II GRAFIČKI RAD ZADATAK 2 1. Za dati rešetkasti nosač i opterećenje odrediti sile u štapovima primenom analitičkih izraza. 2. Usled zadatog optere opterećenja i promene temperature u označenim štapovima sračunati: 2.a) vertikalno pomeranje čvora 0; 2.b) promenu odstojanja između čvorova 3 i (5); 2.c) promenu ugla između štapova 3-(3) i 5-(5). 3. Usled zadatog opterećenja sračunati i nacrtati dijagram vertikalnih pomeranja čvorova gornjeg pojasa 3.a) pomoću elastičnih težina; 3.b) pomoću Vilijotovog plana pomeranja. Zadati su sledeći podaci: a 4,0 m , t 30 oC , E 2 108 kN m 2 , F p 1,2 F , F d F v F 0,01m 2
- GEOMETRIJSKI PODACI:
Tabela 1. m
hm
0 1 2 3 4 5 6 7 8 9
2,8 2,8 2,8 4,8 6,0 6,4 6,0 4,8 2,8 0,0
m
0,00000 0,00000 0,46365 0,29146 0,09967 0,09967 0,29146 0,46365 0,61073
m
m
0 0 0 0 0 0 0 0 0
0,61073 0,61073 0,61073 0,87606 0,98279 0,98279 0,87606 0,61073 -
cos
m
1,00000 1,00000 0,89443 0,95783 0,99504 0,99504 0,95783 0,89443 0,81923
cos m
cos m
1 1 1 1 1 1 1 1 1
0,81923 0,81923 0,81923 0,64018 0,55470 0,55470 0,64018 0,81923 -
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2
1. PRORAČUN SILA U ŠTAPOVIMA - Reakcije oslonaca:
1
20 36 20 20 35 16 35 8 70 kN
V 2
V 9
V 2
V 9 (20 20 35 35) 70 40 110 0
28 1 28
20 8 20 8 35 12 35 20 40 kN
- Momenti savijanja u čvorovima gornjeg i donjeg pojasa:
Tabela 2.
M 0
M 1
M 2
M 3
M 4
M 5
M 6
M 7
M 8
M 9
0
-80
-160
40
240
360
340
320
160
0
- Proračun sila u štapovima korišćenjem analitičkih izraza:
sile u štapovima gornjeg pojasa Om
O1
O3
O5
Om
O6
M m hm cos m
M 1 h1 cos 1 M 3 h3 cos 3 M 5 h5 cos 5
80,00 28,571kN 2,80 1,00
40 4,8 0,89443
6,40 0,99504
hm1 cos m
h5 cos 6
,
,
9,317 kN
360,00
M m1
M 5
za m 1 5
,
,
56,530 kN
O2
O4
M 2 h2 cos 2 M 4 h4 cos 4
160 57,143 kN 2,80 1,00 240,00 6,0 0,95783
,
41,761kN
,
59,162 kN
,
.
za m 6 9
360,00 6,4 0,99504
56,530 kN
, O7
M 6 h6 cos 7
340,00 6,0 0,95783
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3
sile u štapovima donjeg pojasa U m
M ( m1)
U 1
U 3
U 5
U m
U 6
U 8
hm1 cos m M 0 h0 cos 1 M 2 h2 cos 3 M 4 h4 cos 5
h6 cos 6 M 8 h8 cos 8
2,8 1,00
hm cos m M 6
0
M ( m )
za m 1 5
,
160,00 57,143 kN 2,8 1,00 240,00
,
U 2
U 4
M 1 h1 cos 2 M 3 h3 cos 4
80,00 28,571kN 2,80 1,00 40,00 4,8 1,00
340,00 6,0 1,00 160,00 2,8 1,00
56,667 kN
,
U 7
57,143 kN
,
U 9
M 7 h7 cos 7 M 8 h8 cos 9
320,00
4,8 1,00 160,00 2,8 1,00
M m M (m1) , cos m hm hm1
D1
M 1 M 0 cos 1 h1 h0
D2
M 2 M 1 cos 2 h2 h1
0,81923 2,8
D3
M 3 M 2 cos 3 h3 h2
