2,5 2,25 2,5 2,25 2,25 2,25 2,25 2,25 2,5 2,5 2,25 2,5 1,69 65 kNm 10kN/m +20°C -20°C ZADATAK Za zadani sistem metodom sila odrediti M,T,N dijagrame. Grede i stupovi: [ ] 1 5 2 7 1 10 / 10 3 72 / 36 / - - = ⇒ ⋅ = = K m kN E cm h b t α 4 5 3 3 1 2 1 10 74 , 1119 12 72 , 0 36 , 0 12 2592 , 0 72 , 0 36 , 0 m bh I m h b F - ⋅ = ⋅ = = = ⋅ = ⋅ = Štapovi: [ ] 2 8 2 2 3 2 2 / 10 2 10 6 , 3 06 , 0 6 / 6 / m kN E m F cm h b ⋅ = ⋅ = = ⇒ = - 1 5 10 - - = K t α
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2,5 2,25 2,5 2,25 2,25 2,25
2,25
2,25
2,5
2,5
2,25
2,5
1,69
65 kNm10kN/m
+20°C
-20°C
ZADATAK Za zadani sistem metodom sila odrediti M,T,N dijagrame. Grede i stupovi:
[ ]
15
271
10
/103
72/36/
−−=
⇒⋅=
=
K
mkNE
cmhb
tα45
33
1
21
1074,111912
72,036,0
12
2592,072,036,0
mbh
I
mhbF
−⋅=⋅
==
=⋅=⋅=
Štapovi:
[ ]
282
2322
/102
106,306,06/6/
mkNE
mFcmhb
⋅=
⋅==⇒= −
15
10−−= Ktα
X2X2
X1 X1
X2X2
X1X1
X2 X2X1X1
OSNOVNI SISTEMI
10kN/m65 kNm1
,69
2,5
2,25
2,5
2,5
2,25
2,25
2,252,252,252,52,252,5
A B
65 kNm10kN/m
65
65
93,485
4,75 7 2,25
A
A
6,36
7,88
8,50
1,62
65
M
MJ 1cm=50kNm
ODREĐIVANJE DIJAGRAMA Mo
A B
65 kNm10kN/m
4,75 7 2,25
M
N
4,75
10kN/m
A
PROSTA GREDA
REAKCIJE:
0=∑ AM ⇒ )6581,112(75,11
1075,11652/75,410
2 −=⇒=++⋅− BBO
B0=4,069 kN
0=∑ BM ⇒ )31,510(75,11
10375,975,4106575,11
0 =⇒=⋅⋅++− AA
A0=43,431 kN
0=∑ XF ⇒AH=0 kN
Kontrola:
0=∑ YF ⇒ 05,47069,4431,43075,400 =−+⇒=+− BqA
0=0 MOMENT U KARAKTERISTIČNOJ TOČKI
kNm
AM
485,93
2/75,41075,420
=
⋅−=
PROVJES PARABOLE (q=10kN/m')
Raspona 4,75m
kNmql
f 203,288
75,410
8
22
=⋅
==
A
BA
2,5 2,25 2,5 2,25 2,25 2,25
2,25
2,25
2,5
2,5
2,25
2,5
1,69
11
11
1 A'
0,83
0,981
0,78
MJ 1cm=1kNm
m1
m01
1
ODREĐIVANJE DIJAGRAMA m1
B
Ah
Av
1
0,8288
0,5594
2,81
B
B'h
1
1
0,8288
2,25
2,5
B'
C'
B'v
2,5 2,25 2,25
1
1
0,8288
0,8288
0,5594
0,5594
2,25
2,25
2,5
2,5 2,25 2,5 2,25 2,25 2,25
2,5
2,25
2,5
A'B'
C'
Av
Ah
B
A'v
A'h
B'v
B'h
1
23
4
5
6
78
9
1
1 2
2
33
44
ODREĐIVANJE DIJAGRAMA m2
REAKCIJE:
0=∑ XF ⇒ 8288,0=hA
0' =∑cijeli
AM ⇒ 075,48288,075,11'25,0' =⋅++− vh BB
0' =∑desno
CM ⇒)4973,0'488,2('
06875,18288,08125,2'0,7'
+−=⇒
=⋅++vh
hv
BB
BB
-uvrstimo u gornju jednadžbu
A'h
A'v
Av
C'
A'
2,252,52,25
2,25
0,82880,8288
3,06
B=0,0086
B'V=0,3283
B'h=0,3168
štap5=0,4521
9370,375,11')4973,0'48,2(25,0 −=++ vv BB
