Ob 
BEAM OF EECTAAM^ULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
f" sink 2ii 
du iv > 0) 
h;„ = 
» 0 
Jo sink-2;6 — 4?'® 
D'“" cosk 'la + sink 2// 
siiilA -la - -^a? 
dll {y > I). 
The values of the first few odd TT’s are all we shall retjuire. 
ll\ =z - -049, H'g = + -537, H'- = + 1-951. 
We then find, for points along the bottom edge, y” = 0, if)" = 77 / 2 , 
T1 
lev are 
Q = 0, S 0 
^Y 
ttI) 
•^4y-|2!^-537) + ;;rj;(l-951)-l-.. 
M 4 : 
fhis gives therefore a horizontal pressure at the point (O, — h) equal to (‘250), 
and this pressui'e increases at a fairly rapid rate as we move away from the axis of y. 
The stress P, obtained from Boussinesq’s calculation on Stokes’ hypothesis, 
fives for the same point P = — ^ ('657). This value is con¬ 
siderably too high. We gather that Stokes’ hypothesis ceases to give valid results 
for the points in the loAver half of the beam. 
§ 22. Effect of Diatribiitiny the Concentrated Load over a sinad Area instead, 
of a Line. 
In all the above work we have supposed tlie load W concentrated upon a line 
perpendicular to the plane of the strain. This has led us to expressions Avhich make 
the stresses, and one displacement, infinite at the line where the load is applied, 
and the other displacement indeterminate. In practice, hov^ever, owing to the 
elasticity and plasticity of the materials both of the lieam and of the knife-edge, 
contact along a geometrical line is impossible, and the load always distributes itself 
over an area, small but finite. 
In the present section we shall therefore consider the effect of a uniform distrilju- 
lion of load W per unit area (W was formerly load per unit length), extending on 
either side of cc = 0, = h for a distance a. 
Every line element Wc/|^of this load at distance ^ from the middle will produce 
a system of stresses and displacements Pd^, 'd)d^, P^d^, Ydf, sucli as. we have 
just been investigating, except that for x we must write (x ~ ^). 
The stresses and displacements due to the total load are therefore P (x — ^) d^, 
