Post on 06-Sep-2019
1
Konstrukcija linije zatežućih sila
−
⋅ζ⋅
σ=
σ−
σ⋅=∑ = u
au
vv
u
v
aua N
hM1N
zMA)0N(
uau
vaau Nh
MAZ −⋅ζ
=σ⋅=
−⋅+=⋅+= 1uu1auuau a
2dNMyNMM
va1aaauv1a AAZ σ⋅=σ⋅=⇒σ=σ
2
.min
uvaau h9.0
MAZ×
≈σ×=
.min.min h9.0z
9.0.consthh
×≈⇒
≈ζ==
uauu MM0N =⇒=
u.min
auvaau N
h9.0MAZ −×
≈σ×=
Konstrukcija linije zatežućih sila
čisto savijanje:
3
L = 12.0 m
Reakcije oslonaca
Bg = (30×122/2)/9 = 240 kNAg = 30×12 – 240 = 120 kN
Ap1 = Bp1 = 40×9/2 = 180 kN
Bp2 = [(60×3×(9+3/2)]/9Bp2 = 210 kNAp2 = 60×3 – 210 = -30 kN
4
Dijagrami momenata savijanjaoslonac:Mg = 30×32/2 = 135 kNmpolje:xmax = 120/30 = 4.0 mMg = 120×4 – 30×42/2Mg = 240 kNm
polje:Mp1 = 40×92/2 = 405 kNm
oslonac:Mp2 = 60×32/2 = 270 kNm
5
Anvelopa momenata savijanja
( )
( )21
2
1
21
2
1
PP8.1G0.1P8.1G0.1P8.1G0.1
G1.0
PP8.1G6.1P8.1G6.1P8.1G6.1
G6.1
+⋅+⋅
⋅+⋅
⋅+⋅⋅
+⋅+⋅
⋅+⋅⋅+⋅
⋅
6
Anvelopa momenata savijanja
9.0 m
Mu
= 11
09.4
kN
m Mu
= 70
2.0
kNm
4.3 m1.6×G+1.8×P1
1.6×G+1.8×P2
3.0 m
7
Anvelopa momenata savijanja
1.0 m8.0 m
1.0 m8.0 m
1.0×G+1.8×P1
Mu
= 11
09.4
kN
m Mu
= 70
2.0
kNm
4.3 m 1.6×G+1.8×P1
1.0×G+1.8×P2
1.6×G+1.8×P2
1.6×G+1.8×P1 1.0×G+1.8×P1
1.0×G+1.8×P2 1.6×G+1.8×P2
8
Anvelopa momenata savijanja
1.0×G+1.8×P2
1.6×G+1.8×P2
1.0×G+1.8×P1
1.6×G+1.8×P1
nu
lta t
acka
dijagra
ma M
g
MG i MP istog znaka
MG i MP istog znaka MG i MP razlicitog znaka
MG i MP razlicitog znaka
9
Anvelopa momenata savijanja
1.0 m8.0 m
1.0 m8.0 m
4.3 m
Mu
= 11
09.4
kN
m Mu
= 70
2.0
kNm
1.6×G+1.8×P1 1.0×G+1.8×P1
1.0×G+1.8×P2 1.6×G+1.8×P2
10
Dimenzionisanje - presek u polju
aI = 2.5 + 1.0 + 2.5/2 = 4.75 cmaII = 4.75 + 3.0 + 2.5 = 10.25 cm
a1 = (5×4.75 + 4×10.25)/9a1 = 7.19 cm
h = 75 - 7.19 = 67.81 cmh ≈ 68 cm = hrač.
40
21.7
5
4.75
4.75
5.5
4.75
4.75
5RØ25
4RØ25
UØ10/30
30.5
16.2
5
1560
75
2RØ25
2RØ12
22 2RØ12
11
Dimenzionisanje - oslonac
aI = 2.5 + 1.0 + 2.5/2 = 4.75 cmaII = 4.75 + 3.0 + 2.5 = 10.25 cm
a1 = (4×4.75 + 2×10.25)/6a1 = 6.58 cm
h = 75 - 6.58 = 68.42 cmh ≈ 68 cm = hrač.
