Graphics Programs Reference
In-Depth Information
islarger than the functionat the stationary point, leading to the conclusion that the
global minimum occurs at
x
=
0
.
273 494.
EXAMPLE 10.2
c
H
_
y
C
d
x
a
b
b
B
The trapezoid shown is the cross section of abeam. It isformedbyremoving the top
fromatriangle of base
B
60 mm. The problemistofind the
height
y
of the trapezoid that maximizes the sectionmodulus
=
48 mm and height
H
=
S
=
I
x
/
c
where
I
x
is the second moment of the cross-sectional areaabout the axisthat passes
through the centroid
C
of the cross section. By optimizing the sectionmodulus,
we minimize the maximum bending stress
σ
max
=
M
/
S
in the beam,
M
being the
bending moment.
Solution
Considering the area of the trapezoid as a composite of arectangle and
twotriangles, we find the sectionmodulusthrough the following sequence of
computations:
Base of rectangle
a
=
B
(
H
−
y
)
/
H
Base of triangle
b
=
(
B
−
a
)
/
2
Area
A
=
(
B
+
a
)
y
/
2
First momentofareaabout
x
-axis
Q
x
=
(
ay
)
y
/
2
+
2
(
by
/
2
)
y
/
3
Location of centroid
d
=
Q
x
/
A
Distance involvedin
S
c
=
y
−
d
2
by
3
12
ay
3
Second momentofareaabout
x
-axis
I
x
=
/
3
+
/
Ad
2
Parallel axis theorem
I
x
=
I
x
−
Sectionmodulus
S
=
I
x
/
c
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