Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

SolutioN
Centroid location and neutral-axis location for flanged shape
Shape
b
(mm)
d
(mm)
A i
(mm 2 )
r i
(mm)
r o
(mm)
r ci
(mm)
r ci A i
(mm 3 )
⎛r
⎞
b ln⎜
⎟
⎝ r ⎠
o i
(mm)
Inner flange (1) 125 25 3,125 150 175 162.5 507,812.5 19.268835
Web (2) 25 75 1,875 175 250 212.5 398,437.5 8.916874
Outer flange (3) 50 25 1,250 250 275 262.5 328,125.0 4.765509
6,250 1,234,375.0 32.951218
r
r
c
n
3
1, 234,375.0 mm
= = 197.500 mm (from centerofcurvature to centroid)
2
6,250 mm
A
=
⎛ ro
⎞
Σ b ln
⎝
⎜
r ⎠
⎟
i
2
6,250 mm
= = 189.674 mm
32.951218 mm
B C
M
F
A
r c
r i
a
O
P
internal force F and internal moment M at Section A–B
Draw a free-body diagram that cuts through the clamp at Section A–B. We will assume
that a positive (i.e., tension) internal force F acts at the centroid of the cross section. We
will also assume that a positive internal moment M exists at the section. Note that a positive
internal moment creates compression on the inner surface of the curved bar.
We sum forces in the vertical direction to find that the internal force F equals the
clamp force P:
+↑ Σ F = P − F = 0
∴ F =
P
Next, we sum moments about point C, which is located at the centroid of the cross
section:
Σ M = P( r + a) + M = 0
C
c
∴ M = − P( r + a) =− P(197.500 mm + 20 mm) = −P(217.5 mm)
c
Note that moments in configurations such as this are always calculated as the product of
a force and the perpendicular distance between the line of action of the force and the
centroid of the cross section—not the neutral-axis location.
Axial and Bending Stresses
The internal axial force F will create a tensile normal stress σ = F/A that is uniformly
distributed on Section A–B. The bending stress is determined from Equation (8.28). Thus,
the combined axial and bending stress on Section A–B is expressed as
σ =
F
A
Mr ( n − r)
−
rAr ( − r )
c
n
P
= +
A
P(217.5 mm)( r − r)
n
rAr ( − r )
c
n
P ⎡ (217.5 mm)( rn
− r)
⎤
= 1
A
⎢ +
⎣ rr ( c − rn)
⎥
⎦
where r is the distance from the center of curvature to the location where the stress is to
be determined.
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