Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.149
An approximation of impact stresses in the elastic range can be made by neglecting
the inertia of the body struck and the effect of wave propagation and
assuming that the kinetic energy is converted completely into strain energy in that
body. Consider a prismatic bar subjected to an axial impact load in tension. The
energy absorbed per unit of volume when the bar is stressed to the proportional
2
limit is called the modulus of resilience. It is given by ƒ /2E, where ƒy is the yield
stress and E the modulus of elasticity, both in psi.
Below the proportional limit, the unit stress, psi, due to an axial load U, in-lb,
is
y
2UE
ƒ � � (5.260)
AL
where A is the cross-sectional area, in 2 , and L the length of bar, in. This equation
indicates that, for a given unit stress, energy absorption of a member may be improved
by increasing its length or area. Sharp changes in cross section should be
avoided, however, because of associated high stress concentrations. Also, uneven
distribution of stress in a member due to changes in section should be avoided. For
example, if part of a member is given twice the diameter of another part, the stress
under axial load in the larger portion is one-fourth that in the smaller. Since the
energy absorbed is proportional to the square of the stress, the energy taken per
unit of volume by the larger portion is therefore only one-sixteenth that absorbed
by the smaller. So despite the increase in volume caused by doubling of the diameter,
the larger portion absorbs much less energy than the smaller. Thus, energy
absorption would be larger with a uniform stress distribution throughout the length
of the member.
Impact on Short Members. If a static axial load W would produce a tensile stress
ƒ� in the bar and an elongation e�, in, then the axial stress produced in a short
member when W falls a distance h, in, is
2h
ƒ � ƒ� � ƒ� �1 � (5.261)
e�
if ƒ is within the proportional limit. The elongation due to this impact load is
2h
e � e� � e� �1 � (5.262)
e�
These equations indicate that the stress and deformation due to an energy load may
be considerably larger than those produced by the same weight applied gradually.
The same equations hold for a beam with constant cross section struck by a
weight at midspan, except that ƒ and ƒ� represent stresses at midspan and e and e�,
midspan deflections.
According to Eqs. (5.261) and (5.262), a sudden load (h � 0) causes twice the
stress and twice the deflection as the same load applied gradually.
Impact on Long Members. For very long members, the effect of wave propagation
should be taken into account. Impact is not transmitted instantly to all parts of
the struck body. At first, remote parts remain undisturbed, while particles struck
accelerate rapidly to the velocity of the colliding body. The deformations produced