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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16. Zadatak<br />
Za zadano stanje naprezanja odrediti glavna normalna naprezanja, glavna posmi�na naprezanja i<br />
oktaedarska naprezanja.<br />
[ σ ] =<br />
⎢<br />
50 120 0<br />
⎥[<br />
MPa]<br />
ij<br />
⎡− 80<br />
⎢<br />
⎢⎣<br />
σ<br />
−σ<br />
0<br />
50<br />
0<br />
0 ⎤<br />
⎥<br />
60⎥⎦<br />
2<br />
[ −τ<br />
]<br />
( σ −σ<br />
) ( σ −σ<br />
)( σ −σ<br />
)<br />
z<br />
x<br />
( σ −σ<br />
)( σ −σ<br />
)<br />
σ<br />
σ<br />
2<br />
m<br />
m<br />
z<br />
x<br />
yx<br />
zx<br />
m<br />
m<br />
m<br />
m<br />
( σ + σ )<br />
x<br />
y<br />
y<br />
x<br />
y<br />
xy<br />
zy<br />
m<br />
m<br />
m<br />
m<br />
m<br />
1<br />
2<br />
y<br />
2<br />
xy<br />
( σ −σ<br />
)<br />
x<br />
z<br />
z<br />
x<br />
xz<br />
yz<br />
y<br />
m<br />
y<br />
m<br />
= 0<br />
= 0 ⇒<br />
2<br />
xy<br />
2<br />
xy<br />
′ − 80 + 120 1<br />
σ 1,<br />
2 = ±<br />
2 2<br />
2<br />
2<br />
[ ( − 80 −120)<br />
] + 4 ⋅50<br />
= 20 ± 111,<br />
8<br />
′<br />
′<br />
′<br />
σ = 131,<br />
8MPa;<br />
σ = −91,<br />
8MPa;<br />
σ = 60MPa<br />
1<br />
1<br />
2<br />
3<br />
τ<br />
τ<br />
−<br />
σ x + σ y<br />
=<br />
2<br />
σ = 60MPa<br />
τ<br />
σ −σ<br />
τ<br />
±<br />
σ = 131,<br />
8MPa<br />
σ<br />
σ = −91,<br />
8MPa<br />
2<br />
−τ<br />
+ σ σ −τ<br />
2<br />
xy<br />
+ 4τ<br />
x<br />
= 0<br />
′<br />
σ −σ<br />
= 0 ⇒ σ = σ = 60MPa<br />
= σ<br />
τ<br />
τ<br />
σ −σ<br />
= 0 → σ<br />
3<br />
σ −σ<br />
3<br />
τ<br />
yx<br />
0<br />
m<br />
m<br />
y<br />
τ<br />
xy<br />
σ −σ<br />
0<br />
m<br />
σ x + σ y ±<br />
=<br />
z<br />
0<br />
0<br />
σ −σ<br />
2<br />
2<br />
[ − ( σ + σ ) ] − 4(<br />
−τ<br />
+ σ σ )<br />
x<br />
m<br />
= 0<br />
y<br />
2<br />
xy<br />
x<br />
y<br />
21