40
21.7
5
4.75
4.75
5.5
4.75
4.75
4RØ25
2RØ25
UØ10/15
30.5
16.2
5
1560
75
2RØ25
22
2RØ12
2RØ12
12
Analitički izrazi za momente savijanja
2)xL(gM:lx
2xgxAM:lx
2
g
2
gg
−⋅−=≥
⋅−⋅=≤
0M:lx2
xpxAM:lx
1p
21
1p1p
=≥
⋅−⋅=≤
2)xL(pM:lx
xAM:lx2
22p
2p2p
−⋅−=≥
⋅=≤
13Vrednosti momenata savijanja - Excelx Mg Mp1 Mp2
1.6×G+1.8×P1
1.6×G+1.8×P2
1.0×G+1.8×P1
1.0×G+1.8×P2 MAX MIN
0 0 0 0 0 0 0 0 0 00.5 56.25 85 -15 243 63 209.3 29.3 243 0
1 105 160 -30 456 114 393 51 456 01.5 146.3 225 -45 639 153 551.3 65.3 639 0
2 180 280 -60 792 180 684 72 792 02.5 206.3 325 -75 915 195 791.3 71.3 915 0
3 225 360 -90 1008 198 873 63 1008 03.5 236.3 385 -105 1071 189 929.3 47.3 1071 0
4 240 400 -120 1104 168 960 24 1104 04.5 236.3 405 -135 1107 135 965.3 -6 .8 1107 -6 .8
5 225 400 -150 1080 90 945 -45 1080 -455.5 206.3 385 -165 1023 33 899.3 -90 .8 1023 -90 .8
6 180 360 -180 936 -36 828 -144 936 -1446.5 146.3 325 -195 819 -117 731.3 -204.8 819 -204.8
7 105 280 -210 672 -210 609 -273 672 -2737.5 56.25 225 -225 495 -315 461.3 -348.8 495 -348.8
8 0 160 -240 288 -432 288 -432 288 -4328.5 -63 .8 85 -255 51 -561 89.3 -522.8 89.3 -561
9 -135 0 -270 -216 -702 -135 -621 0 -7029.5 -93 .8 0 -188 -150 -487.5 -93 .8 -431.3 0 -487.5
10 -60 0 -120 -96 -312 -60 -276 0 -31210.5 -33 .8 0 -67 .5 -54 -175.5 -33 .8 -155.3 0 -175.5
11 -15 0 -30 -24 -78 -15 -69 0 -7811.5 -3 .75 0 -7 .5 -6 -19 .5 -3 .8 -17 .3 0 -19 .5
12 0 0 0 0 0 0 0 0 0
14
-800
-400
0
400
800
12000 1 2 3 4 5 6 7 8 9 10 11 12
Mu
[kN
m]
1.6×G+1.8×P11.6×G+1.8×P2G+1.8×P1G+1.8×P2
Anvelopa momenata savijanja - Excel
15
Mu
= 11
09.4
kN
m Mu
= 70
2.0
kNm
Z au
≈ 18
12.8 Zau ≈ 1109.4/(0.9×0.68) = 1812.8 kN
Z au
≈ 11
47.1
Zau ≈ 702/(0.9×0.68) = 1147.1
.min
uu
.min
auvaau h9.0
MNh9.0
MAZ×
=−×
≈σ×=
M/z linija
16
17
18
19
Tg
Tp1
Tp2
1,6Tg+1,8(Tp1+Tp2)
20
21
M/z linija
kN3.1964091.4aZ v)1(
a)1(
au =×=σ×=
Z au
≈ 18
12.8
Z au
≈ 11
47.1
Z(1)
au=1
96.3
22
Linija zatežućih silaZ a
u ≈
1812
.8
Z au
≈ 11
47.1
9×Z(1
)au
=176
7.1
23
M/z linija
ζ = 0.964Aa,potr. = 42.33 cm2 (usv. 44.18 cm2)
ζ = 0.894Aa,potr. = 28.85 cm2 (usv. 29.45 cm2)
Z au
≈ 18
12.8
Z au
≈ 11
47.1
9×Z(1
)au
=176
7.1 6×
Z(1)
au=1
178.
