MR. L. N. C4. FILON ON AN APPROXIMATE SOLUTION FOR BENTDING A 
6(j 
Since Lame’s time the problem has been attacked by a large number of 
mathematicians, among them de Saint-Venant, Clebsch, Boussinesq, and more 
recently M. Mathieu, M. Bibiere and Mr. J. H. Michele. Although they have 
not l)een able so far to obtain the solution of the problem as stated quite generally 
above, they have nevertheless made great progress with various particular cases, 
more especially those in udiich some of the dimensions of the block are large 
compared with the rest. 
Fuller references to their work and to the results obtained by them are given in 
the historical summary at the end of this paper. 
§ 2. Object of the Investigcition, 
The object of the present investigation is to obtain the solution for the rectangular 
parallelojiiped under an arbitrary system,of surface loading in two cases, when the 
problem reduces to one of two dimensions, namely :— 
(a) When two of the faces z — oi the bar are constrained to remain plane and 
the stress applied to the other faces is independent of 2 . In this case iv = 0, w and 
V are functions of x and y only. If the breadth 2c of the beam be sufficiently large, 
we may relinquish the constraint along the sides altogether, and we have thus the 
case of a thick plate bent in a plane perpendicular to its own plane. When the plate 
is made indefinitely thick we have two-dimensional strain in an infinite elastic solid 
with a plane boundary. 
(b) When we make the assumption that xz and yz vanish at the boundaries : = c, 
while 22 is actually zero throughout. That this will be very near the truth if c is 
very small is quite evident, so that in any case this condition will hold for a flat beam 
or girder whose height is large com^iared vutli its breadth."^ 
But it seems not improbable that it may continue to hold approximately up to 
a fairly large value of c; we may remember that de Saint-Yen ant, in his solution 
for flexure, assumes both 22 and yy to be zero, in the case where his beam is unstressed 
except at the ends, and his solution is sufficient to satisfy all conditions. Obviously 
vertical pressures and tensions across the faces y = ffi must introduce important 
stresses yy, so that that part of de Saint-Yen ant’s hypothesis, in the generalised 
^u’oblem, must go. Still it appears reasonable to suppose, on the whole, that, even 
for a beam where c and 6 are of the same order, we may, as a first approximation, 
retain’the hypothesis 22 = 0. Of course, eventually, as c increases a stress 22 must 
appear until when c is very large we reach the limiting case of problem (a) when 
this stress is sufficient to ensure the vanishing of the displacement u\ 
If, however, c be not too large, so that we can suppose 22 sensibly zero througliout, 
* September 13, 1902. I have, since writing the above, verified that a solution for rectangular beam.s 
does exist, which fulfils rigidly these conditions. It is, in fact, identical with part of Clebsch’s solution 
for a thick plate. 
