SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA
SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA
46. Zadatak Koriste�i analiti�ki postupak odre�ivanja elasti�ne linije nosa�a odrediti kut zaokreta u to�kama D i F te progib u to�ki E. a = 5m kN q = 10 m b = 2m P = 50kN c = d = 1m EI 5 2 = 8⋅10 Nm Ekvivalentni sustav Reakcije u osloncima ∑ M ∑ M ∑ F z d D A = 0 → F = 0 → q ⋅ 7 ⋅3, 5 + P ⋅8 − F ⋅5 − F ⋅9 = 0 → F = 0 → F A F y ⋅ 2 − P ⋅1 = 0 → F − q ⋅ 7 − P + F F C + F C F = P 50 = = 25kN 2 2 F = 0 → F A C = 11kN = 420 = 84kN 5 70
Momenti savijanja na pojedinim segmentima : ( I ) 0 ≤ x ≤ a → M ( x) = F A 2 x ⋅ x − q ⋅ 2 ( II ) a ≤ x ≤ ( a + b) → M ( x) = F ⋅ x − q ⋅ + F ⋅ ( x − 5) A ( III )( a + b) ≤ x ≤ ( a + b + c) → M ( x) = F ⋅ x − q ⋅ + F ⋅ ( x − 5) ( IV )( a + b + c) ≤ x ≤ ( a + b + c + d ) → M ( x) = F ⋅ x − q ⋅ + F ⋅ ( x − 5) 2 x 2 A C A 2 x 2 C 2 x 2 C + q ⋅ ( x − 7) 2 2 + q ⋅ Iz diferencijalne jednadžbe elasti�ne linije nosa�a slijedi za moment savijanja : 2 d w EI = −M 2 dx 2 d w EI = −F 2 dx 2 d w 2 dx ( I ) ( x) 2 x ⋅ x + q ⋅ 2 2 x 2 ( II ) EI = −F ⋅ x + q ⋅ − F ⋅ ( x − 5) 2 d w 2 dx 2 x 2 ( III ) EI = −F ⋅ x + q ⋅ − F ⋅ ( x − 5) 2 d w 2 dx A A A 2 x 2 ( IV ) EI = −F ⋅ x + q ⋅ − F ⋅ ( x − 5) Za kut zaokreta : 2 3 dw x x ( I ) EI = −FA ⋅ + q ⋅ + C dx 2 6 ( II ) ( III ) ( IV ) ( II ) ( III ) ( IV ) A 2 3 dw x x EI = −FA ⋅ + q ⋅ − FC ⋅ dx 2 6 2 3 dw x x EI = −FA ⋅ + q ⋅ − FC ⋅ dx 2 6 2 3 dw x x EI = −FA ⋅ + q ⋅ − FC ⋅ dx 2 6 Za progib nosa�� : 3 4 x x ( I ) EIw = −FA ⋅ + q ⋅ + C1 ⋅ x + D 6 24 3 4 x x EIw = −FA ⋅ + q ⋅ − FC ⋅ 6 24 3 4 x x EIw = −FA ⋅ + q ⋅ − FC ⋅ 6 24 3 4 x x EIw = −FA ⋅ + q ⋅ − FC ⋅ 6 24 1 C C C ( x − 5) 2 − q ⋅ − q ⋅ 2 ( x − 7) 2 ( x − 7) 2 2 ( x − 5) ( x − 7) 2 2 2 2 + P ⋅ ( x − 8) 2 3 ( x − 5) ( x − 7) ( x − 8) 1 2 ( x − 5) 6 3 + C − q ⋅ − q ⋅ 6 6 3 ( x − 5) ( x − 7) 6 2 3 + C 3 + P ⋅ 3 4 ( x − 5) ( x − 7) ( x − 8) 6 + C − q ⋅ − q ⋅ ⋅ x + D 24 24 2 4 + C 3 + P ⋅ 6 2 ⋅ x + D 3 3 2 + C + C 4 4 ⋅ x + D ( x − 7) 4 2 2 − P ⋅ ( x − 8) 71
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Momenti savijanja na pojedinim segmentima :<br />
( I ) 0 ≤ x ≤ a → M ( x)<br />
= F<br />
A<br />
2<br />
x<br />
⋅ x − q ⋅<br />
2<br />
( II ) a ≤ x ≤ ( a + b)<br />
→ M ( x)<br />
= F ⋅ x − q ⋅ + F ⋅ ( x − 5)<br />
A<br />
( III )( a + b)<br />
≤ x ≤ ( a + b + c)<br />
→ M ( x)<br />
= F ⋅ x − q ⋅ + F ⋅ ( x − 5)<br />
( IV )( a + b + c)<br />
≤ x ≤ ( a + b + c + d ) → M ( x)<br />
= F ⋅ x − q ⋅ + F ⋅ ( x − 5)<br />
