3U]\NáDG8NáDGEHONRZ\]áR*RQ\

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

P i

REFL*HQLHPFLJá\P

q(x)

RQDW*HQLX]PLHQLDMF\PVLOLQLRZR

Oznaczamy podpory literami A, B i C

QDWRPLDVWSRáF]HQLDSU]HJXERZHF\IUDPL

1 i 2.

8ZDOQLDP\XNáDGRGZL]yZL]DVWSXMHP\SRGSRU\UHDNFMDPL

W utwierdzeniu w punkcie A

G]LDáDM QDVWSXMFH UHDNFMH PRPHQW

M

A

RUD] VNáDGRZD

pionowa V

A

i pozioma H

A

NWyUHVRGVLHELHQLH]DOH*QH:SXQNFLH

B

Z\VWSXMHUHDNFMD

R

B

o

P = 3 ql

6 q

.

= 30

o

l

l

3 l

2 l

2 l

= 60

o

6 q

P = 3 ql

.

= 30

o

l

l

3 l

2 l

2 l

= 60

o

A

B

C

2

1

C

R

C

90

o

.

= 30

o

= 60

o

= 90

o

í.

60

o

II

6 q

P = 3 ql

l

l

3 l

2 l

2 l

= 60

o

A

B

C

2

1

= 60

o

H

A

M

A

V

A

R

B

R

C

I

x

y

2

SLRQRZHM OLQLL G]LDáDQLD QDWRPLDVW Z SXQNFLH

C reakcja R

C

G]LDáDMFD Z]GáX* SURVWHM

QDFK\ORQHMGRSR]LRPXSRGNWHP

= 60

o

.

5R]ZL]DQLH ]DGDQLD UR]SRF]QLHP\ RG ]DSLVDQLD UyZQDQLD PRPHQWyZ Z]JOGHP SXQNWX

1

GOD F]FL

I

XNáDGX : UyZQDQLX W\P Z\VWSL GZLH QLHZLDGRPH

M

A

i V

A

, natomiast

SR]RVWDáH VLá\

V

1

, H

1

i H

A

PDM OLQLH G]LDáDQLD SU]HFKRG]FH SU]H] SXQNW

1

D ZLF PRPHQW\

W\FKVLáZ]JOGHPSXQNWX

1

VUyZQH]HUX

∑

=

i

I

i1

M

0 :

í

M

A

í

V

A

Â

l = 0 ( * )

=DSLV]HP\ QDVWSQH UyZQDQLH PRPHQWyZ Z]JOGHP SXQNWX

2

GOD F]FL

II

XNáDGX Z

NWyU\P Z\VWSL WH VDPH QLHZLDGRPH : SXQNFLH

2

ZDUWRü QDW*HQLD REFL*HQLD FLJáHJR

wynosi 4 q.

)LJXUD SRG Z\NUHVHP UR]NáDGX QDW*HQLD REFL*HQLD FLJáHJR PLG]\ SU]HJXEDPL

1 i 2 jest

WUDSH]HP : FHOX XQLNQLFLD NRQLHF]QRFL Z\]QDF]DQLD OLQLL G]LDáDQLD Z\SDGNRZHM WHJR

REFL*HQLD SRáR*HQLD URGND FL*NRFL WUDSH]X SRG]LHOLP\ REFL*HQLH FLJáH G]LDáDMFH QD

W F]ü XNáDGX ]JRGQLH ] SRQL*V]\P U\VXQNLHP :\]QDF]DP\ ZDUWRFL Z\SDGNRZ\FK MDNR

SRODILJXUSRGZ\NUHVDPLUR]NáDGXQDW*HQLDREFL*HQLDFLJáHJR

q

W

II

p

4

=

Â

2 l = 8 ql,

2

1

=

II

t

W

Â

q

Â

l = 2 ql

I

l

A

1

H

A

M

A

V

A

H

1

V

1

II

t

W

II

p

W

1

2

1

2

l

l

l

3

4

3

2

l

4 q

2 q

2 l

II

p

W

II

t

W

II

2 l

l

A

1

H

A

M

A

V

A

H

2

V

2

2

l

l

l

3

4

3

2

l

4 q

2

6 q

l + 3 l = 4 l

2 l

1

3

5yZQDQLHPRPHQWyZZ]JOGHPSXQNWX

2

GODF]FL

II

XNáDGXPDSRVWDü

∑

=

i

II

i2

M

0 :

