SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA

45. Zadatak
Za zadani nosa� i optere�enje odrediti maksimalna normalna i posmi�na naprezanja u visini osi
“z“te smjer i veli�inu glavnih naprezanja u to�ki 3 popre�nog presjeka nad ležajem B.
Položaj težišta popre�nog presjeka nosa�a :
⎛10 ⋅ 60 ⎞
20 ⋅ 60 ⋅ 70 + 2⎜
⎟ ⋅ 40 + 10 ⋅ 60 ⋅30
∑ Ai
yi
2
126000
yT
= =
⎝ ⎠
= = 52,
5cm
∑ Ai
⎛10 ⋅ 60 ⎞
2400
20 ⋅ 60 + 2⎜
⎟ + 10 ⋅ 60
⎝ 2 ⎠
T ( x , y ) = T
T
T
( 30;
52,
5)
Moment površine drugog reda :
I
I
I
z
z
z
= I
z1
+ A y
60 ⋅ 20
=
12
1
3
2
1
+ 2
4
2 ( I + A y )
z2
+ 60 ⋅ 20 ⋅
4
= 110,
5⋅10
cm
Reakcije i unutarnje sile :
∑ F
F
= 0 → F
+ F
= 2F
+ qb − F
2
2
+ I
z3
2 ( 70 − 52,
5)
+ 2⎜
+ ⋅ 60 − 52,
5 ⎟ + + 10 ⋅ 60 ⋅ ( 30 − 52,
5)
8
= 110,
5⋅10
mm
− 2F
− q ⋅b
= 0
4
+ A
3
y
2
3
⎛10
⋅ 60
⎜
⎝
36
b
∑
M B = 0 → FA
⋅3a
− F ⋅ 2a
− F ⋅ a + q ⋅b
⋅ = 0
2
2
b
3Fa
− q
F
2
A =
3a
2
3
3⋅100
⋅ 2 − 40
=
2
3⋅
2
→ FA
= 70kN
B
y
A
A
B
= 2 ⋅100
+ 40 ⋅3
− 70 → F
B
3
10 ⋅ 60
2
= 250kN
⎛ 2
⎜
⎝ 3
⎞
⎟
⎠
2
⎞
⎟
⎠
10 ⋅ 60
12
3
67
2