Environmental Engineering Reference
In-Depth Information
The.shear.modulus.with.the.index.s.can.be.designated.for.the.skin.layers;.
the.shear.modulus.with.index.h.can.be.designated.for.the.sandwich.layers.
The.functional.relationship.between.Cartesian.and.polar.coordinates.can.
be.shown
.
in.Equation.(5.19).
2
2
2
x = r cos ; y = r sin ; r = x + y ;
θ
θ
.
..(5.19)
dr
dx
x
r
;
dr
dy
y
r
d
dx
θ
y
r
2
sin
θ
d
dy
θ
x
r
cos
θ
=
=
cos
θ
=
=
sin ;
θ
=
=
;
=
=
r
2
r
If.we.replace.x.and.y.in.Equation.(5.19),.the.contour.of.the.aviation.proile.
can.be.shown.as.the.following.equation:
..
Ψ=.λr
2
cos
2
θ.-.k(r.sinθ-.α).(r.sinθ-.β).=.0;
The.stress.function.Φ.from.Equation.(5.14).can.be.shown.in.polar.coordi-
nates.as:
Φ=.M(λr
2
cos
2
θ.-.k(r.sinθ-.α).(r.sinθ-.β)..
The. complementary. equation. between. Cartesian. and. polar. coordinates,.
Equation.(5.20),.would.follow:
6
2
2
2
2
d
dr
Φ
1
d
dr
Φ
1
d
d
d
dx
Φ
d
dy
Φ
.
+
+
=
+
.
(5.20)
2
2
2
2
2
r
r
θ
The.left.side.of.the.complementary.Equation.(5.20).represents.the.normal.
radial.stresses.acting.in.the.r.direction,.while.the.outside.forces.represent.a.
combination.of.tension.forces.Fz,.Fy,.and.bending.moments.M
z
r
.and.M
y
r
.
From.the.integrating.stress.function.Φ.=.MΨ.=.M[λx
2
.-k(y-α).(y-β)],.which.
is.relative.to.x.and.y,.we.get.Equation.(5.21).
d
dx
2
Φ
)] ;
d
dy
2
Φ
2
= M[2 - k(y -
λ
α
)(y -
β
=
M[ x - 2k + k( +
λ
β α αβ
-
)]
.
.
(5.21)
2
2
Therefore,. by. adding. this. function,. we. can. ind. coeficient. M. (see. Equation.
(5.22)).
M[2 - k(y -
λ
α
)(y -
β
)] + [ x2 - 2k + k( +
λ
β
α αβ
-
)] = Fz + Fy + Mz + My
Fz + Fy +
Mz + My
.
.
(5.22).
M =
[2 - k(y -
λ
α
)(y -
β
)] + [ x2 - 2k
λ
+ k( +
β α αβ
-
)]
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