40 160 79,924 kN , 0,81923 4,8 2,8
D4
M 4 M 3 cos 4 h4 h3
240 40 49,465 kN , 0,64018 6,0 4,8
D5
M 5 M 4 cos 5 h5 h4
360 240 29,295 kN 0,55470 6,4 6,0
Dm
M m1 M ( m) , cos m hm1 hm
D6
M 5 M 6 cos 6 h5 h6
360 340 0,751kN 0,55470 6,4 6,0
D7
M 6 M 7 cos 7 h6 h7
340 320 15,621kN 0,64018 6,0 4,8
1
za m 1 5
1
1 0,81923 2,8
1
1
1
1
1
80 0 34,876 kN
1
1
1
za m 6 9 1
1
,
160 80 34,876 kN
1
1
1
,
za m 6 9
sile u dijagonalama Dm
8,333 kN
40,000 kN
6,0 1,00
,
,
0,000
,
66,667 kN 57,143 kN
,
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4
sile u vertikalama M 1
20,000 kN ,
V m
1 h Pm M ( m) m1 M ( m1) , hm
V 1
80
V 0
4,0
za m 1 4
h 1 1 2,8 P1 M (1) 0 M (0) 0 80 0 20,000 kN 4 2,8 h1
1 1 2,8 h P2 M ( 2) 1 M (1) 70 160 80 50,000 kN 4 2,8 h2 1 1 2,8 h 160 45,833 kN V 3 P3 M (3) 2 M (2) 0 40 4 4,8 h3 h 1 1 4,8 40 18,000 kN V 4 P4 M ( 4) 3 M (3) 20 240 4 6,0 h4 V 2
M 5
tg 5 tg 6
360
2 0,1 11,250 kN
V 5
V m
h 1 Pm M ( m) m1 M ( m1) , hm
V 6
V 7
V 8
h5
6,4
za m 6 7
1 1 4,8 h P6 M (6) 7 M (7) 0 340 320 12,000 kN 4 6,0 h6 h 1 1 2,8 P7 M (7) 8 M (8) 35 320 160 28,333 kN 4 4,8 h7 P8 0,00
- Sile u štapovima So : hm
M m
[ m]
[ kNm]
0
2,8
0
1
2,8
2
Tabela 3. Om
U m
Dm
V m
[kN ]
[ kN ]
[kN ]
[kN ]
-
-
-
-
20,000
1
0,81923
28,571
0,00
-34,876
20,000
1,00000
1
0,81923
57,143
-28,571
-34,876
-50,000
40
0,89443
1
0,81923
-9,317
-57,143
79,924
-45,833
6,0
240
0,95783
1
0,64018
-41,761
8,333
49,465
-18,000
5
6,4
360
0,99504
1
0,55470
-56,530
40,000
29,295
11,250
6
6,0
340
0,99504
1
0,55470
-56,530
56,667
-0,751
12,000
7
4,8
320
0,95783
1
0,64018
-59,162
66,667
-15,621
28,333
8
2,8
160
-74,535
57,143
11,625
0,000
cos m
cos m
-
-
-80
1,00000
2,8
-160
3
4,8
4
m
cos
m
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5
2. PRORAČUN POMERANJA USLED ZADATOG OPTERE ĆENJA I PROMENE TEMPERATURE U OZNA ČENIM ŠTAPOVIMA: 2.a) Vertikalno pomeranje čvora 0: - Sile u štapovima S usled generalisane sile P 1 :
V 2
1 28
9 1 36 1,286 , 7
V 9
1 28
2 1 8 0,286 , 7
V 2
V 9 1 1,286 0,286 1 0
M 0
M 1
M 2
M 3
M 4
M 5
M 6
M 7
M 8
M 9
0
-4
-8
-48/7
-40/7
-32/7
-24/7
-16/7
-8/7
0
O1
O3
O5
O6
O8
U 1
U 3
U 5
M 1
h1 cos 1
4,571 0,718 6,4 0,99504
,
O7
2,286 0,532 4,8 0,89443
,
O9
,
U 2
8 2,857 , 2,8 1,00
U 4
5,714 0,952 6,0 1,00
h5 cos 6 M 7 h7 cos 8
h4 cos 5
M 5
M 4
O4
4,571 0,718 6,40 0,99504
h5 cos 5
h2 cos 3
M 5
M 2
O2
6,857 1,597 4,8 0,89443
h3 cos 3
h0 cos 1
,
M 3
M 0
4 1,429 2,80 1,00
0 2,8 1,00
0,000
,
,
M 2 h2 cos 2 M 4 h4 cos 4
