Environmental Engineering Reference
In-Depth Information
We. already. know. the. stress. function. Φ. in. polar. coordinates. (Equation.
(5.14)),.so.we.can.ind.all.the.components.necessary.to.determine.the.normal.
and.shear.stresses,.Equation.(5.23).
d
dr
Φ
2
=
2
M r
λ
cos
θ
- k
(sin
θ α
-
)(sin
θ β
-
)
;
2
d
dr
Φ
2
=
2
M
λ
cos
θ
- k
(sin
θ α
-
)(sin
θ β
) ;
-
2
.
.
(5.23)
d
d
Φ
θ
[
]
=
2
M r
2
λ
cos
θ
- k
(sin
θ α
-
)(sin
θ
-
β
) ;
2
d
dr
Φ
[
]
=
2
M -
2
λ
r
sin
θ
+
(sin
k
θ α
-
)(
sin
θ β
-
) ;
2
Now,.for.the.normal.and.shear.stresses.we.ind.(Equation.(5.24)):
.
σ
r
.=.2M(λcos
2
θ-.2λ/r.sinθ-.ρω
2
r/3cosθ)
.
σ
θ
=.2M[λcos
2
θ-.k(sinθ-.α)(sinθ-.β)].-.ρω
2
r
3
/3sinθ.
(5.24)
.
τ
rθ
=.2M[2λcosθ-.k/r(.cosθ.-.α)(.cosθ-.β)]
In. the. case. of. torsion. rotor. blades,. the. equation. of. compatibility. can. be.
shown.as.Equation.(5.25).
d
dy
τ
-
d
dx
τ
xz
xz
.
=
(
-2Gxz - 2Gyz
)
.
(5.25)
Here,.the.shear.stresses.are.represented.as.Equation.(5.26).
d
dy
Φ
d
dx
Φ
.
.
τ
=
;
τ
=
(5.26).
xz
yz
The.stress.function.is.represented.in.Equation.(5.14).
In.this.case,.the.necessary.components.look.like:
d
dy
Φ
2
.
=
M [ x - k(2y -
λ
β α αβ
.
-
+
)]
(5.27)
1
By.substituting.Equation.(5.27),.we.ind.coeficient.M
1
:
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