SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA
SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA
SKRIPTA RIJEÅ ENIH ZADATAKA IZ OTPORNOSTI MATERIJALA
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20. Zadatak<br />
U to�ki A napregnutog elementa izmjerene su dužinske deformacije u smjerovima x, y i n �xx = -<br />
8·10 -4 , �yy = 4·10 -4 , �nn = 6·10 -4 . Odrediti promjene pravog kuta izme�u osi x i y, te veli�inu i<br />
smjerove glavnih deformacija.<br />
ε<br />
nn<br />
⇒ ε<br />
ε<br />
γ<br />
xy<br />
xy<br />
= ε<br />
xy<br />
xx<br />
= 2ε<br />
xy<br />
cos<br />
2<br />
1<br />
=<br />
sin 2ϕ<br />
= 12,<br />
7 ⋅10<br />
ϕ + ε<br />
−4<br />
2<br />
2 1<br />
−4<br />
−4<br />
−4<br />
( ε − ε cos ϕ − ε sin ϕ)<br />
= ( 6 ⋅10<br />
+ 8⋅10<br />
cos30°<br />
− 4 ⋅10<br />
sin 30°<br />
)<br />
nn<br />
yy<br />
= 25,<br />
4 ⋅10<br />
Glavne deformacije :<br />
1,<br />
2<br />
1,<br />
2<br />
1<br />
2<br />
ε + ε 1<br />
2 2<br />
⎛ − 8 + 4 1<br />
= ⎜ ±<br />
⎝ 2 2<br />
−4<br />
−4<br />
−4<br />
sin<br />
−4<br />
xx<br />
2<br />
rad<br />
ϕ + ε<br />
( ε − ε )<br />
xy<br />
sin 2ϕ<br />
⇒<br />
xx yy<br />
2 2<br />
ε1, 2 = ± xx yy + 4ε<br />
xy<br />
ε<br />
ε<br />
= −2<br />
⋅10<br />
ε = 12 ⋅10<br />
ε = −16<br />
⋅10<br />
Kontrola :<br />
+ ε = ε + ε<br />
ε xx yy<br />
− 8⋅10<br />
− 4 ⋅10<br />
−4<br />
−4<br />
1<br />
+ 4 ⋅10<br />
± 14 ⋅10<br />
2<br />
−4<br />
= − 4 ⋅10<br />
−4<br />
yy<br />
2 ( − 8 − 4)<br />
+ 4(<br />
12,<br />
7)<br />
−4<br />
= 12 ⋅10<br />
−4<br />
Smjerovi glavnih deformacija :<br />
tgϕ<br />
tgϕ<br />
ϕ<br />
01<br />
01<br />
02<br />
ε xy<br />
=<br />
ε − ε<br />
=<br />
+ ϕ<br />
02<br />
1<br />
2<br />
ε<br />
xy<br />
yy<br />
ε − ε<br />
yy<br />
−16<br />
⋅10<br />
−4<br />
12,<br />
7 ⋅10<br />
= −4<br />
12 ⋅10<br />
− 4 ⋅10<br />
−4<br />
−4<br />
12,<br />
7 ⋅10<br />
=<br />
−4<br />
−16<br />
⋅10<br />
− 4 ⋅10<br />
−4<br />
= 57,<br />
8°<br />
+ 32,<br />
4°<br />
= 90,<br />
2°<br />
≈ 90°<br />
2<br />
⎞<br />
⎟ ⋅10<br />
⎠<br />
−4<br />
sin 60°<br />
= 1,<br />
588 ⇒ ϕ = 57,<br />
8°<br />
−4<br />
01<br />
= −0,<br />
635 ⇒ ϕ<br />
02<br />
= −32,<br />
4°<br />
25