BEAM OF EECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 83 
strain in which cross-sections originally plane remain plane, and the curvature of the 
elastic line is at all points j)roportional to the bending moment. 
(n.) The part dx of U. This corresponds to the lateral contraction of the 
material under tensions Q, and is the same as if each strip of thickness clx and height 
26 were independently stretched. 
(iii.) The terms — ^ S 
[ o\' 4/x y" 
]_ 6/z (X 4- /^) h " 
+ 4/^ ] 
10/A (X /a) J 
of U and 
^ J 13Xf -f- 16/a ^ X' ?/2-i 
86 [ 10/A (X^ -j- /a) 2/a (X^ /a) ^ 
These correspond to a distortion of the cross-sections and to a parabolic distribution 
of shear. 
In the particular case, where the load reduces to a central isolated weio-ht W and 
the two symmetrical support reactions, the additional terms (iii.) in V are of the 
form (omitting the constant) 
g 'Wx [ loX' + 16/a] 
^~yLb [20 (V-f /a)J 
4 - 4 
r 8 
7jW (J — x) ?/" 
E63 
for > 0 
and 
g W.r fl3X' + 16/a] 
^ /a 6 [20 (V -f /a)] 
+ 
^ E63 
for X <. 0, 
2 Z being the distance between the suj^ports. 
It might have been supposed that this particular problem would have been 
capable of solution by breaking up the beam in the middle and treating it as two 
inverted cantilevers, to each of which we could apjjly de Saint-Venant’s solution. 
This, I believe, is often done by engineers. 
Now such an attempt is, in strictness, bound to fail, because de Saint-Venant’s 
solution implies distortion of the cross-section at the fixed end, whereas in the 
present problem the central cross-section of the beam must necessarily remain 
plane, from symmetry. 
Moreover, we are left in doubt as to the condition of fixing to be adopted. Are 
we to suppose, with de Saint-Venant, the central element of the terminal cross- 
section to remain vertical, or, with Professor Love (‘Theory of Elasticity,’ vol. 1, 
pp. 179-180), the elastic line to he horizontal at the built-in end ? In the case of a 
cantilever the difference is quite immaterial, as it merely amounts to a rigid body 
displacement. But here w'e must remember the cantilevers are only fictitiously 
severed, and the above difference corresponds to an actual sharp bend of the beam 
in the middle. 
It is interesting to compare the true solution with those obtained in this way. 
M 2 