3167,0'
3282,0'
0613,4'3722,12
=
−=
−=
h
v
v
B
B
B
0' =∑cijeli
BM ⇒ 075,48288,075,11'25,0' =⋅+−− vh AA
0' =∑ljevo
CM ⇒ hvvh AAAA '6447,0'075,4'0625,3' −=⇒=−−
-uvrstimo u gornju jednadžbu
3283,0'
5091,0'
7298,3'3256,7
=
−=
−=
v
h
h
A
A
A
SILE AV, B, štap 1, štap 5 uvjet:
→→→→
+=+ BštapBB vh5''
očitano(AutoCAD):
↓=
=
0086,0
)(4521,05
B
tlakŠ
štap1=0,7201
AV=0,1809
A'v=0,3283
A'h=0,5091
štap4=0,3295
štap9=0,2398 štap5=0,4521
štap2
štap1=0,7201 štap6=0,3818
1
1Ah
Av2,5
M
T
N
uvjet:
→→→→
+=+ vvh AštapAA 1'' očitano(AutoCAD):
↓=
=
1809,0
)(7201,01
vA
vlakŠ
SILE 4,6,8,9
uvjet(ravnoteža čvora):
→→→→
=++ 0954
očitano(AutoCAD):
)(3295,04
)(2398,09
tlakŠ
tlakŠ
=
=
uvjet(ravnoteža čvora):
→→→→
=++ 0621
očitano(AutoCAD):
)(3818,06 vlakŠ =
MOMENTI U KARAKTERISTIČNIM TOČKAMA Presjek 1-1 4522,05,21 −=−= vAM
2,252,5
2
2
Ah
Av
S6 MN
T
2,25 2,252,25
33
B
S9MN
T
2,252,2544
BT
NM
m2
MJ 1cm=1kNm
2,252,252,52,252,5 2,25B
1
Av
0,4522
0,5783
0,0194
Presjek 2-2 (kontrola)
00025,2675,42 =⇒=⋅+−= SAM v
Presjek 3-3
5783,025,295,43 −=⋅−−= SBM
Presjek 4-4
0194,025,24 −=−= BM
DIJAGRAM m2
t
2,5 2,25 2,5 2,25 2,25 2,25
m1
0,781
0,98
0,83
2,5 2,25 2,5 2,25 2,25 2,25
0,0194
0,5783
0,4522
m2
2,5 2,25 2,5 2,25 2,25
UTJECAJ TEMPERATURE (dijagram κt)
451056,5
72,0
)20(2010
−− ⋅=−−
=∆
=h
ttt ακ
KOEFICIJENTI POPUSTLJIVOSTI Koeficijent δ11
dxIE
m∫=
11
21
11δ
+
+
+
+⋅
⋅+
⋅
⋅=3
83,0
3
2
2
25,2
3
1
3
283,0
2
25,283,0
3
283,0
2
5,283,0
1
11
11 IEδ
+
+⋅
⋅+
+⋅
⋅+
+⋅
⋅3
78,0
3
92,02
2
25,298,0
3
1
3
298,0
2
5,298,098,0
3
1
3
21
2
5,21
( ++=
⋅
⋅+
+⋅
⋅ 82792,057408,0
178,0
3
2
2
25,278,098,0
3
1
3
278,0
2
25,278,0
11IE
)11
12668,74563,074879,000695,120866,124166,106125,1
IE=+++++
Koeficijent δ22¸
2,252,252,52,252,5
m2
0,4522
0,5783
0,0194
m1
0,83
1 0,98
0,78
dxIE
m∫=
11
22
22δ
+
⋅
⋅+
⋅
⋅=3
24522,0
2
25,24522,0
3
24522,0
2
5,24522,0
1
11
22 IEδ
+
⋅+⋅
⋅+
⋅
⋅ 0194,0
3
1
3
25783,0
2
25,25783,0
3
25783,0
2
5,25783,0
=
⋅
⋅+
⋅+⋅
⋅3
20194,0
2
25,20194,05783,0
3
1
3
20194,0
2
25,20194,0
1111
86226,0)00029,000449,02550,027869,015336,017041,0(
1
IEIE=+++++
Koeficijent δ12= δ21
21
11
12
11
2112 δδ === ∫∫ dx
IE
mmdx
IE
mm
+
⋅
⋅+
⋅
⋅=3
24522,0
2
25,283,0
3
24522,0
2
5,283,0
1
11
12 IEδ
⋅
⋅+
⋅
⋅+
⋅
⋅3
25783,0
2
5,298,0
3
15783,0
2
5,21
3
14522,0
2
25,21
⋅+⋅
⋅+
⋅+⋅
