Building Design and Construction Handbook - Merritt - Ventech!

5.150 SECTION FIVE
move through the struck body in the form of elastic waves. The waves travel with
a constant velocity, ft/s,
where E � modulus of elasticity, psi
p � density of the struck body, lb/ft 3
E
c � 68.1 (5.263)
��
If an impact imparts a velocity v, ft/s, to the particles at one end of a prismatic
bar, the stress, psi, at that end is
v
ƒ � E � 0.0147v �Ep � 0.000216pcv (5.264)
c
if ƒ is in the elastic range. In a compression wave, the velocity of the particles is
in the direction of the wave. In a tension wave, the velocity of the particles is in
the direction opposite the wave.
In the plastic range, Eqs. (6.263) and (5.264) hold, but with E as the tangent
modulus of elasticity. Hence, c is not a constant and the shape of the stress wave
changes as it moves. The elastic portion of the stress wave moves faster than the
wave in the plastic range. Where they overlap, the stress and irrecoverable strain
are constant.
(The impact theory is based on an assumption difficult to realize in practice—
that contact takes place simultaneously over the entire end of the bar.)
At the free end of a bar, a compressive stress wave is reflected as an equal
tension wave, and a tension wave as an equal compression wave. The velocity of
the particles there equals 2v.
At a fixed end of a bar, a stress wave is reflected unchanged. The velocity of
the particles there is zero, but the stress is doubled, because of the superposition
of the two equal stresses on reflection.
For a bar with a fixed end struck at the other end by a moving mass weighing
W m lb, the initial compressive stress, psi, is
ƒ � 0.0147v �Ep (5.265)
o o
where v o is the initial velocity of the particles, ft/s, at the impacted end of the bar
and E and p, the modulus of elasticity, psi, and density, lb/ft 3 , of the bar. As the
velocity of W m decreases, so does the pressure on the bar. Hence, decreasing compressive
stresses follow the wave front. At any time t � 2L/c, where L is the length
of the bar, in, the stress at the struck end is
�2�t / �
ƒ � ƒoe (5.266)
where e � 2.71828, � is the ratio of W b, the weight of the bar, to W m, and � �
2L/c.
When t � �, the wave front with stress ƒ o arrives back at the struck end, assumed
still to be in contact with the mass. Since the velocity of the mass cannot change
suddenly, the wave will be reflected as from a fixed end. During the second interval,
� � t � 2�, the compressive stress is the sum of two waves moving away from the
struck end and one moving toward this end.
Maximum stress from impact occurs at the fixed end. For � greater than 0.2,
this stress is