Przykład 2.5.

3U]\NáDG8NáDGEHONRZ\]áR*RQ\
3ROHFHQLHZ\]QDF]\üUHDNFMHSRGSRURZHRUD]RGG]LDá\ZDQLDZSRáF]HQLXSU]HJXERZ\PZ
WHOHVNRSLH L Z WXOHL GOD SRQL*V]HJR XNáDGX REFL*RQHJR REFL*HQLHP FLJá\P R QDW
*HQLX
]PLHQLDMF\PVL
OLQLRZRLVLáVNXSLRQ
P = ql.
q
P = ql
2l
2l
2l
l
3l
Przyjmujemy oznaczenia dla podpór: A, B, i C
teleskopowego 2 oraz dla tulei 3.
RUD] GOD SRáF]H SU]HJXERZHJR
1,
q
1
A
2
B
3
C
P = ql
2l
2l
2l
l
3l
8ZDOQLDP\ XNáDG RG ZL
]yZ ]DVW
SXMF SRGSRU\ UHDNFMDPL 3RGSRUD R]QDF]RQD OLWHU
XWZLHUG]HQLHP D ZL
F Z SXQNFLH
A jest
A G]LDáDM GZLH QLH]DOH*QH RG VLHELH VNáDGRZH UHDNFML
pionowa VA i pozioma HA oraz moment MA. W miejscu podpory przegubowej przesuwnej B
G]LDáD UHDNFMD SLRQRZD RB SURVWRSDGáD GR NLHUXQNX PR*OLZHJR SU]HVXZX 3UDZD SRGSRUD
R]QDF]RQDOLWHU CMHVWSRGSRUSU]HJXERZQLHSU]HVXZQ]DWHPZSXQNFLH CG]LDáDMGZLH
QLH]DOH*QH RG VLHELH VNáDGRZH UHDNFML SLRQRZD VC i pozioma HC 2EFL*HQLH FLJáH
]DVW
SXMHP\GZLHPDZ\SDGNRZ\PL W1 i W2:DUWRFLZ\SDGNRZ\FKREFL*HQLDFLJáHJRV
UyZQHSRORPILJXUSRGZ\NUHVHPUR]NáDGXQDW
*HQLDREFL*HQLDLZ\QRV]
W1 = W2 =
1
2
Â
q Âl =
l
2l
3
ql.
2
l
W1
2l
W2
MA
y
HA
1
A
2
B
C HC
3
P = ql
RB
VA
x
2l
3l
l
VC
2l
2l
7ZRU]\P\VFKHPDWSUDF\XNáDGXEHONRZHJRG]LHOFJRQDWU]\SRGXNáDG\&]
ü XNáDGX
I
w miejscu przegubu 1 i teleskopu 2 MHVW SRáF]RQD ] VVLHGQLPL SRGXNáDGDPL 1LH MHVW
QDWRPLDVW SRáF]RQD ] SRGáR*HP &]
ü II XNáDGX RSLHUD VL
QD SRGáR*X Z PLHMVFX
sztywnego zamocowania w punkcie A &]
ü III XNáDGX SRáF]RQD MHVW ] SRGáR*HP Z
PLHMVFX Z\VW
SRZDQLD SRGSRU\ SU]HJXERZHM SU]HVXZQHM B oraz podpory przegubowej
nieprzesuwnej C : GDOV]HM F]
FL UR]ZL]DQLD ]DGDQLD ]DVWRVXMHP\ GRGDWNRZ\ SRG]LDá
F]
FLIIIXNáDGX]HZ]JO
GXQDWXOHM
ZSXQNFLH3.
l
2l
W1
I
H1
1
M2
2
R2
V1
III
W2
II
MA
HA
V1
2
A
1
H1
R2
B
P = ql
M2
RB
VA
2l
VC
2l
2l
l
3l
C HC
3
:SRáF]HQLXSU]HJXERZ\P
1G]LDáDMGZLHQLH]DOH*QHVNáDGRZHRGG]LDá\ZDQLDSLRQRZD V1
i pozioma H11DWRPLDVWZSRáF]HQLXWHOHVNRSRZ\P2Z\VW
SXMHSR]LRPHRGG]LDá\ZDQLHR2
RNLHUXQNXSURVWRSDGá\PGRRNáDGHNWHOHVNRSXRUD]PRPHQWM2.
