Mat Cat Ngang

ac trng hnh hoc mat cat ngang Khoa Xay Dng & C Hoc ng Dung

Bai tap sc ben vat lieu trang 1 06/2013

Chng 04

AC TRNG HNH HOC MAT CAT NGANG

I) TOM TAT LY THUYET

1) Dien tch hnh phang:

F

F dF (4.1)

2) Mo men tnh cua hnh phang oi vi truc Ox va Oy : .

.

x

F

y

F

S y dF

S x dF

(4.2)

3) Momen tnh cua hnh phang oi vi mot truc nao o bang khong, truc o c goi la truc trong tam, giao iem cua hai truc trung tam la trong tam hnh phang. Goi C la trong tam mat cat, t C ke he truc toa o o ox Cy song song vi he truc xOy t

o ta co: 00

C

C

x x x

y y y

,

Thay vao (4.2) ta co 0 00

. . 0x C C x C xF

S y y dF y F S y F S

Vay ta co: .

.x C

y C

S y F

S x F

, neu hnh phang la nhieu hnh ghep lai: 1

1

.

.

i

i

n

x C ii

n

y C ii

S y F

S x F

(4.3)

Trong o n la so mien ma ta biet trc dien tch va trong tam.

xHnh 4.1

y

x

y dF

oy

oxC

Cx

Cyox

oy

O



4) Trong tam hnh phang:

1

1

1

1

.

.

i

i

n

C iy i

C n

ii

n

C ix i

C n

ii

x FS

xF

F

y FS

yF

F

(4.4)

5) Cac momen quan tnh cua hnh phang oi vi truc Ox va Oy :

2

2

.

.

x

F

y

F

J y dF

J x dF

(4.5)

6) Momen quan tnh ly tam cua hnh phang oi vi he truc Ox va Oy : . .xyF

J x y dF

7) Momen quan tnh cc cua hnh phang oi vi tam O : 2.F

J dF (4.7)

x yJ J J

8) Momen quan tnh ly tam cua hnh phang oi vi he truc trong o co mot truc oi xng bang khong, he truc o c goi la he truc quan tnh chnh, neu he truc quan tnh chnh i qua trong tam hnh phang c goi la he truc quan tnh chnh trung tam.

9) Ban knh quan tnh cua hnh phang oi vi truc Ox va Oy :

xx

yy

Ji

F

Ji

F

(4.8)

10) Mo men quan tnh cua mot so hnh thng gap:

Hnh 4.2

x

y

C

b

h

y

x

d

Cx

b

h C



Hnh ch nhat:

3

3

12

12

x

y

bhJ

hbJ

Hnh tron ac:

44

44

0,0564

2 0,132

x y

x

dJ J d

dJ J d

Hnh tam giac:3

36xbh

J

11) Cong thc chuyen truc song song:

Momen quan tnh cua hnh phang oi vi truc Ox : 2 .x

F

J y dF , trong o 0 1y y y

1 1

2 2 2 20 1 0 1 0 1 0 0. . . 2 . . 2x x x

F F F F

J y y dF y dF y dF y y dF y F J y S

Vay ta co: 1 1

1 1

20 0

20 0

2

2

x x x

y y y

J J y F y S

J J x F x S

(4.8)

Neu O C (O la trong tam mat cat), ta co 1 1

0x yS S

1

1

20

20

x x

y y

J J y F

J J x F

(4.9)

II) V DU V du 1: Tnh cac momen quan tnh hnh ch nhat canh b h oi vi cac truc ,x y nh hnh V.4.1

xHnh 4.2

y

x

y dF

1y

1xO

Ox

Oy1x

1y

O



Momen quan tnh cua hnh ch nhat oi vi cac truc trung tam: 1 1

3 3

;12 12x ybh hb

J J

S dung cong thc chuyen truc song song ta co: 1

1

2 233

2 233

1. .

2 12 2 3

1. .

2 12 2 3

x x

y y

h bh hJ J bh bh bh

b hb bJ J bh bh hb

V du 2: Tnh cac momen quan tnh hnh tam giac oi vi truc x nh hnh V.4.2

Momen quan tnh cua tam giac oi vi truc trung tam: 1

3

36xbh

J

S dung cong thc chuyen truc song song ta co:

1

2 23 31 1. .

3 2 36 3 2 12x xh bh h bh

J J bh bh

V du 3: Tnh cac momen quan tnh chnh trung tam cua hnh phang nh hnh V.4.3a He truc oi xng xy la he truc quan tnh chnh trung tam cua hnh phang. Chia hnh phang thanh 3 hnh nh hnh V.4.3b, ta co 1 2 3x x x xJ J J J

Hnh V.4.2

xb

h1x

b

h C

x

/ 3h

Hnh V.4.1

1x

1y

C

b

hC

b

h

x

y



1

2 3

3

423

30.900

1276626

450.30 900 30.450.30

12 2 2

x

x

x x

J

J cm

J J

1 2 3

3 34900.30 30.4502 45765

12 12y y y yJ J J J cm

Chia hnh phang thanh 3 hnh nh hnh V.4.3c, ta co 1 2 3x x x xJ J J J (hnh (1) la hnh ac, hnh (2), hnh (3) la hai hnh ch nhat rong)

33

4

450 30.900450.960 22 76626

12 12xJ cm

1 2 3

3 34900.30 30.4502 45765

12 12y y y yJ J J J cm

V du 4: Xac nh trong tam, tnh cac momen quan tnh chnh trung tam cua hnh phang nh hnh V.4.4a

Chon he truc toa o 1x y nh hnh V.4.4b ( y la truc oi xng cua mat cat ngang), chia hnh phang thanh hai hnh gom hnh ac (1) va hnh rong (2).

b

b7b

15b

b

b7b

15b

1x

y

Hnh V.4.4

)a )b

xC

cy

Hnh V.4.3

900

40

30450

450 30

)a )b

900

30

30450

450 30

x

y

3

1

2

900

30

30450

450 30

x

y

1

2

3

)c



Trong tam hnh phang: 2

12

1

0

3,5 .15 .7 3 .13 .6 89

15 .7 13 .6 18

i

C

C ii

C

ii

x

y Fb b b b b b

y bb b b b

F

He truc xy la he truc quan tnh chnh trung tam cua hnh phang.

