53 
4 
140, The solution ‘has also been given for an elliptically shaped panel of 
plating, In this oase, if the mjor and minor axes of the ellipse are taken 
as oo-ordinate axes Ox, Oy, it is found that the principal stresses in the 
plane of the plate at any point are 6x and 6y parallel to the axes, Both 
6x and 6y are separately constant over the panel, but are not in general 
equal to one another, their ratio depending on the ratio of the minor and 
major axes of the ellipse. However, although strictly 6x %6,, if it be 
assumed as an approximation that the elliptical panel behaves as a soap 
bubble with 6x= 64 =8,, it can be shown that the energy absorbed by the 
plate according to this approximation differs by only a small fraction from 
the energy calculated by the more exact solution. Using equation 51 this 
error varies from 0 to &% as the ellipse varies between the extremes cases 
of a circle and an infinitely long panel, whilst if equation 52 is used, the 
corresponding error varies from 0 to 15% 
141.  Yhis result for the elliptical panel illustrates a result which can be 
shown to be generally independent of the shape of the panel. Thus, for any 
element of plating subjected to a two-dimensional principal stress system 
» apd 62 producing principal strains e,, e, in the plane of the plate, 
the energy U absorbed per unit volume by plastic stretching oan be 
evaluated by using either equation 51 or equation 52 together with 
equations 54 and 55 and these values of U oan be compared with the soap 
bubble approximation 6, = 6, = S,which gives 
U =B,(e, + e2) eee cece eco eee coe (57) 
Fig. 26 shows the resulting 
comparison for different ratios 
2/6, of the two tensile strains 
IE | | in the plane of the plate (The 
ie remaining principal strain 
EQUATION 52 63 ==— 4, — 6,corresponds to 
ee thinning of the plate). Fora 
== EQUATION 51 given state of strain, not only 
does the soap bubble approximation 
give an energy absorption 
ne Ua differing by relatively small 
EQUATION (56 | amounts from that deduced from 
equations 54 or 52 but it also 
givea an intermediate estimate. 
o8 
oO6 
2S80e, 
142. The assumtion that the 
plate behaves as a soap bubble 
Pig. 26 - Energy absorbed in plastic 
stre tohing of thin plate 
exerting uniform tension g, in 
all directions thus appears to be 
a reasonable averages approximation 
to equations 54 and 52. In 
particular, the practically 
important case of a reotangular 
panel, which has yet to be solved 
by using equation 51 or, equation 
52 oan be easily solved by 
LOAD 
using the soap bubble approxi- DEFLECTION 
mation. Provided the central 
deflection is not excessive, 
that is less than one sixth of Fig, 27 - Plastic streto of panel b 
Lateral load 
