3U]\NáDG%HOND]HVNUDWRZDQLHP

3ROHFHQLH Z\]QDF]\ü UHDNFMH SRGSRURZH GOD SRQL*V]HM EHONL ]H VNUDWRZDQLHP REFL*RQHM

REFL*HQLHPFLJá\PRVWDá\PQDW *HQLX

q.

Przyjmujemy oznaczenia podpór: A i B

 1DWRPLDVW SRáF]HQLD SU]HJXERZH R]QDF]DP\

cyframi: 1, 2, 3 i 4.

5R]ZL]DQLH ]DGDQLD UR]SRF]\QDP\ RG XZROQLHQLD XNáDGX RG ZL ]yZ ]DVW SXMF SRGSRU\

UHDNFMDPL*yUQDSRGSRUDR]QDF]RQDOLWHU

A

MHVWSRGSRUWHOHVNRSRZ=DWHPZSXQNFLH

A

G]LDáDUHDNFMDSR]LRPD

R

A

SURVWRSDGáDGRNLHUXQNXPR*OLZHJRSU]HVXZXRUD]PRPHQW

M

A

.

'ROQD SRGSRUD R]QDF]RQD OLWHU

B

 MHVW SRGSRU SU]HJXERZ QLHSU]HVXZQ : SXQNFLH

B

Z\VW SXMGZLHQLH]DOH*QHRGVLHELHVNáDGRZHUHDNFMLSLRQRZD

V

B

i pozioma H

B

2EFL*HQLH

FLJáHRVWDá\PQDW *HQLX

q

]DVW SXMHP\GZLHPDZ\SDGNRZ\PL

W

1

= W

2

= q

Â

l = 2 ql.

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czterech niewiadomych: M

A

, R

A

, V

B

i H

B

-HG\QLHUHDNFM 

V

B

PR*HP\Z\]QDF]\ü]UyZQDQLD

l

l

l

2 l

q

l

l

l

l

l

2 l

q

l

l

A

B

1

2

3

4

B

I

4

W

2

l

l

l

l

l

l

l

A

1

2

3

M

A

H

B

R

A

V

B

x

y

W

1

2

U]XWyZVLáQDRSLRQRZGODFDáHJRXNáDGX

0

=

∑

i

iy

P

: V

B

í

q

Â

l = 0

⇒

V

B

= 4 ql

1DVW SQHUyZQDQLHUyZQRZDJL]DSLV]HP\GOD

I

F] FLXNáDGX

=UyZQDQLDPRPHQWyZZ]JO GHPSXQNWX

2

GODF] FL

I

XNáDGXZ\]QDF]\P\ZDUWRüVLá\

S

1

 G]LDáDMFHM Z SU FLH

B

í

4

 0RPHQW\ RGG]LDá\ZD

V

2

i H

2

 Z]JO GHP SXQNWX

2

 V UyZQH

]HUX JG\* OLQLH G]LDáDQLD W\FK VLá SU]HFKRG] SU]H] SXQNW

2

 =DSLVXMF UyZQDQLH PRPHQWyZ

Z]JO GHPSXQNWX

2

GODF] FL

I

XNáDGXVNRU]\VWDP\]WZLHUG]HQLD9DULJQRQD:W\PFHOX

]DVW SXMHP\VLá 

S

1

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5yZQDQLHPRPHQWyZZ]JO GHPSXQNWX

2

GODF] FL

I

XNáDGXPDSRVWDü

∑

=

i

I

i2

M

0 :

í

S

1

ÂFRV.

Â

l

í

W

2

Â

l = 0

⇒

S

1

=

í

3

10

2

ql

.ROHMQHUyZQDQLH]DSLV]HP\GODZ ]áDSRGSRURZHJR

B.

B

4

2

3

H

2

V

2

S

1

ÂVLQ.

I

l

l

l

W

2

S

1

ÂFRV.

sin

.

=

10

1

cos

.

=

10

3

4

B

I

l

l

l

2

3

H

2

V

2

W

2

S

1

l

3 l

.

10 l

.

V

B

H

B

.

S

1



S

2





90

o

sin





=

5

1

cos





=

5

2



2l

l

5 l

3

3U]\MPXMHP\ R  SURVWRSDGá GR OLQLL G]LDáDQLD VLá\

S

2

 Z\VW SXMFHM Z SU FLH

B

í  :

UyZQDQLXU]XWyZVLáQDW  R GOD Z ]áD SRGSRURZHJR

B

 MHG\Q QLHZLDGRP E G]LH VNáDGRZD

pozioma H

B

 : FHOX XSURV]F]HQLD REOLF]HQLD U]XWX VLá\

S

1

 QD R  ]DVW SXMHP\ VLá 

S

1

jej

VNáDGRZ\PLSLRQRZLSR]LRP

=DSLVXMHP\UyZQDQLHU]XWyZVLáQDRGODZ ]áDSRGSRURZHJR

B.

0

=

∑

i

B

i



P

: V

B

ÂVLQ

í

H

B

ÂFRV

í

S

1

ÂFRV.

ÂFRV

+ S

1

ÂVLQ.

ÂVLQ

= 0

⇒

H

B

=

3

11

ql

3R]RVWDáH UHDNFMH Z\]QDF]\P\ ] UyZQD UyZQRZDJL ]DSLVDQ\FK GOD FDáHJR XNáDGX 5HDNFM

SR]LRP

R

A

REOLF]\P\NRU]\VWDMF]UyZQDQLDU]XWyZVLáQDRSR]LRP

0

=

∑

i

ix

P

: R

A

+ H

B

= 0

⇒

R

A

=

í

3

11

ql

Moment podporowy M

A

 Z PLHMVFX SRGSRU\ WHOHVNRSRZHM PR*HP\ Z\]QDF]\ü ] UyZQDQLD

PRPHQWyZGODFDáHJRXNáDGXZ]JO GHPSXQNWX

B.

5yZQDQLHWRPDSRVWDü

∑

=

i

iB

M

0 :

í

R

A

Â

l

í

M

A

í

W

Â

l = 0

⇒

M

A

=

í

3

14

ql

2

6SUDZG]HQLH SRSUDZQRFL SU]HSURZDG]RQ\FK REOLF]H Z\NRQDP\ ]DSLVXMF UyZQDQLH

UyZQRZDJL ] NWyUHJR ZF]HQLHM QLH NRU]\VWDOLP\ 0R*H WR E\ü QD SU]\NáDG UyZQDQLH

PRPHQWyZGODFDáHJRXNáDGXZ]JO GHPSXQNWX

C.

0

=

∑

i

iC

M

: H

B

Â

l

í

V

B

Â

l

í

M

A

+ W

Â

l =

3

11

ql

Â

l

í

ql

Â

l

í

(

í

3

14

ql

2

)

+ 4 ql

Â

l =

=

3

22

ql

2

í

ql

2

+

3

14

ql

2

+ 8 ql

2

= 0

5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR

S

2

S

1

ÂFRV.

S

1

ÂVLQ.

V

B

H

B

.







90

o

sin





=

5

1

cos





=

5

2



2l

l

5 l

4

W = W

1

+ W

2

= 4 ql

l

l

l

l

l

l

l

A

1

2

3

M

A

H

B

=

3

11

ql

R

A

=

í

3

11

ql

V

B

= 4 ql

B

C