⋅+ 5783,03
1
3
20194,0
2
25,278,00194,0
3
1
3
25783,0
2
25,298,0
++++=
⋅
⋅ 24096,016957,028149,031277,0(1
3
20194,0
2
25,278,0
11IE
M
651,62
8,50
7,88
6,36
0,780,98
1
0,83
m1
2,252,252,52,252,5 2,25
11
1011,2)011349,018050,043218,047228,0
IE=++++
SLOBODNI ČLANOVI Član δ10
dxmdxIE
Mmt
tv∫∫ +=+= κδδδ 1
11
01101010
Član δv10 (utjecaj zadanog opterećenja):
dxIE
Mmv∫=
11
0110δ
+
−−
⋅⋅+
⋅−
⋅⋅=2
83,0
2
1
3
225,236,6
2
183,0
3
25,288,7
1
11
10 IEvδ
⋅−−
⋅+
⋅+⋅
⋅+
⋅+
⋅ 78,03
2
3
98,0
2
25,262,178,0
3
1
3
298,0
2
25,25,8
3
298,0
3
1
2
5,25,8
++−−=
⋅−
⋅+
⋅−
⋅+ 483,10729,8450,5(
178,0
3
1
2
25,265
3
278,0
2
5,262,1
11IE
11
466,16)012,19948,0543,1734,8
IE
−=−−−
t
2,252,5 2,25 2,5 2,25 2,25
m1
0,83
0,780,98
1
651,62
6,36
7,88
8,50
M
0,0194
0,5783
0,4522
m2
2,252,252,252,52,252,5
Član δt10 (utjecaj temperature):
dxm tt
∫= κδ 110 451056,5
72,0
)20(2010
−− ⋅=−−
=∆
=h
ttt ακ
+⋅+⋅+⋅+⋅+
⋅+⋅=2
25,298,0
2
5,298,0
2
5,21
2
25,21
2
25,283,0
2
5,283,010 t
t κδ
( )2875,01025,1225,125,1125,1933,0037,12
25,278,0
2
25,278,0 ⋅++++++=
⋅+⋅ tκ
429,8⋅= tκ
Član δ10= tIEκ429,8
466,16
11
+−
Član δ20
dxmdxIE
Mmt
tv∫∫ +=+= κδδδ 2
11
02202020
Član δv20 (utjecaj zadanog opterećenja):
dxIE
Mmv∫=
11
0220δ
2,5 2,25 2,5 2,25 2,25 2,250,0194
0,5783
0,4522
m2
t
+
⋅−
⋅⋅+
⋅−
⋅⋅= 4522,02
1
3
225,236,6
2
14522,0
3
25,288,7
1
11
20 IEvδ
+
⋅+⋅
⋅+
⋅
⋅ 0194,0
3
1
3
25783,0
2
25,25,8
3
25783,0
2
5,25,8
⋅−
⋅+
⋅−⋅−
⋅ 0194,03
1
2
25,2655783,0
3
1
3
20194,0
2
5,262,1
)4728,03749,07485,30962,41569,29694,2(1
11
−−++−−=IE
11
8706,1
IE=
Član δt20 (utjecaj temperature):
dxm tt
∫= κδ 220 451056,5
72,0
)20(2010
−− ⋅=−−
=∆
=h
ttt ακ
⋅+⋅+⋅+
⋅+⋅=2
25,20194,0
2
25,25783,0
2
5,25783,0
2
25,24522,0
2
5,24522,010 t
t κδ
( )0218,06506,07228,05087,05653,0 ++++= tκ
4693,2⋅= tκ
Član δ20= tIEκ4693,2
8706,1
11
+
JEDNADŽBE KONTINUITETA
0
0
20222121
10212111
=++
=++
δδδ
δδδ
XX
XX
08706,14693,286226,01011,2
0466,16429,81011,212668,7
1121
1121
=+⋅++
=−⋅++
t
t
IEXX
IEXX
κ
κ
8706,18306,46086226,01011,2
466,16054,15731011,212668,7
21
21
−−=+
+−=+
XX
XX
21 4104,0218,220 XX −−= - iz druge jednadžbe, uvrstimo u prvu
kNmX
kNX
XX
8,213)58,15(4104,0218,220
58,158237,0
833,12
59,15561011,29248,2423,1569
1
2
22
−=−⋅−−=
−=−
=
−=+−−
Rješenje: X1=-213,8kNm; X2=-15,58kN
Av
Ah
B
65 kNm
2,252,252,252,52,252,5
