7/17/2019 Curs 13 Aplicatii

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CURS 13

APLICAŢII

1. GRINZI DREPTE

2. GRINZI COTITE

3. CADRE TRIPLU ARTICULATE

4. GRINZI CU ZABRELE (rezolvae !r"# $eo%a &e'"#"lor)

7/17/2019 Curs 13 Aplicatii

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2 m 2 m 2 m 2 m 2 m

5 t10 t 5 t 5 t

VA+ VB = 25 t

5 x 2 + 10 x 4 + 5 x 6 + 5 x 10 = VB x 8

VB = 16,25 t;

VA= 25 t – 16,25 t = 8,75 t

 

VA= 8,75 t VB = 16,25 t

8,75 t5 t

3,75 t

11,25 t

6,25 t

5 t

10 t

5 t

16,25 t

5 t

T

++

*

1. GRINZI DREPTE

7/17/2019 Curs 13 Aplicatii

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2 m 2 m 2 m 2 m 2 m

5 t 10 t 5 t 5 t

MD = 8,75 t x 2 m = 17,5 tm

ME = 8,75 t x 4 m - 5 t x 2 m = 25 tm

ME = 8,75 t x 6 m - 5 t x 4 m – 10 t x 2 m

 = 12,5 tm

MB= 8,75 t x 8 m - 5 t x 6 m – 10 t x 4 m

 – 5 t x 2 m = - 10 tm

 

VA= 8,75 t VB = 16,25 t

A

+

B

E CFD

17,5 tm

25 tm

12,5 tm

10 tm

+

*

1. GRINZI DREPTE

7/17/2019 Curs 13 Aplicatii

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8 m 2 m

5 t/m5 t

VA= 18,75 tVB = 26,25 t

A

T

BC

18,75 t

21,25 t

5 t10 t

+

*

VA+ VB = 5 t/m x 8 m + 5 t = 45 t

5 x 8 x 4 + 5 x 10 = VB x 8

VB = 26,25 t;

