55 
it progresses. A second result for the plastic wave in a wire is that no 
plastic wave can transmit a particle velocity greater than a certain 
critical value; thus if a long wire is hung from a fixed point anda 
weight is dropped on to a yoks at the lower end of the wire it is found 
that if the weight is dropped from above a certain height there is little or 
no plastic strain in the wire except very close to the yoke where it would 
break, This result is more especially relevant to the question of rupture 
discussed later. 
146. For lateral loads on beams, one case which has been solved, inoluding 
strain hardening effects, is that of an infinitely long beam subjected to 
a concentrated dynamic load at one seotion. The more practical case of a 
beam of finite length presents greater difficulties. However, using the 
preceding simplified theory that the beam yields under a constant bending 
moment, it is possible to solve some simple special cases. Consider, 
for example, a simply supported beam given an initial triangular impulsive 
distribution of velocity which is zero at each end and rises linearly to a 
maximum at the centre. In this case the deflection at any tim is also of 
triangular shape, that is, of the shape given in fig.25, and all the 
initial kinetio energy is ultimately absorbed by plastic bending at the 
centre. For a given amount of energy commmnicated as initial velocity 
in this particular way, the maximum central deflection of the beam can 
easily be calculated by equating this given energy to the energy absorbed 
in central bending, which is simply Go times the angle of bend at the centre; 
this angle is thus determined and hence the central deflection can be 
calculated. 
147. For a panel of plating dished plastically by lateral loads, it has 
been seen that the assumption of a uniform tension in all directions in ths 
plane of the plate is a reasonable approximation for static loads. This 
approximation can also be applied to solve certain special cases of dynamic 
loading. In particular, if 
a circular panel of plating 
fixed round its circumference 
is given an initial distri- 
bution of uniform velocity 
corresponding to a very large 
uniform pressure acting for a 
negligibly short tims, then 
the cross section of the 
deformed panel at any time t 
is shown in fige29. Here 
the outer annulus AB, A'Bt 
(in section) has been 
deformed into the cone 
frustrum AC, A'C', (in 
section) which is at rest 
whilst the central portion 
BB' has moved to CC- and is 
still undeformed and moving 
with its initial velocity v. 
Fig.29 - Section through deformed circular At a later tim the deformed 
panel shape is similarly ADD‘ a’ 
where DD' is undeformed and 
moving with velocity v whilst AD, A'D' is stretched and at rest. 
148. Effectively, a plastic wave travels inwards from the fixed edge 
leaving behind it a stretched portion at rest whilst the central portion 
ahead of it is undeforme This plastic wave travels in fact witha 
constant velocity 2, See where Sp is a yield stress as defined 
previously and © is mass density. For steel, c, is about 570 ft. per seo. 
and is therefore not only small compared with the velocity of elastic 
waves in steel (about 17000 ft. per sec.) but also in comparison with the 
velocity of the pressure pulse in water (about 5000 ft. per sec.).* 
a This result is relevant to the question of whether cavitation in the 
water occurs with a finite panel of plating, 
