Environmental Engineering Reference
In-Depth Information
Here:. R,. θ,. Z. are. the. current. polar. coordinates;. ε,ϒx,ϒ
R
,ϒ
θ
. are. the. current.
deformation.existing.from.the.acting.forces.Fz,.Fy,.Fx.and.bending.moments.
Mz,.My,.and.torsion.moment.Mx..Q
ki
.is.the.coeficient.of.stiffness.in.the.polar.
coordinates.and.Q
ki.
=.Q
ik
.(k,.I.=.ε,.x,.R,.θ).
This.system.follows.the.theory.of.Kerhgofa-Klebsha.and.has.been.trans-
formed.on.ive.independent.relationships.represented.in:
R
R
Fz
E S
Fy
E S
My
E I
X
G T
M
.
ε
z
=
;
ε
y
=
;
=
;
=
.
R
θ
1 1
2
2
1 R
xz x
Here:
E
1
,.E
2.
are.the.modulus.of.elasticity;
S
1
,.S
2.
are.the.sections.area.;
I
z
,.I
R
.are.the.moments.of.inertia.for.axes.x,.R;
G
xz
.is.a.shear.module;
T
x
.is.a.geometrical.stiffness.for.torsion;
T
x
.is.a.moment.of.inertia.for.axes.x.
These.parameters.can.be.determined.in:
.
I
z
.=.⌠y
2
dS;.I
R.
=.⌠z
2
dS;.
The.geometrical.stiffness.for.torsion.can.be.determined.as:
. T
x.
=.⌠
s
(R
2
θdS
If. we. assume. that. the. stress. components. in. the. top. section. can. be. found.
using.Equations.designated.in.the.polar.coordinates,.we.get:
2
1
r
d
dr
Φ
1
r
d
d
Φ
σ
=
×
+
×
+
Z(r,
θ
)
r
2
2
θ
2
d
dr
Φ
σ
θ
=
+
Y(r,
θ
)
.
.
(5.12)
2
1
r
d
d
Φ
-
1
r
d
2
Φ
d
dr
1
d
d
Φ
θ
τ
=
×
×
=
×
r
θ
2
θ
2
r
drd
Here:.Φ.is.the.stress.function.and.a.function.of.variables.r.and.θ.
We.also.assume.that.Z(r,θ).=.Y(r,θ).=.-⌠Rdr.where.R.is.a.body.force.R
r.
=.Fz.
and.R
θ.
=.Fy;.therefore,.Equation.(5.12).will.result.in.the.following:
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