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.

Przyjmujemy oznaczenia dla podpór: A, B, i C

 RUD] GOD SRáF]H SU]HJXERZHJR

1,

teleskopowego 2 oraz dla tulei 3.

8ZDOQLDP\XNáDGRGZL ]yZ]DVW SXMF SRGSRU\UHDNFMDPL 3RGSRUD R]QDF]RQD OLWHU

A jest

XWZLHUG]HQLHP D ZL F Z SXQNFLH

A

 G]LDáDM GZLH QLH]DOH*QH RG VLHELH VNáDGRZH UHDNFML

pionowa V

A

i pozioma H

A

oraz moment M

A

. W miejscu podpory przegubowej przesuwnej B

G]LDáD UHDNFMD SLRQRZD

R

B

SURVWRSDGáD GR NLHUXQNX PR*OLZHJR SU]HVXZX 3UDZD SRGSRUD

R]QDF]RQDOLWHU

C

MHVWSRGSRUSU]HJXERZQLHSU]HVXZQ]DWHPZSXQNFLH

C

G]LDáDMGZLH

QLH]DOH*QH RG VLHELH VNáDGRZH UHDNFML SLRQRZD

V

C

i pozioma H

C

 2EFL*HQLH FLJáH

]DVW SXMHP\GZLHPDZ\SDGNRZ\PL

W

1

i W

2

:DUWRFLZ\SDGNRZ\FKREFL*HQLDFLJáHJRV

UyZQHSRORPILJXUSRGZ\NUHVHPUR]NáDGXQDW *HQLDREFL*HQLDLZ\QRV]

W

1

= W

2

=

2

1

Â

q

Â

l =

2

3

ql.

q

P = ql

2 l

2 l

2 l

3 l

l

q

P = ql

2 l

2 l

2 l

3 l

l

3

2

A

C

B

1

2 l

l

P = ql

2 l

2 l

2 l

3 l

l

3

2

A

C

B

1

W

1

W

2

V

A

H

A

H

C

V

C

M

A

R

B

2 l

l

x

y

2

7ZRU]\P\VFKHPDWSUDF\XNáDGXEHONRZHJRG]LHOFJRQDWU]\SRGXNáDG\&] ü

I

XNáDGX

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] FL

III

XNáDGX]HZ]JO GXQDWXOHM ZSXQNFLH

3.

:SRáF]HQLXSU]HJXERZ\P

1

G]LDáDMGZLHQLH]DOH*QHVNáDGRZHRGG]LDá\ZDQLDSLRQRZD

V

1

i pozioma H

1

1DWRPLDVWZSRáF]HQLXWHOHVNRSRZ\P

2

Z\VW SXMHSR]LRPHRGG]LDá\ZDQLH

R

2

RNLHUXQNXSURVWRSDGá\PGRRNáDGHNWHOHVNRSXRUD]PRPHQW

M

2

.

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\ZDQLH

V

1

.

∑

=

i

I

iy

P

0 : V

1

í

W

1

= 0

⇒

V

1

=

2

3

ql

W celu wyznaczenia momentu M

2

 ]DSLV]HP\ UyZQDQLH PRPHQWyZ Z]JO GHP SXQNWX

1. W

UyZQDQLX W\P QLH Z\VWSL RGG]LDá\ZDQLD

V

1

, H

1

oraz R

2

 JG\* OLQLH G]LDáDQLD W\FK VLá

SU]HFKRG]SU]H]SXQNW

1.

