BEAM OF EECTANGITLAE CROSS-SECTIOISr UNDER ANY SYSTEM OF LOAD. 139 
§ 36, Case ivhere the Shear is spread over an Area instead of a Line. 
As in § 22, we may consider the effect of distributing the concentrated shear over 
an area, instead of over a line. This is all the more imj)ortant because, although 
we can, in practice, approximate to a line-distribution of pressure by means of a 
knife-edge, we cannot in the same way approximate to a line distrilmtion of sliear— 
shear being usually transmitted by means of projecting collars, wliich have a certain 
thickness. It is true that a thin notch miglit l^e cut into the material and an edo-e 
inserted in it which might be pulled sideways. But the cutting of such a notcli 
would seriously weaken the material, l)esides altering the coiiditions so much as to 
render our solution inapplicable. 
If we suppose our shear spread over a length 2cd of the upper edge, and if we 
adhere to tlie notation on p. 104, we find easily, L now denoting shearing force per 
unit area :— 
U= - 
^ ^ j log — {x — a') lo. 
2 tt + 
p- 
^ h 
2(d 
+ y' {4'i — i*-:) 
_ (A A \ 2L/; « 
7r(M + /^)7 
sin ( 2 v + 2 ) sin ( 2 p + 2) <^3 
( 2 v + 2)! //"+'“ 
I ^ p^/_L^U-''^^shi(2^-(-3)(^i-?A''+bsin(2z/ + 3)(/).3 
TT \\' + /J, /X f h 
2L?/” sin ( 2 v + 1) sin ( 2 p + 1 ) 
TTfX 0 
]p+i i^2v + 1) ! 
H.., 
2 L//' - sin f 2 z. + 2 )ch,- sin ( 2 . + 2 ) </>„ ^ . 
+ 'TTfx -o - (2v + 2) ! -=(H,,,i-H3„) 
L 1_ _ 
27r + /X 
2U 
OOf - )■ — 
& .. ( 9 
+ -:| ,i-+Pr'“ 
TT X -(- yU- jJL i Q 
{;x + o') cj), ~ {X - a) - y 
rf +-cos { 2 v + 2)</)i - 7V.+2 cos { 2 v + 2)0. 
4/' 
log ' 
TTu ^ r. 
//-"+- (2i' -f 2) ! 
H 
2.- 
2 L/z 
—v 
,y2,/+i cos { 2 v + 1) 0^ — ?y*'+* cos (2i^ + 1) 0. 
+ 
7r(A'-f/Li)7 (2r + 1)! 
2 L//'« 9y‘'+2 cos { 2 p + 2 ) 0 ^ —7y‘'+'“ cos ( 2 p + 2 ) 0 . 
- H„_,) 
TTfX 0 
{2v + 2) \ 
2L|f“ r-,-‘'+i cos (2i' + l)0j —cos (2j'-fl)0. 
're IX 1 
T 2 
/;-•'+! (2i^ + 1)! 
