3U]\NáDG8NáDGEHONRZRNUDWRZ\

:\]QDF]\üUHDNFMH

30 kN/m

80 kN

5,0 m

4,0 m

4.0 m

5,0 m

3.0 m

5R]ZL]DQLH

8ZDOQLDP\ XNáDG ] ZL ]yZ ZSURZDG]DMF LFK ZáDFLZH UHDNFMH L RWU]\PXMHP\ XNáDG VLá

przedstawiony na rysunku.

30 kN/m

80 kN

H

A

V

A

H

B

V

B

M

B

M

A

A

B

1

2

1LH]QDQ\FK MHVW  VNáDGRZ\FK RSLVXMF\FK UHDNFMH SRGSyU : FHOX LFK Z\]QDF]HQLD

SRG]LHOP\ NRQVWUXNFM  Z\MFLRZ QD F] FL Z WHQ VSRVyE DE\ Z\G]LHOLü XNáDG

WUyMSU]HJXERZ\&] ü,Z\G]LHORQDSU]HJXEDPLLVWDQRZLWDNLXNáDG

30 kN/m

80 kN

H

A

V

A

H

B

V

B

M

B

M

A

A

B

1

2

R

1

R

1

R

2

R

2

3

I

II

III

=DXZD*P\ *H GRNáDGQLHM PDP\ WX GR F]\QLHQLD ] XNáDGHP SU WyZ SURVW\FK REFL*RQ\FK

W\ONR QD NRFDFK L SRáF]RQ\FK SU]HJXEHP F]\OL ] NUDWRZQLF R MHGQ\P Z (OH 0R*HP\

]DWHP XVWDOLü NLHUXQNL Z]DMHPQHJR RGG]LDá\ZDQLD Z SU]HJXEDFK  L  L GR SROLF]HQLD

RGG]LDá\ZDZHZQ WU]Q\FK5

1

i R

2

Z\NRU]\VWDüUyZQDQLDUyZQRZDJLZ ]áD

80 kN

y

R

1

R

2

x

α

α

I

=SRGDQ\FKRGOHJáRFLZ\QLND*HGODR]QDF]RQHJRQDU\VXQNXNWD

α

:

cos

α

= 4/5 ,

cos(90-

α

) = sin

α

=3/5.

2

5yZQDQLDUyZQRZDJLZ ]áD

0

P

i

I

ix

=

∑

i

0

P

i

I

iy

=

∑

SU]\MPXMSRVWDü

R

1

cos

α

+ 80 kN – R

2

cos

α

= 0 i R

1

cos(90-

α

) + R

2

cos(90-

α

) = 0,

LVWG5

1

= - 50 kN, R

2

= 50 kN.

:GDOV]HMF] FLUR]ZL]DQLDXZ]JO GQLRQ\E G]LHU]HF]\ZLVW\]ZURWUHDNFML5

1

.

:\NRU]\VWDP\REOLF]RQHZLHONRFL5

1

i R

2

GRZ\]QDF]HQLDUHDNFMLSRGSRURZ\FKUR]ZD*DMF

UyZQRZDJ F] FL,,L,,,

30 kN/m

H

A

V

A

M

A

II

50 kN

x

y

α

H

B

V

B

M

B

x

α

50 kN

y

III

=DSLVXMFNROHMQRUyZQDQLD

0

P

i

II

ix

=

∑

⇒

H

A

+ 4/5(50 kN) = 0

0

P

i

II

iy

=

∑

⇒

V

A

+ 3/5(50 kN) – 30 kN/m

ÂP 

0

M

i

II
iA

=

∑

⇒

M

A

+3/5(50kN)

ÂP±N1PÂPÂP 

RWU]\PDP\ZDUWRFL

H

A

= - 40 kN, V

A

= 120 kN, M

A

= 225 kNm.

$QDORJLF]QLHGODF] FL,,,PDP\UyZQDQLD

0

P

i

III

ix

=

∑

⇒

-H

B

+ 4/5(50 kN) = 0

0

P

i

III

iy

=

∑

⇒

V

B

– 3/5(50 kN) = 0

0

M

i

III
iB

=

∑

⇒

-M

B

+3/5(50kN)

ÂP 

]NWyU\FKRWU]\PXMHP\ZDUWRFL

H

B

= 40 kN, V

B

= 30 kN, M

B

= 150 kNm.

2VWDWHF]QLHUR]ZL]DQLHPDSRVWDü

5,0 m

4,0 m

5,0 m

30 kN/m

80 kN

40 kN

120 kN

40 kN

30 kN

150 kNm

225 kNm

1

2

4.0 m

3.0 m