8 2,857 2,80 1,00
5,714 0,994 6,0 0,95783
, ,
, M 6 h6 cos 7 M 8 h8 cos 9 M 1 h1 cos 2 M 3 h3 cos 4
3,429 6,0 0,95783
0,597
1,143 0,498 . 2,8 0,81923
4 1,429 2,80 1,00
6,857 1,429 4,8 1,00
, ,
,
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6
M 8
M 1 M 0 cos 1 h1 h0
D2
M 2 M 1 1 8 4 1,744 cos 2 h2 h1 0,81923 2,8
D3
M 3 M 2 cos 3 h3 h2
6,857 8 1,744 , 0,81923 4,8 2,8
D4
M 4 M 3 cos 4 h4 h3
5,714 6,857 0,744 , 0,64018 6,0 4,8
D5
M 5 M 4 cos 5 h5 h4
4,571 5,714 0,429 , 0,55470 6,4 6,0
D6
M 5 M 6 cos 6 h5 h6
4,571 3,429 0,257 , 0,55470 6, 4 6,0
D7
M 6 M 7 cos 7 h6 h7
3,429 2,286 0,149 , 0,64018 6,0 4,8
D8
M 7 M 8 cos 8 h7 h8
2,286 1,143 0,083 . 0,81923 4,8 2,8
V 0
V 1
h 1 1 2,8 P1 M (1) 0 M (0) 0 4 0 1,000 , 4 2,8 h1
V 2
1 P2 M ( 2)
V 3
1 P3 M (3)
h2
V 4
1 P4 M ( 4)
h3
V 5
1
1 0,81923 2,8
1
1
1
1
1
1
4 4,0
,
1
1
4 0 1,744 ,
1
1
M 1
h8 cos 9
1,143 0,408 . 2,8 1,00
D1
U 9
M 8
h8 cos 8
1,143 0,408 , 2,8 1,00
U 8
1
1
1
1,000 ,
h1 h2
h3
1 2,8 M (1) 9 7 8 4 0,286 , 4 2,8 1 48 2,8 M ( 2) 0 8 1,000 , 4 7 4,8
1 40 4,8 48 M (3) 20 0,571 , 4 7 6,0 7 h4 32 7 M 5 tg 5 tg 6 2 0,1 0,143 , 6,4
h5
V 6
1 1 24 4,8 16 h P6 M (6) 7 M (7) 0 0,114 , 4 7 6,0 7 h6
V 7
h 1 1 16 2,8 8 P7 M (7) 8 M (8) 0 0,048 , 4 7 4,8 7 h7
V
P
0 000
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7
- Proračun pomeranja: EF c v0
S S F F l EF t S l o
o
c
c
s
štap O1 O2 O3 O4 O5 O6 O7 O8 O9 U1 U2 U3 U4 U5 U6 U7 U8 U9 D1 D2 D3 D4 D5 D6 D7 D8 V0 V1 V2 V3
t
s
Fc l F
toSl
l
Fc
F
So
S
t o
[m]
[m2]
[m2]
[kN ]
[1]
[ C ]
[kNm]
[ C m ]
0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010
28,571 57,143 -9,317 -41,761 -56,530 -56,530 -59,162 -74,535 -69,752 0,000 -28,571 -57,143 8,333 40,000 56,667 66,667 57,143 57,143 -34,876 -34,876 79,924 49,465 29,295 -0,751 -15,621 11,625 20,000 20,000 -50,000 -45,833
1,429 2,857 1,597 0,994 0,718 0,718 0,597 0,562 0,498 0,000 -1,429 -2,857 -1,429 -0,952 -0,571 -0,476 -0,408 -0,408 -1,744 -1,744 1,744 0,744 0,429 -0,257 -0,149 -0,083 1,000 1,000 -0,286 -1,000
30 30 30 30 30 30 30 30 30 -
136,093 544,192 -55,450 -144,456 -135,972 -135,972 -122,913 -156,105 -141,349 0,000 136,093 544,192 -39,693 -126,933 -107,856 -105,778 -77,714 -77,714 297,002 297,002 680,629 229,939 90,625 1,392 14,542 -4,711 56,000 56,000 40,040 219,998
0,000 -171,480 -342,840 -171,480 -114,240 -68,520 -57,120 -48,960 -48,960 -48,960 -48,960 -