1
24
Linija zatežućih silaZ a
u ≈
1812
.8
Z au
≈ 11
47.1
v=0.51
v=0.51
v=0.51v=0.51
POMERANJE LINIJE ZATEŽUCIH SILA:v = 0.75×h = 0.75×68 = 51 cm
M/z linije
25
Linija pokrivanja armaturom
5RØ25
2RØ25
4RØ25
7Ø25
2Ø25
4Ø25
2Ø25
>4Ø25 >4Ø25
>7Ø25 >7Ø25
26
Linija pokrivanja armaturom
5RØ25
2RØ25
5Ø25
2Ø25
2Ø25
>4Ø25 >4Ø25
>7Ø25 >7Ø25
2Ø25
2Ø25
2Ø25
>5Ø >5Ø
>2Ø>2Ø
27
Linija pokrivanja armaturom
3RØ25
2RØ25
3Ø25
2Ø25
2Ø25
>4Ø25 >4Ø25
>7Ø25 >7Ø25
2Ø25
2Ø25
2Ø25
>5Ø >5Ø
>2Ø>2Ø
>3Ø >3Ø
2Ø25
28
Linija pokrivanja armaturomZ a
u ≈
1812
.8Z a
u ≈
1147
.1
A B
POMERENA LINIJA ZATEŽUCIH SILA LINIJA ZATEŽUCIH SILA
LINIJA POKRIVANJA ARMATUROM
v=51
v=51
v=51v=51
POMERANJE LINIJE ZATEŽUCIH SILA:v = 0.75×h = 0.75×68 = 51 cm
(M, N)(M, N, T)
29Z a
u ≈
1812
.8Z a
u ≈
1147
.1
A
B
3
1
1
POMERENA LINIJA ZATEŽUCIH SILA LINIJA ZATEŽUCIH SILA
LINIJA POKRIVANJA ARMATUROM
v=51
v=51
v=51v=51
UKUPNA DUŽINA ŠIPKE (700 cm)
STATICKI POTREBNA DUŽINA ŠIPKE (460 cm)
2
1
5
2Ø25
1Ø25
2Ø25
2Ø25
2Ø25
69
35
65
125
205
25
85
165
180280
DUŽINA SIDRENJA:
Ls1 = 79.4 ˜ 80 cm (dobra adhezija)Ls2 = 119 ˜ 120 cm (loša adhezija)
POMERANJE LINIJE ZATEŽUCIH SILA:v = 0.75×h = 0.75×68 = 51 cm
NOSIVOST JEDNE ŠIPKE ARMATURE:
Z(1)au = a(1)
a ×σv = 4.91×40 = 196.3 kN
Zau ≈ Mau / (0.9×h) - NuMau = Mu + Nu × (d 2 - a1)Nu = 0: Mau = Mu ; Zau ≈ Mu / (0.9×h)
(M, N, T) (M, N)M/Z linija
30
31
32
Sidrenje armature
Ø75.3175.18.1
4004Ø
8.14ØL
p
vs ×=
××=
τ×σ
×=
donja zona (dobra adhezija)
22Ø...cm70cm8.692.275.31L 1s ≈=×=25Ø...cm80cm4.795.275.31L 1s ≈=×=
gornja zona (loša adhezija)
22Ø...cm105cm705.1L 2s =×=
1s1s2p
1p2s1p2p L5.1LL
32
×=×ττ
=⇒τ=τ
25Ø...cm120cm805.1L 2s =×=
33
Sidrenje armature1Ø
25v=51 v=51
L.Z.S. LINIJA M z
v=51 v=51 305205
LØ25s1 = 80
STATICKI POTREBNADUŽINA ŠIPKE (390 cm)
UKUPNA DUŽINA ŠIPKE (550 cm)
LØ25s1 = 80
5 5
A B
p
vs 8.14
ØLτ×
σ×=
34
35
36
37
38
1215
6015
5
15 15
BA
1
1
1-1
3
3
3-32-2
2
2
75
39
1215
