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

This strain-energy density is represented by area OAB of the stress–strain
diagram. The strain-energy density of the 25 mm diameter segment is roughly
one-fourth the strain-energy density of the 18 mm diameter segment. When the
25 mm diameter is reduced to 18 mm, the volume of segment (2) is roughly
halved. However, the strain energy absorbed by each unit volume of the remaining
material is roughly quadrupled (i.e., area OCD compared with area OAB),
resulting in a net gain in energy-absorbing capacity. For the rod considered here,
this gain amounts to about a 40 percent increase in the allowable collar mass for
the constant-diameter rod.
σ
240 MPa
124.4 MPa
O
A
B
1,777 με
C
D
3,429 με
ε
ExAmpLE 17.8
The cantilever post AB consists of a steel pipe that has an outside diameter of
33 mm and a wall thickness of 3 mm. A 30 kg block moving horizontally with
a velocity v 0 strikes the post squarely at B. What is the maximum velocity v 0 for
which the largest normal stress in the post does not exceed 190 MPa? Assume that
E = 200 GPa for the steel pipe.
Plan the Solution
Calculate the maximum dynamic moment from the allowable normal stress and
the section properties of the post. Then determine the maximum allowable
dynamic load and the corresponding horizontal deflection of the post at B. Use
conservation of energy to equate the work done on the post to the kinetic energy
of the block, and solve for the maximum velocity v 0 .
850 mm
B
A
v 0
30 kg
SolutioN
The moment of inertia of the pipe is
I
π = [ (33 mm) − (27 mm) ] = 32,127.7 mm
64
4 4 4
The maximum dynamic moment that can be applied to the post at A without exceeding
the 190 MPa limit is
M
max
2 4
σ maxI
(190 N/mm )(32,126.7 mm )
= = = 369,944 Nmm ⋅
c
33 mm/2
Since the cantilever post has a span of 850 mm, the maximum allowable dynamic
load is
P
max =
369,944 Nmm ⋅
850 mm
= 435.2 N
From Appendix C, the maximum horizontal deflection of the post at B can be calculated
as
v
max
PmaxL
(435.2 N)(850 mm)
= 3 = 3
= 13.865 mm
2 4
3EI 3(200,000 N/mm )(32,127.7 mm )
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