4,000 4,000 4,472 4,176 4,020 4,020 4,176 4,472 4,883 4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,000 4,883 4,883 4,883 6,248 7,211 7,211 6,248 4,883 2,800 2,800 2,800 4,800
0,010
o
So S
o
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8
2.b) Promena odstojanja izme đu čvorova 3 i (5): - Sile u štapovima S usled generalisane sile P 1 :
O4
U 4
M 4
h3 cos 4
M ( 4 ) M 4
M (5) M 5
-3,7482
-2,1864
0
H 3
P cos 0,7809
0
2,4988
4,9976
V 3
P sin 0,6247
0,435 ,
O5
U 5
h4 cos 4 M (3)
M ( 3) M 3
2,4988 6,0 0,95783
3,7482 0,781 4,8 1,00
,
M 5 h5 cos 5 M ( 4)
h4 cos 5
4,9976 6,40 0,99504
0,785
2,1864 0,364 6,0 1,00
,
M 4 M (3) 2,499 3,748 1 H 3 0,781 0,651 , cos 4 h4 4,8 h3 0,64018 6,0 4,998 2,186 1 M 5 M (4) 1 H 3 0,781 0,657 , D5 cos 5 h5 6,0 h4 0,55470 6,4 h 1 1 4,8 3,748 0,500 , V 3 P3 0,625 , V 4 P4 M (4) 3 M (3) 0 2,186 4 6,0 h4 1
D4
V 5
P5v
M 5 h5
tg 5
0,625
- Proračun pomeranja:
EF c l
4,998
6,4
0,10 0,547
,
S S F F l EF t S l o
s
c
o
c
t
s
,
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9
EF c l
S S F F l EF t S l 633,751 2 10 o
o
c
c
8
t
s
0,01 10 5 (137,400) 2.114,249 kNm
s
Promena odstojanja između čvorova 3 i (5): l
EF c l EF c
2.114,249 0,00106 m 1,06 mm 2 108 0,01
2.c) Promena ugla između štapova 3-(3) i 5-(5): - Sile u štapovima S usled generalisanog opterećenja sile (dva jedinična sprega suprotnih smerova):
O4
U 4
D4
M 4 h4 cos 4 M (3)
h3 cos 4
M (3) M 3
M (4 ) M 4
M (5) M 5
1
1
0
H 3 0
0
1
1
V 3 0
1 6,0 0,95783 1
4,8 1,00
0,208
M 4 M (3) H 3 cos 4 h4 h3 1 M M ( 4) 1
0,174 m 1 , m
1
,
O5
U 5
M 5 h5 cos 5 M ( 4)
h4 cos 5
1 6,40 0,99504 1
6,0 1,00
1 1 0 0,065 m 1 , 0,64018 6,0 4,8 1
1
1
1
1
0,167
0,157 m 1 , m
1
,
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10
štap O4 O5 U4 U5 D4 D5 V3 V4 V5
l
Fc
F
So
[m]
[m2]
[m2]
[kN ]
0,010
0,012 0,012 0,012 0,012 0,010 0,010 0,010 0,010 0,010
-41,761 -56,530 8,333 40,000 49,465 29,295 -45,833 -18,000 11,250
4,176 4,020 4,000 4,000 6,248 7,211 4,800 6,000 6,400
t o
S [m
1
]
-0,174 -0,157 0,208 0,167 -0,065 -0,019 0,000 0,050 0,0156
o
S o S
s
F c F
l EF c t
t S l 54,684 2 10 o
8
s
Promena ugla između štapova 3-(3) i 5-(5):
EF c EF c
954,684 2 10
8
0,01
0,00047734 rad 0o 1 38
Fc l F
toSl o
[ C ]
[kN ]
[ C ]
30 30 -
25,287 29,732 5,778 22,267 -20,089 -4,014 0,000 -5,400 1,123 54,684
24,960 20,040 45,000
EF c
So S
0,01 10 5 45,000 954,684 kN
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11