6015
5
15 15
BA
1
1
1-1
3
3
5
75
5 54x7.5=30
40
15
5 530
40
3-3
5
75
15 15
5 530
40
2-2
5
75
2
2
1 2RØ25 6 2RØ25
6 2RØ25
35
1
1 2RØ25
6
75
1 2RØ251 2RØ25
40
1215
6015
5
15 15
BA
1
1
1-1
3
3
5
75
5 54x7.5=30
40
15
5 530
40
3-3
5
75
15 15
5 530
40
2-2
5
75
2
2
1 2RØ25 6 2RØ253 2RØ25
6 2RØ25
3 2RØ25 L=925 (2)
35
890
35
1
2
1 2RØ25, 3 2RØ25
6
1 2RØ25
75
1 2RØ25
3
2 1RØ25 L=860 (1)
35
825
2 1RØ251 2RØ25
3 2RØ25
2
3
2 1RØ25
41
1215
6015
55
15 15
BA
1
1
1-1
3
3
5.5
5
75
5 54x7.5=30
40
15
5 530
40
3-3
5
75
15 15
5 530
40
2-2
5
75
2
2
1 2RØ25
4 2RØ25
6 2RØ253 2RØ25
6 2RØ25
4 2RØ25 L=760 (2)760
3 2RØ25 L=925 (2)
35
890
35
4 2RØ25
1
2
5
5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25
6
5 1RØ255 1RØ25 1 2RØ25
75
1 2RØ25
5 1RØ25
4 2RØ25
5
3
4 4
2 1RØ25 L=860 (1)
35
825
2 1RØ251 2RØ25
3 2RØ25
2
3
2 1RØ25
42
1215
6015
55
17015 15 25 375 25 25 175 10275
BA
1
1
1-1
3
3
55.
522
521
.5
75
5 54x7.5=30
40
30 55
15
5 530
40
7.5 55 15 7.53-3
16
55.
522
521
.5
75
16
15 15
5 530
40
7.5 55 15 7.52-2
55.
522
521
.5
75
16
2
2
1 2RØ25
4 2RØ25
11 2RØ12
6 2RØ25
7 2RØ2210 2RØ22
9 2RØ22
8 2RØ22
3 2RØ25
11 2RØ12
11 2RØ12
11 2RØ12
7 2RØ22
9 2RØ22
8 2RØ22
11 2RØ12
11 2RØ12
10 2RØ22 10 2RØ227 2RØ22, 8 2RØ227 2RØ22, 9 2RØ22 7 2RØ22, 8 2RØ22
11 ±1RØ126 2RØ25
4 2RØ25 L=760 (2)760
3 2RØ25 L=925 (2)
35
890
359 2RØ22 L=395 ( 2)
395
8 2RØ22 L=500 (2)500
7 2RØ22 L=865 (2)830
35
10 2RØ22 L=480 (2)
35
445
11 ±1RØ12 L=640 (8)
20
620 11 ±1RØ12 L=640 (8)
20
620
7 2RØ22
4 2RØ25
11 ±1RØ12
11 ±1RØ12
11 ±1RØ12
75
1
2
5
5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25
6
5 1RØ25
7
8
99
8
10
8 2RØ22
5 1RØ25 1 2RØ25
75
1 2RØ25
5 1RØ25
4 2RØ25
5
3
4 4
2 1RØ25 L=860 (1)
35
825
2 1RØ251 2RØ25
3 2RØ25
2
3
2 1RØ25
43
1215
6015
55
17015 15 25 375 25 25 175 10275
BA
1
1
1-1
3
3
55.
522
521
.5
75
5 54x7.5=30
40
30 55
15
5 530
40
7.5 55 15 7.53-3
16
55.
522
521
.5
75
16
15 15
5 530
40
7.5 55 15 7.52-2
55.