5R]ZL]DQLH ]DGDQLD UR]SRF]\QDP\ RG Z\]QDF]HQLD RGG]LDá\ZD GOD F]
FL I XNáDGX =
UyZQDQLDU]XWyZVLáQDRSLRQRZZ\]QDF]\P\RGG]LDá\ZDQLHV1.
3
V1 íW1 = 0
⇒
V1 =
ql
∑i PiyI = 0 :
2
W celu wyznaczenia momentu M2 ]DSLV]HP\ UyZQDQLH PRPHQWyZ Z]JO
GHP SXQNWX 1. W
UyZQDQLX W\P QLH Z\VWSL RGG]LDá\ZDQLD V1, H1 oraz R2 JG\* OLQLH G]LDáDQLD W\FK VLá
SU]HFKRG]SU]H]SXQNW1.
M2 íW1 Âl = 0
⇒
M2 = 3 ql2
∑ M i1I = 0 :
i
= WU]HFKUyZQD UyZQRZDJL GOD F]
FL XNáDGX SR]RVWDáR QDP GR Z\NRU]\VWDQLD MHGQR D
I
ZL
F QLH PR*HP\ Z\]QDF]\ü SR]RVWDá\FK GZX QLHZLDGRP\FK
H1 i R2. Do obliczenia tych
RGG]LDá\ZDSR]LRP\FKSRZUyFLP\ZGDOV]HMF]
FLUR]ZL]DQLD]DGDQLD
MA
HA
V1 =
A
1
VA
2l
2
H1
3
ql
2
II
1DSRZ\*V]\PU\VXQNXVLáD
V1R]QDF]RQDMHVWNRORUHPF]DUQ\PMDNRZLHONRü]QDQD
II XNáDGX NRU]\VWDMF ] ZDUWRFL
=DSLV]HP\ WHUD] UyZQDQLD UyZQRZDJL GOD F]
FL RGG]LDá\ZDQLDSLRQRZHJR
V15yZQDQLHU]XWyZVLáQDRSLRQRZPDSRVWDü
3
VA íV1 = 0
VA =
ql
⇒
∑i PiyII = 0 :
2
=UyZQDQLDPRPHQWyZZ]JO
GHPSXQNWX AREOLF]\P\ZDUWRüPRPHQWXZXWZLHUG]HQLX MA.
:UyZQDQLXW\PQLHZ\VWSLRGG]LDá\ZDQLH H1RUD]VNáDGRZHUHDNFMLSLRQRZD VA i pozioma
HAJG\*OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNWA.
MA = íql2
⇒
∑ M iAII = 0 : íMA íV1 Âl = 0
i
= UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD F]
FL SR]RVWDá\FKGZXQLHZLDGRP\FK
II XNáDGX QLH PR*HP\ Z\]QDF]\ü
H1 i HA.
: GDOV]HM NROHMQRFL ]DMPLHP\ VL
UyZQRZDJ F]
FL XNáDGXPRPHQW
III XNáDGX 1D U\VXQNX WHM F]
FL
M2R]QDF]RQ\MHVWNRORUHPF]DUQ\PSRQLHZD*MHJRZDUWRüMHVWMX*]QDQD
III
W2
R2
2
B
M2 = 3 ql2
C HC
3
P = ql
RB
l
VC
2l
2l
3LHUZV]\P UyZQDQLHP UyZQRZDJL GOD F]
FL III XNáDGX E
G]LH UyZQDQLH PRPHQWyZ
Z]JO
GHP SXQNWX B 0RPHQW\ VLá R2, RB, W2 i HC Z]JO
GHP SXQNWX B V UyZQH ]HUX JG\*
OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNWB.
5
VC Âl íM2 íP Âl = 0
⇒
VC =
ql
∑i M iBIII = 0 :
4
:GDOV]HMNROHMQRFL]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSLRQRZGODF]
FLIIIXNáDGX
5
⇒
RB =
ql
∑i PiyIII = 0 : RB + VC íP íW2 = 0
4
3ROHFHQLH GR ]DGDQLD REHMPXMH UyZQLH* Z\]QDF]HQLH UHDNFML SR]LRP\FK HA,
HC i
RGG]LDá\ZD SR]LRP\FK H1, R2 RUD] RGG]LDá\ZD Z WXOHL M3, V3. W tym celu podzielimy
F]
üIIIXNáDGXQDF]
üOHZIVLSUDZV ).