VA= 45 t – 26,25 t = 18,75 t

 TB = 18,75 t – 5 x 8 t = - 21,25 t

x / 8- x = 18,75/21,25

21,25 x = 18,75 (8 – x)x (21,25 + 18,75) = 150

x = 150/40 = 3,75 m

+

26,25 tx = 3,75 m

1. GRINZI DREPTE

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8 m 2 m

5 t/m5 t

VA= 18,75 t VB = 26,25 t

A

+

BC

35,156 tm

10 tm

+

*

 MMA = 18,75 t x 3,75 m – 5 x 3,75 x 3,75/2 t m

= 35,156 tm

MB !"#$%t$ = - (5 t x 2 m) = - 10 tm

x = 3,75

m

1. GRINZI DREPTE

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VA+ VB = 10 t

&A= 2 t/m x 5 m = 10 tMA= 0 = 2 t/m x 5 m + 10 t x 2 m – 4 x VB

VB = 7,5 t

VA= 10 t - VB = 2,5 t

VA= 2,5 t 4 m

5 m

2 t/m10 t

&A= 10 t

VB = 7,5 t2 m

2. GRINZI COTITE

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VA= 2,5 t 4 m

5 m

2 t/m10 t

&A= 10 t

VB = 7,5 t

2 m T

+

&A= 10 t

VA= 2,5 t 10 t

VB = 7,5 t

MC= 10 t x 5 m – 2 t/m x 5 mx 5/2 m = 25 tm

MD= - ( - 7,5t x 2 m) = 15 tm

MC= 25 tm

MD= 15 tm+

+

+

-

D

C

2. GRINZI COTITE

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VA= 2,5 t 4 m

5 m

2 t/m10 t

&A= 10 t

VB = 7,5 t

2 m  N

VA= 2,5 t

*

'00

D

C

2. GRINZI COTITE

+

MC= 25 tm

MD= 15 tm

+

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VA= 5 t 4 m

5 m

1 t/m5 t

&A= 5 t

2 mC

B

VA= 5 t

&A= 1 t/m x 5 m = 5 t

MA= 1 t/m x 5 mx 2,5 m + 5 t x 2 m =22,5 tm

MA= 22,5 tm

2. GRINZI COTITE

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VA= 5 t

5 m

1 t/m5 t

&A= 5 t

2 mC

B

MA= 22,5 tm

2. GRINZI COTITE

C

B

5 t

*  N

C

5 t

+

T

B+

A= VA= 5 t

B = - 5 t +1 t/m x 5 m=0

TA= &A = 5 t

TB$% *t%= 5 t - 1 t/m x 5 m = 0

TB$% ".!$

= 5 t

T= 5 t - 5 t = 0

5 t

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VA= 5 t

5 m

1 t/m5 t

&A= 5 t

2 mC

B

MA= 22,5 tm

2. GRINZI COTITE

CB

22,5 tm

*+

C

,

B

MA= - 22,5 tm

MB$% *t% = - 22,5 tm +1 t/m x 5 m x 2,5 m = -10tm

MB$% ".!$ = -( 5t x 2m) = - 10 tm

10 tm *'00

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5 m 5 m

6 m

5t/m

2t/m

5 m 5 m

6 m

5t/m

3. CADRE TRIPLUARTICULATE

VA

&B

VB

&A

C

BA

ED

  VA + VB = 5 t/m x 10 m = 50 t 2 t/m x 6 m – &A – &B = 0

&A + &B = 2 t/m x 6 m = 12 t

MA= 2t/m x 6 m x 3 m + 5 t/m x 10 m x 5 m

 – VB x 10 m = 0VB x 10 m = 36 tm + 225 tm = 261 tm

VB = 26,1 t

VA= 50 t - 26,1 t = 23,' t VA= 23,' t

M,*t.$ = VA x 5 m + &A x 6 m – 2 t/m x 6 m x 3 m

 – 5 t/m x 5 m x 2,5 m = 0

&A x 6 m = 36 t m + 62,5 tm – 23,8 t x 5 m = - 1' tm

&A = - 3,167 t

&B = 12 t – (- 3,167 t) = 15,167 t

2t/m

7/17/2019 Curs 13 Aplicatii

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5 m 5 mVA= 23,' t VB = 26,1 t

&A = 3,167 t

C

BA

ED

6 m

5t/m

2t/m

&B = 15,167 t

23,' t

A= 23,' t

D =3,167 t + 2 t/m x 6 m = 15,167 t

E =23,' t – 5 t/m x 10 m = - 26,1 t

26,1 t

15,167 t

* *

*

 N

3. CADRE TRIPLUARTICULATE

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5 m 5 mVA= 23,' t VB = 26,1 t

&A = 3,167 t

C

BA

ED

6 m

5t/m

2t/m

&B = 15,167 t

3,167 t

TA= 3,167 t

TD$% *t% = 3,167 t + 2 t/m x 6 m = 15,167 t; TD$% ".!$ = 23,' t

TE $% ".!$= 23,' t – 5 t/m x 10 m = - 26,1 t

TE $% *t%= 3,167 t + 2 t/m x 6 m = 15,167 t

TC = 23,' t – 5t/m x 5 m = - 1,1 t

26,1 t

15,167 t

*

*

+

T

23,' t 15,167 t1,1 t

+

3. CADRE TRIPLUARTICULATE

7/17/2019 Curs 13 Aplicatii

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3,167 t

26,1 t

15,167 t

*

*

+

T

23,' t 1,1 t

+

15,167 t 23,' t

1,1 t

26,1 t

5 m5 m

x

x / 5-x = 1,1 / 23,'

23,' x = 5,5 – 1,1 x 

25 x = 5,5

x = 5,5 / 25 = 0,22 m

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5 m 5 mVA= 23,' t VB = 26,1 t

&A = 3,167 t

C

BA

ED

6 m

5t/m

2t/m

&B = 15,167 t

MA= 0

MD$% *t% = - 3,167 t x 6 m - 2 t/m x 6 m x 3 m = - 55,002 tm;

MD$% ".!$ = - 55,002 tm

M = 0; ME $% *t%= - ( 15,167 t x 6 m ) = - '1,002 tm

Mm$x = 23,' t x 4,78 m – 3,167 t x 6 m – 2 t/m x 6 m x 3 m – 5 t/m x 4,78 m x 4,78m/2 =114,242 tm – 1',002 + 36 + 57,121 tm = 2,11' tm

