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

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ENERgy METHOdS
Significance: In the two cases just described, impact forces are imparted to an elastic spring
by a freely falling weight and by a weight in motion. For these investigations, it has been
assumed that the materials behave elastically and that no dissipation of energy (in the form
of heat or sound or permanent deformation) takes place at the point of impact. The inertia of
the resisting system has been neglected, and perfect rigidity of the block was implicit.
At first glance, the behavior of a spring system may not seem particularly useful or
relevant. After all, mechanics of materials is the study of deformable solid materials such
as axial members, shafts, and beams. However, the elastic behavior of these types of components
is conceptually equivalent to the behavior of a spring. Therefore, the two models
discussed previously can be seen as general cases that are widely applicable to common
engineering components.
In this investigation, the deflection of a system is directly proportional to the magnitude
of the applied force, regardless of whether that force is statically or dynamically
applied. The block-and-spring models analyzed here show that the maximum dynamic
response of a deformable solid can be determined from the product of its static response
and an appropriate impact factor. Once the maximum deflection D max due to impact has
been determined, the maximum dynamic force can be found from F max = kD max . The
maximum dynamic load F max can be considered an equivalent static load (a slowly applied
load that will produce the same maximum deflection as the dynamic load). Provided that
the assumption of identical material behavior under both static and dynamic loads is valid
(a valid assumption for most mechanical-type loadings), the stress–strain diagram for any
point in the loaded system does not change. Consequently, the stress and strain distributions
produced by the equivalent static load will be the same as those produced by the
dynamic load.
It is also significant that we have assumed no dissipation of energy during impact.
There will always be some energy dissipation in the form of sound, heat, local deformations,
and permanent distortions. Because of dissipation, less energy must be stored by the elastic
system, and therefore, the actual maximum deflection due to impact loads is reduced. Given
the assumptions delineated here, the actual impact factor will have a value somewhat less
than that predicted by Equations (17.23) and (17.26); thus, the equivalent static load
approach will be conservative. All in all, the equivalent static load approach provides the
engineer a conservative, rational analysis of the stresses and strains produced by impact
loading, using only the familiar equations found in mechanics of materials theory.
ExAmpLE 17.5
750 mm
B
1,200 N collar
30 mm
A
The 1,200 N collar shown is released from rest and slides without friction downward a
distance of 30 mm, where it strikes a head fixed to the end of the rod. The AISI 1020
cold-rolled steel [E = 200 GPa] rod has a diameter of 15 mm and a length of 750 mm.
Determine
(a) the axial deformation and the normal stress in the rod under static conditions; that is,
the collar is gradually lowered until it contacts the rod head and comes to rest without
impact.
(b) the maximum axial deformation of the rod if the collar is dropped from the height of
30 mm.
(c) the maximum dynamic force exerted on the rod by the collar.
(d) the maximum normal stress in the rod due to the dynamic force.
(e) the impact factor n.