II

p

W

Â

l +

II

t

W

Â

3

4

l

í

M

A

í

V

A

Â

l = 0 ( ** )

: UyZQDQLX W\P QLH Z\VWSXM QLHZLDGRPH

H

A

, V

2

i H

2

/LQLH G]LDáDQLD W\FK VLá SU]HFKRG]

SU]H] SXQNW ZLF LFK PRPHQW\ Z]JOGHP SXQNWX V UyZQH ]HUX 3R SRGVWDZLHQLX GR

UyZQDQLDZDUWRFLZ\SDGNRZ\FK

II

p

W i

II

t

W

UR]ZL]XMHP\XNáDGUyZQD

í

M

A

í

V

A

Â

l = 0 ( * )

í

M

A

í

V

A

Â

l +

3

32

Â

ql

2

= 0 ( ** )

=UR]ZL]DQLDXNáDGXUyZQDRWU]\PXMHP\

M

A

=

í

3

16

ql

2

, V

A

=

3

16

ql.

:\]QDF]RQHUHDNFMHQDVFKHPDFLHXNáDGXR]QDF]RQHVNRORUHPF]DUQ\P

3R]RVWDáHUyZQDQLDUyZQRZDJL]DSLV]HP\GODFDáHJRXNáDGX

3U]HG ]DSLVDQLHP UyZQD UyZQRZDJL Z\]QDF]\P\ Z\SDGNRZ REFL*HQLD FLJáHJR RUD]

VNáDGRZSLRQRZLSR]LRPVLá\VNXSLRQHMSU]\áR*RQHMGRSUDZHJRNRFDXNáDGX

:UyZQDQLXVXP\PRPHQWyZZ]JOGHPSXQNWX

C

Z\VWSLW\ONRMHGQDQLHZLDGRPD

R

B

JG\*

OLQLHG]LDáDQLDUHDNFML

H

A

i R

C

SU]HFKRG]SU]H]SXQNW

C

DZLFPRPHQW\Z\PLHQLRQ\FKVLá

Z]JOGHPWHJRSXQNWXVUyZQH]HUX

0

=

∑

i

iC

M

:

í

M

A

í

V

A

Â

l

í

R

B

Â

l

í

P

y

Â

l + W

Â

l + l ) = 0

⇒

R

B

=

3

37

ql

=DSLV]HP\WHUD]UyZQDQLHVXP\U]XWyZVLáQDRSLRQRZGODFDáHJRXNáDGX

6 q

P = 3 ql

l

l

3 l

2 l

2 l

= 60

o

A

B

C

2

1

= 60

o

H

A

M

A

=

í

3

16

ql

2

V

A

=

3

16

ql

R

B

R

C

P

x

= 3 ql

ÂFRV

=

2

3

ql

l

l

3 l

2 l

2 l

A

B

C

2

1

= 60

o

H

A

M

A

=

í

3

16

ql

2

V

A

=

3

16

ql

R

B

R

C

P

y

= 3 ql

ÂVLQ

=

2

3

ql

W =

2

1

Â

q

Â

l + l + 3 l ) = 18 ql

4

∑

=

i

iy

P

0 : V

A

+ R

B

+ R

C

ÂVLQ

í

W

í

P

y

= 0

⇒

R

C

=

9

3

11

ql

6XPD U]XWyZ VLá QD R SR]LRP GOD FDáHJR XNáDGX XPR*OLZL Z\]QDF]HQLH RVWDWQLHM

niewiadomej H

A

.

0

=

∑

i

ix

P

: H

A

í

R

C

ÂFRV

í

P

x

= 0

⇒

H

A

=

9

3

10

ql.

6SUDZG]LP\ SRSUDZQRü Z\NRQDQ\FK REOLF]H ]DSLVXMF UyZQDQLH UyZQRZDJL ZF]HQLHM

niewykorzystane.