213,8kNm
12,92
10kN/m8,71
15,58kN
B'v
C'
B'
2,5
2,25
2,252,252,5
B'h
B
2,81
65 kNm15,58kN
12,92
213,8kNm
432
1
98
7
6
5
43
2
1
B'h
B'v
A'h
A'v
B
Ah
Av
C'
B'A'
2,5
2,25
2,5
2,252,252,252,52,252,5
2,5
2,25
2,25
8,71
12,92
15,58kN8,71
12,92
213,8kNm 65 kNm10kN/m
Rješavanje osnovnog sistema opterećenog zadanim opterećenjem i silamaX1,X2 REAKCIJE:
0=∑ XF ⇒ kNAh92,12−=
0' =∑cijeli
AM ⇒ 02
75,4106575,11'25,0'
2
=⋅−++− vh BB
3,06
2,25
2,25 2,5 2,25
A'
C'
A'v
A'h
10kN/m
213,8kNm12,9212,92
0' =∑desno
CM ⇒658,60'488,2'
08,213656875,192,128125,2'0,7'
+−=⇒
=−+⋅−+vh
hv
BB
BB
-uvrstimo u gornju jednadžbu
812,4775,11')658,60'48,2(25,0 =+− vv BB
kNB
kNB
B
h
v
v
989,47'
09,5'
977,62'3722,12
=
=
=
0' =∑cijeli
BM ⇒ 0375,975,4106575,11'25,0' =⋅⋅++−− vh AA
0' =∑ljevo
CM ⇒
76,68'6447,0'02
75,4108,21375,4'0625,3'
2
+−=⇒=⋅++−− hvvh AAAA
-uvrstimo u gornju jednadžbu
kNA
kNA
A
v
h
h
566,42'
628,40'
623,297'3256,7
=
=
−=−
B'h=47,99kN
B'V=5,09kN
B=45,69kNštap5=50,42kN
A'h=40,63kN
A'v=42,56kN
štap1=68,67kN
AV=86,01kN
Š2=50,05kN
Š1=68,67kNŠ6=36,42kN
Š5=50,41kNŠ9=26,73kN
Š4=36,74kN
SILE AV, B, štap 1, štap 5 uvjet:
→→→→
+=+ BštapBB vh5''
očitano(AutoCAD):
↑=
=
kNB
tlakkNŠ
69,45
)(42,505
uvjet:
→→→→
+=+ vvh AštapAA 1'' očitano(AutoCAD):
↑=
=
kNA
tlakkNŠv
01,86
)(67,681
SILE Š2, Š3, Š4, Š6, Š7, Š8, Š9 uvjet(ravnoteža čvora):
→→→→
=++ 0621 ŠŠŠ
očitano(AutoCAD):
)(42,366
)(05,502
tlakŠ
tlakŠ
=
=
uvjet(ravnoteža čvora):
→→→→
=++ 0954 ŠŠŠ
očitano(AutoCAD):
)(73,269
)(74,364
tlakkNŠ
tlakkNŠ
=
=
Š4=36,74kN
Š8=8,91kN
Š3=35,65kN
15,58kN
Š2=50,05kN
Š3=35,65kN
Š7=3,43kN
34
B
87
15,58kN
2
6
Av
Ah
1
9
5
uvjet(ravnoteža čvora):
→→→→
=++ 0843 ŠŠŠ
očitano(AutoCAD):
)(91,88
)(65,353
tlakkNŠ
tlakkNŠ
=
=
uvjet(ravnoteža čvora):
→→→→→
=+++ 0732 1XŠŠŠ
očitano(AutoCAD):
)(43,37 tlakkNŠ =
SILE U ŠTAPOVIMA:
kNŠ
kNŠ
kNŠ
kNŠ
kNŠ
kNŠ
kNŠ
kNŠ
kNŠ
73,269
91,88
43,37
42,366
41,505
74,364
65,353
05,502
67,681
=
=
=
=
=
=
=
=
=
T
MN
2,5
12,92
1
10kN/m
86,01kN
T
NM
2
2,5 2,25
10kN/m 36,42kN
12,92
86,01kN213,8kNm
T
NM 26,73kN
45,69kN3
2,25 2,252,25
65 kNm8,91kN
8,71kN
MOMENTI U KARAKTERISTIČNIM TOČKAMA Presjek 1-1
kNTT
kNT
kNN
kNmM
LD
L
59,2442,36
01,612501,86
92,12
78,1832
5,2105,201,86
2
=−=
=−=
=
=−⋅=
Presjek 2-2
kNTT
kNT
kNN
M
LD
L
34,143,3
09,242,365,4701,86