3 32 21 2 4

3 3

1 2 4

15 . 7 13 . 689 893,5 .15 .7 3 .13 .6 118,917b

12 18 12 18

7 . 15 6 . 13870,25

12 12

x x x

y y y

b b b bJ J J b b b b b b b b

b b b bJ J J b

V du 5: Xac nh trong tam, tnh cac momen quan tnh chnh trung tam cua hnh phang nh hnh V.4.5a

Chon he truc toa o 1x y nh hnh V.4.5b ( y la truc oi xng cua mat cat ngang), chia hnh phang thanh hai hnh gom hnh.

Trong tam hnh phang: 2

2 21

2 2 2

1

0

4 .8 8,5 .76,1

8 7

i

C

C ii

C

ii

x

y Fb b b b

y bb b

F

He truc xy la he truc quan tnh chnh trung tam cua hnh phang.

3 32 21 2 2 2 4

3 3

1 2 4

. 8 7 .4 6,1 .8 8,5 6,1 .7 118,85b

12 12

8 . . 729,25

12 12

x x x

y y y

b b b bJ J J b b b b b b

b b b bJ J J b

V du 6: Tnh cac momen quan tnh chnh trung tam cua hnh phang nh hnh V.4.6a

b

b

8b

7b

b

b

8b

7b

1x

1y

x

(1)

(2)

Hnh V.4.5

)a )b

cy



He truc oi xng xy la he truc quan tnh chnh trung tam cua hnh phang. Chia hnh phang thanh 2 tam giac nh hnh V.4.6b

3

4

22 .

22

12 12x y

bb

bJ J

V du 6: Tnh cac momen quan tnh chnh trung tam cua hnh phang nh hnh V.4.6a.

He truc oi xng xy la he truc quan tnh chnh trung tam cua hnh phang nh hnh V.4.6b. Chia hnh phang thanh 3 hnh gom hnh ch nhat ac (1) va hai tam giac rong (2).

3 34

23 34

75.70 12,5.204 2110416,667

12 12

70.75 40.12,5 100 12 40.12,5 1901041,667

12 36 3 2

x

y

J cm

J cm

50cm

75cm

40cm70cm

50cm

75cm

40cm70cm x

y

)a )b

Hnh V.4.7

b b b b

x

y

(1)

(2)

)a )b

Hnh V.4.6



III) BAI TAP 4.1. Tnh cac momen quan tnh chnh trung tam cua mat cat ngang nh hnh B.4.1. 4.2. Xac nh trong tam, tnh cac momen quan tnh chnh trung tam cua mat cat ngang nh

hnh B.4.2. Khi tnh bo qua cac goc ln (xem tai v tr cac goc lng la goc vuong). So sanh ket qua tnh c vi tra bang thep 70 70 6L .

4.3. Xac nh trong tam, tnh cac momen quan tnh chnh trung tam cua cac mat cat ngang nh hnh B.4.3.

4.4. Tnh cac momen quan tnh chnh trung tam cua mat cat ngang nh hnh B.4.4. Mt ct ngang gm hai thp ch 70 70 6L ghep lai.

14

16

76

70

30

120Hnh B.4.3

30

150

70

70

6

Hnh B.4.2

6

53014 14

360

9

Hnh B.4.1



4.5. Xac nh trong tam, tnh cac momen quan tnh chnh trung tam cua cac mat cat ngang

nh hnh B.4.5.

6b

b

4b

6b

b

4b

Hnh B.4.5

6

6

30

22

5 5b b2b

610

5

48

18

3

6

5

70

6

Hnh B.4.4

6

70



4.6. Mt ct ngang ca mt dm thp c t hp t bn thp gc u cnh s hiu 40 40 5L

v bn tm thp c kch thc 7 70 nh hnh B.4.6. Tnh cc mmen qun tnh chnh trung tm ca mt ct.

4.7. Mt ct ngang ca mt dm thp c t hp t bn thp gc u cnh s hiu 40 40 5L

v tm thp c kch thc 174 6 nh hnh B.4.7. Tnh cc mmen qun tnh chnh trung tm ca mt ct.

41 98

183

60

20 60

100

160

Hnh B.4.7

6

40 40 5L

174

70

Hnh B.4.6

7

40 40 5L

50cm75cm

40cm

70cm

25cm

5

48

18

3

6

5

348

5 5b b2b

2b



4.8. Mt ct ngang ca mt dm thp c t hp t hai thp gc s hiu 18I v thp gc s hiu 22I nh hnh B.4.8. Tnh cc mmen qun tnh chnh trung tm ca mt ct.

4.9. Mt ct ngang ca mt dm thp c t hp t hai thp gc s hiu 18I v hai tm thp

thp c kch thc 400 12 nh hnh B.4.9. Tnh cc mmen qun tnh chnh trung tm ca mt ct.

Hnh B.4.10

180 70 5

18I

y

y

x x

180 70 5

Hnh B.4.10

180 70 5

y

y

x x

180 70 5

6 180

240

Hnh B.4.9

18I 18I

12

400Hnh B.4.8

18I18I

22I

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Mat Cat Ngang