∑

=

i

I

i1

M

0 : M

2

í

W

1

Â

l = 0

⇒

M

2

= 3 ql

2

=WU]HFKUyZQDUyZQRZDJLGODF] FL

I

XNáDGXSR]RVWDáRQDPGRZ\NRU]\VWDQLDMHGQRD

ZL F QLH PR*HP\ Z\]QDF]\ü SR]RVWDá\FK GZX QLHZLDGRP\FK

H

1

i R

2

. Do obliczenia tych

RGG]LDá\ZDSR]LRP\FKSRZUyFLP\ZGDOV]HMF] FLUR]ZL]DQLD]DGDQLD

2 l

2 l

2 l

3 l

l

1

W

1

2

2 l

l

H

1

M

2

V

1

R

2

I

A

V

A

H

A

M

A

1

H

1

V

1

II

W

2

P = ql

3

2

C

B

H

C

V

C

R

B

M

2

R

2

III

2 l

II

A

V

A

H

A

M

A

1

H

1

V

1

=

2

3

ql

3

1DSRZ\*V]\PU\VXQNXVLáD

V

1

R]QDF]RQDMHVWNRORUHPF]DUQ\PMDNRZLHONRü]QDQD

=DSLV]HP\ WHUD] UyZQDQLD UyZQRZDJL  GOD F] FL 

II

  XNáDGX NRU]\VWDMF ] ZDUWRFL

RGG]LDá\ZDQLDSLRQRZHJR

V

1

5yZQDQLHU]XWyZVLáQDRSLRQRZPDSRVWDü

∑

=

i

II

iy

P

0 : V

A

í

V

1

= 0

⇒

V

A

=

2

3

ql

=UyZQDQLDPRPHQWyZZ]JO GHPSXQNWX

A

REOLF]\P\ZDUWRüPRPHQWXZXWZLHUG]HQLX

M

A

.

:UyZQDQLXW\PQLHZ\VWSLRGG]LDá\ZDQLH

H

1

RUD]VNáDGRZHUHDNFMLSLRQRZD

V

A

i pozioma

H

A

JG\*OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNW

A.

∑

=

i

II

iA

M

0 :

í

M

A

í

V

1

Â

l = 0

⇒

M

A

=

í

ql

2

= UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD F] FL 

II

  XNáDGX QLH PR*HP\ Z\]QDF]\ü

SR]RVWDá\FKGZXQLHZLDGRP\FK

H

1

i H

A

.

: GDOV]HM NROHMQRFL ]DMPLHP\ VL  UyZQRZDJ F] FL 

III

  XNáDGX 1D U\VXQNX WHM F] FL

XNáDGXPRPHQW

M

2

R]QDF]RQ\MHVWNRORUHPF]DUQ\PSRQLHZD*MHJRZDUWRüMHVWMX*]QDQD

3LHUZV]\P UyZQDQLHP UyZQRZDJL GOD F] FL 

III

  XNáDGX E G]LH UyZQDQLH PRPHQWyZ

Z]JO GHP SXQNWX

B

 0RPHQW\ VLá

R

2

, R

B

, W

2

i H

C

 Z]JO GHP SXQNWX

B

 V UyZQH ]HUX JG\*

OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNW

B.

0

=

∑

i

III

iB

M

: V

C

Â

l

í

M

2

í

P

Â

l = 0

⇒

V

C

=

4

5

ql

:GDOV]HMNROHMQRFL]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSLRQRZGODF] FL

III

XNáDGX

∑

=

i

III

iy

P

0 : R

B

+ V

C

í

P

í

W

2

= 0

⇒

R

B

=

4

5

ql

3ROHFHQLH GR ]DGDQLD REHMPXMH UyZQLH* Z\]QDF]HQLH UHDNFML SR]LRP\FK

H

A

, H

C

i

RGG]LDá\ZD SR]LRP\FK

H

1

, R

2

 RUD] RGG]LDá\ZD Z WXOHL

M

3

, V

3

. W tym celu podzielimy

F] ü

III

XNáDGXQDF] üOHZ

IV

LSUDZ

V ).

