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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