∑

=

i

iA

M

0 :

í

M

A

í

W

Â

l

í

P

y

Â

l + R

B

Â

l + R

C

ÂVLQ

Â

l =

=

í

(

í

3

16

Â

ql

2

)

í

ql

Â

l

í

2

3

ql

Â

l +

3

37

ql

Â

l +

9

3

11

ql

Â

2

3

Â

l =

= ql

2

(

3

16

íí

2

27

+

3

148

+

6

77

)

= 0

5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR

3U]HGVWDZLP\RWU]\PDQHZ\QLNLZSRVWDFLOLF]EG]LHVLWQ\FK

M

A

=

í

3

16

ql

2

=

í

ql

2

, V

A

=

3

16

ql = 5,33 ql, H

A

=

9

3

10

ql = 1,92 ql

R

B

=

3

37

ql = 12,33 ql, R

C

=

9

3

11

ql = 2,12 ql

=DSLV]HP\MHV]F]HUD]UyZQDQLHPRPHQWyZZ]JOGHPSXQNWX

A

GODFDáHJRXNáDGX

∑

=

i

iA

M

0 :

í

M

A

í

W

Â

l

í

P

y

Â

l + R

B

Â

l + R

C

ÂVLQ

Â

l =

=

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\QLN QLH ZLDGF]\ R SRSHáQLHQLX EáGX MHVW QDWRPLDVW NRQVHNZHQFM

]DRNUJOHQLDZDUWRFLUHDNFML

W tym samym zadaniu wyznaczymy reakcje

EH] NRQLHF]QRFL UR]ZL]\ZDQLD XNáDGX GZX

UyZQD ] GZLHPD QLHZLDGRP\PL 5R]ZD*DQ\ XNáDG EHONRZ\ SRG]LHOLP\ QD SRMHG\QF]H

EHONL]ZDQHSRGXNáDGDPL

l

l

3 l

2 l

2 l

P

x

=

2

3

ql

A

B

C

2

1

= 60

o

H

A

=

9

3

10

ql

M

A

=

í

3

16

ql

2

V

A

=

3

16

ql

R

B

=

3

37

ql

R

C

=

9

3

11

ql

P

y

=

2

3

ql

W =

2

1

Â

q

Â

l + l + 3 l ) = 18 ql

5

3RQL*V]\U\VXQHNSU]HGVWDZLDVFKHPDWSUDF\XNáDGXEHONRZHJR

5R]ZL]\ZDQLH]DGDQLDUR]SRF]\QDP\RG F]FL

II ), która w punkcie 1 i 2

SRáF]RQD MHVW

SU]HJXERZR]VVLHGQLPLEHONDPLQDWRPLDVWQLHMHVWSRáF]RQD]SRGáR*HP

W przegubach 1 i 2

QD F]ü

II

XNáDGX G]LDáDM QLH]QDQH RGG]LDá\ZDQLD

H

1

, V

1

, H

2

i V

2

.

2GG]LDá\ZDQLD SLRQRZH

V

1

i V

2

PR*HP\ Z\]QDF]\ü ]DSLVXMF UyZQDQLD UyZQRZDJL GOD

F]FL

II

: UyZQDQLX PRPHQWyZ Z]JOGHP SXQNWX

1

GOD F]FL

II

Z\VWSL W\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

1.

∑

=

i

II

i1

M

0 : V

2

Â

l

í

II

p

W

Â

l

í

II

t

W

Â

3

2

l = 0

⇒

V

2

=

3

14

ql

: FHOX Z\]QDF]HQLD VLá\

V

1

PR*HP\ ]DSLVDü UyZQDQLH U]XWyZ VLá QD R SLRQRZ OXE

UyZQDQLHPRPHQWyZZ]JOGHPSXQNWX

2

GODF]FL

II

5yZQDQLHU]XWyZVLáQDRSLRQRZ

PDSRVWDü

0

=

∑

i

II

iy

P

: V

1

+ V

2

í

II

p

W

í

II

t

W = 0

⇒

V

1

=

3

16

ql

1

2

H

1

H

2

V

2

V

1

II

p

W

l

l

II

t

W

l

3

4

3

2

l

2 l

II

II

6 q

4 q

1

2

H

1

H

2

V

2

V

1

I

4 q

= 60

o

l

l

3 l

2 l

2 l

P = 3 ql

= 60

o

1

A

H

A

M

A

V

A

B

C

2

R

B

R

C

H

1

H

2

V

2

V

1

III

6

.RU]\VWDMF]Z\]QDF]RQHMZDUWRFL

V

1

QDVFKHPDFLHVLáDR]QDF]RQDMHVWF]DUQ\PNRORUHP

]DSLV]HP\UyZQDQLDUyZQRZDJLGODF]FL

I

XNáDGX

=UyZQDQLDU]XWyZVLáQDRSLRQRZGODF]FL

I

REOLF]\P\ZDUWRüUHDNFML

V

A

.