III

W

2

P = ql

3

2

C

B

H

C

V

C

R

B

M

2

= 3 ql

2

R

2

l

2 l

2 l

V

3

P = ql

3

C H

C

V

C

=

4

5

ql

2 l

M

3

V

IV

l

2 l

W

2

=

2

3

ql

2

B

M

2

= 3 ql

2

R

2

3

M

3

V

3

R

B

=

4

5

ql

4

:WXOHLZ\VW SXMMDNRRGG]LDá\ZDQLDVLáDSLRQRZD

V

3

oraz moment M

3

. Reakcje R

B

i V

C

V

MX*]QDQHDZL FQDU\VXQNXR]QDF]RQH]RVWDá\NRORUHPF]DUQ\P=UyZQDQLDU]XWyZ VLá QD

RSR]LRPGODF] FL

V

XNáDGXZ\]QDF]\P\VNáDGRZSR]LRP

H

C

.

0

=

∑

i

V

ix

P

: H

C

= 0

.RU]\VWDMF ] UyZQDQLD U]XWyZ VLá QD R SLRQRZ REOLF]\P\ ZDUWRü RGG]LDá\ZDQLD

pionowego V

3

w tulei 3.

0

=

∑

i

V

iy

P

: V

3

+ V

C

í

P = 0

⇒

V

3

=

í

4

1

ql

Moment M

3

G]LDáDMF\ZWXOHL

3

Z\]QDF]\P\]UyZQDQLDPRPHQWyZZ]JO GHPSXQNWX

3 dla

( V

  F] FL XNáDGX

 : UyZQDQLX W\P QLH Z\VWSL VLá\

P, V

3

i H

C

 SRQLHZD* LFK OLQLH

G]LDáDQLDSU]HFKRG]SU]H]WHQSXQNW

0

=

∑

i

V

i3

M

: V

C

Â

l

í

M

3

= 0

⇒

M

3

=

2

5

ql

2

'ODF] FL

IV

XNáDGX]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSR]LRP

∑

=

i

IV

ix

P

0 : R

2

= 0

3R]RVWDáHUyZQDQLDGODF] FL

IV

XNáDGXPXV]E\üVSHáQLRQHWR*VDPRFLRZRJG\*]QDQH

VMX*ZV]\VWNLHVLá\G]LDáDMFHQDW F] üXNáDGX

: FHOX Z\]QDF]HQLD RGG]LDá\ZDQLD SR]LRPHJR

H

1

 VNRU]\VWDP\ ] UyZQDQLD U]XWyZ VLá QD R

SR]LRPGODF] FL

I

XNáDGX

0

=

∑

i

I

ix

P

: H

1

í

R

2

= 0

⇒

H

1

= 0

6NáDGRZ SR]LRP

H

A

 REOLF]\P\ ] UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD F] FL 

II )

XNáDGX

∑

=

i

II

ix

P

0 : H

A

í

H

1

= 0

⇒

H

A

= 0

3RSUDZQRü SU]HSURZDG]RQ\FK REOLF]H VSUDZG]LP\ ]DSLVXMF UyZQDQLH UyZQRZDJL GOD

FDáHJR XNáDGX QD SU]\NáDG UyZQDQLH PRPHQWyZ Z]JO GHP SXQNWX

A

 2EFL*HQLH FLJáH

]DVWSLP\ MHGQ Z\SDGNRZ

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 = W

1

+ W

2

= 2

Â

2

1

Â

q

Â

l = 3 ql

∑

=

i

iA

M

0 : R

B

Â

l + V

C

Â

l

í

M

A

í

W

Â

l

í

P

Â

l =

=

4

5

ql

Â

l +

4

5

ql

Â

l

í

(

í

ql

2

)

í

ql

Â

l

í

ql

Â

l =

(

2

15

+

2

25

+ 3

íí

)

ql

2

= 0

5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR

P = ql

2 l

2 l

2 l

3 l

l

3

2

A

C

B

1

W = 3 ql

V

A

=

2

3

ql

H

A

= 0

H

C

= 0

V

C

=

4

5

ql

M

A

=

í

ql

2

R

B

=

4

5

ql