3. PRORAČUN I KONSTRUKCIJA DIJAGRAMA VERTIKALNIH POMERANJA ČVOROVA GORNJEG POJASA USLED ZADATOG OPTERE ĆENJA 3.a) Pomoću elastičnih težina: Dijagram vertikalnih pomeranja čvorova gornjeg pojasa, usled zadatog opterećenja, dobićemo kao dijagram momenata savijanja fiktivnog nosača, na potezu čvorova gornjeg pojasa, opterećenog elastičnim težinama. - Fiktivan nosač:
0 (0) 0 v( 0 )
0 ( 2) 0 v( 2 )
f
0
M ( 2)
f
0
T ( 2)
M (0) T (0)
f
f
0 9 0 v9
0
M 9
f
0
0
T 9
f
0
Fiktivan nosač je jednom statički neodređen. Osnovni sistem fiktivnog nosača:
Statičku nepoznatu fiktivnog nosača odredićemo kao vertikalno pomeranje čvora (2) zadatog nosača usled datog opterećenja:
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12
- Elastične težine fiktivnog nosača:
1
M 1
M (1) 4,0 1,0 [1] ,
O1
U 2
D2
V 2
4
M 1
h1 cos 1 M (1)
h1 cos 2
1
M (1)
cos 2
h1
M (1)
1,0 4
1,0 2,80 1,00 1,0
2,80 1,00
M (0)
0,357 m 1 ,
0,357 m 1 ,
0 1,0 0,81923 2,8
0,25 m 1
M 2 M ( 2) 0
0,436 m 1 ,
O2
D1
V 1
M 2 h2 cos 2
1
M 1
cos 1 h1 h M (1) 0 h1
1
0 2,80 1,00 1,0
0,81923 2,8 1
2,8
4
2,8
1,0
0, 0,436 m 1
,
0,25 m 1 ,
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13
M 3 M (3)
O3
U 4
D4
V 3
1
4,0 1,0 4
M 3 h3 cos 3 M (3)
h3 cos 4
[1] , M 4 1,0
4,8 0,89443
1,0 4,8 1,00
1
h M (3) 2 h3
1 4
1,0
0,233 m 1 ,
0,208 m 1 ,
M 4 M (3) cos 4 h4 h3 1
M ( 4) 0 O4
D3
M 4 h4 cos 4
1
M 3
cos 3 h3
0 6,0 0,95783 1,0
0,81923 4,8
1,0 0,325 m 1 , 0,64018 4,8
2,8 4,8
0,146 m 1 ,
V 4
1
M (3)
1 4
1,0 0,25 m 1
0
0,254 m 1 ,
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14
M 5
1
M (5) 4,0 1,0 [1] 4
O5
O6
D5
D6
V 5
P(5)
M 5 h5 cos 5
1
M 5
cos 5 h5 M 5 h5
1,0 6,40 0,99504 1,0
0,55470 6,4
0,157 m 1 ,
0,282 m 1 ,
tg 5 tg 6 0,5
1,0 6,4
2 0,1 0,469 m 1
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15
M 6
M (6) 0 ,
O7
U 7
D7
V 6
M 7
M 6
h6 cos 7 M ( 7 )
h7 cos 7
1
M (7 )
cos 7
h7
1
M (7 )
1 4
4
0 6,0 0,95783
1,0 4,8 1,00
1
M (7) 4,0 1,0 [1] 0,
O8
D8
V 7
M 7 h7 cos 8
1,0 4,8 0,89443
0,233 m 1 ,
0,208 m 1 ,
1,0 0,325 m 1 , 0,64018 4,8
1,0 0,250 m 1
,
1
M 7
cos 8 h7 1
h M (7 ) 8 h7
1
1,0
0,81923 4,8
1 4
1,0
2,8 4,8
0,254 m 1 ,
0,146 m 1
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16
štap O1 O2 O3 O4 O5 U1 U2 U3 U4 U5 D1 D2 D3 D4 D5 V0 V1 V2 V3 V4 V5
l
F
So
S1
S2
S3
S4
[m]
[m2]
[kN ]
[m-1]
[m-1]
[m-1]
[m-1]
4,000 4,000 4,472 4,176 4,020 4,000 4,000 4,000 4,000 4,000 4,883 4,883 4,883 6,248 7,211 2,800 2,800 2,800 4,800 6,000 6,400
0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010