522
521
.5
75
16
2
212 18RUØ10/ 10 12 16RUØ10/ 25 12 19RUØ10/ 15 12 6RUØ10/ 15 12 8RUØ10/ 25
1 2RØ25
4 2RØ25
11 2RØ12
6 2RØ25
7 2RØ2210 2RØ22
9 2RØ22
8 2RØ22
3 2RØ25
11 2RØ12
11 2RØ12
11 2RØ12
13 9RUØ10/ 15
7 2RØ22
9 2RØ22
8 2RØ22
11 2RØ12
11 2RØ12
10 2RØ22 10 2RØ227 2RØ22, 8 2RØ227 2RØ22, 9 2RØ22 7 2RØ22, 8 2RØ22
11 ±1RØ126 2RØ25
4 2RØ25 L=760 (2)760
3 2RØ25 L=925 (2)
35
890
359 2RØ22 L=395 (2)
395
8 2RØ22 L=500 (2)500
7 2RØ22 L=865 (2)830
35
10 2RØ22 L=480 (2)
35
445
11 ±1RØ12 L=640 (8)
20
620 11 ±1RØ12 L=640 (8)
20
620
12 RUØ10/15
7 2RØ22
4 2RØ25
11 ±1RØ12
11 ±1RØ12
11 ±1RØ12
12 RUØ10/25
12 RUØ10/15
13 RUØ10/15
12 19RUØ10/ 15 L=232 ( 67)
35
70
35
70
1 11 1
75
1
2
5
5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25
6
5 1RØ25
7
8
99
8
10
8 2RØ22
5 1RØ25 1 2RØ25
75
1 2RØ25
5 1RØ25
4 2RØ25
5
3
4 4
2 1RØ25 L=860 (1)
35
825
2 1RØ251 2RØ25
3 2RØ25
2
3
2 1RØ25
44
1215
6015
55
17015 15 25 375 25 25 175 10275
BA
1
1
1-1
3
3
55.
522
521
.5
75
5 54x7.5=30
40
30 55
15
5 530
40
7.5 55 15 7.53-3
16
55.
522
521
.5
75
16
15 15
5 530
40
7.5 55 15 7.52-2
55.
522
521
.5
75
16
2
212 18RUØ10/ 10 12 16RUØ10/ 25 12 19RUØ10/ 15 12 6RUØ10/ 15 12 8RUØ10/ 25
1 2RØ25
4 2RØ25
11 2RØ12
6 2RØ25
7 2RØ2210 2RØ22
9 2RØ22
8 2RØ22
3 2RØ25
11 2RØ12
11 2RØ12
11 2RØ12
13 9RUØ10/ 15
7 2RØ22
9 2RØ22
8 2RØ22
11 2RØ12
11 2RØ12
10 2RØ22 10 2RØ227 2RØ22, 8 2RØ227 2RØ22, 9 2RØ22 7 2RØ22, 8 2RØ22
11 ±1RØ126 2RØ25
4 2RØ25 L=760 (2)760
3 2RØ25 L=925 (2)
35
890
359 2RØ22 L=395 (2)
395
8 2RØ22 L=500 (2)500
7 2RØ22 L=865 (2)830
35
10 2RØ22 L=480 (2)
35
445
11 ±1RØ12 L=640 (8)
20
620 11 ±1RØ12 L=640 (8)
20
620
12 RUØ10/15
7 2RØ22
4 2RØ25
11 ±1RØ12
11 ±1RØ12
11 ±1RØ12
12 RUØ10/25
12 RUØ10/15
13 RUØ10/15
12 19RUØ10/ 15 L=232 ( 67)
35
70
35
70
1 11 1
75
1
2
5
5 1RØ254 2RØ25, 1 2RØ25, 3 2RØ25
6
5 1RØ25
7
8
99
8
10
BETON (MB 30):
ARMATURA (RA 400/500):
589.473.645
KOLICINE MATERIJALA:
11.04.2011.420x580R 1:25/10MB 30 RA 400/500
GODIŠNJI ZADATAK - LIST br.1
PLAN ARMATURE POS 1 (var.1)M. STOJANOVIC
3.645 m3
589.47 kg
= 161.7 kg/m3 betona
8 2RØ22
5 1RØ25 1 2RØ25
75
1 2RØ25
5 1RØ25
4 2RØ25
5
3
4 4
2 1RØ25 L=860 (1)
35
825
2 1RØ251 2RØ25
3 2RØ25
2
3
2 1RØ25