7. MATRI^EN METOD NA

DEFORMACII 7.1 KOORDINATI NA KONSTRUKCIJA I KOORDINATI

NA ELEMENTI

( 7.1). , . , . , . , , . . , .

102

7.1

, (), - .

. , ( 7.1), . . , , . ( 7.1), , . , . , , P, , U. , P,U , p,u .

3 1

12

34

a b

2

X

Y

1

2

1

23

103

(i) 7.2 (ii)

( 7.2) , . , . 7.2 MATRICA NA KRUTOST NA KONSTRUKCIJA I

MATRICA NA KRUTOST NA ELEMENTI

, , (6.2). (6.2) : [ ] { } { } [ ] { } { }00 PUK0PUK ==+ (7.1) :

[ ]

=

nn2n1n

n22221

n11211

kkk

kkkkkk

K

L

M

L

L

1

2 3

4

u1,p1 u2,p2

u3,p3

u4,p4

element

4

3

2

1element

4

3

2

1

pppp

,

uuuu

104

- .

{ }

=

n

2

1

U

UU

UM

{ }

=

0n

20

10

0

P

PP

PM

{ }oP . (7.1), , . , , , (7.1) : [ ] { } { }0PUK = (7.2) [ ]K .

.

.

vektor na sili vo dopolnitelnite vrski na ki-nemati~ki opredelen si-stem predizvikani od nadvore[ni tovari

vektor na nepoznatite deforma-cii na sistemot

105 7.2.1 MATRICI NA KRUTOST NA KARAKTERISTI^NI

ELEMENTI

i.

, , ( 7.3). [ ]elk 44 .

(iii) 7.3

. ,

0u,0u,0u,1u 4321 ==== . ( 7.3) , 6.4.1. , ( 7.3) 0u,0u,1u,0u 4321 ==== . , :

1 2

3 4

l

EI

lEI4

lEI2

2lEI6

2lEI6b

u2=1

3lEI12

2lEI6

2lEI6

b

u1=1

3lEI12

a b

v

106

[ ]

=

lEI4

lEI6

lEI2

lEI6

lEI6

lEI12

lEI6

lEI12

lEI2

lEI6

lEI4

lEI6

lEI6

lEI12

lEI6

lEI12

k

22

2323

22

2323

el (7.3)

[ ]

=

22

22

3el

l4l6l2l6l612l612

l2l6l4l6l612l612

lEIk (7.4)

ii.

, , ( 7.4). [ ]elk 33 .

(iv) 7.4

3lEI3

2lEI3

u1=1

3lEI3

1

2 3

l

EI

lEI3

2lEI3

2lEI3

u2=1

a)

b)

v)

107

0u,0u,1u 321 === . ( 7.4) . , ( 7.4) 0u,1u,0u 321 === . , :

[ ]

=

=3l33

l3l3l33l33

lEI

lEI3

lEI3

lEI3

lEI3

lEI3

lEI3

lEI3

lEI3

lEI3

k 23

323

22

323

el (7.5)

(v) 7.5

7.2.2 FORMIRAWE NA MATRICATA NA KRUTOST NA SISTEM

SO PRIMENA NA METODOT NA KODNI BROEVI

[ ]elk , .. . . [ ]K

[ ]

[ ] [ ] [ ] [ ] sT kK =

1 2

3

l

EI [ ]

=

23

el

l3l3l3l333

l333

lEIk

108

[ ]

[ ][ ]

[ ]

=

eln

el2

el1

s

k

kk

kO

(7.6)

[ ]sk , . , . , .

[ ]K .

.

. . Primer 7.1

( 7.6) .

(b) , .

P

P

c.

1 2

109

7.6 , .

7.7

( 7.7).

7.3 VEKTOR NA TOVARI

, 7.2, , { }0P . . ,

[ ]

=

44434241

34333231

24232221

14131211

el1

aaaaaaaaaaaaaaaa

k

02 1 0

1

2

3

1 2

3

4

[ ]

=

333231

232221

131211el2

bbbbbbbbb

k

0 1 0

[ ] ( )

+=

1121

212222

aaaba

K 0

2

0

1

0

0

1

110 . , ; { }elop . , , . . . Primer 7.2

( 7.6), . ( 7.6) P , . ; ( 7.8). :

{ } { } { }ekvivalentojazoloo PPP +=

{ }

=

+

=

2P3

8ql

8Pl

2P

8ql

8Pl

P0

P

2ekvivalent

2jazol

o

8ql2

2P

8Pl

8Pl

8ql5

8ql3

{ }

=

8Pl

2P8Pl2P

p el10

1

2 3

1 2

34

{ }

=

8ql38

ql8ql5

p2

el20

8ql5

2

P

8ql2

Pl { }

=P

8ql

8Pl

P

2

ekvivalento

111

7.8

[ ]K { }oP (7.2), { }U : { } [ ] { }o1 PKU = (7.7) [ ]F . . { }U , , { }elu . , :

{ } [ ] { } { }eloeleleldef pukp += (7.8) .

112 7.3 ( 7.9) . 1.

; ( 7.10 ).

2. ( 7.10). 3.

[ ]elk .

(i) 7.9

10KN/m

40K

1 2

j.

31

23

12

34

1 2

3 4

4

1 3

2

113

(ii) 7.10

4.

[ ]K :

[ ]

=

241818189027182790

27EIK

5.

.

