ee ce ne ll cl 
EQUILIBRIUM OF ELASTIC SOLIDS. 115 
the amount of sliding is determined by the linear elasticity of the substance. The 
deflection of the beam thus arises partly from the bending of the whole beam, and 
partly from the sliding of the planks; and since each of these deflections is small 
compared with the length of the beam, the total deflection will be the sum of the 
deflections due to bending and sliding. 
Let A=Mc=Efzy?dy. . (65.) 
A is the stiffness of the beam as found in Case V., the equation of the trans- 
verse section being expressed in terms of 2 and y, y being measured from the 
neutral surface. 
Let a horizontal beam, whose length is 2 7, and whose weight is 2 7, be sup- 
ported at the extremities and loaded at the middle with a weight W. 
Let the deflection at any point be expressed by 0, y, and let this quantity be 
small compared with the length of the beam. 
At the middle of the beam, 0, y is found by the usual methods to be 
eg se a 
8 9=% (qe? +ZPW) See i(BBRY 
Let Bos rdy= (sectional area). . - (67.) 
B is the resistance of the beam to the sliding of the planks. The deflection 
of the beam arising from this cause is 
iy =a qwt+W). . 2... 68) 
The quantity is small compared with 6, y, when the depth of the beam is 
small compared with its length. 
The whole deflection a y=6,7+6,y 
® /5 Z 
pe GUA Ge ae w) top (w+ W) 
Dye peated) e 1/7 
ay=n (sr t55 +W(sa+55) - (69.) 
Case XIII. 
When the values of the compressions at any point have been found, when 
two different sets of forces act on a solid separately, the compressions, when the 
forces act at the same time, may be found by the composition of compressions, 
because the small compressions are independent of one another. 
It appears from Case I., that if a cylinder be twisted as there described, the 
compressions will be inversely proportional to the square of the distance from 
the centre. 
VOL. XX. PART I. 2H 
