Układ belkowy złożony 2.pdf

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3U]\NáDG8NáDGEHONRZ\]áR * RQ\
3ROHFHQLHZ\]QDF]\üUHDNFMHSRGSRURZHGODSRQL * V]HMEHONLREFL * RQHMVLá VNXSLRQ
P i
q(x) RQDW * HQLX]PLHQLDM F\PVL OLQLRZR
REFL * HQLHPFL Já\P
6 q
P = 3 ql
= 60 o
. = 30 o
l
2 l
l
3 l
2 l
Oznaczamy podpory literami A , B i C QDWRPLDVWSRá F]HQLDSU]HJXERZHF\IUDPL 1 i 2 .
6 q
P = 3 ql
A
1
2
B
C
= 60 o
. = 30 o
l
2 l
l
3 l
2 l
8ZDOQLDP\XNáDGRGZL ]yZL]DVW SXMHP\SRGSRU\UHDNFMDPL
M A
6 q
P = 3 ql
A
1
2
B
C
= 60 o
y
H A
R C
I
II
R B
V A
= 60 o
x
l
2 l
l
3 l
2 l
C
90 o
R C
. = 30 o = 60 o
= 90 o í . 60 o
W utwierdzeniu w punkcie A G]LDáDM QDVW SXM FHUHDNFMHPRPHQW M A RUD]VNáDGRZD
pionowa V A i pozioma H A NWyUHV RGVLHELHQLH]DOH * QH:SXQNFLH B Z\VW SXMHUHDNFMD R B o
89765480.020.png 89765480.021.png 89765480.022.png 89765480.023.png 89765480.001.png
 
SLRQRZHMOLQLLG]LDáDQLDQDWRPLDVWZSXQNFLH
C reakcja R C G]LDáDM FDZ]GáX * SURVWHM
= 60 o .
5R]ZL ]DQLH]DGDQLDUR]SRF]QLHP\RG]DSLVDQLDUyZQDQLDPRPHQWyZZ]JO GHPSXQNWX
QDFK\ORQHMGRSR]LRPXSRGN WHP
1
I XNáDGX:UyZQDQLXW\PZ\VWSLGZLHQLHZLDGRPH M A i V A , natomiast
SR]RVWDáHVLá\
GODF]FL
V 1 , H 1 i H A PDMOLQLHG]LDáDQLDSU]HFKRG]FHSU]H]SXQNW 1 DZLFPRPHQW\
W\FKVLáZ]JOGHPSXQNWX
1 VUyZQH]HUX
M A
V 1
A
H A
1
H 1
V A
l
I
å
M
I
i1
=
0 : í M A í V A Â l = 0 ( * )
i
=DSLV]HP\QDVWSQHUyZQDQLHPRPHQWyZZ]JOGHPSXQNWX
2 GODF]FL II XNáDGXZ
2 ZDUWRüQDW*HQLDREFL*HQLDFLJáHJR
NWyU\PZ\VWSLWHVDPHQLHZLDGRPH:SXQNFLH
wynosi 4 q .
6 q
4 q
1
2 l
2
l + 3 l = 4 l
1 i 2 jest
WUDSH]HP:FHOXXQLNQLFLDNRQLHF]QRFLZ\]QDF]DQLDOLQLLG]LDáDQLDZ\SDGNRZHMWHJR
REFL*HQLDSRáR*HQLDURGNDFL*NRFLWUDSH]XSRG]LHOLP\REFL*HQLHFLJáHG]LDáDMFHQD
WF]üXNáDGX]JRGQLH]SRQL*V]\PU\VXQNLHP:\]QDF]DP\ZDUWRFLZ\SDGNRZ\FKMDNR
SRODILJXUSRGZ\NUHVDPLUR]NáDGXQDW*HQLDREFL*HQLDFLJáHJR
W II
p
=
4
q
2 l = 8 ql ,
W
II
=
1
q  l = 2 ql
Â
Â
2
2 q
W
II
4 q
2 l
4
2 l
l
3
3
W
II
II
W
II
p
1
2
2 l
4
l
l
l
3
3
M A
H 2
1
A
2
W
II
p
H A
1
2
V A
V 2
l
2 l
l
l
2
)LJXUDSRGZ\NUHVHPUR]NáDGXQDW*HQLDREFL*HQLDFLJáHJRPLG]\SU]HJXEDPL
89765480.002.png 89765480.003.png 89765480.004.png 89765480.005.png
5yZQDQLHPRPHQWyZZ]JOGHPSXQNWX
2 GODF]FL II XNáDGXPDSRVWDü
4 l í M A í V A Â l = 0 ( ** )
å
M
II
i2
=
0 :
W Â l +
II
p
W Â
II
3
i
H A , V 2 i H 2 /LQLHG]LDáDQLDW\FKVLáSU]HFKRG]
SU]H]SXQNWZLFLFKPRPHQW\Z]JOGHPSXQNWXVUyZQH]HUX3RSRGVWDZLHQLXGR
UyZQDQLDZDUWRFLZ\SDGNRZ\FK II
p
W i
W UR]ZL]XMHP\XNáDGUyZQD
II
í
M A í V A Â l = 0 ( * )
32 Â ql 2 = 0 ( ** )
í
M A í V A Â l +
3
=UR]ZL]DQLDXNáDGXUyZQDRWU]\PXMHP\
M A = í
16 ql 2 , V A =
3
16 ql .