IV
W2 =
3
ql
2
M2 = 3 ql2
M3
2
M3
3
B
C HC
3
R2
5
RB =
ql
4
l
V
V3
P = ql
V3
2l
2l
3
VC =
5
ql
4
: WXOHL Z\VW
SXM MDNR RGG]LDá\ZDQLD VLáD SLRQRZD
V3 oraz moment M3. Reakcje RB i VC V
MX* ]QDQH D ZL
F QD U\VXQNX R]QDF]RQH ]RVWDá\ NRORUHP F]DUQ\P = UyZQDQLD U]XWyZ VLá QD
RSR]LRPGODF]
FL
VXNáDGXZ\]QDF]\P\VNáDGRZSR]LRPHC.
HC = 0
∑ PixV = 0 :
i
.RU]\VWDMF ] UyZQDQLD U]XWyZ VLá QD R SLRQRZ REOLF]\P\ ZDUWRü RGG]LDá\ZDQLD
pionowego V3 w tulei 3.
∑P
= 0:
V
iy
⇒
V3 + VC íP = 0
V3 = í
i
1
ql
4
Moment M3 G]LDáDMF\ZWXOHL 3Z\]QDF]\P\]UyZQDQLDPRPHQWyZZ]JO
GHPSXQNWX 3 dla
( V F]
FL XNáDGX : UyZQDQLX W\P QLH Z\VWSL VLá\ P, V3 i HC SRQLHZD* LFK OLQLH
G]LDáDQLDSU]HFKRG]SU]H]WHQSXQNW
∑M
V
i3
= 0:
⇒
VC Âl íM3 = 0
M3 =
i
'ODF]
FL
5 2
ql
2
IVXNáDGX]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSR]LRP
R2 = 0
∑ PixIV = 0 :
i
3R]RVWDáHUyZQDQLDGODF]
FL
IVXNáDGXPXV]E\üVSHáQLRQHWR*VDPRFLRZRJG\*]QDQH
VMX*ZV]\VWNLHVLá\G]LDáDMFHQDW
F]
üXNáDGX
: FHOX Z\]QDF]HQLD RGG]LDá\ZDQLD SR]LRPHJR
H1 VNRU]\VWDP\ ] UyZQDQLD U]XWyZ VLá QD R
SR]LRPGODF]
FL XNáDGX
I
∑P
I
ix
=0:
⇒
H1 íR2 = 0
H1 = 0
i
6NáDGRZ SR]LRP
HA REOLF]\P\ ] UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD F]
FL II )
XNáDGX
∑P
II
ix
= 0:
HA
í
⇒
H1 = 0
HA = 0
i
3RSUDZQRü SU]HSURZDG]RQ\FK REOLF]H VSUDZG]LP\ ]DSLVXMF UyZQDQLH UyZQRZDJL GOD
FDáHJR XNáDGX QD SU]\NáDG UyZQDQLH PRPHQWyZ Z]JO
GHP SXQNWX
]DVWSLP\ MHGQ Z\SDGNRZ
A 2EFL*HQLH FLJáH
W NWyUHM ZDUWRü UyZQD MHVW SROX ILJXU\ SRG Z\NUHVHP
UR]NáDGX QDW
*HQLD REFL*HQLD )LJXUD WD PD R V\PHWULL QD NWyUHM OH*\ MHM URGHN FL
*NRFL
/LQLDG]LDáDQLDZ\SDGNRZHMSU]HFKRG]LSU]H]URGHNFL
*NRFLRPDZLDQHMILJXU\
W = W1 + W2 = 2 Â
MA = íql2
HA = 0
5
ql
4
l
RB =
3
VA =
ql
2
2l
∑M
iA
3l
= 0:
Â
q Âl = 3 ql
W = 3 ql
2 B
1
A
1
2
C HC = 0
3
P = ql
2l
VC =
5
ql
4
2l
RB Âl + VC Âl íMA íW Âl íP Âl =
i
=
5
5
15
25
ql Âl + ql Âl í(íql2) íql Âl íql Âl = (
+
+ 3 íí) ql2 = 0
4
4
2
2
5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR
4