Układ belkowy złożony 2, Budownictwo Studia, Mechanika(1)

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*
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
 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
Â
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
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
Â
q
Â
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
Â
l
í
R
B
Â
l
í
P
y
Â
l
+
W
Â
l
+
l
) = 0
Þ
R
B
=
37
ql
3
=DSLV]HP\WHUD]UyZQDQLHVXP\U]XWyZVLáQDRSLRQRZGODFDáHJRXNáDGX
3
:UyZQDQLXW\PQLHZ\VWSXMQLHZLDGRPH
 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
Â
q
Â
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
Â
l
í
P
y
Â
l
+
R
B
Â
l
+
R
C
ÂVLQ
Â
l
=
i
=
í
(
í
16
Â
ql
2
)
í
ql
Â
l
í
3
ql
Â
l
+
37
ql
Â
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
Â
l
í
P
y
Â
l
+
R
B
Â
l
+
R
C
ÂVLQ
Â
l
=
i
=
í
(
í
ql
2
)
í
ql
Â
l
í
ql
Â
l
+ 12,33
ql
Â
l
+ 2,12
ql
Â
0,866
Â
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
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
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