2<br />
x<br />
2<br />
A<br />
C<br />
A<br />
2<br />
x<br />
2<br />
C<br />
2<br />
x<br />
2<br />
C<br />
+ q ⋅<br />
( x − 7)<br />
2<br />
2<br />
+ q ⋅<br />
Iz diferencijalne jednadžbe elasti�ne linije nosa�a slijedi za moment savijanja :<br />
2<br />
d w<br />
EI = −M<br />
2<br />
dx<br />
2<br />
d w<br />
EI = −F<br />
2<br />
dx<br />
2<br />
d w<br />
2<br />
dx<br />
( I )<br />
( x)<br />
2<br />
x<br />
⋅ x + q ⋅<br />
2<br />
2<br />
x<br />
2<br />
( II ) EI = −F<br />
⋅ x + q ⋅ − F ⋅ ( x − 5)<br />
2<br />
d w<br />
2<br />
dx<br />
2<br />
x<br />
2<br />
( III ) EI = −F<br />
⋅ x + q ⋅ − F ⋅ ( x − 5)<br />
2<br />
d w<br />
2<br />
dx<br />
A<br />
A<br />
A<br />
2<br />
x<br />
2<br />
( IV ) EI = −F<br />
⋅ x + q ⋅ − F ⋅ ( x − 5)<br />
Za kut zaokreta :<br />
2 3<br />
dw x x<br />
( I ) EI = −FA<br />
⋅ + q ⋅ + C<br />
dx 2 6<br />
( II )<br />
( III )<br />
( IV )<br />
( II )<br />
( III )<br />
( IV )<br />
A<br />
2 3<br />
dw x x<br />
EI = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
dx 2 6<br />
2 3<br />
dw x x<br />
EI = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
dx 2 6<br />
2 3<br />
dw x x<br />
EI = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
dx 2 6<br />
Za progib nosa�� :<br />
3 4<br />
x x<br />
( I ) EIw =<br />
−FA<br />
⋅ + q ⋅ + C1<br />
⋅ x + D<br />
6 24<br />
3 4<br />
x x<br />
EIw = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
6 24<br />
3 4<br />
x x<br />
EIw = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
6 24<br />
3 4<br />
x x<br />
EIw = −FA<br />
⋅ + q ⋅ − FC<br />
⋅<br />
6 24<br />
1<br />
C<br />
C<br />
C<br />
( x − 5)<br />
2<br />
− q ⋅<br />
− q ⋅<br />
2<br />
( x − 7)<br />
2<br />
( x − 7)<br />
2<br />
2 ( x − 5)<br />
( x − 7)<br />
2<br />
2<br />
2<br />
2<br />
+ P ⋅<br />
( x − 8)<br />
2<br />
3<br />
( x − 5)<br />
( x − 7)<br />
( x − 8)<br />
1<br />
2<br />
( x − 5)<br />
6<br />
3<br />
+ C<br />
− q ⋅<br />
− q ⋅<br />
6<br />
6<br />
3 ( x − 5)<br />
( x − 7)<br />
6<br />
2<br />
3<br />
+ C<br />
3<br />
+ P ⋅<br />
3<br />
4<br />
( x − 5)<br />
( x − 7)<br />
( x − 8)<br />
6<br />
+ C<br />
− q ⋅<br />
− q ⋅<br />
⋅ x + D<br />
24<br />
24<br />
2<br />
4<br />
+ C<br />
3<br />
+ P ⋅<br />
6<br />
2<br />
⋅ x + D<br />
3<br />
3<br />
2<br />
+ C<br />
+ C<br />
4<br />
4<br />
⋅ x + D<br />
( x − 7)<br />
4<br />
2<br />
2<br />
− P ⋅<br />
( x − 8)<br />
71
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