Problem 9.9
Given:
Curved beam is built up by welding together rectangular and elliptical cross section curved beams.
The center of curvature is located 20 mm from B.
The curved beam is subjected to a positive bending
moment M
x
.
Find:
the the stress at B and C in terms of M
x
.
Solution:
For the ellipse
b
1
7.5 mm
⋅
:=
h
1
15 mm
⋅
:=
A
1
π
h
1
⋅
b
1
⋅
353.429 mm
2
=
:=
R
1
50 mm
⋅
:=
A
m1
2
π
⋅
b
1
⋅
h
1
R
1
R
1
2
h
1
2
−
−
⎛
⎝
⎞
⎠
⋅
7.235 mm
=
:=
For the rectangle
b
2
30 mm
⋅
:=
a
2
20 mm
⋅
:=
c
2
35 mm
⋅
:=
A
2
b
2
c
2
a
2
−
()
⋅
450 mm
2
=
:=
R
2
a
2
c
2
+
2
27.5 mm
=
:=
A
m2
b
2
ln
c
2
a
2
⎛
⎜
⎝
⎞
⎟
⎠
⋅
16.788 mm
=
:=
Total section
AA
1
A
2
+
803.429 mm
2
=
:=
A
m
A
m1
A
m2
+
24.024 mm
=
:=
R
R
1
A
1
⋅
R
2
A
2
⋅
+
A
37.398 mm
=
:=
N'
0
:=
Let
M
x
1N
⋅
m
⋅
:=
At point B
r2
0
m
m
⋅
:=
σ
θθ
N'
A
M
x
Ar
A
m
⋅
−
()
⋅
Ar
⋅
RA
m
⋅
A
−
()
⋅
+
0.212 MPa
=
:=
or
σ
θθ
0.212 M
x
⋅
=
MPa
At point C
r6
5
m
m
⋅
:=
σ
θθ
N'
A
M
x
Ar
A
m
⋅
−
()
⋅
Ar
⋅
RA
m
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 Spring '10
 MechanicsofSolids
 mechanics, Stress, Second moment of area, New York City Subway, Conic section

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