3
:\]QDF]RQHUHDNFMHQDVFKHPDFLHXNáDGXR]QDF]RQHVNRORUHPF]DUQ\P
3R]RVWDáHUyZQDQLDUyZQRZDJL]DSLV]HP\GODFDáHJRXNáDGX
M A = í
16 ql 2
6 q
P = 3 ql
3
A
1
2
B
C
= 60 o
H A
R C
16 ql
R B
V A =
3
= 60 o
l
2 l
l
3 l
2 l
3U]HG]DSLVDQLHPUyZQDUyZQRZDJLZ\]QDF]\P\Z\SDGNRZREFL*HQLDFLJáHJRRUD]
VNáDGRZSLRQRZLSR]LRPVLá\VNXSLRQHMSU]\áR*RQHMGRSUDZHJRNRFDXNáDGX
W = 2
1  q  l + l + 3 l ) = 18 ql
16 ql 2
3 ql
M A = í
3
P y = 3 ql ÂVLQ = 2
A
1
2
B
C
3 ql
H A
R C
P x = 3 ql ÂFRV =
16 ql
R B
2
V A =
3
= 60 o
l
2 l
l
3 l
2 l
:UyZQDQLXVXP\PRPHQWyZZ]JOGHPSXQNWX
C Z\VWSLW\ONRMHGQDQLHZLDGRPD R B JG\*
H A i R C SU]HFKRG]SU]H]SXQNW C DZLFPRPHQW\Z\PLHQLRQ\FKVLá
Z]JOGHPWHJRSXQNWXVUyZQH]HUX
OLQLHG]LDáDQLDUHDNFML
å i
M
iC
=
0
: í M A í V A  l í R B  l í P y  l + W  l + l ) = 0
Þ
R B =
37 ql
3
=DSLV]HP\WHUD]UyZQDQLHVXP\U]XWyZVLáQDRSLRQRZGODFDáHJRXNáDGX
3
:UyZQDQLXW\PQLHZ\VWSXMQLHZLDGRPH
89765480.006.png 89765480.007.png 89765480.008.png 89765480.009.png 89765480.010.png
 
å
P
=
0 : V A + R B + R C ÂVLQ í W í P y = 0
Þ
R C =
11
3
ql
iy
9
i
6XPDU]XWyZVLáQDRSR]LRPGODFDáHJRXNáDGXXPR*OLZLZ\]QDF]HQLHRVWDWQLHM
niewiadomej H A .
å i
P
=
0
: H A í R C ÂFRV í P x = 0
Þ
H A =
10
3
ql .
ix
9
W = 2
1  q  l + l + 3 l ) = 18 ql
16 ql 2
3 ql
M A = í
3
P y = 2
3 ql
H A =
10
3
ql
A
1
2
B
C
P x =
2
9
11
3
R C =
ql
16 ql
37 ql
9
= 60 o
V A =
R B =
3
3
l
2 l
l
3 l
2 l
6SUDZG]LP\SRSUDZQRüZ\NRQDQ\FKREOLF]H]DSLVXMFUyZQDQLHUyZQRZDJLZF]HQLHM
niewykorzystane.
