51 
436. A third basic assumption is that under plastic yielding the steel is 
incompressible, that is, there is no change in volum of the steel so that 
to the first order in strains 
6, + Sa + 63z=0 eee eee coe coe (55) 
This assumption is reasonably accurate. Equations 54, 55 and either 
equations 51 or 52 give three basic equations for the simple theory of 
plastic yielding in steel. Together with the three equations of 
equilibrium in a statio problem or the three equations of motion in a 
dynamic problem a total of six equations is thus obtained to determine the 
six unknowns G,, 6; 5 S35 %,5 5, z- ‘In addition, there are the usual 
relations depending only on geometry which conneot the strains 6,, 62, 63, 
With displacements. These latter can be introduced as convenient in any 
specific problem if the equations are thereby simplified, The hull 
structure of a ship may be considered broadly as composed of two simple 
units, namely, a beam and a panel of plating. To illustrate the 
implications of the preceding theory of plestioc yielding, two relatively 
simple statio problems will first be considered, 
Plastic bending of a beam 
137. Pig. 2) represents a beam of symmetrical seotion bent in a plane of 
symmetry. Assum, as in the simple theory of elastic bending, that plane 
sections remain plane and that the beam may be considered as composed of 
fibres each of which is 
stressed either in simple 
tension or simple compression, 
Srey In this case, either equation 
51 or equation 52 gives the 
result that the fibre stress is 
So for a fibre in tension or - 
&q for a fibre in compression. 
Henos, neglecting any elastic 
region, the stress is 65 
everywhere above the neutral 
Leet “NEUTRAL axis and-€, below this axis, 
Aas For pure bending with no re- 
Sultaot axial foros the area 
above the neutral axis must be 
equal to the area of the cross— 
section below this axis; this 
determines ths position of the 
neutral axis for any given shape 
of cross section bent plastio-— 
ally. Having fixed this axis, 
the resultant moment G due to 
Fig, 2) - Symmetrical seotion of beam the fibre stresses can be shown= 
to depend only on Bp and the 
Shape of the cross-seotion:G will be independent of the amount of bending, 
Thus, on this simple theory which neglects elastic effects, the plastic 
resisting moment for any given beam is a constant G. 
138. Consider in particular that the beam is simply supported on a span 
of length 1 and subjeoted to a central concentrated load W. The bending 
moment in the beam is then W1/4 at the centre decreasing linearly to zero 
at each end, Henos, if W is increased steadily from zero, then while 
w< 4G,/1 the bending moment is everywhere less than the value Gp necessary 
m= The resultant bending momemt G is given by:- 
d, ° 
Ge 8,b(x) x dx - Bob(x) x dx 
° -d, 
where the symbols have the meaning indicated in fig. 2) 
