Civil Engineering Reference
In-Depth Information
The principal axis directions
y
,
z
are defined by the condition that
I
yz
=
0.
(5.60)
Thesedirectionscanbefoundbyconsideringarotationofthecentroidalaxesfrom
y
1
,
z
1
to
y
2
,
z
2
throughanangle
α
,asshowninFigure5.36a.The
y
2
,
z
2
coordinates
ofanypointarerelatedtoits
y
1
,
z
1
coordinatesasdemonstratedinFigure5.36aby
y
2
=
y
1
cos
α
+
z
1
sin
α
z
2
=−
y
1
sin
α
+
z
1
cos
α
(5.61)
The second moments of area about the
y
2
,
z
2
axes are
z
2
d
A
=
I
y
1
cos
2
α
−
2
I
y
1
z
1
sin
α
cos
α
+
I
z
1
sin
2
α
I
y
2
=
A
y
2
d
A
=
I
y
1
sin
2
α
+
2
I
y
1
z
1
sin
α
cos
α
+
I
z
1
cos
2
α
I
z
2
=
(5.62)
A
y
2
z
2
d
A
=
1
I
y
2
z
2
=
2
(
I
y
1
−
I
z
1
)
sin2
α
+
I
y
1
z
1
cos2
α
A
The principal axis condition of equation 5.60 requires
I
y
2
z
2
=
0, so that
tan2
α
=−
2
I
y
1
z
1
/(
I
y
1
−
I
z
1
)
.
(5.63)
In this topic,
y
and
z
are reserved exclusively for the principal centroidal axes.
y
1
cos
I
y
z
z
1
sin
y
2
y
1
sin
C
z
1
cos
(
I
y
1
,I
y
1
z
1
)
y
1
z
2
2
I
y
,I
z
z
1
I
y
y
2
1
z
(
y
1
, z
1
)
z
2
z
1
(
y
2
, z
2
)
(
I
y
1
,-I
y
1
z
1
)
(a) Rotation of axes
(b) Mohr's circle construction
Figure 5.36
Calculation of principal axis properties.
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