c
0
Ea.
fa Ea fp Ep
fa 1, 2 fp 0, 86
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Fmin
Fmin Fdop = 1, 1 2, 0
W
W
N
S
S
T
T
T =N tg [kN/m]
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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
-
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F = T
S+Pj
A
H
A
4, 5 H O
A
1 2
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O
O1
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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)
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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
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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
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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
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350
sr 20
kNm3
F
1, 2
1 :m
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