å
M
iA
=
0 : í M A í W  l í P y  l + R B  l + R C ÂVLQ  l =
i
= í ( í
16  ql 2 ) í ql  l í
3 ql  l +
37 ql  l +
11
3
ql Â
3 Â l =
3
2
3
9
2
16 íí
27 +
148 +
77 ) = 0
= ql 2 (
3
2
3
6
5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR
3U]HGVWDZLP\RWU]\PDQHZ\QLNLZSRVWDFLOLF]EG]LHVLWQ\FK
M A = í
16 ql 2 = í ql 2 , V A =
16 ql = 5,33 ql , H A =
10
3
ql = 1,92 ql
3
3
9
R B =
37 ql = 12,33 ql , R C =
11
3
ql = 2,12 ql
3
9
A GODFDáHJRXNáDGX
=DSLV]HP\MHV]F]HUD]UyZQDQLHPRPHQWyZZ]JOGHPSXQNWX
å
M
iA
=
0 : í M A í W  l í P y  l + R B  l + R C ÂVLQ  l =
i
= í ( í ql 2 ) í ql  l í ql  l + 12,33 ql  l + 2,12 ql  0,866  l =
= 0,0014 ql 2 §
2WU]\PDQ\Z\QLNQLH ZLDGF]\RSRSHáQLHQLXEáGX MHVWQDWRPLDVWNRQVHNZHQFM
]DRNUJOHQLDZDUWRFLUHDNFML
W tym samym zadaniu wyznaczymy reakcje EH]NRQLHF]QR FLUR]ZL]\ZDQLDXNáDGXGZX
UyZQD]GZLHPDQLHZLDGRP\PL5R]ZD*DQ\XNáDGEHONRZ\SRG]LHOLP\QDSRMHG\QF]H
EHONL]ZDQHSRGXNáDGDPL
4
89765480.011.png 89765480.012.png 89765480.013.png 89765480.014.png 89765480.015.png 89765480.016.png
3RQL*V]\U\VXQHNSU]HGVWDZLDVFKHPDWSUDF\XNáDGXEHONRZHJR
6 q
4 q
II
H 1
1
2
H 2
V 1
V 2
III
M A
I
4 q
P = 3 ql
= 60 o
A
1
2
B
C
H A
H 1 H 2
R C
R B
V 1
V 2
V A
= 60 o
l
2 l
l
3 l
2 l
II ), która w punkcie 1 i 2 SRáF]RQDMHVW
SU]HJXERZR]VVLHGQLPLEHONDPLQDWRPLDVWQLHMHVWSRáF]RQD]SRGáR*HP
W
II
2 l
4
l
3
3
W
II
p
II
l
l
H 1
1
2
H 2
V 1
2 l
V 2
W przegubach 1 i 2 QDF]ü II XNáDGXG]LDáDMQLH]QDQHRGG]LDá\ZDQLD H 1 , V 1 , H 2 i V 2 .
2GG]LDá\ZDQLDSLRQRZH
V 1 i V 2 PR*HP\Z\]QDF]\ü]DSLVXMFUyZQDQLDUyZQRZDJLGOD
II :UyZQDQLXPRPHQWyZZ]JOGHPSXQNWX 1 GODF]FL II Z\VWSLW\ONR
niewiadoma V 2 QDWRPLDVWPRPHQW\VLá H 1 , V 1 i H 2 Z]JOGHPSU]HJXEX 1 VUyZQH]HUXJG\*
LFKOLQLHG]LDáDQLDSU]HFKRG]SU]H]SXQNW
F]FL
1 .
å
M
II
i1
=
0 : V 2 Â l í II
p
W Â l í II
W Â
2 l = 0
Þ
V 2 =
14 ql
i
V 1 PR*HP\]DSLVDüUyZQDQLHU]XWyZVLáQDRSLRQRZOXE
UyZQDQLHPRPHQWyZZ]JOGHPSXQNWX
2 GODF]FL II 5yZQDQLHU]XWyZVLáQDRSLRQRZ
PDSRVWDü
å i
P
II
iy
=
0
: V 1 + V 2 í II
p
W í II
W = 0
Þ
V 1 =
16 ql
5
5R]ZL]\ZDQLH]DGDQLDUR]SRF]\QDP\RGF]FL
3
3
:FHOXZ\]QDF]HQLDVLá\
3
89765480.017.png 89765480.018.png 89765480.019.png

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