55,002 tm

*

*

-

+

'1,002 tm

+

3. CADRE TRIPLUARTICULATE

-

x = 0,22 m4,78 m

Mm$x = 2,11' tm

7/17/2019 Curs 13 Aplicatii

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10 t 10 t 10 t 10 t 10 t5 t 5 t

3 m 3 m 3 m 3 m 3 m3 m

3 m

18 m

4. GRINZI CU Z-BRELEDETER+INAREA EORTURILOR DIN BARE

UTILIZÎND METODA SECŢIUNILOR 

AB

VA= 30 t VB = 30 t

10 t 10 t 10 t 10 t 10 t5 t 5 t

3 m 3 m 3 m 3 m 3 m3 m

3 m

18 m

AB

VA= 30 t

CD

$ !

 = 0  VA + VB = 5 x 10 t + 2 x 5 t = 60 t

VA = VB = 30 t

 = 0 - C + + D x * 450= 0

M = 30 t x ' m – 5 t x ' m – 10 t x 6 m – 10 t x 3 m

 – C x 3 m = 0

C = 30 x 3 – 5 x 3 – 10 x 2 – 10 = 45 t; C = 45 t

M$ = 30 t x 6 m – 5 t x 6 m – 10 t x 3 m –  x 3 m = 0

 = 30 x 2 – 5 x 2 – 10 = 40 t; = 40 t

D x * 450= C - = 5 t; D = 5 2 t

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10 t 10 t 10 t 10 t 10 t5 t 5 t

3 m 3 m 3 m 3 m 3 m3 m

3 m

AB

10 t 10 t 10 t 10 t 10 t

5 t5 t

VA= 30 t VB = 30 t

25 t

25 t

15 t

15 t

5 t5 t

=450

TD

T = D * α

D = 5 t / * 45 0= 5 2 t

T

4. GRINZI CU Z-BRELEDETERMINAREA EFORTURILOR DIN

DIAGONALE

D = T/cos α

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10 t 10 t 10 t 10 t 10 t5 t 5 t

3 m 3 m 3 m 3 m 3 m3 m

AB

10 t 10 t 10 t 10 t 10 t

5 t5 t

VA= 30 t VB = 30 t$ 1

M1 = 25 t x 3 m = 75 tmM$ = 25 t x 6 m – 10 t x 3 m = 120 tm

M = 25 t x ' m – 10 t x 6 m – 10 t x 3 m= 135 tm

M1 = 75 tm

M$ = 120 tm

M = 135 tm

C

D

$

M$

M$ = x

3 m

M$ = 120 tm = x 3 m

= 40 t

 I = Ma/h

4. GRINZI CU Z-BRELEDETERMINAREA EFORTURILOR DIN TALPA

INFERIOARA

+

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10 t 10 t 10 t 10 t 10 t5 t 5 t

3 m 3 m 3 m 3 m 3 m3 m

AB

10 t 10 t 10 t 10 t 10 t

5 t5 t

VA= 30 t VB = 30 t$ 1

M = 25 t x ' m – 10 t x 6 m – 10 t x 3 m= 135 tm

M1 = 75 tm

M$ = 120 tm

M = 135 tm

C

D

$

MC

MC = C x

3 m

M = 135 tm = C x 3 m

C = 45 t

 C = Mc/h

4. GRINZI CU Z-BRELEDETERMINAREA EFORTURILOR DIN TALPA

SUPERIOARA

+

7/17/2019 Curs 13 Aplicatii

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4 m 4 m 4 m4 m10 m 10 m

6 m

6 m

6t/m

10t/m10t/m

/. STRUCTURI CO+BINATE

10 m 10 m

6 m

6t/m

VA= 60 t VB = 60 t

VA + VB = 6 t/m x 20 m = 120 t

VA = VB = 120 t/ 2 = 60 t

&A=&B = %2 / 8 = 6t/m x 202 / 8 x 6 =50 t

&A=&B = 50 t

&A= 50 t &B = 50 t

7/17/2019 Curs 13 Aplicatii

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VB = 60 t

&B = 50 t

10t/m

4 m 4 m

6 m