∑

=

i

I

iy

P

0 : V

A

í

V

1

= 0

⇒

V

A

=

3

16

ql

-DNRNROHMQH]DSLV]HP\UyZQDQLHPRPHQWyZZ]JOGHPSXQNWX$GODF]FL

I

XNáDGX

∑

=

i

I

iA

M

0 :

í

M

A

í

V

1

Â

l = 0

⇒

M

A

=

í

3

16

ql

2

Niewiadome reakcje R

B

i R

C

Z\]QDF]\P\NRU]\VWDMF]UyZQDUyZQRZDJLGODF]FL

III )

XNáDGX:DUWRüRGG]LDá\ZDQLD

V

2

MHVWMX*]QDQDDZLFQDVFKHPDFLHVLáDWDMHVWR]QDF]RQD

kolorem czarnym.

=DVWSLP\ REFL*HQLH FLJáH G]LDáDMFH QD F]ü

III

XNáDGX VLá Z\SDGNRZ

III

t

W

,

QDWRPLDVWVLáVNXSLRQ

P

MHMVNáDGRZ\PLSLRQRZLSR]LRP

1

A

H

A

M

A

V

A

H

1

V

1

=

3

16

ql

I

l

l

3 l

2 l

B

C

2

= 60

o

R

B

R

C

H

2

P

x

= 3 ql

ÂFRV

=

2

3

ql

P

y

= 3 ql

ÂVLQ

=

2

3

ql

V

2

=

3

14

ql

3

8

l

3

4

l

III

t

W

=

2

1

Â

q

Â

l = 8 ql

III

l

3 l

2 l

B

C

2

4 q

= 60

o

P = 3 ql

= 60

o

R

B

R

C

H

2

V

2

=

3

14

ql

III

7

= UyZQDQLD PRPHQWyZ Z]JOGHP SXQNWX

C

GOD F]FL

III

Z\]QDF]\P\ UHDNFM

R

B

.

0RPHQW\SR]RVWDá\FKQLHZLDGRP\FKVLá

H

2

i R

C

Z]JOGHPSXQNWX

C

VUyZQH]HUXJG\*LFK

OLQLHG]LDáDQLDSU]HFKRG]SU]H]WHQSXQNW

0

=

∑

i

III

iC

M

:

III

t

W

Â

3

8

l + V

2

Â

l

í

P

y

Â

l

í

R

B

Â

l = 0

⇒

R

B

=

3

37

ql

5HDNFM

R

C

Z\]QDF]\P\]UyZQDQLDU]XWyZVLáQDRSLRQRZGODF]FL

III

XNáDGX

0

=

∑

i

III

iy

P

: R

B

í

V

2

í

P

y

+ R

C

ÂVLQ

í

III

t

W

= 0

⇒

R

C

=

9

3

11

ql

1LHZLDGRP

H

A

PR*HP\Z\]QDF]\ü]UyZQDQLDU]XWyZVLáQDRSR]LRPGODFDáHJRXNáDGX

0

=

∑

i

ix

P

: H

A

í

R

C

ÂFRV

í

P

x

= 0

⇒

H

A

=

9

3

10

ql.

:\]QDF]HQLH UHDNFML GUXJLP VSRVREHP MHVW NRU]\VWQLHMV]H JG\* QLH ZL*H VL ]

NRQLHF]QRFL UR]ZL]\ZDQLD XNáDGX GZX UyZQD ] GZLHPD QLHZLDGRP\PL 1DOH*\ MHGQDN

GRGDWNRZR Z\]QDF]\ü RGG]LDá\ZDQLD SLRQRZH

V

1

i V

2

w przegubach 1 i 2

FKRFLD* Z

SROHFHQLX GR ]DGDQLD QLHZLDGRPH WH QLH E\á\ Z\PLHQLRQH 1DWRPLDVW QLH Z\]QDF]DP\

RGG]LDá\ZD SR]LRP\FK

H

1

i H

2

JG\* Z *DGQ\P UyZQDQLX UyZQRZDJL Z\NRU]\VW\ZDQ\P

GR UR]ZL]DQLD ]DGDQLD QLH SRMDZLDM VL WH QLHZLDGRPH =H Z]JOGX QD LGHQW\F]QH Z\QLNL

UR]ZL]DQLD ]DGDQLD SU]\ ]DVWRVRZDQLX REX VSRVREyZ SRPLMDP\ SRZWyUQH Z\NRQDQLH

sprawdzenia.