92,12
0008,21325,242,362
75,41075,401,86
2
−=−=
=−−=
=
=⇒=−⋅−−⋅=
Presjek 3-3
kNTT
kNT
kNN
kNmM
LD
L
96,1891,871,8
34,169,4573,2691,871,8
92,12
46,21025,273,265,469,4565
−=−−=
−=−++=
=
=⋅−⋅+=
MN
T4
2,25 2,25
65 kNm
45,69kN
26,73kN
T
NM
45,69kN
65 kNm
2,25
Presjek 4-4
kNTT
kNT
N
kNmM
LD
L
69,4573,26
964,1869,4573,26
0
8,16725,269,4565
−=−=
−=−=
=
=⋅+=
Presjek 5-5
kNTT
kNT
N
kNmM
LD
L
069,45
69,45
0
65
=+=
−=
=
=
-20°C
+20°C
10kN/m65 kNm1
,69
2,5
2,25
2,5
2,5
2,25
2,25
2,252,252,252,52,252,5
DIJAGRAMI M,T,N METODOM JEDINIČNE SILE
183,78kN
m
213,8kN
m
210,46kN
m
167,8kN
m
65kN
mM
T
86,01kN
61,06
24,59kN
2,09
1,34
18,96
45,69
12,92kN
N
-68,67kN
-50,05kN-35,65kN
-36,74kN
-50,41kN
-36,42kN-3,43kN
-15,58kN-8,91kN
-26,73kN
45,69kN
DIJAGRAMI M,T,N u DIM-u
Degrees of Freedom: 30 Nodal displacements: nd u_i v_i phi_i -------------------------------------------------------- 1: 0 0 0.000709341 2: 2.45579e-06 0.000604506 -1.82111e-05 3: 4.66601e-06 0.000525235 -1.87187e-05 4: 7.1218e-06 0.000557396 3.61233e-05 5: 7.1218e-06 0.000579215 -5.65953e-05 6: 7.1218e-06 -1.28493e-05 -0.00057007 7: 7.1218e-06 -0.00221197 -0.0013847 8: 0 0 -2.84872e-06 9: 0 0 0.000149143 10: 0.000159837 0.000511308 0 11: 2.44215e-05 0.00051402 0 12: -8.71267e-05 0.000538573 0 13: -0.000188414 0.00050392 0 14: 0 0 -0.000138467 Element end forces: el N_ij T_ij M_ij N_ji T_ji M_ji ------------------------------------------------------------------------------------------------ G1: -7.6385 79.4238 0 7.6385 -54.4238 167.309 G2: -7.6385 24.6005 -167.309 7.6385 -2.10051 197.348 G3: -7.6385 -2.6846 -197.348 7.6385 2.6846 190.637 G4: 0 -15.8721 -190.637 0 15.8721 154.925 G5: 0 -39.9665 -154.925 0 39.9665 65 G6: 0 0 -65 0 0 65 S: 39.9665 0 0 -39.9665 0 0 Š1: 56.2353 0 0 -56.2353 0 0 Š2: 40.9882 0 0 -40.9882 0 0 Š3: 32.1259 0 0 -32.1259 0 0 Š4: 33.1146 0 0 -33.1146 0 0 Š5: 45.4328 0 0 -45.4328 0 0 Š6: 29.8233 0 0 -29.8233 0 0 Š7: 4.78511 0 0 -4.78511 0 0 Š8: 8.03147 0 0 -8.03147 0 0 Š9: 24.0944 0 0 -24.0944 0 0 X2: 9.21579 0 0 -9.21579 0 0 Reactions: nd R_x R_y M -------------------------------------------------------- 1: -7.6385 79.4238 8: 0 39.9665 9: 39.7644 -39.7644 14: -32.1259 -32.1259
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