[ ]

=

=

3618181818121812

1818361818121812

27EI

l4l6l2l6l612l612

l2l6l4l6l612l612

lEIk

22

22

31

0

3

0

1

0013

[ ]

=

=

4.55.137.25.135.135.135.135.4

7.25.134.55.135.135.45.135.4

27EI

l4l6l2l6l612l612

l2l6l4l6l612l612

lEI3k

22

22

32

2

0

1

0

[ ]

=

=

3618181818121812

1818361818121812

27EI

l4l6l2l6l612l612

l2l6l4l6l612l612

lEIk

22

22

33

0

3

0

2

2qa 2

202P=

158Pl

=

8Pl

3012ql2

=

302ql

= 2ql

3012ql2

=

2 010

0023

114

(iii) 7.11

{ }

=

201015

Po

6. , (7.7),

{ }

=

=

15201520

8Pl

2P8Pl2P

p el10 { }

=

=

30303030

12ql2ql12ql2ql

p

2

2

el20

0

3

0

1

2

0

1

0

115

{ } [ ] { }

==

201015

054166.0008333.0008333.0008333.001349.0002381.0008333.0002381.001349.0

EI27PKU o

1

{ }

=

12499.1003967.0

3928.0

EI27U

7. . , , , . , , .

{ }

=

003928.0

12499.1

EI27u el1 { }

=

003967.003928.00

EI27u el2

{ }

=

00

003967.012499.1

EI27u el3

8. , (7.8), .

{ } [ ] { } { }1o111def pukp +=

0

3

0

2

2

0

1

0

0

3

1

0

116

{ }

=

+

=

178.2843.26

893.857.13

15201520

003928.0

12499.1

EI27

3618181818121812

1818361818121812

27EIp 1def

{ } =

+

=

30303030

003967.003928.00

EI27

4.55.137.25.135.135.135.135.4

7.25.134.55.135.135.45.135.4

27EIp 2def

{ }

=

393.40248.35893.8

75.24

p 2def

{ }

=

+

=

321.2057.13

393.2057.13

0000

00

003967.012499.1

EI27

3618181818121812

1818361818121812

27EIp 3def

, ( 7.12). .

=

=

=

0M

0Y

0X

jazol

28.126.4

24.7 55.2

28.113.5

8.89

40.39320

20.393

117

7.12. :

==== 0393.2020393.40M,0893.8893.8M 21

==

==

025.5575.248qY

057.1343.2640X

7.4 MATRICI NA TRANSFORMACII

, , () . , . , , . .

118

() .

. Primer 7.4

( 7.13), .

(iv) (v) 7.13

1. . , , .

20KN/m 40KN

1 2

p.

1

23

12

3 4

1

2

119

7.14

2.

[ ]

=

=

8.024.04.024.024.0096.024.0096.04.024.08.024.024.0096.024.0096.0

EI

l4l6l2l6l612l612

l2l6l4l6l612l612

lEIk

22

22

3lokal1

[ ]

=

=048.024.0048.0

24.020.124.0048.024.0048.0

EI3l33

l3l3l33l33

lEIk 23

lokal2

3.

7.15

q

1 1 2

1 2

1 2

3

4

1

2

120 4.

,

7.16 1u2 =

5.

{ } [ ] { }global11lokal1 uTu =

global

14

3

2

1lokal

14

3

2

1

uuuu

1000025.10000100001

uuuu

=

{ } [ ] { }global22lokal2 uTu =

global

22

1

lokal

23

2

1

uu

0010075.0

uuu

=

6. ,

, ,

[ ] [ ] [ ] [ ]1lokal1T1global1 TkTk =

1

0.75 1.25

1u2 =

121

[ ]

=

1000025.10000100001

8.024.04.024.024.0096.024.0096.04.024.08.024.0

24.0096.024.0096.0

EI

1000025.10000100001

k global1

[ ]

=

8.03.04.024.03.015.03.012.04.03.08.024.0

24.012.024.0096.0

EIk global1

[ ]

=

0010075.0

048.024.0048.024.020.124.0048.024.0048.0

EI0100075.0

k global2

[ ]

=

2.118.018.0027.0

EIk global2

7.

:

[ ]

=

177.012.012.000.2

EIK

8.

0 0 1 2

1

2

0

0

2 1

2

1

12

3 4

18ql5

8ql2

8ql3

2 1 240KN

122

7.17 9.

:

{ } [ ] { }lokal2oT2global2o pTp =

{ }

=

=

5.62875.46

5.375.625.62

0100075.0

p global2o

{ }

=875.46

5.62p ekv

10.

:

{ } { } { }

=

+

=+=875.6

5.62875.46

5.62400

ppP ekvjaz

11.

:

{ } [ ] { }

=

==

549.62989.34

EI1

875.65.62

889.5353.0353.0521.0

EI1PKU 1

{ }

=

=5.375.625.62

8ql38

ql8ql5

p2

lokal20

2

1

123 12.

:

{ }

=

989.34549.6200

EI1u global1

{ }

=989.34

549.62EI1u global2

13.

:

{ } [ ] { }

=

==

989.34186.78

00

EI1

989.34549.6200

EI1

1000025.10000100001

uTu global11lokal1

{ } [ ] { }

=

==

0989.34912.46

EI1

989.34549.62

EI1

0010075.0

uTu global22lokal2

14.

:

{ } [ ] { }lokal1lokal1lokal1 ukp =

{ }

=

=

227.9891.0769.4891.0

989.34186.78

00

EI1

8.024.04.024.024.0096.024.0096.04.024.08.024.0

24.0096.024.0096.0

EIp lokal1

{ } [ ] { }lokal2lokal2lokal2 ukp =

12

0

0

2

1

124

{ }

=

=649.10246.53649.10

0989.34912.46

EI1

048.024.0048.024.020.124.0048.024.0048.0

EIp lokal2

15.

{ } { } { }

=

+

=+=

149.48254.9851.51

649.10246.53649.10

5.375.625.62

ppp lokal2lokal2o2

7.18

7.19

51.851

9.254

48.149

2

0.891 4.769

0.891 9.227

1

Mdef

9.22

4.796