Dane:
P = 20 kN
M = 30 kNm
q = 5 kN/m
L = 3 m
E = 2 * 10

5

MPa

J = 1000000 cm

4

Szukane:

Obliczyć przemieszczenie w punkcie C.


Rozwiązanie:

Σ

M

F

= V

A

*7L – q*2L*6L – P*4L – M – q*2L*L

V

A

= (14q*L

2

+ 4P*L + M)/7L

M(x) = V

A

*x – ½*q * x

2



AF

+ ½*q*(x-2L)

2



BF

– P*(x-3L)



CF

– M*(x-4L)

0



DF

–

- ½*q*(x-5L)

2



EF

EJ*w

’’

= - M(x)

EJ*w

’’

= - V

A

*x + ½*q * x

2



AF

- ½*q*(x - 2L)

2



BF

+ P*(x - 3L)



CF

+ M*(x - 4L)

0



DF

+

+ ½*q*(x - 5L)

2



EF

EJ*w

’

= C – ½*V

A

*x

2

+ 1/6*q * x

3



AF

– 1/6*q*(x - 2L)

3



BF

+ ½*P*(x –3 L)

2



CF

+

+ M*(x - 4L)



DF

+ 1/6*q*(x - 5L)

3



EF

EJ*w = D + C*x – 1/6*V

A

*x

3

+ 1/24*q * x

4



AF

– 1/24*q*(x - 2L)

4



BF

+

1/6*P*(x –3 L)

3



CF

+ ½*M*(x - 4L)

2



DF

+ 1/24*q*(x - 5L)

4



EF

warunki brzegowe:

1. x = 0 ⇒ w =0
2. x = 7L ⇒ w =0

z 1.

D = 0

z 2. w(7L) = 0

0 = C*7L – 1/6*V

A

*(7L)

3

– 1/24*q*(7L – 2L)

4

+1/6*P*(7L-3L)

3

+ ½*M*(7L – 4L)

2

+

+ 1/24*q*(7L – 5L)

4

C = 343/42*V

A

*L

2

+ 625/168*q*L

3

– 64/42*P*L

2

– 8/7*M*L – 2/21*q*L

3

w(x) = 1/EJ * [(343/42*V

A

*L

2

+ 625/168*q*L

3

–64/42*P*L

2

– 8/7*M*L – 2/21*q*L

3

)*x –

1/6*V

A

*x

3

+ 1/24*q*x

4



AF

– 1/24*q*(x – 2L)

4



BF

+ 1/6*P*(x – 3L)

3



CF

+

+ ½*M*(x – 4L)

2



DF

+ 1/24*q*(x – 5L)

4



EF

w(9m) = 1/(2*10

8

*0,01) * [(343/42*42,857*3

2

+ 625/168*5*3

3

– 64/42*20*3

2

– 8/7*30*3 –

2/21*5*3

3

) * 9 – 1/6*42,857*9

3

+ 1/24*5*9

4

– 1/24*5*(9 – 2*3)

4

]

w(9m) = 0,0128m = 1,28cm = 12,8mm