28,571 57,143 -9,317 -41,761 -56,530 0,000 -28,571 -57,143 8,333 40,000 -34,876 -34,876 79,924 49,465 29,295 20,000 20,000 -50,000 -45,833 -18,000 11,250
-0,357
O5
F c
F [kN ] -33,999
l
-0,357
S 0 S 2
F c
F [kN ]
l S 0 S 3
F c
F [kN ]
-0,233
0,357
l
25,287
-33,999 0,357
-68,000 0,208
5,778 0,167
-0,250 0,250
F [kN ]
8,090 -0,174
0,436 -0,436
F c
l S 0 S 4
-68,000
0,436 -0,436
-0,250 0,250
22,267 -74,251 74,251
0,254 -0,325
-74,251 -170,157
-14,000 -35,000 -0,146 0,250
99,128 -100,444
0,260 -0,300
35,000 -55,000
32,120 -27,000
-0,200 0,250
EF cW m
štap
S 0 S 1
l
F
So
S5
S6
S7
S8
[m]
[m2]
[kN ]
[m-1]
[m-1]
[m-1]
[m-1]
4,020
0,012
-56,530
-0,157
-116,999
S 0 S 5
F c F
[kN ] 29,732
l
-400,408
S 0 S 6 [kN ]
F c F
17,672 17,672
l S 0 S 7 [kN ]
F c F
80,355 -63,374
21,600 18,000 104,135
l S 0 S 8
F c
[kN ]
F
l
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17
Fiktivni nosač opterećen elastičnim težinama:
- Proračun reakcija i presečnih sila:
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18
Proračun na delu od preseka 2 do preseka 9 m
m
EF cW m
EF cW m m
T m
f T m
M m
[kN ]
[kN ]
[kN ]
[kNm]
[kNm]
[mm]
140,000
0,070
f
f
vm
M m f
EF c 10 3
2
1,000
0,000
0,000
235,354
3
0,857
17,672
15,147
235,354
941,416
1081,416
0,541
4
0,714
104,135
74,382
217,682
870,728
1952,144
0,976
5
0,571
83,740
47,851
113,547
454,188
2406,332
1,203
6
0,429
47,120
20,194
29,807
119,228
2525,560
1,263
7
0,286
155,225
44,350
-17,313
-69,252
2456,308
1,228
8
0,143
269,001
38,429
-172,538
-690,152
1766,156
0,883
9
0,000
0,000
0,000
-441,539
-1766,156
0,000
0,000
676,893
240,354
-441,539
9
W
V 2
EF cW m m
m 2
9
240,354 kN ,
W
V 9
EF W c
m 2
m
V 2W 676,893 240,354 436,539 kN
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19
3.b) Pomoću Vilijotovog plana pomeranja: - Promene dužina štapova isled dejstva zadatog opterećenja: štap O1 O2 O3 O4 O5 O6 O7 O8 O9 U1 U2 U3 U4
l
F 2
So
Δl
S ol
EF 10 3
[m]
[m ]
[kN ]
[mm]
4,000 4,000 4,472 4,176 4,020 4,020 4,176 4,472 4,883 4,000 4,000 4,000 4,000
0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012 0,012
28,571 57,143 -9,317 -41,761 -56,530 -56,530 -59,162 -74,535 -69,752 0,000 -28,571 -57,143 8,333
0,0476 0,0952 -0,0174 -0,0727 -0,0947 -0,0947 -0,1029 -0,1389 -0,1419 0,0000 -0,0476 -0,0952 0,0139
l 76 3,62 7,24 -1,32 -5,52 -7,20 -7,20 -7,82 -10,56 -10,79 0,00 -3,62 -7,24 1,06
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20
Promene dužina štapova upisane su duž odgovarajućih štapova kako bi konstrukcija Viliotovog plana bila olakšana.